**Lemma 2.14.1.** — We have $\Gamma(Y, i^{-1}\mathcal{O}_X) = S^{-1}A$ with $S = 1 + I$.

*Proof.* — Suppose we have a finite covering $Y = \cup_i D(\bar{g}_i)$ with $g_i \in A$ and $\bar{g}_i$ is its reduction modulo $I$. Let $S_i = 1 + I[\frac{1}{g_i}]$ and $S_{ij} = 1 + I[\frac{1}{g_i g_j}]$. Suppose given elements $x_i \in S_i^{-1}A[\frac{1}{g_i}]$ such that $x_i$ and $x_j$ have the same image in $S_{ij}^{-1}A[\frac{1}{g_i g_j}]$.

For any index $i$, we can write $x_i$ under the form

$$x_i = \frac{y_i}{g_i^{n_i} + a_i}$$

with $y_i \in A$, $a_i \in I$ and $n_i \in \mathbb{N}$ that we can chose as big as we want. We can thus find a collection of elements $(y_i)_i$ of elements of $A$, $(a_i)_i$ of $I$ and $n \in \mathbb{N}$ such that for all $i$,

$$x_i = \frac{y_i}{g_i^n + a_i}.$$

Suppose now that $A$ is an integral domain. Since $x_i$ and $x_j$ have the same image in $S_{ij}^{-1}A[\frac{1}{g_i g_j}] \subset \text{Frac}(A)$, we have for all indices $i$ and $j$,

$$(g_i^n + a_i)y_j = (g_j^n + a_j)y_i.$$

Now, since $Y = \cup_i D(\bar{g}_i)$, we can find a collection $(\lambda_i)_i$ of elements of $A$ such that

$$\sum_i \lambda_i g_i^n \in 1 + I.$$

Let us note

$$\mu = \sum_i \lambda_i (g_i^n + a_i) \in 1 + I.$$

and

$$\xi = \sum_i \lambda_i y_i \in A.$$

We have for all $j$,

$$\xi(g_j^n + a_j) = \mu y_i.$$

We deduce that

$$\mu^{-1}\xi \in S^{-1}A$$

has image $x_i$ in $S_i^{-1}A[\frac{1}{g_i}]$ for all indices $i$. This proves the result when $A$ is an integral domain.

Now, for any $A$, the fact that $x_i$ and $x_j$ have the same image in $S_{ij}^{-1}A[\frac{1}{g_i g_j}]$ is translated into the existence of an integer $m$ and a collection of elements $(b_{ij})_{i,j}$ of $A$ such that the following relations

$$(g_i^m g_j^m + b_{ij})(g_i^n + a_i)y_j = (g_i^m g_j^m + b_{ij})(g_j^n + a_j)y_i$$

are satisfied in $A$ for all indices $i, j$. We can now replace $A$ by the sub-$\mathbb{Z}$-algebra generated by $(g_i)_i, (b_{ij})_{i,j}, (y_i)_i, (a_i)_i$ and $I$ by its intersection with this sub-ring and suppose that $A$ is Noetherian. Now, since $A$ is Noetherian, it has a finite filtration by ideals

$$(0) = J_r \subset J_{r-1} \subset \cdots \subset J_0 = A$$

with $J_k/J_{k+1} \simeq A/\mathfrak{p}_k$, $0 \le k \le r-1$, where $\mathfrak{p}_k$ is a prime ideal of $A$. The case when $A$ is an integral domain shows that if $M$ is an $A$-module isomorphic to $A/\mathfrak{p}$. Then,

$$S^{-1}M \xrightarrow{\sim} \Gamma(Y, i^{-1}\widetilde{M}).$$

Now, if we have an exact sequence of $A$-modules

$$0 \to M' \to M \to M'' \to 0$$
this gives rise to a diagram with exact rows

<figure><img src="image_30.png" /></figure>

We deduce the result for $(A/J_k)_{0 \le k \le r}$ by induction on $k$ using the snake lemma. $\square$

**2.14.1.3. Étale localization around a closed subset.** —

**Definition 2.14.2.** — For $(A, I)$ a couple we define

$$A^h = \Gamma(Y, i^{-1}\mathcal{O}_{X_{\text{ét}}}),$$

the henselization of $(A, I)$, where $i: Y = V(I) \hookrightarrow \text{Spec}(A) = X$.

One has, by definition,

$$\Gamma(Y, \mathcal{P}^s i^{-1} \mathcal{O}_{X_{\text{ét}}}) = \lim_{\underset{\mathcal{C}}{\rightharpoonup}} \mathcal{O}(U)$$

where $\mathcal{P}^s i^{-1}$ means the reciprocal image as a presheaf on $Y_{\text{ét}}$ and where $\mathcal{C}$ is the category of diagrams

<figure><img src="image_31.png" /></figure>

Let us rewrite this diagram as

<figure><img src="image_32.png" /></figure>

Since $U \times_X Y \to Y$ is étale the section $s$ is an open embedding. Thus, up to replacing $U$ by an open subscheme we can suppose that in our colimit we have an isomorphism

$$U \times_X Y \xrightarrow{\sim} Y.$$

We change the definition of the category $\mathcal{C}$ accordingly. Now, the section $s$ is a closed immersion. One has to be careful that, à priori, in the preceding $U \to X$ may not be separated in a neighborhood of $Y$. Nevertheless, let us recall that we want to compute the sections on $Y$ of the sheaf associated to the presheaf $\mathcal{P}^s p^{-1}\mathcal{O}_{X_{\text{ét}}}$. We are thus allowed to Zariski localize on $Y$. For each point $y$ of $Y$ we can choose an affine neighborhood $U_y$ of $s(y)$ in $U$. There is then a diagram

<figure><img src="image_33.png" /></figure>

It is thus natural to add the following condition in the definition of $\mathcal{C}$: $U$ is affine. But now we can replace $U$ by

$$\lim_{s(Y) \subset V \subset U} V$$

where $V$ goes through the set of neighborhoods of $s(Y)$ as in Section 2.14.1.2. We have reached the following definition.

**Definition 2.14.3 ([46, Definition 5-Chapter XI]).** — A local-étale neighborhood of $Y$ in $X$ is a morphism $U \to X$ with $U$ of the form $\text{Spec}(S^{-1}B)$ where $\text{Spec}(B) \to X$ is étale, $\text{Spec}(B) \times_X Y \xrightarrow{\sim} Y$ and $S = 1 + BI$.
Thus, for $U \to X$ a local-étale neighborhood of $Y$ in $X$, $U \to X$ is only pro-étale and not étale in general. The advantage of the Zariski localization we did on $U$ is the following.

**Lemma 2.14.4.** — The category of local-étale neighborhoods of $Y$ in $X$ is equivalent to a poset i.e. there is at most one morphism between two objects (and it has a collection of objects that from a set such that any object in the category is isomorphic to one of those objects).

*Proof.* — Let $U \to X$ and $U' \to X$ be two local-étale neighborhoods of $Y$. Suppose $U = \lim_{\leftarrow Y \subset W \subset V} W$ and $U' = \lim_{\leftarrow Y \subset W' \subset V'} W'$ with $V \to X$ and $V' \to X$ affine étale satisfying $V \times_X Y \xrightarrow{\sim} Y$ and $V' \times_X Y \xrightarrow{\sim} Y$. Since $V \to X$ and $V' \to X$ are morphisms of finite presentation one has

$$ \operatorname{Hom}_X(U', U) = \lim_{\substack{\leftarrow Y \subset W \subset V \\ \rightarrow Y \subset W' \subset V'}} \lim_{\rightarrow W} \operatorname{Hom}_X(W', W). $$

Consider now two morphisms of $X$-schemes

$$ W' \xrightarrow{\underset{g}{f}} W $$

This corresponds to a morphism of étale schemes over $X$, $W' \to W \times_X W$. When restricted to $Y \subset W'$ this factorizes through $\Delta_{W/X}$ that is open in $W \times_X W$ since $W \to X$ is étale. We deduce that up to shrinking $W'$ one has $f = g$ and we deduce the result. $\square$

We can now prove the main result of this section.

**Proposition 2.14.5.** — We have

$$ \Gamma(Y, i^{-1}\mathcal{O}_{X_{\acute{e}t}}) = \lim_{\substack{\xrightarrow{U \to X} \\ \text{local-\'etale nbd of } Y}} \mathcal{O}(U). $$

*Proof.* — Let $\mathcal{F}$ be the presheaf on the principal open subsets of $Y_{Zar}$ defined by

$$ \mathcal{F}(V) = \lim_{\substack{\xrightarrow{U \to W} \\ \text{local-\'etale nbd of } V}} \mathcal{O}(U) $$

where $W$ is any principal open subset of $X$ such that $W \cap Y = V$. One clearly has using some finite presentation arguments,

$$ \forall y \in Y, \quad \mathcal{F}_y = \mathcal{O}_{X,y}^h, $$

the usual henselization of the local ring $\mathcal{O}_{X,y}$ that is equal to

$$ (\nu_* i^{-1} \mathcal{O}_{X_{\acute{e}t}})_y $$

where $\nu : X_{\acute{e}t} \to X_{Zar}$ is the projection. There is in fact an evident natural morphism of presheaves

$$ \mathcal{F} \to \nu_* i^{-1} \mathcal{O}_{X_{\acute{e}t}}. $$

If we can prove that $\mathcal{F}$ is a Zariski sheaf this will prove that $\mathcal{F} = \nu_* i^{-1} \mathcal{O}_{X_{\acute{e}t}}$. Using some finite presentation arguments this is reduced to the case when the ring $A$ is a finite type $\mathbb{Z}$-algebra and thus in particular excellent. Let us now remark that since $A$ is Noetherian,

$$ \mathcal{F} \subset \mathcal{O}_{\tilde{X}/Y} $$

as presheaves on $Y_{Zar}$. This is in fact a consequence of Krull's intersection theorem: if $R$ is a Noetherian ring, $J$ an ideal of $R$, $S = 1 + J$, $\tilde{R}$ the $J$-adic completion of $R$ then $S^{-1}R \subset \tilde{R}$. We can now use [46, Corollaire 1-Section 3-Chapitre XI]. Let $B = \tilde{A}^\wedge$ (integral closure). Note $\tilde{X} = \text{Spec}(B)$ and $\tilde{Y} = V(B \cap I\tilde{A})$. We note $\tilde{i} : \tilde{Y} \hookrightarrow \tilde{X}$ and $\pi : \tilde{Y} \to \tilde{X}$. Let $Y = \cup_i D(\tilde{g}_i)$ be a finite cover of $Y$ by principal open subsets where $g_i \in A$ and $\tilde{g}_i$ is its reduction modulo $I$. Let
$(x_i)_i \in \tilde{H}^0((D(\bar{g}_i))_i, \mathcal{F})$ with $x_i \in \mathcal{F}(D(\bar{g}_i))$. Since $\mathcal{O}_{\widehat{X}/Y}$ is a sheaf it gives rise to an element $x \in \widehat{A}$ such that for all index $i$, $x_i = x$ in $\Gamma(D(\bar{g}_i), \mathcal{O}_{\widehat{X}/Y})$. Now we have if

$$C_i = S_i^{-1} \overline{A[\frac{1}{g_i}]}^{\widehat{A}(\frac{1}{g_i})}$$

where $S_i = 1 + (\overline{A[\frac{1}{g_i}]}^{\widehat{A}(\frac{1}{g_i})} \cap I\widehat{A}(\frac{1}{g_i}))$,

$$x_i \in C_i \cap \mathrm{Im}(\widehat{A} \to \widehat{A}(\frac{1}{g_i})).$$

This implies that

$$x \in H^0(Y, \pi_* \bar{i}^{-1} \mathcal{O}_{\bar{X}})$$

that is equal to $\mathcal{F}(Y)$ according to Lemma 2.14.1. $\square$

## 2.14.2. Zariskian and Henselian structure sheaves. —

## 2.15. Étale morphisms of Noetherian analytic adic spaces

To explain the definition of an étale morphism of perfectoid spaces we need to explain first a basic result by Huber about the structure of étale morphisms of Noetherian analytic adic spaces.

**2.15.1. Definition.** — Here we work in the framework of what we call *Noetherian analytic adic spaces*. This means that we consider adic spaces that are locally the spectrum of a Tate strongly Noetherian affinoid ring. This contains as a particular case the “classical” Tate quasi-separated rigid spaces.

There are different equivalent definitions of étale morphisms of Noetherian analytic adic spaces. For a morphism of schemes $f: X \to Y$ we have the following equivalent definitions of an étale morphism:

1. $f$ is flat locally of finite presentation and unramified in the sense that for all $x \in X$,

* $\mathrm{m}_{f(x)}\mathcal{O}_{Y,f(x)} = \mathfrak{m}_x \subset \mathcal{O}_{X,x}$,

* the extension $k(x)|k(f(x))$ is finite degree separable.

2. $f$ is flat locally of finite presentation, locally quasi-finite with reduced geometric fibers.

3. $f$ is locally of finite presentation and formally étale.

4. If $U = \mathrm{Spec}(B)$ is an affine open subset of $X$ satisfying $f(U) \subset V = \mathrm{Spec}(A)$ an affine open subset of $Y$ then, $f^*: A \to B$ is such that

$$B \simeq A[X_1, \dots, X_n]/(f_1, \dots, f_n)$$

for some $n \ge 1$ and $f_1, \dots, f_n \in A[X_1, \dots, X_n]$ such that if

$$J = \det \left( \frac{\partial f_i}{\partial X_j} \right)_{1 \le i,j \le n} \in A[X_1, \dots, X_n]$$

it satisfies

$$J \mod (f_1, \dots, f_n) \in (A[X_1, \dots, X_n]/(f_1, \dots, f_n))^\times.$$

According to Huber ([39, Section 1.7]), such a type of definition extends to the case of Noetherian analytic adic spaces. More precisely, a morphism of Noetherian analytic adic spaces $f: X \to Y$ is called locally of finite type if locally on $X$ and $Y$ it is of the form $\mathrm{Spa}(B, B^+) \to \mathrm{Spa}(A, A^+)$ with

1. $(A, A^+)$ and $(B, B^+)$ complete Tate strongly Noetherian affinoid rings
2. $(B, B^+)$ is topologically of finite type over $(A, A^+)$ in the sense that there exists a surjection

$$A\langle X_1, \dots, X_n \rangle \to B$$

such that $B^+$ is the integral closure of the image of $A^+\langle X_1, \dots, X_n \rangle$.

This is a well behaved definition according to the following result.

**Proposition 2.15.1 ([36, Satz 3.8.15]).** — Suppose $(A, A^+)$ and $(B, B^+)$ are complete Tate strongly Noetherian rings and $f: \mathrm{Spa}(B, B^+) \to \mathrm{Spa}(A, A^+)$ is locally of finite type. Then, $(B, B^+)$ is of topologically finite type over $(A, A^+)$ in the sense that there exists a surjection $A\langle X_1, \dots, X_n \rangle \to B$ such that $B^+$ is the integral closure of the image of $A^+\langle X_1, \dots, X_n \rangle$.

We then have the following result by Huber.

**Proposition 2.15.2 ([39, Section 1.7]).** — For a locally of finite type morphism $f: X \to Y$ between Noetherian analytic adic spaces the following are equivalent:

1. $f$ is locally quasi-finite in the sense that for all $y \in Y$, the topological space $f^{-1}(y)$ is locally discrete on $X$ and for all $x \in X$ the morphism of local rings $\mathcal{O}_{Y,f(x)} \to \mathcal{O}_{X,x}$ is flat unramified (i.e. $\mathfrak{m}_{f(x)}\mathcal{O}_{X,x} = \mathfrak{m}_x$ and $k(x)|k(f(x))$ is finite degree separable),

2. $f$ is formally étale in the sense that for all $(R, R^+)$ a complete Tate strongly Noetherian affinoid ring equipped with

* a square zero ideal $I \subset R$,

* a morphism $\mathrm{Spa}(R, R^+) \to Y$,

one has

$$X(R, R^+) \xrightarrow{\sim} X(R/I, (R/I)^+).$$

3. If $f(U) \subset V$ with $U = \mathrm{Spa}(B, B^+)$ and $V = \mathrm{Spa}(A, A^+)$, with $A$ and $B$ strongly Noetherian complete Tate rings, one can write $B = A\langle X_1, \dots, X_n \rangle/(f_1, \dots, f_n)$ with $B^+$ the integral closure of the image of $A^+\langle X_1, \dots, X_n \rangle$ and such that the image of

$$J = \det \left( \frac{\partial f_i}{\partial X_j} \right)_{1 \le i,j \le n} \in A\langle X_1, \dots, X_n \rangle$$

in $B$ lies in $B^\times$.

**2.15.2. Algebraization of étale morphisms à la Elkik.** — The following result is a particular case of a deeper result by Elkik ([25, Théorème 7] in the Noetherian case, see [3, Theorem 1.16.23] for some non-Noetherian case) that can be proven directly in an elementary manner as Huber does in [39, Proposition 1.7.1 (iii)].

**Proposition 2.15.3.** — Let $(A, A^+)$ and $(B, B^+)$ be strongly Noetherian complete affinoid Tate rings. Let $\mathrm{Spa}(B, B^+) \to \mathrm{Spa}(A, A^+)$ be an étale morphism. There exists $n \ge 1$ and

$$f_1, \dots, f_n \in A[X_1, \dots, X_n]$$

such that

$$B \simeq A\langle X_1, \dots, X_n \rangle/(f_1, \dots, f_n)$$

with $B^+$ the integral closure of the image of $A^+\langle X_1, \dots, X_n \rangle$ and such that the image of

$$J = \det \left( \frac{\partial f_i}{\partial X_j} \right)_{1 \le i,j \le n} \in A[X_1, \dots, X_n]$$

in $A\langle X_1, \dots, X_n \rangle/(f_1, \dots, f_n)$ is invertible.
**2.15.3.** A key result by Huber. — Zariski's main theorem says that any quasi-finite separated morphism of schemes $f: X \to Y$ with $Y$ quasi-compact quasi-separated admits a factorization

<figure><img src="image_34.png" /></figure>

This is in particular the case if $f$ is qc separated and étale. Nevertheless, when $f$ is separated étale one can not in general find such a factorization with $\bar{X} \to Y$ finite étale. For example, let $f: X \to Y$ be a dominant morphism of proper smooth algebraic curves over a field $k$. Suppose that $f$ is generically étale i.e. $k(X)|k(Y)$ is separable. Let $U \subset X$ be the biggest open subset such that $f|_U$ is étale. Then, $f|_U$ admits a factorization

$$ U \xrightarrow{\text{open immersion}} \bar{U} \xrightarrow{\text{finite étale}} f(U) $$

if and only if $f^{-1}(f(U)) = U$.

Nevertheless, recall that if $(A, \mathfrak{m})$ is an Henselian local ring and $X \to \text{Spec}(A)$ is quasi-compact étale then one can split

$$ X = U \coprod U' $$

with

$$ U \to \text{Spec}(A) \setminus \{\mathfrak{m}\} $$

and

$$ U' \xrightarrow{\text{finite étale}} \text{Spec}(A). $$

As a consequence, if $f: X \to Y$ is a quasi-compact étale morphism of schemes and $y \in Y$, up to replacing $Y$ by an étale neighborhood of $y$, we can split $X = X' \coprod X''$ with $y \notin f(X')$ and $X'' \to Y$ is finite étale.

The local rings of analytic adic spaces are Henselian (Proposition 2.7.4) and one can go even further for étale morphisms of analytic Noetherian adic spaces.

**Proposition 2.15.4** ([39, Lemma 2.2.8]). — Let $(A, A^+)$ and $(B, B^+)$ be strongly Noetherian complete affinoid Tate rings and

$$ f: X = \text{Spa}(B, B^+) \to \text{Spa}(A, A^+) = Y $$

be an étale morphism. Any point of $Y$ has a neighborhood $V$ such that the étale morphism $f^{-1}(V) \to V$ has a factorization

<figure><img src="image_35.png" /></figure>

*Proof.* — We apply Proposition 2.15.3. With the notations of this proposition, let

$$ B' = A[X_1, \dots, X_n]/(f_1, \dots, f_n) $$

with Jacobian $J \in A[X_1, \dots, X_n]$ whose image in $B = A\langle X_1, \dots, X_n \rangle/(f_1, \dots, f_n)$ is a unit. Let $S$ be the image of $1 + A^{\circ\circ}[X_1, \dots, X_n]$ in $B'$. The image of $J$ in $S^{-1}B'$ is thus a unit. We deduce
that, up to replacing $B'$ by $B'' = B[\frac{1}{s}]$ for some $s \in S$, we can suppose that we have an étale morphism of schemes

$$g : \mathrm{Spec}(B'') \to \mathrm{Spec}(A)$$

that induces $f : X \to Y$ in the following sense. There is an analytification functor

$$(-)^{ad} : \{\text{finite type schemes}/\mathrm{Spec}(A)\} \to \{\text{locally of finite type adic spaces}/\mathrm{Spa}(A, A^+)\}$$

that sends $A_{\mathrm{Spec}(A)}^n$ to the adic affine space $A_{\mathrm{Spa}(A, A^+)}^{n,ad}$. Let $t_1, \dots, t_n \in B''$ be the image of $T_1, \dots, T_n$. One then has

$$X = \{x \in \mathrm{Spec}(B'')^{ad} \mid |t_1(x)| \le 1, \dots, |t_n(x)| \le 1\} \xrightarrow{g^{ad}} \mathrm{Spa}(A, A^+) = Y.$$

Let $y \in Y$. Since $\mathcal{O}_{Y,y}$ is an Henselian local ring, the étale morphism

$$\mathrm{Spec}(B'') \times_{\mathrm{Spec}(A)} \mathrm{Spec}(\mathcal{O}_{Y,y}) \to \mathrm{Spec}(\mathcal{O}_{Y,y})$$

splits as a disjoint union of a finite étale morphism and an étale morphism whose image is contained in $\mathrm{Spec}(\mathcal{O}_{Y,y}) \setminus \{\mathfrak{m}_y\}$. Since $g$ is of finite presentation we deduce that, up to replacing $\mathrm{Spa}(A, A^+)$ by a rational localization that is a neighborhood of $y$, we can suppose that $g$ splits as a disjoint union

$$g = g_1 \sqcup g_2 : \mathrm{Spec}(B_1'') \sqcup \mathrm{Spec}(B_2'') \to \mathrm{Spec}(A)$$

with $g_1 : \mathrm{Spec}(B_1'') \to \mathrm{Spec}(A)$ finite étale and

$$\mathrm{supp}(y) \notin \mathrm{Im}(g_2).$$

The set $\mathrm{Im}(g_2)$ is Zariski open in $\mathrm{Spec}(A)$. We deduce that $\mathrm{Im}(g_2^{ad}) \subset Y$ is Zariski open. It contains the quasi-compact open subset $V = g_2^{ad}(X \cap \mathrm{Spec}(B_2'')^{ad})$. Since $V$ is quasi-compact open in $Y$, $\bar{V}$ is its set of specializations. The specializations of a given point of $Y$ have the same support in $\mathrm{Spec}(A)$ and we deduce that $y \notin \bar{V}$. Up to replacing $Y$ by a rational localization that is a neighborhood of $y$, we can thus suppose that $\mathrm{Spec}(B_2'')^{ad} \cap X = \emptyset$. We thus have a factorization of $f$ as

$$f : X \xrightarrow{\text{open immersion}} \mathrm{Spec}(B_1'')^{ad} \xrightarrow{\text{finite étale}} Y. \quad \square$$

## 2.16. Vector bundles on analytic adic spaces
<figure><img src="image_36.png" /></figure>
# CHAPTER 3

## PERFECTOID SPACES

### 3.1. Perfectoid rings

In this text we emphasize perfectoid rings over any base i.e. without a base. They do not contain a field in general contrary to Scholze's original article [49].

**3.1.1. Generalities.** — Let us start by giving the definition of a perfectoid ring.

**Definition 3.1.1.** — A perfectoid ring $A$ is a complete Tate ring satisfying

1. $A$ is uniform i.e. $A^\circ$ is bounded,

2. There exists a pseudo-uniformizer $\varpi$ satisfying $\varpi^p|p$ and such that Frob: $A^\circ/\varpi \to A^\circ/\varpi^p$ is an isomorphism.

An affinoid perfectoid ring is an affinoid ring $(A, A^+)$ such that $A$ is perfectoid.

For $A$ a perfectoid ring, since $p \in A^{\circ\circ}$, $A$ is automatically a $\mathbb{Z}_p$-algebra that is $p$-adically separated complete.

**Example 3.1.2.** — 1. Let $K|\mathbb{Q}_p$ be an arithmetically profinite algebraic extension of $\mathbb{Q}_p$, for example $K|\mathbb{Q}_p$ Galois of infinite degree with Gal[$K|\mathbb{Q}_p$] a $p$-adic Lie group ([50]). Then, the main result of Fontaine and Wintemberger ([54]) says that $\widehat{K}$ is a perfectoid field. This is for example the case for $\widehat{\mathbb{Q}}_p(\zeta_{p^\infty})$ or $\widehat{\mathbb{Q}}_p(p^{1/p^\infty})$.

2. If $K$ is a complete algebraically closed non-Archimedean field then $K$ is perfectoid.

3. Any characteristic $p$ perfect complete non-Archimedean field like $\mathbb{F}_p((T^{1/p^\infty}))$.

4. If $K$ is a perfectoid field then $K\langle X_1^{1/p^\infty}, \dots, X_d^{1/p^\infty} \rangle$, the completion for the Gauss norm of the $K$-algebra $\bigcup_{n \ge 0} K\langle X_1^{1/p^n}, \dots, X_d^{1/p^n} \rangle$, is perfectoid.

5. More generally, if $A$ is perfectoid and $I$ a set let us consider

$$A\langle X_i^{1/p^\infty} \rangle_{i \in I}$$

that is $A^\circ\langle X_i^{1/p^\infty} \rangle_{i \in I}[\frac{1}{\varpi}]$ where $\varpi$ is a pseudo-uniformizer of $A$ and $A^\circ\langle X_i^{1/p^\infty} \rangle_{i \in I}$ is the $\varpi$-adic completion of $A^\circ[X_i^{1/p^\infty}]_{i \in I} = \bigcup_{n \ge 0} A[X_i^{1/p^n}]_{i \in I}$. This is a perfectoid ring with

$$A\langle X_i^{1/p^\infty} \rangle_{i \in I}^{\circ} = A^\circ\langle X_i^{1/p^\infty} \rangle_{i \in I}.$$

6. If $A$ is a perfectoid ring and $P$ is a profinite topological space then $\mathscr{C}(P, A)$ is perfectoid with $\mathscr{C}(P, A)^\circ = \mathscr{C}(P, A^\circ)$.
**Remark 3.1.3.** — Since we teased it to the reader, consider

$$A = \mathbb{Z}_p[[T^{1/p^\infty}]]\langle X^{1/p^\infty} \rangle [\frac{1}{T}]/(TX - p)$$

where $\mathbb{Z}_p[[T^{1/p^\infty}]]\langle X^{1/p^\infty} \rangle$ is the $(T, p)$-adic completion of $\cup_{n \ge 0} \mathbb{Z}_p[T^{1/p^n}, X^{1/p^n}]$. We will prove later (Example 3.4.11) that this is a perfectoid ring that does not contain a field and whose adic spectrum as a spectral topological space is connected, identified with the one of the adic closed ball $\mathbb{B}_{\mathbb{F}_p((T))}^1$ (if one wants a non-connected example it suffices to consider $K_1 \times K_2$ where $K_1$, resp. $K_2$, is a perfectoid field of characteristic $p$, resp. 0).

The following lemma says that we can in fact shorten the definition of a perfectoid ring and that, hopefully, this does not depend on the choice of the pseudo-uniformizer.

**Lemma 3.1.4.** — 1. In Definition 3.1.1 the injectivity of Frob is automatic.

2. For A perfectoid, the Frobenius of $A^\circ/p$ is surjective.

3. For A perfectoid and $\varpi$ any pseudo-uniformizer satisfying $\varpi^p|p$, Frob : $A^\circ/\varpi \to A^\circ/\varpi^p$ is an isomorphism.

*Proof.* — Point (1) is deduced from the fact that $A^\circ$ is integrally closed in A.

For point (2), let $\varpi$ be such that $\varpi^p|p$ and Frob : $A^\circ/\varpi \to A^\circ/\varpi^p$ is bijective. It suffices to prove by induction that for all $n \ge 1$,

$$\text{Frob} : A^\circ/(p, \varpi^n) \to A^\circ/(p, \varpi^{pn})$$

is surjective. The case $n = 1$ is immediate. Now if $a \in A^\circ$ satisfies

$$a = b^p + \lambda p + \mu \varpi^{pn}$$

with $\lambda, \mu \in A^\circ$, write $\mu = c^p + \nu \varpi^p$. One obtains

$$a = b^p + (c\varpi^n)^p + \lambda p + \nu\varpi^{p(n+1)}$$

which is congruent to $(b + c\varpi^n)^p \mod (p, \varpi^{p(n+1)})$.

Point (3) is an immediate consequence of points (1) and (2). $\square$

**3.1.2. The characteristic $p$ case.** — In characteristic $p$, perfectoid rings are simple to describe. What is remarkable in the next proposition is that the uniformity of our ring is automatic. This is in fact deduced from Banach's open mapping theorem.

**Proposition 3.1.5.** — A complete Tate $\mathbb{F}_p$-algebra is perfectoid if and only if it is a perfect ring.

*Proof.* — Let $A$ be an $\mathbb{F}_p$-perfectoid algebra. We use point (2) of Lemma 3.1.4 to deduce that for all $n \ge 1$,

$$\text{Frob} : A^\circ/\varpi^n \xrightarrow{\sim} A^\circ/\varpi^{pn}.$$

By taking the projective limit when $n \ge 1$ varies we deduce that $A^\circ$ and thus $A$ is perfect.

Reciprocally, let $A$ be a perfect complete Tate $\mathbb{F}_p$-algebra. Let $A_0$ be a ring of definition of $A$ and $\varpi \in A_0$ be a pseudo-uniformizer. The Frobenius

$$\text{Frob} : A \xrightarrow{\sim} A$$

is a surjective map of $\mathbb{F}_p((\varpi^{1/p^\infty}))$-Banach spaces. By Banach's open mapping theorem, $\varpi^N A_0 \subset A_0^p$ for some $N \gg 0$. Thus, $A_0^{1/p} \subset \varpi^{-N/p} A_0$. By induction we deduce that

$$A_0^{1/p^k} \subset \varpi^{-N(\frac{1}{p} + \frac{1}{p^2} + \cdots + \frac{1}{p^k})} A_0.$$

From this we deduce that $A_0^{1/p^\infty}$, the perfection of $A_0$ inside $A$, is bounded. We can thus suppose, up to replacing $A_0$ by its perfection, that $A_0$ is perfect. Now, if $a \in A^\circ$, there exists an integer $n \in \mathbb{Z}$ such that for all $k \ge 0$, $a^{p^k} \in \varpi^n A_0$ and thus, since $A_0$ is perfect, $a \in \varpi^{n/p^k} A_0$. This being
true for all $k \ge 0$, we deduce that $a \in \bar{\omega}^{-1}A_0$. Thus, $A^\circ \subset \bar{\omega}^{-1}A_0$ and is thus bounded. We deduce that $A$ is uniform and it is immediate to conclude that $A$ is perfectoid. $\square$

**Remark 3.1.6.** — The proof gives more: it says that if $A_0$ is a ring of definition of the perfectoid $\mathbb{F}_p$-algebra $A$, then the inclusion $A_0^{1/p^\infty} \subset A^\circ$ is in fact an almost equality

$$A_0^{1/p^\infty} \underset{\text{almost}}{=} A^\circ.$$

This allows us to construct plenty of affinoid perfectoid $\mathbb{F}_p$-algebras without having to verify that it is uniform and to almost compute their ring of integral elements.

**Example 3.1.7.** — Let $(A, A^+)$ be an affinoid perfectoid $\mathbb{F}_p$-algebra. Let $f_1, \dots, f_n \in A$ that generate $A$ as an ideal and $g \in A$. Then the ring $A\langle\frac{f_1\dots f_n}{g}\rangle$ is perfect since $A\langle\frac{f_1\dots f_n}{g}\rangle = A\langle\frac{f_1^p\dots f_n^p}{g^p}\rangle$. It is thus perfectoid with

$$A\langle\frac{f_1, \dots, f_n}{g}\rangle^+ \underset{\text{almost}}{=} \bigcup_{k \ge 0} A^+\langle\frac{f_1^{1/p^k}, \dots, f_n^{1/p^k}}{g^{1/p^k}}\rangle.$$

Thus, any rational subset of $\text{Spa}(A, A^+)$ is affinoid perfectoid.

The following corollary will be used later to see that for any $\mathbb{F}_p$-adic space $X$, one can define $\lim_{\leftarrow\text{Frob}} X$ as a perfectoid space. This will give us plenty of examples of perfectoid spaces for free.

**Corollary 3.1.8.** — Let $A$ be a Tate $\mathbb{F}_p$-algebra. Then the completion of its perfection $\lim_{\to\text{Frob}} A$,

$$\widehat{\lim}_{\to\text{Frob}} A_0[\frac{1}{\bar{\omega}}],$$

where $\bar{\omega}$ is a pseudo-uniformizer in the ring of definition $A_0$ and the completion is the $\bar{\omega}$-one, is perfectoid.

## 3.2. Tilting

**3.2.1. Any $p$-adic ring.** — In the following, the word $p$-adic means $p$-adically separated complete.

**Definition 3.2.1.** — For $A$ a $p$-adic $\mathbb{Z}_p$-algebra we set

$$A^b = \lim_{x \to x^p} A$$

as a set. For $x \in A^b$, $x = (x^{(n)})_{n \ge 0}$, $(x^{(n+1)})^p = x^{(n)}$, we set $x^\sharp = x^{(0)}$. We equip $A^b$ with a multiplicative monoid structure by setting $(xy)^{(n)} = x^{(n)}y^{(n)}$.

The application $x \mapsto x^\sharp$ is multiplicative,

$$(xy)^\sharp = x^\sharp y^\sharp.$$

The following lemma is well known and essentially due to Fontaine.

**Lemma 3.2.2.** — The reduction modulo $p$ induces a bijection

$$A^b \underset{\text{Frob}}{\sim} \lim_{\leftarrow} A/pA.$$

An inverse sends $(x_n)_{n \ge 0}$ to $(\lim_{k \to +\infty} (\widetilde{x}_{n+k})^{p^k})_{n \ge 0}$ where for $x \in A/pA$, $\tilde{x}$ is any lift to $A$.

*Proof.* — Use the fact that for $a, b \in A$, $a \equiv b \pmod{p^k}$ implies $a^p \equiv b^p \pmod{p^{k+1}}$. $\square$
This gives $A^b$ the structure of a perfect $\mathbb{F}_p$-algebra. The multiplication rule in this algebra is simple, $(xy)^{(n)} = x^{(n)}y^{(n)}$, but the addition rule is more complicated, this is given by a renormalization process:

$$ (x + y)^{(n)} = \lim_{k \to +\infty} (x^{(n+k)} + y^{(n+k)})^{p^k}. $$

which we write with the $\sharp$ notation as

$$ (x + y)^{\sharp} = \lim_{k \to +\infty} \left( (x^{1/p^k})^{\sharp} + (y^{1/p^k})^{\sharp} \right)^{p^k} $$

**Example 3.2.3.** — Let $A = \mathbb{Z}_p \langle T^{1/p^\infty} \rangle$, the $p$-adic completion of $\bigcup_{n \ge 0} \mathbb{Z}_p [T^{1/p^n}]$, one has $A^b = \mathbb{F}_p [X^{1/p^\infty}]$ with $X^\sharp = T$. One then has

$$ (1 + X)^{\sharp} = \sum_{\alpha \in \mathbb{N}[\frac{1}{p}]/[0,1]} \lim_{k \to +\infty} \binom{p^k}{p^k \alpha} T^{\alpha}. $$

We refer to [21] for a study of the $p$-adic numbers $\lim_{k \to +\infty} \binom{p^k}{p^k \alpha} \in \mathbb{Z}_p$.

**3.2.2. Perfectoid rings.** — Let us first verify that we can change the definition of the tilting for a perfectoid ring.

**Lemma 3.2.4.** — For A a perfectoid ring and $\varpi$ a pseudo-uniformizer of A satisfying $\varpi|p$, reduction modulo $\varpi$ induces an isomorphism

$$ A^{\circ, \flat} = \varinjlim_{\text{Frob}} A^{\circ}/p \xrightarrow{\sim} \varinjlim_{\text{Frob}} A^{\circ}/\varpi. $$

*Proof.* — There is an exact sequence of projective systems of abelian groups

$$ 0 \to (A^\circ/\varpi^{-1}pA^\circ)_{n \ge 0} \to (A^\circ/pA^\circ)_{n \ge 0} \to (A^\circ/\varpi A^\circ)_{n \ge 0} \to 0 $$

where the transitions maps are resp. $\varpi^{p-1}$Frob, Frob and Frob. The left hand projective system is essentially 0 and thus

$$ \varinjlim_{n \ge 0} A^\circ/\varpi^{-1}pA^\circ = 0 $$

$$ R^1 \varinjlim_{n \ge 0} A^\circ/\varpi^{-1}pA^\circ = 0. $$

We deduce that

$$ \varinjlim_{\text{Frob}} A^\circ/pA^\circ \xrightarrow{\sim} \varinjlim_{\text{Frob}} A^\circ/\varpi A^\circ. $$

□

Let us now verify that, up to an integral unit, any pseudo-uniformizer is a sharp.

**Lemma 3.2.5.** — For A a perfectoid ring, for any pseudo-uniformizer $\varpi$ there exists $\lambda \in (A^\circ)^\times$ such that $\lambda\varpi = x^\sharp$ for some $x \in A^{\circ, \flat}$.

*Proof.* — Let us chose $\varpi$ a pseudo-uniformizer satisfying $\varpi^p|p$. From the surjectivity of

$$ \text{Frob} : A^\circ/\varpi^p \to A^\circ/\varpi^p $$

and Lemma 3.2.4 we deduce that there exists $x \in A^{\circ, \flat}$ such that $x^\sharp \equiv \varpi$ modulo $\varpi^p$. This implies the result for $\varpi$ since then $\frac{x^\sharp}{\varpi} \in 1 + \varpi^{p-1}A^\circ \subset (A^\circ)^\times$.

Now for any pseudo-uniformizer $\varpi$, let us chose $\varpi' \in A^{\circ, \flat}$ such that $\varpi'^\sharp$ is a pseudo-uniformizer. Up to replacing $\varpi'$ by $\varpi'^{1/p^n}$ for $n \gg 0$, we can find $k \in \mathbb{N}$ such that $\varpi'' = \varpi'^{\sharp, -k}\varpi$ satisfies $\varpi'' \in A^\circ$ is a pseudo-uniformizer and $(\varpi'')^p|p$ (this type of result is typically deduced from Proposition 2.1.15). We deduce the assertion for $\varpi$ by application of the preceding case to $\varpi''$. □
**Remark 3.2.6.** — In general, the question to say something non-trivial about the image of the application $(-)^{\sharp} : A^{\flat} \to A$ for $A$ perfectoid is a difficult one, see Proposition 3.8.8 and Corollary 3.13.2 for example.

**Proposition 3.2.7.** — Let $A$ be a perfectoid ring. Define $A^{\flat} = \lim_{\leftarrow x \to x^p} A$ with the multiplication rule $(xy)^{(n)} = x^{(n)}y^{(n)}$ and equipped with the projective limit topology.

1. If $\varpi \in A^{\circ, \flat}$ is such that $\varpi^{\sharp}$ is a pseudo-uniformizer of $A$ then $A^{\flat} = A^{\circ, \flat}[\frac{1}{\varpi}]$ and the ring structure on $A^{\circ, \flat}[\frac{1}{\varpi}]$ corresponds to the ring structure on $A^{\flat}$ defined by the formula $(x+y)^{(n)} = \lim_{k \to +\infty} (x^{(n+k)} + y^{(n+k)})^{p^k}$.

2. $A^{\flat}$ is an $\mathbb{F}_p$-perfectoid algebra satisfying $A^{\flat, \circ} = A^{\circ, \flat}$ and $\varpi \in A^{\flat}$ is a pseudo-uniformizer of $A^{\flat}$ if and only if $\varpi^{\sharp}$ is a pseudo-uniformizer of $A$.

3. For $\varpi \in A^{\flat}$ a pseudo-uniformizer such that $\varpi^{\sharp}|_p$ there is a canonical isomorphism $A^{\flat, \circ}/\varpi \cong A^{\circ}/\varpi^{\sharp}$.

*Proof.* — Point (1) is easy. For point (2), if $\|\cdot\| : A \to \mathbb{R}_+$ is a power multiplicative norm defining the topology of $A$ then if we set $\|x\| = \|x^{(0)}\|$ for $x \in A^{\flat}$, $\|\cdot\|$ is a power multiplicative norm defining the topology of $A^{\flat}$ which is thus a uniform Tate ring satisfying $A^{\flat, \circ} = A^{\circ, \flat}$.

For point (3), the projection $x \mapsto x^{(0)}$ from $A^{\circ, \flat}$ to $A^{\circ}/p$ induces a surjective morphism

$$A^{\circ, \flat}/\varpi \to A^{\circ}/(p, \varpi^{\sharp}) = A^{\circ}/\varpi^{\sharp}.$$

The injectivity is deduced from the easy fact that if $x \in A^{\circ}$ satisfies $x^{p^n} \in A^{\circ} \varpi^{\sharp}$ then $x \in A^{\circ} \varpi^{1/p^n, \sharp}$ since $A^{\circ}$ is integrally closed in $A$. $\square$

### 3.3. Integral perfectoid rings

There is a notion of integral perfectoid rings that is useful in proofs.

**Proposition 3.3.1 (Integral perfectoid ring).** — Let $R$ be a ring and $\varpi \in R$. Suppose

1. $\varpi$ is a non-zero divisor,

2. $R$ is $\varpi$-adically separated complete,

3. $\varpi^p|_p$ and Frob : $R/\varpi \to R/\varpi^p$.

Then $R[\frac{1}{\varpi}] = \lim_{\to \times \varpi} R$ equipped with the inductive limit topology is a perfectoid ring such that $R$ is almost equal to $R[\frac{1}{\varpi}]^{\circ}$.

*Proof.* — By definition, $R$ is a ring of definition of the Huber ring $R[\frac{1}{\varpi}]$. Let

$$\frac{a}{\varpi^k} \in R[\frac{1}{\varpi}]^{\circ}$$

with $a \in R$ and $k \ge 1$. There exists $N \ge 1$ such that for all $n \ge 0$, $(\frac{a}{\varpi^k})^n \in \varpi^{-N}R$ that is to say $a^n \in \varpi^{kn-N}R$. Replacing $n$ by $p^n$ we obtain for all $n \ge 0$,

$$a^{p^n} \in \varpi^{kp^n-N}R.$$

Thus, for $n \gg 0$,

$$a^{p^n} \in \varpi^{(k-1)p^n}R.$$

The injectivity of

$$\text{Frob: } R/\varpi \to R/\varpi^p,$$

implies that for any $x \in R$, $(\frac{x}{\varpi})^p \in R \Rightarrow \frac{x}{\varpi} \in R$. From this we deduce by induction that

$$\forall x \in R, \forall i, j \ge 0, \left(\frac{x}{\varpi^j}\right)^{p^i} \in R \implies \frac{x}{\varpi^j} \in R.$$
We can apply this to deduce that

$$a \in \bar{\omega}^{k-1} R$$

since $(\frac{a}{\bar{\omega}^{k-1}})^p \in R$. We have thus proven that

$$R[\frac{1}{\bar{\omega}}]^{\circ} \subset \bar{\omega}^{-1} R$$

and it is thus bounded.

Now, as in Lemma 3.2.5, we can suppose, up to multiplying $\bar{\omega}$ by a unit in $R$, that there exists $\bar{\omega}^b \in R^b$ such that $(\bar{\omega}^b)^\sharp = \bar{\omega}$. The preceding result is true if replace $\bar{\omega}$ by $(\bar{\omega}^{b,1/p^n})^\sharp$ for all $n$. We deduce that $R$ is almost equal to $R[\frac{1}{\bar{\omega}}]^\circ$.

Finally one has to check that the fact that

$$\text{Frob} : R[\frac{1}{\bar{\omega}}]^{\circ}/\bar{\omega} \to R[\frac{1}{\bar{\omega}}]^{\circ}/\bar{\omega}^p$$

is almost surjective implies it is surjective. We can suppose that we can write $\bar{\omega} = \bar{\omega}^{b,\sharp}$. Let $x \in R[\frac{1}{\bar{\omega}}]^\circ$. One has $\bar{\omega}x \in R$ and thus there exists $y \in R$ such that $\bar{\omega}x \equiv y^p \text{ modulo } \bar{\omega}^p R$. We deduce that $x \equiv z^p \text{ modulo } \bar{\omega}^{p-1} R$ with $z \in R[\frac{1}{\bar{\omega}}]^\circ$ since $z^p \in R$. We have thus proven that is $\bar{\omega}' = (\bar{\omega}^b, \frac{p-1}{p})^\sharp$ then

$$\text{Frob} : R[\frac{1}{\bar{\omega}}]^{\circ}/\bar{\omega}' \to R[\frac{1}{\bar{\omega}}]^{\circ}/\bar{\omega}'^p$$

is surjective. The result is deduced from the independence of the choice of a pseudo-uniformizer in the definition of a perfectoid ring, see point (3) of Lemma 3.1.4. $\square$

Here is a typical example of an integral perfectoid ring.

**Proposition 3.3.2.** — Let $(A, A^+)$ be a Huber pair with $A$ perfectoid. Then for any pseudo-uniformizer $\bar{\omega}$ satisfying $\bar{\omega}^p|_p$, Frob : $A^+/\bar{\omega} \xrightarrow{\sim} A^+/\bar{\omega}^p$ and $A^+$ is an integral perfectoid ring.

*Proof.* — Let us begin with the injectivity. If $x \in A^+$ satisfies $x^p \in A^+\bar{\omega}^p$ it then satisfies $x^p \in A^\circ\bar{\omega}^p$. This implies $x = \lambda\bar{\omega}$ with $\lambda \in A^\circ$. But then, $\lambda^p\bar{\omega}^p \in \bar{\omega}^p A^+$ which implies $\lambda^p \in A^+$ and thus $\lambda \in A^+$ since $A^+$ is integrally closed.

For the surjectivity. For any $x \in A^+$ one can write $x = a^p + \lambda\bar{\omega}^p$ with $a, \lambda \in A^\circ$. This implies $a^p \in A^+$ and thus $a \in A^+$. We deduce that

$$\text{Frob} : A^+/\bar{\omega} \to A^+/\bar{\omega}^p A^\circ$$

is surjective. Using that the ring $A^\circ/A^+$ is perfect we conclude. $\square$

## 3.4. Untilting

The tilting functor $(-)^b$, due to Fontaine, has in fact an adjoint and the adjunction map is Fontaine's $\theta$ map. This adjoint to tilting will allow us to untilt.

**Proposition 3.4.1.** — The functors

perfect $\mathbb{F}_p$-algebras $\xrightarrow{W(-)} \xleftarrow{(-)^b}$ p-adically separated complete rings

are adjoint, where $W(-)$ is a left adjoint of $(-)^b$. The adjunction maps are given by

$$\theta : W(R^b) \to R \\ \sum_{n \ge 0} [a_n] p^n \mapsto \sum_{n \ge 0} a_n^\sharp p^n.$$
and

$$ \begin{aligned} R & \to W(R)^{\flat} \\ x & \mapsto ([x^{p^{-n}}]_{n \ge 0}. \end{aligned} $$

Thus, for $R$ a perfect $\mathbb{F}_p$-algebra and $x \in R$ seen as $W(R)^{\flat}$,

$$ x^{\sharp} = [x]. $$

Let's dig further the structure of $\theta$ for perfectoid rings.

**Lemma 3.4.2.** — For $A$ a perfectoid ring, $\theta : W(A^{\flat, \circ}) \to A^{\circ}$ is surjective.

*Proof.* — Since $A^{\circ}$ is $p$-adically complete it suffices to check that the reduction modulo $p$ of $\theta$ is surjective. But this is identified with the projection onto the first component $\lim_{\leftarrow\text{Frob}} A^{\circ}/p \to A^{\circ}/p$. $\square$

We are now going to classify all possible kernels for $\theta$. The following definition showed up in [27] for perfectoid fields. This is inspired by the theory of Weierstrass factorization of power series; we see Witt vectors as “holomorphic functions of the variable $p$” (see [27, Chapitre 1]).

**Definition 3.4.3.** — Let $A$ be a characteristic $p$ perfectoid ring. An element

$$ \xi = \sum_{n \ge 0} [a_n] p^n \in W(A^{\circ}) $$

is distinguished of degree 1 if $a_0 \in A^{\circ\circ}$ and $a_1 \in (A^{\circ})^{\times}$. We note $\mathcal{D}_1(A)$ the set of degree 1 distinguished elements in $W(A^{\circ})$.

Thus, the element $\xi$ is primitive of degree 1 if and only if $\xi \mod W(A^{\circ\circ})$ is in $pW(\tilde{A})^{\times}$. From this we deduce that $W(A^{\circ})^{\times}$ acts by multiplication on $\mathcal{D}_1(A)$. Moreover,

$$ \mathcal{D}_1(A)/W(A^{\circ})^{\times} \to \{\text{Cartier divisors of } \text{Spec}(W(A^{\circ}))\}. $$

Those are in fact a particular type of *Cartier divisors on Spec(W(A°))* thanks to the following lemma.

**Lemma 3.4.4.** — Any degree 1 distinguished element $\xi \in W(A^{\circ})$ is a non-zero divisor.

*Proof.* — Write $\xi = \sum_{n \ge 0} [a_n] p^n$. Let $\varpi$ be a pseudo-uniformizer such that $a_0 \in \varpi A^{\circ}$. The image of $\xi$ in $W(A^{\circ})/[\varpi]$ is of the form $p \times \text{unit}$. The result is deduced from the fact that $W(A^{\circ})/[\varpi]$ has no $p$-torsion and $W(A^{\circ})$ is $[\varpi]$-adically separated without $[\varpi]$-torsion. $\square$

**Remark 3.4.5.** — The point of view of distinguished elements comes from [27] where the authors use the terminology “primitive”. This point of view is the starting point of the work [23] in the non-perfect prismatic case.

We will need the following result.

**Lemma 3.4.6.** — Let $A$ be a characteristic $p$ perfectoid ring, $\varpi \in A$ a pseudo-uniformizer and $\xi \in \mathcal{D}_1(A)$.

1. $W(A^{\circ})$ is $[\varpi]$-adically separated complete.

2. $\times \xi : W(A^{\circ}) \hookrightarrow W(A^{\circ})$ is strict with respect to the $[\varpi]$-adic topology on $W(A^{\circ})$.
*Proof.* — The ([ω], *p*)-adic topology on *W*(*A*°) is the weak topology of uniform convergence of the Teichmüller coefficients i.e.

$$ (A^{\circ})^{\mathbb{N}} \xrightarrow{\sim} W(A^{\circ}) \\ (a_n)_{n \ge 0} \mapsto \sum_{n \ge 0} [a_n]p^n. $$

is an homeomorphism where the left hand side is equipped with the product topology and the right hand one with the ([ω], *p*)-adic one. We deduce from this observation that *W*(*A*°) is ([ω], *p*)-adically separated complete. It is thus [ω]-adically separated complete (if a ring is *I*-adically separated complete for an ideal *I* then it is *J*-adically separated complete for any sub-finite type ideal *J* ⊂ *I*).

Let us now prove point (2). Write $\xi = \sum_{n \ge 0} [a_n]p^n$. Since $a_0 \in A^{\circ\circ}$, up to changing $\varpi$ we can suppose that $\varpi|a_0$ in $A^\circ$. We then have $\xi \equiv p \times \text{unit modulo } [\varpi]$. Since $W(A^\circ)/[\varpi]$ is without $p$-torsion, if for some $f \in W(A^\circ)$, $\xi f \in ([\varpi])$ then $f \in ([\varpi])$. From this we deduce by induction that $(\xi) \cap ([\varpi^n]) = (\xi[\varpi^n])$ as ideals of $W(A^\circ)$. $\square$

**Lemma 3.4.7.** — For A a perfectoid ring,

1. If $\xi \in W(A^{b,\circ})$ is distinguished of degree 1 and satisfies $\theta(\xi) = 0$ then $\ker \theta = (\xi)$.

2. The kernel of $\theta$ is generated by a degree 1 distinguished element.

*Proof.* — Let us first construct a generator of $\ker \theta$. Let $\varpi \in A^b$ be a pseudo-uniformizer satisfying $\varpi^\sharp|p$. Using the surjectivity of $\theta$ one c"an write

$$ p = \theta(a)\varpi^{\sharp}. $$

We deduce that

$$ \xi = p - a[\varpi] \in \ker \theta. $$

Let us prove this is a generator. If we equip $W(A^{b,\circ})$ with the [ω]-adic topology, the inclusion

$$ \times \xi : W(A^{b,\circ}) \hookrightarrow W(A^{b,\circ}) $$

is strict (Lemma 3.4.6). Since $W(A^{b,\circ})$ is [ω]-adically separated complete we deduce that $(\xi) \subset W(A^{b,\circ})$ is a closed ideal for the [ω]-adic topology. Thus,

$$ W(A^{b,\circ})/\xi $$

is [ω]-adically separated complete. It thus suffices to prove that

$$ \theta : W(A^{b,\circ})/\xi \to A^{\circ} $$

is injective when reduced modulo [ω]. But the reduction modulo [ω] of this morphism is

$$ A^{b,\circ}/\varpi \to A^{\circ}/\varpi^{\sharp} $$

that is an isomorphism (Proposition 3.2.7).

Let now $\xi' \in W(A^{b,\circ})$ be another degree 1 distinguished element such that $\theta(\xi') = 0$. Write

$$ \xi' = x\xi \text{ with } x = \sum_{n \ge 0} [x_n]p^n \in W(A^{b,\circ}). $$

One can find a pseudo-uniformizer $\varpi'$ of $A^b$ such that $\xi'$ and $\xi$ are $p \times \text{unit in } W(A^{b,\circ})/[\varpi']$. Since $W(A^{b,\circ})/[\varpi']$ has no $p$-torsion, we deduce that $x$ modulo $[\varpi']$ is a unit and thus $x$ is a unit since $W(A^{b,\circ})$ is $[\varpi']$-adic. $\square$

We can now state the main result.
**Theorem 3.4.8.** — Let $A$ be a characteristic $p$ perfectoid ring. There is a bijection

$$ \mathcal{D}_1(A)/W(A^\circ)^\times \xrightarrow{\sim} \{\text{Untilts of } A \text{ over } \mathbb{Z}_p\} $$

that sends $(\xi)$ to $W(A^\circ)[\frac{1}{[\varpi]}]/(\xi)$ where a ring of definition is the image of $W(A^\circ)$. An inverse sends $A^\sharp$ to the kernel of $\theta : W(A^\circ) \to A^{\sharp, \circ}$.

*Proof.* — It remains to prove that for $\xi$ of degree 1, $W(A^\circ)/\xi$ is integral perfectoid (see Section 3.3). Let $\varpi \in A$ be a pseudo-uniformizer such that if $\xi = \sum_{n \ge 0} [a_n] p^n$ then $\varpi^p|_{a_0}$. We already saw in the proof of Lemma 3.4.7 that $W(A^{p,\circ})/\xi$ is $[\varpi]$-adically separated complete. Moreover, $([\varpi], \xi) = ([\varpi], p)$ and $([\varpi^p], \xi) = ([\varpi^p], p)$. The result is easily deduced. $\square$

**Corollary 3.4.9.** — The category of perfectoid rings is equivalent to the category of couples $(A, I)$ where $A$ is a characteristic $p$ perfectoid ring and $I$ a Cartier divisor of $\text{Spec}(W(A^\circ))$ generated by a degree one distinguished element, $I \in \mathcal{D}_1(A)/W(A^\circ)^\times$.

**Remark 3.4.10.** — The subcategory of $\mathbb{Q}_p$-perfectoid rings corresponds to degree 1 distinguished elements $\xi = \sum_{n \ge 0} [a_n] p^n$ such that $a_0 \in A^\times$, that is to say $a_0$ is a pseudo-uniformizer.

**Example 3.4.11.** — We can now explain Remark 3.1.3. Let $R = \mathbb{F}_p((U^{1/p^\infty}))\langle Y^{1/p^\infty} \rangle$, an $\mathbb{F}_p$-perfectoid algebra with $\varpi = U$. There is an identification $W(R^\circ) = \mathbb{Z}_p[\mathbb{I}^{1/p^\infty}]\langle X^{1/p^\infty} \rangle$ since this last ring is $p$-adic without $p$-torsion with reduction modulo $p$ the perfect ring $R^\circ$. Via this identification, $T = [U]$ and $X = [Y]$. Let now $\xi = [\varpi X] - p \in \mathcal{D}_1(R)$. The resulting untilt of $R$ over $\mathbb{Z}_p$ is the ring $A$ of Remark 3.1.3. This perfectoid ring does not contain a field since if this field where of characteristic $p$ then $A$ would be a perfectoid $\mathbb{F}_p$-algebra and thus $\xi \equiv 0 \pmod p$, if this where of characteristic 0 then $A$ would be a $\mathbb{Q}_p$-perfectoid algebra and thus $\xi \pmod p$ would be in $R^\times$. The identification of $\text{Spa}(A, A^\circ)$ with $\text{Spa}(R, R^\circ)$ as topological spaces is Proposition 3.7.2.

**Remark 3.4.12.** — With the fancy point of view of [48], the preceding example 3.4.11 proves the existence of a morphism $\mathbb{B}_{\mathbb{F}_p((T^{1/p^\infty}))}^{1,1/p^\infty} \to \text{Spa}(\mathbb{Z}_p)^\diamond$ such that the reciprocal image of $\text{Spa}(\mathbb{Q}_p)^\diamond \hookrightarrow \text{Spa}(\mathbb{Z}_p)^\circ$ is the punctured perfectoid ball $\mathbb{B}_{\mathbb{F}_p((T^{1/p^\infty}))}^{1,1/p^\infty} \setminus \{0\}$ and the reciprocal image of $\text{Spa}(\mathbb{F}_p)^\diamond$ is the origin $\{0\}$. From this we deduce that the $v$-sheaf $\text{Spa}(\mathbb{Z}_p)^\diamond$ is spatial with topological space $\{s, \eta\}$ with $s \le \eta$. If all perfectoid rings contained a field then $|\text{Spa}(\mathbb{Z}_p)^\diamond|$ would be a discrete set with two elements.

**Corollary 3.4.13 (Tilting equivalence).** — If $A$ is a perfectoid ring, tilting induces an equivalence between the category of perfectoid $A$-algebras and the category of perfectoid $A^b$-algebras.

Finally, let us make a definition.

**Definition 3.4.14.** — A perfectoid algebra $A$ is a perfectoid field if $A$ is a field and $A^\circ$ is a rank 1 valuation ring.

The following is easy and left to the reader.

**Proposition 3.4.15.** — The tilting equivalence identifies perfectoid fields: $A$ is a perfectoid field if and only if $A^b$ is a perfectoid field.

**Example 3.4.16.** — Take $K = \mathbb{F}_p((T^{1/p^\infty}))$. The untilt over $\mathbb{Q}_p$ corresponding to $\xi = [T] - p$ is $\widetilde{\mathbb{Q}}_p(p^{1/p^\infty})$. Let $\varepsilon = 1 + T$. The one corresponding to $\xi = 1 + [\varepsilon^{\frac{1}{p}}] + \dots + [\varepsilon^{\frac{p-1}{p}}]$ is $\widetilde{\mathbb{Q}}_p(\zeta_{p^\infty})$.

For the next example, recall ([10, Section 2.4.4]) that a complete non-archimedean field $(K, |\cdot|)$ is spherically complete if any decreasing set of closed balls has non-empty intersection (if the radius
of the balls goes to zero this is automatic by completeness). This is equivalent to saying that if $(a_n)_{n \in \mathbb{N}}$ is a decreasing sequence of ideals of $\mathcal{O}_K$ then

$$R^1 \underset{n \ge 0}{\operatorname{lim}} a_n = 0.$$

According to Kaplansky ([42]) this is equivalent to saying that $(K, |\cdot|)$ is maximally complete: if $(L, |\cdot|)$ is an extension of $(K, |\cdot|)$ with the same residue field and the same value group, i.e. $|L| = |K|$ inside $\mathbb{R}$, then $L = K$.

**Example 3.4.17 (Untilting spherically complete fields ([44]))**

Let $\Gamma$ be a non-trivial subgroup of $(\mathbb{R}, +)$. Let $k$ be a characteristic $p$ field. To this datum one associates

$$k((\Gamma)) = \left\{ \sum_{\gamma \in \Gamma} a_{\gamma} T^{\gamma} \mid a_{\gamma} \in k, \{\gamma \mid a_{\gamma} \neq 0\} \text{ is well ordered} \right\}.$$

This is a spherically complete valued field with residue field $k$ and valuation group $\Gamma$. If $k$ is perfect of characteristic $p$ and $\Gamma$ is divisible by $p$ this is a perfectoid field. When $\Gamma = \mathbb{Z}[\frac{1}{p}]$ this is the spherical closure of the perfectoid field $k((T^{1/p^\infty}))$. If moreover $k$ is algebraically closed and $\Gamma$ is divisible i.e. is a sub-$\mathbb{Q}$-vector space of $\mathbb{R}$, this is algebraically closed. For $k$ perfect and $\Gamma$ divisible by $p$ one has

$$W(k[[\Gamma]]) = \left\{ \sum_{\gamma \in \Gamma_+} a_{\gamma} X^{\gamma} \mid a_{\gamma} \in W(k), \forall n \{\gamma \mid v_p(a_{\gamma}) = n\} \text{ is well ordered} \right\}$$

where $X^{\gamma} = [T^{\gamma}]$. Now, the untilt $K|_{\mathbb{Q}_p}$ corresponding to $\xi = [T] - p$ is the maximally complete value field extension of $\mathbb{Q}_p$ with residue field $k$ and valuation group $\Gamma$ described in [44]. Noting $p^{\gamma} = (T^{\gamma})^{\sharp}$ one has

$$\mathcal{O}_K = \left\{ \sum_{\gamma \in \Gamma_+} [a_\gamma] p^\gamma \mid a_\gamma \in k, \{\gamma \mid a_\gamma \neq 0\} \text{ is well ordered} \right\}$$

where $[a_\gamma] = a_\gamma^\sharp$. For example, when $\Gamma = \mathbb{Q}$ and $k = \mathbb{F}_p$ this is the spherical completion of $\mathbb{C}_p$.

### 3.5. Affinoid perfectoid rings

In this section we upgrade the preceding results for perfectoid rings to affinoid perfectoid rings i.e. we replace $A^\circ$ by $A^+$. Let us begin by an evident definition.

**Definition 3.5.1.** — An affinoid perfectoid ring is a Huber pair $(A, A^+)$ where $A$ is a perfectoid ring. It is called an affinoid perfectoid field if $A$ is a field and $A^+$ a valuation subring.

Given a perfectoid ring $A$, the different possible $A^+$ are in bijection with the integrally closed subrings of $\tilde{A} = A^\circ/A^{\circ\circ}$. Since $\tilde{A} = \tilde{A}^b$, we deduce that the tilting equivalence extends to affinoid perfectoid rings. Using Proposition 3.3.2 and the preceding results one deduces the following where we replace $A^\circ$ by any $A^+$ and we have $A^{b,+} = A^{+,b}$.

**Theorem 3.5.2.** — 1. Let $(A, A^+)$ be an affinoid perfectoid ring. The tilting functor induces an equivalence between $(A, A^+)$-affinoid perfectoid algebras and $(A^b, A^{b,+})$-affinoid perfectoid algebras.

2. For $(A, A^+)$ an $\mathbb{F}_p$-affinoid perfectoid algebra let us note

$$\mathcal{D}_1(A, A^+) = \left\{ \sum_{n \ge 0} [a_n] p^n \in W(A^+) \mid a_0 \in A^{\circ\circ} \text{ and } a_1 \in (A^+)^\times \right\}.$$

Then the tilting equivalence induces an equivalence between

* triples $(A, A^+, I)$ where $(A, A^+)$ is affinoid perfectoid of characteristic $p$ and $I \subset W(R^+)$ an ideal generated by an element of $\mathcal{D}_1(A, A^+)$ i.e. $I \in \mathcal{D}_1(A, A^+)/W(A^+)^\times$,
• the category of affinoid perfectoid algebras.

3. The equivalence puts in correspondence affinoid perfectoid fields.

*Proof.* — It essentially suffices to verify that $\mathcal{D}_1(A, A^+)/W(A^+) \times \xrightarrow{\sim} \mathcal{D}_1(A, A^\circ)/W(A^\circ)^\times$. This is an easy exercise. □

3.6. The cotangent complex point of view

In this section we explain the cotangent complex proof of the tilting equivalence as originally presented in [49].

3.6.1. Vanishing of the cotangent complex. — Let us begin with a lemma.

**Lemma 3.6.1.** — Let $A$ be an $\mathbb{F}_p$-algebra and $B$ an $A$-algebra. Then, the morphism induced by the Frobenius, Frob : $(A \to B) \to (A \to B)$, on the complex $\mathbb{L}_{B/A}$ is homotopic to zero.

*Proof.* — According to Lemma A.2.13, Frob : $(A \to B) \to (A \to B)$ induces a morphism of complexes $P_{\bullet, B/A} \to P_{\bullet, B/A}$ that is homotopic to $\text{Frob}_{P_{\bullet, B/A}}$. The result is deduced since by applying $\Omega^1$, $d(x^p) = px^{p-1}dx$ when $x \in P_{\bullet, B/A}$ and this is zero since $P_{\bullet, B/A}$ is an $A$-module. □

**Proposition 3.6.2.** — For any morphism $A \to B$ of perfect $\mathbb{F}_p$-algebras, $\mathbb{L}_{B/A} \simeq 0$ in the derived category of $B$-modules.

*Proof.* — By functoriality of the cotangent complex, the Frobenius of $B$ induces a semi-linear automorphism of $\mathbb{L}_{B/A}$. But this automorphism is zero according to Lemma 3.6.1. □

**Proposition 3.6.3.** — For any morphism $(A, A^+) \to (B, B^+)$ of affinoid perfectoid algebras and $\varpi$ a pseudo-uniformizer of $A$ satisfying $\varpi|_p$,

$$ \mathbb{L}_{(B^+/\varpi)/(A^+/\varpi)} \simeq 0 $$

in $D(B^+/\varpi)$.

*Proof.* — Suppose first that $\varpi^p|_p$. The isomorphism

$$ \begin{array}{ccc} B^+/\varpi & \xrightarrow{\text{Frob}} & B^+/\varpi^p \\ \uparrow & & \uparrow \\ A^+/\varpi & \xrightarrow{\text{Frob}} & A^+/\varpi^p \end{array} $$

induces an isomorphism

$$ \mathbb{L}_{(B^+/\varpi)/(A^+/\varpi)} \xrightarrow{\sim} \mathbb{L}_{(B^+/\varpi^p)/(A^+/\varpi^p)}. $$

Since this isomorphism is homotopic to 0 we deduce that $\mathbb{L}_{(B^+/\varpi^p)/(A^+/\varpi^p)}$ is acyclic.

For any $\varpi$ satisfying $\varpi|_p$, up to multiplying $\varpi$ by a unit in $A^+$ one can suppose that it has a $p$-root. This implies the result in general. □

3.6.2. The tilting equivalence via the cotangent complex. —
### 3.7. Identification of the adic spectra as topological spaces

Let $(A, A^+)$ be affinoid perfectoid.

**Proposition 3.7.1.** — 1. For $x \in \text{Spa}(A, A^+)$ and $f \in A^b$, the formula

$$|f(x^b)| = |f^\sharp(x)|$$

defines an element $x^b \in \text{Spa}(A^b, A^{b,+})$.

2. This induces a bijection

$$(-)^b : |\text{Spa}(A, A^+)| \xrightarrow{\sim} |\text{Spa}(A^b, A^{b,+})|.$$

*Proof.* — The only thing to verify in point (1) is that

$$|(f + g)(x^b)| \le \sup\{|f(x^b)|, |g(x^b)|\}.$$

This is deduced from the continuity of the valuation $x^b$ that implies

$$|(f + g)(x^b)| = \lim_{k \to +\infty} |(f^{1/p^k, \sharp} + g^{1/p^k, \sharp})(x)|^{p^k}$$

where here the limit is taken in the valuation group associated to $x$.

For point (2) we need to define an inverse. Let $x \in \text{Spa}(A^b, A^{b,+})$. The field $K = \text{Frac}(A^b/\text{supp}(x))$ is perfect and its valuation subring $K^+$ too. The completion $(\tilde{K}, \tilde{K}^+)$ is an affinoid perfectoid field and there is a natural morphism $(A^b, A^{b,+}) \to (\tilde{K}, \tilde{K}^+)$. Using the tilting correspondence we deduce an affinoid perfectoid field $(L, L^+)$ together with a morphism $(A, A^+) \to (L, L^+)$ that gives the preceding by tilting. Let us note $x^\sharp$ for the image of the closed point of $\text{Spa}(L, L^+)$ via $\text{Spa}(L, L^+) \to \text{Spa}(A, A^+)$. One easily verifies that $x \mapsto x^\sharp$ defines an inverse to $x \mapsto x^b$. $\square$

**Proposition 3.7.2.** — The preceding bijection

$$|\text{Spa}(A, A^+)| \xrightarrow{\sim} |\text{Spa}(A^b, A^{b,+})|$$

is in fact an homeomorphism.

*Proof.* — Let $f_1, \dots, f_n \in A^b$ generate $A^b$ as an ideal and $g \in A^b$. For each $x \in \text{Spa}(A, A^+)$, there exists $i \in \{1, \dots, n\}$ such that $|f_i(x^b)| \ne 0$ and thus $|f_i^\sharp(x)| \ne 0$. We deduce that $f_1^\sharp, \dots, f_n^\sharp$ do not vanish simultaneously on $\text{Spa}(A, A^+)$ and thus generate $A$ as an ideal. Now one verifies immediately that the pullback via our bijection of the rational subset $\text{Spa}(A^b, A^{b,+})(\frac{f_1 \dots f_n}{g})$ is

$$\text{Spa}(A, A^+)(\frac{f_1^\sharp \dots f_n^\sharp}{g^\sharp}).$$

Our map is thus continuous. It is moreover generalizing. In fact, for $x \in \text{Spa}(A^b, A^{b,+})$, the affinoid field $(K(x), K(x)^+)$ is perfectoid. The morphism

$$(A^b, A^{b,+}) \to (K(x), K(x)^+)$$

untilts to a morphism

$$(A, A^+) \to (K(x)^\sharp, K(x)^{\sharp,+})$$

where $(K(x)^\sharp, K(x)^{\sharp,+})$ is an affinoid perfectoid field. The image of the closed point of $\text{Spa}(K(x)^\sharp, K(x)^{\sharp,+})$ is $x^\sharp \in \text{Spa}(A, A^+)$. Using the identifications

$$|\text{Spa}(A^b, A^{b,+})|_x = |\text{Spa}(K(x), K(x)^+)| = |\text{Spa}(K(x)^\sharp, K(x)^{\sharp,+})|$$

we conclude. We can now apply Lemma 2.11.6 to conclude that our map is an homeomorphism. $\square$
## 3.8. Sheafiness and acyclicity

We now come to one of the main results of perfectoid spaces that allows us to glue affinoid perfectoid rings and work locally. This localization process is at the heart of the proof of the purity theorem 3.11.1 by Scholze and was typically missing in the work of Faltings in relative p-adic Hodge theory. It was missing in the work [26] too where in fact the constructed uniform Tate rings are perfectoid.

**Theorem 3.8.1.** — Let $(A, A^+)$ be an affinoid perfectoid ring.

1. The pair $(A, A^+)$ is sheafy.

2. Any rational subset is affinoid perfectoid.

3. For $X = \text{Spa}(A, A^+)$ one has

* $H^i(X, \mathcal{O}_X) = 0$ for $i > 0$,

* $H^i(X, \mathcal{O}_X^+)$ is almost zero for $i > 0$.

**3.8.1. The characteristic p case.** — Let us prove Theorem 3.8.1 when $A$ is an $\mathbb{F}_p$-perfectoid algebra. For any $U$ a rational subset of $X = \text{Spa}(A, A^+)$, $(\mathcal{O}_X(U), \mathcal{O}_X^+(U))$ is affinoid perfectoid (see Example 3.1.7). Let us fix $\varpi$ a pseudo-uniformizer of $A$. Now, we can write

$$A^+ = \widehat{\lim_{\underset{i}{\longrightarrow}} A_i^+}$$

where the colimit is filtered and $(A_i, A_i^+)$ is an affinoid Tate $\mathbb{F}_p((\varpi))$-algebra such that $A_i$ is of topological finite type over $\mathbb{F}_p((\varpi))$. Since $A^+$ is perfect we can suppose that for any index $i$, there exists $j \ge i$ such that $A_i \to A_j$ factorizes through Frob.

Let $(U_\alpha)_\alpha$ be a finite covering of $\text{Spa}(A, A^+)$ by rational subsets. There exists an index $i_0$ and a rational covering $(U_{i_0, \alpha})_\alpha$ of $\text{Spa}(A_{i_0}, A_{i_0}^+)$ whose pull back to $\text{Spa}(A, A^+)$ is $(U_\alpha)_\alpha$ (use [37, Proposition 3.10]). We note $(U_{i, \alpha})_\alpha$ the pullback of this covering to $X_i = \text{Spa}(A_i, A_i^+)$ when $i \ge i_0$. We then have, using the universal property of rationally localized affinoid rings,

$$\mathcal{O}_X^+(U_\alpha) = \widehat{\lim_{\underset{i \ge i_0}{\longrightarrow}}} \mathcal{O}_{X_i}^+(U_{i, \alpha})$$

where the completion is the $\varpi$-adic one. The same goes on for the $U_{\alpha_1} \cap \cdots \cap U_{\alpha_r}$, the finite intersections of elements of our covering. Let $C_i^\bullet$ be the Čech complex of the sheaf $\mathcal{O}_{X_i}^+$ relative to the covering $(U_{i, \alpha})_\alpha$. We have $H^0(C_i^\bullet) = A_i^+$ and $H^q(C_i^\bullet)$, $q > 0$, is $\varpi^\infty$-torsion. We deduce that $\widehat{\lim_{\underset{i \ge i_0}{\longrightarrow}}} H^0(C_i^\bullet) = \widehat{\lim_{\underset{i \ge i_0}{\longrightarrow}}} A_i^+$ and $\widehat{\lim_{\underset{i \ge i_0}{\longrightarrow}}} H^q(C_i^\bullet)$, $q > 0$, is almost zero i.e. killed by $\varpi^{1/p^n}$ for all $n \ge 0$.

Let $D^\bullet$ be the Čech complex of the presheaf $\mathcal{O}_X^+$ with respect to the covering $(U_\alpha)_\alpha$. We have

$$D^\bullet = \widehat{\lim_{\underset{i \ge i_0}{\longrightarrow}}} C_i^\bullet.$$

From this we deduce the existence of exact sequences for $q \ge 0$

$$0 \to R^1 \widehat{\lim_{\underset{n \ge 0}{\longrightarrow}}} [H^{q-1}(\widehat{\lim_{\underset{i \ge i_0}{\longrightarrow}}} C_i^\bullet)]/\varpi^n \to H^q(D^\bullet) \to R^1 \widehat{\lim_{\underset{n \ge 0}{\longrightarrow}}} [H^q(\widehat{\lim_{\underset{i \ge i_0}{\longrightarrow}}} C_i^\bullet)]/\varpi^n \to 0.$$

The result is deduced from the fact that $\widehat{\lim_{\underset{n \ge 0}{\longrightarrow}}} R^1 \widehat{\lim_{\underset{n \ge 0}{\longrightarrow}}}$ send projective systems of almost zero modules to almost zero modules.
**3.8.2. Some integral perfectoid rings and applications.** — Before proving Theorem 3.8.1 we need some preliminary result about untilted rational domains.

**Proposition 3.8.2.** — Let $A$ be a perfectoid ring. Let $f_1, \dots, f_n \in A^{b, \circ}$ that generate an open ideal and $g \in A^{b, \circ}$. Then the $\varpi$-adic completion of the sub-ring

$$ \bigcup_{k \ge 0} A^{\circ} \left[ \frac{\left( \frac{f_1^{1/p^k}}{g^{1/p^k}} \right)^{\sharp}}{\left( \frac{f_n^{1/p^k}}{g^{1/p^k}} \right)^{\sharp}} , \dots, \frac{\left( \frac{f_1^{1/p^k}}{g^{1/p^k}} \right)^{\sharp}}{\left( \frac{f_n^{1/p^k}}{g^{1/p^k}} \right)^{\sharp}} \right] $$

of $\bigcup_{k \ge 0} A \left[ \frac{1}{(g^{1/p^k})^{\sharp}} \right]$ is integral perfectoid.

*Proof.* — Let $\varpi \in A^b$ be a pseudo-uniformizer satisfying $(\varpi^\sharp)^p|p$ (and thus we take the $\varpi^\sharp$-adic completion in the statement of the proposition). Let $R$ be the ring in the statement. Since $f_1, \dots, f_n$ generate an open ideal, for $k \gg 0$, $\varpi \in A^{b, \circ} f_1^{1/p^k} + \dots + A^{b, \circ} f_n^{1/p^k}$. Using the identification $A^{b, \circ}/\varpi^p \xrightarrow{\sim} A^\circ/(\varpi^\sharp)^p$, we deduce that for $k \gg 0$, $(f_1^{1/p^k})^\sharp, \dots, (f_n^{1/p^k})^\sharp$ generate an open ideal and thus this is true for any $k \ge 0$. Thus, for $k \ge 0$,

$$ A^{\circ} \left[ \frac{\left( \frac{f_1^{1/p^k}}{g^{1/p^k}} \right)^{\sharp}}{\left( \frac{f_n^{1/p^k}}{g^{1/p^k}} \right)^{\sharp}} , \dots, \frac{\left( \frac{f_1^{1/p^k}}{g^{1/p^k}} \right)^{\sharp}}{\left( \frac{f_n^{1/p^k}}{g^{1/p^k}} \right)^{\sharp}} \right] = \varinjlim_{k \ge 0} A^{\circ} [T_1, \dots, T_n] / \left( (g^{1/p^k})^\sharp T_1 - (f_1^{1/p^k})^\sharp, \dots, (g^{1/p^k})^\sharp T_n - (f_n^{1/p^k})^\sharp \right) $$

where the transition maps are given by $T_i \mapsto T_i^p$, $1 \le i \le n$. The ring $R$ is without $\varpi^\sharp$-torsion, and thus its $\varpi^\sharp$-adic completion is without $\varpi^\sharp$-torsion too. Moreover,

$$ R/\varpi^\sharp R = \varinjlim_{k \ge 0} (A^{b, \circ}/\varpi)[T_1, \dots, T_n]/(g^{1/p^k} T_1 - f_1^{1/p^k}, \dots, g^{1/p^k} T_n - f_n^{1/p^k}) $$

where the transition maps are given by $T_i \mapsto T_i^p$, $1 \le i \le n$. This can be rewritten as

$$ (A^{b, \circ}/\varpi)[T_1^{1/p^\infty}, \dots, T_n^{1/p^\infty}]/(T_1^{1/p^k} g^{1/p^k} - f_1^{1/p^k}, \dots, T_n g^{1/p^k} - f_n^{1/p^k})_{k \ge 0} $$

The same type of formula holds for the reduction of $R$ modulo $(\varpi^\sharp)^p$. One easily deduces that
Frob : $R/\varpi^\sharp \xrightarrow{\sim} R/(\varpi^\sharp)^p$. $\square$

**Corollary 3.8.3.** — Let $X = \text{Spa}(A, A^+)$ with $(A, A^+)$ affinoid perfectoid.

1. For $f_1, \dots, f_n \in A^b$ that generate the unit ideal and $g \in A^b$, the rational subset

$$ X \left( \frac{f_1^\sharp, \dots, f_n^\sharp}{g^\sharp} \right) $$

is affinoid perfectoid.

2. Via the homeomorphism $|\text{Spa}(A, A^+)| \xrightarrow{\sim} |\text{Spa}(A^b, A^{b,+})|$, if $U$ is a rational subset of $|\text{Spa}(A^b, A^{b,+})|$ and $U^\sharp$ the corresponding one in $|\text{Spa}(A, A^+)|$ then

(a) $U^\sharp$ is affinoid perfectoid with tilting $U$,

(b) If $\varpi^\sharp|p$ there is an identification

$$ \mathcal{O}(U)^+/\varpi = \mathcal{O}(U^\sharp)^+/\varpi^\sharp. $$

We will prove later that any rational domain is affinoid perfectoid but let us already note the following corollary.

**Corollary 3.8.4.** — For $(A, A^+)$ affinoid perfectoid,

1. $\text{Spa}(A, A^+)$ has a base of its topology made of affinoid perfectoid subsets.

2. The completed residue fields $(K(x), K(x)^+)$ for $x \in \text{Spa}(A, A^+)$ are affinoid perfectoid fields.

**3.8.3. Stable uniformity of affinoid perfectoid rings and applications.** —
**3.8.3.1.** A general result. —

**Lemma 3.8.5.** — Let $(A, A^+)$ be an affinoid Tate ring and let $X = \text{Spa}(A, A^+)$. Suppose that is has a base $\mathcal{B}$ of rational subsets such that for each $U \in \mathcal{B}$, $\mathcal{O}_X(U)$ is uniform. Then $(A, A^+)$ is stably uniform.

*Proof.* — This is immediately reduced to proving that $(A, A^+)$ is uniform. Let $A_0$ be a ring of definition of $A$ and $\varpi \in A_0$ a pseudo-uniformizer. Let $(U_i)_i$ be a finite rational cover of $\text{Spa}(A, A^+)$ such that for all $i$, $\mathcal{O}(U_i)$ is uniform. According to Lemma 2.9.10 one can find $f_1, \dots, f_n \in A$ generating the unit ideal and such that the associated standard rational cover satisfies

$$ \forall j \in \{1, \dots, n\}, \exists i, \text{Spa}(A, A^+) \left( \frac{f_1, \dots, f_n}{f_j} \right) \subset U_i. $$

We can suppose that $f_1, \dots, f_n \in A_0$. Consider now

$$ \pi : \tilde{X} \to X = \text{Spec}(A_0) $$

the blow up of the ideal $(f_1, \dots, f_n)$. We note $X_\eta = \text{Spec}(A)$. There is a diagram

<figure><img src="image_37.png" /></figure>

For each $j \in \{1, \dots, n\}$, the morphism $A^\circ \to A(\frac{f_1, \dots, f_n}{f_j})^\circ$ factorizes through $\mathcal{O}(U_i)^\circ$ for some index $i$. From this we deduce that there exists $N \ge 0$ such that for all $j \in \{1, \dots, n\}$, $A^\circ \to \varpi^{-N} A_0(\frac{f_1, \dots, f_n}{f_j})$ and thus

$$ A^\circ \to \varpi^{-N} A_0 \left[ \frac{f_1}{f_j}, \dots, \frac{f_n}{f_j} \right]. $$

We thus have

$$ A^\circ \subset \varpi^{-N} H^0(\tilde{X}, \mathcal{O}_{\tilde{X}}). $$

Now, since $\tilde{X} \to X$ is proper of finite presentation, $H^0(\tilde{X}, \mathcal{O}_{\tilde{X}})$ is a finite $A_0$-module and it is thus bounded. We deduce the result. $\square$

**3.8.3.2. Applications.** — We can now put together Corollary 3.8.4 and Lemma 3.8.5 to obtain the following result.

**Theorem 3.8.6.** — For $(A, A^+)$ affinoid perfectoid and $X = \text{Spa}(A, A^+)$:

1. A is stably uniform and thus sheafy with $\mathcal{O}_X$ acyclic.

2. For $i > 0$, $H^i(X, \mathcal{O}_X^+)$ is almost zero.

*Proof.* — It only remains to prove the second point. Via the homeomorphism $|X| \tilde{\to} |X^b|$ this is reduced to $H^i(X^b, \mathcal{O}_X^+) = 0$. But we have, as a sheaf on $X^b$, $\mathcal{O}_X^+ / \varpi^\sharp = \mathcal{O}_{X^b}^+ / \varpi$ if $\varpi^\sharp | p$. We thus have for $i > 0$, $H^i(X, \mathcal{O}_X^+ / \varpi^\sharp)$ is almost zero. Replacing $\varpi$ by $\varpi^{1/p^n}$ for all $n \ge 0$ one easily deduces that $H^i(X, \mathcal{O}_X^+)$ is almost zero. $\square$

**3.8.4. Approximation of rational open subsets.** — We have proven everything in Theorem 3.8.1 but point (2).

**Proposition 3.8.7.** — Let $(A, A^+)$ affinoid perfectoid. Then, for any rational open subset $U \subset \text{Spa}(A, A^+)$ there exists $f_1, \dots, f_n \in A^b$ that generate $A$ and $g \in A^b$ such that $U = \text{Spa}(A, A^+) (\frac{f_1, \dots, f_n}{g^2})$.
*Proof.* — Write $U = \text{Spa}(A, A^+)(\frac{a_1, \dots, a_n}{b})$ where $Aa_1 + \dots + Aa_n = A$. Let's us chose a pseudo-uniformizer $\varpi \in A$ such that $\varpi b_U^{-1} \in \mathcal{O}(U)^+$. Then,

$$U = \bigcap_{i=1}^{n} \text{Spa}(A, A^{+}) \left( \frac{a_i, \varpi}{b} \right).$$

Since rational open subsets are stable under finite intersections, it is sufficient to prove the result for a rational domain of the form $\text{Spa}(A, A^+)(\frac{a, \varpi}{b})$. We now use Proposition 3.8.8. Up to multiplying $\varpi$ by a unit we can suppose $\varpi = (\varpi^b)^\sharp$ for some $\varpi^b \in A^b$. Using the proposition we can chose $\alpha \in A^b$ such that $\text{forall } x \in \text{Spa}(A, A^+)$,

$$|a(x) - \alpha^{\sharp}(x)| < \sup\{|a(x)|, |\varpi(x)|\}$$

and thus

$$\sup\{|a(x)|, |\varpi(x)|\} = \sup\{|\alpha^{\sharp}(x)|, |\varpi(x)|\}.$$

Now, if we chose $\beta \in A^b$ such that for all $x$,

$$|b(x) - \beta^{\sharp}(x)| < \sup\{|b(x), |\varpi|\}|$$

we find that

$$\text{Spa}(A, A^{+})\left(\frac{a, \varpi}{b}\right) = \text{Spa}(A, A^{+})\left(\frac{\alpha^{\sharp}, (\varpi^{b})^{\sharp}}{\beta^{\sharp}}\right). \quad \square$$

For the proof of the next proposition we follow [43, Lemma 5.5] rather than [49, Corollary 6.7]. This approximation result is a key point in the theory.

**Proposition 3.8.8.** — For $f \in A^\circ$ and $\varpi$ a pseudo-uniformizer, there exists $h \in A^{b, \circ}$ such that for all $x \in \text{Spa}(A, A^+)$,

$$|f(x) - h^{\sharp}(x)| < \sup\{|f(x)|, |\varpi(x)|\}|$$

*Proof.* — We can suppose that $\varpi = \varpi^{b, \sharp}$ for some pseudo-uniformizer $\varpi^b$ of $A^b$. Let $\xi \in W(A^{b,+})$ be a degree 1 distinguished element generating the kernel of $\theta : W(A^{b,+}) \to A^+$. Up to multiplying $\xi$ by a unit of $W(A^{b,+})$ we can suppose

$$\xi \in p + [a]W(A^{b,+})$$

with $a \in A^{\circ\circ}$. Write $f = \theta(g)$ with

$$g = \sum_{k \ge 0} [g_k] p^k \in W(A^{b,+}).$$

For $h = \sum_{k \ge 0} [h_k] p^k \in W(A^{b,+})$, we note

$$\tau(h) = \frac{h - [h_0]}{p} \in W(A^{b,+}).$$

Let us now define by induction the sequence $(g_i)_{i \ge 0}$ by setting

$$\begin{aligned} g_0 &= g \\ g_{i+1} &= g_i - \tau(g_i)\xi. \end{aligned}$$

Let $n \in \mathbb{N}$ be such that $\varpi^b|a^n$ in $A^{b,+}$. Write for any $i \ge 0$,

$$g_i = \sum_{k \ge 0} [g_{i,k}] p^k.$$

Let now $x \in \text{Spa}(A^b, A^{b,+})$. Let

$$N = \inf\{ i \ge 0 \mid |g_{i,0}(x)| > |a(x)|^{i+1} \} \in \mathbb{N} \cup \{+\infty\}.$$

Let us prove by induction on $i$ that for $0 \le i \le N$ one has

$$|g_i(x_G)| \le |a(x)|^i.$$
The case $i=0$ is evident. Now if we suppose proven the inequality for $i < N$ we have

$$g_{i+1} \in [g_{i,0}] + [a] \sum_{k \ge 0} [g_{i,k+1}] p^k W(A^{b,+})$$

and thus

$$|g_{i+1}(x_G)| \le |a(x)|^{i+1}$$

which completes the induction.

Suppose $N < +\infty$. From the inequality $|g_N(x_G)| \le |a(x)|^N$ and $|g_{N,0}(x)| > |a(x)|^{N+1}$ we get

$$\forall k > 0, |g_{N+1,k}(x)| < |g_{N+1,0}(x)|.$$

By induction on $i > N$, starting with $i = N + 1$, we find that the preceding inequality extends to

$$\forall i > N, \forall k > 0, |g_{i,k}(x)| < |g_{i,0}(x)|.$$

Now we have

$$|f(x^\sharp) - g_{n,0}^\sharp(x^\sharp)| \le \sup_{k>0} |g_{n,k}(x)| \cdot |p(x^\sharp)|^k.$$

If $n \le N$ this is strictly less than

$$|a(x)|^{n+1} < |\varpi(x)|$$

since $\varpi|a^n$. If $n > N$ this is strictly less than

$$|g_{n,0}(x)| = |g_{n,0}^\sharp(x^\sharp)|.$$

We can thus take $h = g_{n,0}$ and this concludes the proof. $\square$

**Corollary 3.8.9.** — For $(A, A^+)$ affinoid perfectoid, any rational subset of $\text{Spa}(A, A^+)$ is affinoid perfectoid. The tilting correspondence induces a bijection between rational subsets of $\text{Spa}(A, A^+)$ and rational subsets of $\text{Spa}(A^b, A^{b,+})$.

We have thus proven Theorem 3.8.1 in its entirety.

## 3.9. Perfectoid spaces

By definition a perfectoid space is an adic space that is locally isomorphic to the adic spectrum of an affinoid perfectoid ring, the so-called affinoid perfectoid spaces. We now focus on giving examples of perfectoid spaces and some constructions we can do with them that may not exist in the category of Noetherian adic spaces.

**3.9.1. The tilting equivalence.** — We can now upgrade the tilting equivalence of Corollary 3.4.13 to spaces.

**Proposition 3.9.1.** — For $X$ be a perfectoid space tilting induces an equivalence

$$(-)^b : \text{Perfectoid spaces}/X \xrightarrow{\sim} \text{Perfectoid spaces}/X^b.$$

**3.9.2. The perfectoid space associated to an $\mathbb{F}_p$-adic analytic space.** — Let us now collect a natural consequence of the results of Section 3.1.2.

**Proposition 3.9.2.** — The inclusion of the category of $\mathbb{F}_p$-perfectoid spaces in the category of $\mathbb{F}_p$-adic analytic spaces has a right adjoint, $X \mapsto \lim_{\leftarrow \xrightarrow{\text{Frob}} X} X$.
*Proof.* — This is an easy consequence of Corollary 3.1.8. More precisely, for $(A, A^+)$ an $\mathbb{F}_p$-affinoid Tate sheafy ring,

$$ \lim_{\underset{\text{Frob}}{\leftarrow}} \mathrm{Spa}(A, A^+) = \mathrm{Spa}(R, R^+) $$

where $R^+ = \widehat{\lim}_{\underset{\text{Frob}}{\leftarrow}} A^+$ ($\varpi$-adic completion) and $R = R^+ [\frac{1}{\varpi}]$. In fact, if $A_0$ is a ring of definition of $A$ contained in $A^+$, for any $x \in A^+$, since $x$ is power bounded there exists an integer $N \ge 1$ such that for all $k \ge 0$, $x^{p^k} \in \varpi^{-N} A_0$. From this we deduce that $\lim_{\underset{\text{Frob}}{\rightarrow}} A_0$ is almost equal to $\lim_{\underset{\text{Frob}}{\rightarrow}} A^+$ and thus their $\varpi$-adic completions are almost equal. $\square$

**3.9.3. Cofiltered limits of affinoid perfectoids.** — The next operation on perfectoid spaces does not exists in general in the category of Noetherian adic spaces. This will be a basic element in the definition of the pro-étale topology on perfectoid spaces.

**Proposition 3.9.3.** — In the category of perfectoid spaces cofiltered limits of affinoid perfectoid spaces exist and are affinoid perfectoid.

*Proof.* — Consider $\lim_{\underset{\text{Frob}}{\leftarrow}}_{i \in I} \mathrm{Spa}(A_i, A_i^+)$ a cofiltered limit of affinoid perfectoids. Fix $i_0 \in I$ and $\varpi \in A_{i_0}$ a pseudo-uniformizer. We still note $\varpi$ for its image in $A_i$ when $i \ge i_0$. Then one verifies that if

$$ R^+ = \widehat{\lim}_{\underset{i}{\leftarrow}} A_i^+ $$

($\varpi$-adic completion) then $R^+$ is integral perfectoid and

$$ \lim_{\underset{i \in I}{\leftarrow}} \mathrm{Spa}(A_i, A_i^+) = \mathrm{Spa}(R^+ [\frac{1}{\varpi}], R^+) $$

This proves the statement. $\square$

**Example 3.9.4.** — Let $X$ be a perfectoid space. Then for $x \in X$,

$$ \begin{aligned} X_x &= \lim_{\substack{U \ni x \\ \text{nbd of } x}} U \\ &= \mathrm{Spa}(K(x), K(x)^+) \end{aligned} $$

that is perfectoid.

**3.9.4. The example of profinite sets.** — The possibility to include profinite sets in the theory of perfectoid spaces allows us to speak about pro-étale torsors under a profinite group.

**Lemma 3.9.5.** — Let $P$ be a profinite set. Note $\underline{P}$ for the functor on perfectoid spaces that sends $X$ to $\mathscr{C}(|X|, P)$. Then for any perfectoid space $X$, $X \times \underline{P}$ is representable by a perfectoid space.

*Proof.* — This is reduced to the affinoid perfectoid case. In this case, if $X = \mathrm{Spa}(A, A^+)$ and $P = \lim_{\underset{\text{Frob}}{\leftarrow}} P_i$ with $P_i$ finite, then

$$ \begin{aligned} X \times \underline{P} &= \lim_{\underset{i}{\leftarrow}} X \times \underline{P_i} \\ &= \lim_{\underset{i}{\leftarrow}} \mathrm{Spa}(\mathscr{C}(P_i, A), \mathscr{C}(P_i, A^+)) \\ &= \mathrm{Spa}(\mathscr{C}(P, A), \mathscr{C}(P, A^+)). \end{aligned} $$

This proves the result. $\square$
**3.9.5. The example of perfect formal schemes.** — Let us here give an example that again relies on Proposition 3.1.5.

**Proposition 3.9.6.** — Let $R$ be a perfect $\mathbb{F}_p$-algebra that is $I$-adic with $I$ an ideal of finite type. Then, $\text{Spa}(R, R)_a = \text{Spa}(R, R) \setminus V(I)$ is a perfectoid space.

**3.9.6. Zariski closed subsets.** — If $X$ is a perfectoid space and $Z \subset |X|$ is a subset we can look at the functor on perfectoid spaces that sends $Y$ to the set of morphisms $Y \to X$ such that $|Y| \to Z$. If this is representable by a perfectoid space we will say that $Z$ is represented by a perfectoid space.

Let $X = \text{Spa}(A, A^+)$ be affinoid perfectoid. Let $I$ be an ideal of $A$.

**Lemma 3.9.7.** — The closed subset $V(I) = \{x \in X \mid \forall f \in I, |f(x)| = 0\} \subset |X|$ is represented by an affinoid perfectoid ring.

*Proof.* — Fix $\varpi$ a pseudo-uniformizer. We have

$$V(I) = \varinjlim_{\{f_1, \dots, f_k\} \subset I, \text{finite subset } n \ge 1} X\left(\frac{f_1, \dots, f_k}{1}\right)$$

and is thus a cofiltered limit of affinoid perfectoids. $\square$

The characteristic $p$ case is simpler for the following reason. Suppose $X = \text{Spa}(A, A^+)$ is an $\mathbb{F}_p$-affinoid perfectoid space and $I \subset A$ is an ideal. Then, if $V(I) = \text{Spa}(R, R^+)$ one has

$$R = \varinjlim_{\text{Frob}} A/I,$$

see Section 3.1.2. Let $I^{1/p^\infty}$ be the perfection of $I$, an ideal of $A$. One has moreover, see Remark 3.1.6,

$$R^\circ = \varprojlim_{\text{almost}} A^\circ / A^\circ \cap I^{1/p^\infty}$$

where the completion if the $\varpi$-adic one. Thus, at the end $A^+ \to R^+$ is almost surjective. This is still the case in general but the proof is more complicated, see Section 3.13.

**3.9.7. Connected components.** — If $X$ is a qcqs perfectoid space then $\pi : |X| \to \pi_0(X)$. Then any connected component $\pi^{-1}(c)$, $c \in \pi_0(X)$, is affinoid perfectoid. In fact,

$$\pi^{-1}(c) = \varinjlim_{U \ni c} \pi^{-1}(U)$$

where $U$ goes through the set of open/closed neighborhoods of $c$ in the profinite space $\pi_0(X)$. For such a $U$, $\pi^{-1}(U)$ is open/closed in $X$ and thus this limit exists.

**3.9.8. Fiber products.** — One of the reasons perfectoid spaces are good objects is the next proposition.

**Proposition 3.9.8.** — The category of perfectoid spaces has fiber products. Moreover fiber products commute with tilting and the fiber product of affinoid perfectoid spaces is affinoid perfectoid.

*Proof.* — Via the tilting equivalence of Proposition 3.9.1 it is enough to prove the existence of $X \times_Y Z$ for $X, Y, Z \mathbb{F}_p$-perfectoid spaces. Now, for $\mathbb{F}_p$-perfectoid spaces the result is a consequence of Proposition 3.1.5. $\square$
**Remark 3.9.9.** — Let $\mathbb{Q}_p^{cyc}$ be the completion of the cyclotomic extension of $\mathbb{Q}_p$, a perfectoid field. The topological ring $\mathbb{Q}_p^{cyc} \hat{\otimes}_{\mathbb{Q}_p} \mathbb{Q}_p^{cyc}$ is not perfectoid. Nevertheless, the fiber product $\text{Spa}(\mathbb{Q}_p^{cyc}) \times \text{Spa}(\mathbb{Q}_p^{cyc})$ exists in the category of $\mathbb{Q}_p$-perfectoid spaces and is equal to

$$ \text{Spa}(\mathbb{Q}_p^{cyc}) \times \text{Gal}(\mathbb{Q}_p^{cyc} | \mathbb{Q}_p) = \text{Spa} \left( \mathcal{C}(\text{Gal}(\mathbb{Q}_p^{cyc} | \mathbb{Q}_p), \mathbb{Q}_p^{cyc}), \mathcal{C}(\text{Gal}(\mathbb{Q}_p^{cyc} | \mathbb{Q}_p), \mathbb{Z}_p^{cyc}) \right) $$

but this is not a fiber product in the category of adic spaces. One has to be careful in which category one takes the fiber products.

Next result is more surprising but an essential tool in [28].

**Proposition 3.9.10.** — Let $k$ be a characteristic $p$ discrete perfect field. Let $X$ and $Y$ be $k$-perfectoid spaces. Then the fiber product $X \times_{\text{Spa}(k)} Y$ exists in the category of adic spaces and is a perfectoid space.

*Proof.* — Suppose $X = \text{Spa}(A, A^+)$ and $Y = \text{Spa}(B, B^+)$ are affinoid perfectoid. Consider $A^+ \hat{\otimes}_k B^+$ equipped with the $(\varpi_A \otimes 1, 1 \otimes \varpi_B)$-adic topology. We can apply Proposition 3.9.6 to conclude that

$$ X \times_k Y = \text{Spa}(A^+ \hat{\otimes}_k B^+, A^+ \hat{\otimes}_k B^+) \setminus V(\varpi_A \otimes \varpi_B) $$

is perfectoid.

□

One has to be careful that in the preceding situation, the fiber product of affinoid perfectoid spaces is not affinoid and not even quasi-compact anymore!

**Example 3.9.11.** — One has

$$ \text{Spa}(k((X^{1/p^\infty}))) \times_{\text{Spa}(k)} \text{Spa}(k((Y^{1/p^\infty}))) = \text{Spa}(k[[X^{1/p^\infty}, Y^{1/p^\infty}]] \setminus V(XY)). $$

## 3.10. Étale morphisms of perfectoid spaces

### 3.11. Purity

The purity theorem is inspired by the following evident result: let $X$ be a perfect $\mathbb{F}_p$-scheme and $Y \to X$ be an étale morphism. Then $Y$ is a perfect scheme.

**Theorem 3.11.1.** — For $(A, A^+)$ affinoid perfectoid, a Huber pair $(B, B^+)$ and a morphism $(A, A^+) \to (B, B^+)$ such that $A \to B$ is finite étale and $B^+$ is the integral closure of $A^+$ in $B$, $(B, B^+)$ is affinoid perfectoid.

*Proof.* — The case of a perfectoid field is treated in [27]. Let $X = \text{Spa}(A, A^+)$. The characteristic $p$-case is deduced from Proposition 3.1.5.

Since $B$ is a projective $A$-module of finite type, it defines a sheaf of $\mathcal{O}_X$-algebras $\mathcal{B} = B \otimes_A \mathcal{O}_X$. Now, we use the equivalence of 2.7.6, the case of a perfectoid field, the characteristic $p$ case and the tilting equivalence to deduce that over a rational covering $(U_i)_i$ of $X$, $\mathcal{B}(U_i)$ is a perfectoid ring. According to Proposition 4.1.3, $B$ is stably uniform and thus sheafy. Thus, $\text{Spa}(B, B^+) = \text{a perfectoid space that is affinoid. But the morphism } \text{Spa}(B, B^+)^b \to \text{Spa}(A^b, A^{b,+}) \text{ is finite locally free étale and thus } \text{Spa}(B, B^+)^b \text{ is affinoid perfectoid. Via the tilting equivalence we deduce that } (B, B^+) \text{ is affinoid perfectoid.}$ □

**Corollary 3.11.2.** — Let $X$ be an Noetherian analytic adic space and $X' \to X$ be a morphism with $X'$ perfectoid. For any $U \to X$ étale one can define $U \times_X X'$ as an étale perfectoid space over $X'$. This defines a continuous morphism of sites

$$ X'_{\text{ét}} \to X_{\text{ét}}. $$
### 3.12. A criterion for a morphism to be an isomorphism

**Proposition 3.12.1.** — Let $f: X \to Y$ be a $qc$ $qs$ morphism of perfectoid spaces. The following are equivalent:

1. $f$ is an isomorphism.

2. For all affinoid perfectoid fields $(K, K^+)$, the map $X(K, K^+) \to Y(K, K^+)$ is bijective.

3. For all algebraically closed affinoid perfectoid fields $(K, K^+)$, the map $X(K, K^+) \to Y(K, K^+)$ is bijective.

*Proof.* — It is clear that (1) implies (2) and (3) and that (2) implies (3). Let us verify (2) ⇒ (1). Let us first verify that $|f|: |X| \to |Y|$ is bijective. For this it is enough to verify that for all $y \in Y$,

$$X_y := X \times_Y \mathrm{Spa}(K(y), K(y)^+) \to \mathrm{Spa}(K(y), K(y)^+)$$

is an isomorphism. By hypothesis this morphism has a section $\mathrm{Spa}(K(y), K(y)^+) \to X_y$. Let $x \in X_y$ be the image via this section of the closed point $y$ of $\mathrm{Spa}(K(y), K(y)^+)$. Consider now the morphism $\mathrm{Spa}(K(x), K(x)^+) \hookrightarrow X_y$. The composite

$$\mathrm{Spa}(K(x), K(x)^+) \hookrightarrow X_y \to \mathrm{Spa}(K(y), K(y)^+)$$

has a section (the preceding section sends $y$ to $x$ and thus the generalizations of $y$ to the generalizations of $x$). We deduce that $K(y) = K(x)$ and since $x \mapsto y$, $K(x)^+ = K(y)^+$. We thus have

$$\mathrm{Spa}(K(x), K(x)^+) \xrightarrow{\sim} \mathrm{Spa}(K(y), K(y)^+).$$

But now the morphism

$$X_y \to \mathrm{Spa}(K(y), K(y)^+)$$

induces a bijection for all affinoid fields $(K, K^+)$

$$X_y(K, K^+) \xrightarrow{\sim} \mathrm{Spa}(K(y), K(y)^+)(K, K^+).$$

We deduce that for all $(K, K^+)$,

$$\mathrm{Spa}(K(x), K(x)^+)(K, K^+) \xrightarrow{\sim} X_y(K, K^+).$$

But for such an affinoid field $(K, K^+)$ and an element of $\mathrm{Spa}(K(x), K(x)^+)$, the image of the morphism

$$\mathrm{Spa}(K, K^+) \to \mathrm{Spa}(K(x), K(x)^+) \to X_y$$

is contained in the generalizations of $x$ in $X_y$. Since

$$\bigcup_{\substack{(K, K^+) \\ u: \mathrm{Spa}(K, K^+) \to X_y}} \mathrm{Im}(u) = |X_y|$$

we deduce that $|X_y|$ is the set of generalizations of $x$ i.e.

$$\mathrm{Spa}(K(x), K(x)^+) \xrightarrow{\sim} X_y.$$

We thus have a bijection $|f|: |X| \xrightarrow{\sim} |Y|$ satisfying: for all $x \in X$,

$$(K(f(x)), K(f(x))^+) \xrightarrow{\sim} (K(x), K(x)^+).$$

We can now apply Corollary 2.11.5 to deduce that $|f|$ is an homeomorphism. Let us now replace $Y$ by an affinoid perfectoid open subset and fix $\varpi$ a pseudo-uniformizer of $\mathcal{O}(Y)$. According to Lemma 2.11.1,

$$f^{-1}(\mathcal{O}_Y^+ / \varpi) \to \mathcal{O}_X^+ / \varpi$$

is an isomorphism. We can now replace $\varpi$ by $\varpi^n$ for all $n$ to deduce that

$$f^{-1}\mathcal{O}_Y^+ = \varinjlim_{n \ge 0} f^{-1}(\mathcal{O}_Y^+ / \varpi^n) \xrightarrow{\sim} \varinjlim_{n \ge 0} \mathcal{O}_X^+ / \varpi^n = \mathcal{O}_X^+.$$
This implies that $f$ is an isomorphism and we have proven (2) ⇒ (1).

Point (3) is deduced from point (2) using the equality for any affinoid perfectoid field $(K, K^+)$,

$$X(K, K^+) = X(\widehat{K}, \widehat{K}^+) \stackrel{\mathrm{Gal}(\bar{K}|K)}{=} \square$$

**3.13. André lemma and Zariski closed subsets**

**Lemma 3.13.1 (André).** — Let $(A, A^+)$ be an affinoid perfectoid ring and $f \in A$. There exists a morphism $(A, A^+) \to (B, B^+)$ of affinoid perfectoid rings such that

1. The image of $f$ in $B$ is of the form $g^\sharp$ for some $g \in B^\flat$.

2. The morphism $A^+/\varpi \to B^+/\varpi$ is almost faithfully flat.

More precisely, we can take $\mathrm{Spa}(B, B^+) = V(T-f) \subset \mathrm{Spa}(A\langle T^{1/p^\infty} \rangle, A^+\langle T^{1/p^\infty} \rangle)$.

*Proof.* — □

The following corollary is worth noting although we won't use it.

**Corollary 3.13.2.** — For $(A, A^+)$ an affinoid perfectoid ring there exists a morphism $(A, A^+) \to (B, B^+)$ of affinoid perfectoid rings such that

1. $A^+/\varpi \to B^+/\varpi$ is almost faithfully flat.

2. For any $f \in B$ there exists $g \in B^\flat$ such that $f = g^\sharp$.

*Proof.* — For any finite subset $S = \{f_1, \dots, f_n\} \subset A$ let
$$ \mathrm{Spa}(A_S, A_S^+) = V(T_1 - f_1, \dots, T_n - f_n) \subset \mathrm{Spa}(A\langle T_1^{1/p^\infty}, \dots, T_n^{1/p^\infty} \rangle, A^+\langle T_1^{1/p^\infty}, \dots, T_n^{1/p^\infty} \rangle). $$

We can now consider

$$ \mathrm{Spa}(A_1, A_1^+) = \lim_{\leftarrow\, S} \mathrm{Spa}(A_S, A_S^+). $$

This construction gives by induction a projective system

$$ \mathrm{Spa}(A, A^+) \leftarrow \mathrm{Spa}(A_1, A_1^+) \leftarrow \dots \leftarrow \mathrm{Spa}(A_n, A_n^+) \leftarrow \mathrm{Spa}(A_{n+1}, A_{n+1}^+) \leftarrow \dots $$

One verifies easily that

$$ \mathrm{Spa}(B, B^+) = \lim_{\underset{n \ge 1}{\leftarrow}} \mathrm{Spa}(A_n, A_n^+) $$

satisfies the expected properties.

□
# CHAPTER 4

## THE PRO-ÉTALE TOPOLOGY

### 4.1. Pro-étale perfectoid covers before Scholze

**4.1.1. Colmez and Faltings remarks.** — The construction of pro-étale perfectoid covers in relative p-adic Hodge theory goes back to the work of numerous authors. Let us cite some constructions.

**Proposition 4.1.1 (Colmez, [19, Section 4.4]).** — Let A be a complete uniform Tate $\mathbb{Q}_p$-algebra. There exists a filtered system of finite étale uniform A-algebras $(A_i)_{i \in I}$ such that

$$ \widehat{\lim}_{i \in I} A_i $$

is perfectoid.

*Proof.* — Consider the $\mathbb{F}_p$-vector space

$$ V = 1 + A^{\circ\circ} / (1 + A^{\circ\circ})^p. $$

For $S \subset V$ a finite dimension sub-vector space we note $\tilde{S}$ for its reciprocal image inside $1 + A^{\circ\circ}$. For such an $S$,

$$ A_S = A \otimes_{\mathbb{Z}[\tilde{S}], [s] \mapsto [s^p]} \mathbb{Z}[\tilde{S}]. $$

If $x_1, \dots, x_d \in \tilde{S}$ induce a basis of $S$ then

$$ A[X_1, \dots, X_d] / (X_1^p - x_1, \dots, X_d^p - x_d) \xrightarrow{\sim} A_S $$

and thus $A \to A_S$ is finite étale. According to Proposition 4.1.3, $A_S$ is Tate uniform. We can then look at

$$ A_1 = \widehat{\lim}_{S} A_S $$

Let us note that $A_1$ is uniform. In fact, if we fix $\varpi$ and $\beta \in ]0, 1[$ then the transition maps in the inductive limit $\widehat{\lim}_{S} A_S$ are isometric for the spectral norm associated to $\varpi$ and $\beta$, see Theorem 2.1.16 and thus the topology of $A_\infty$ is induced by a power multiplicative norm. We now reiterate the operation to obtain a sequence

$$ A = A_0 \to A_1 \to \dots \to A_n \to \dots. $$

We note

$$ A_\infty = \widehat{\lim}_{n \ge 0} A_n $$

that is the completion of $\widehat{\lim}_{n \ge 0} A_n$ for the spectral norm or, said in another way, $\widehat{\lim}_{n \ge 0} \widehat{A_n^\circ[\frac{1}{\varpi}]}$ where the completion is the $\varpi$-adic one. We now have

$$ 1 + A_\infty^{\circ\circ} / (1 + A_\infty^{\circ\circ})^p = \widehat{\lim}_{n \ge 0} 1 + A_n^{\circ\circ} / (1 + A_n^{\circ\circ})^p. $$
But the transition maps in this inductive limit are zero and thus any element of $1 + A_{\infty}^{\circ\circ}$ is a $p$-power. Now, if $\bar{\omega}$ is a pseudo-uniformizer and $x_n$ a $p^n$-th root of $1 + \bar{\omega}$, for $n \gg 0$, $x_n - 1$ is a pseudo-uniformizer satisfying $(x_n - 1)^p|p$. It is moreover immediate that for such a uniformizer $\bar{\omega}$, the Frobenius $A_{\infty}^{\circ}/\bar{\omega} \to A_{\infty}^{\circ}/\bar{\omega}^p$ is surjective. $\square$

**Remark 4.1.2.** — The $C$-perfectoid rings $A$, $C|\mathbb{Q}_p$ complete algebraically closed, for which any element of $1 + A^{\circ\circ}$ has a $p$-power are called sympathetic by Colmez. Those are exactly those for which the logarithm $\log : 1 + A^{\circ\circ} \to A$ is surjective. Another characterization is that those are exactly the one satisfying $H_{\text{et}}^1(\text{Spa}(A, A^{\circ}), \mathbb{Q}_p) = 0$.

**Proposition 4.1.3.** — Let $A$ be a uniform Tate ring and $B$ a finite étale $A$-algebra. Then $B$ is a uniform Tate ring and $B^{\circ}$ is the integral closure of $A^{\circ}$ in $B$.

*Proof.* — Let $R$ be the integral closure of $A^{\circ}$ in $B$. We can find a finite étale $B$-algebra $B'$ such that $A \to B'$ is Galois with Galois group $G$ and $B \to B'$ is Galois with Galois group $H$. One can for example take

$$ \mathrm{Spec}(B') = \mathrm{Isom}_{\mathrm{Spec}(A)}(\mathrm{Spec}(A) \times \{1, \dots, n\}, \mathrm{Spec}(B)) $$

if the rank of $B$ is constant equal to $n$, $G = \mathfrak{S}_n$ and $H = \mathfrak{S}_{n-1}$. For any $x \in R$,

$$ \mathrm{Tr}_{B/A}(x) = \sum_{[\tau] \in G/H} \tau(x) $$

which is thus integral over $A^{\circ}$ and thus in $A^{\circ}$ since $A^{\circ}$ is integrally closed in $A$. We thus have

$$ \mathrm{Tr}_{B/A}(R) \subset A^{\circ}. $$

Let $M$ be a finite type sub-$A^{\circ}$-module of $B$ such that $M[\frac{1}{\bar{\omega}}] = B$. Up to replacing $M$ by $\bar{\omega}^k M$ with $k \gg 0$, we can suppose that $M \subset R$. Now, if

$$ M^{\vee} = \{x \in B \mid \forall m \in M, \mathrm{Tr}_{B/A}(xm) \in A^{\circ}\}, $$

we have

$$ R \subset M^{\vee}. $$

Now, if we fix a diagram of morphisms of $A$-modules

$$ A^d \xrightarrow{\substack{s \\ \varepsilon}} B $$

with $\varepsilon s = \mathrm{Id}$ and $\varepsilon((A^{\circ})^d) = M$, we have $M^{\vee} \subset s^{\vee}((A^{\circ})^d)$ where $s^{\vee}: A^d = (A^d)^* \xrightarrow{s^*} B^* \xrightarrow{\frac{\mathrm{Tr}_{B/A}^{-1}}{\sim}} B$. We deduce from this that $R$ is contained in a finite type sub-$A^{\circ}$-module of $A$. We have $B = R[\frac{1}{\bar{\omega}}]$ and $B$ with its canonical topology as a finite type $A$-module is a Tate ring with $R$ as a ring of definition. Now, if $x \in B^{\circ}$, there exists $N \ge 0$ such that for all $k \ge 0$, $x^k \in \bar{\omega}^{-N}R$. But since $R$ is contained in a finite type $A^{\circ}$-module, $\bar{\omega}^{-N}R$ is contained in a finite type $A^{\circ}$-module and thus $A^{\circ}[x]$ too. We deduce that $x \in R$. $\square$

Here is an example of uniform Tate algebras where one can construct some more explicit pro-étale perfectoid covers.

**Example 4.1.4.** — Let $K$ be a characteristic zero non-Archimedean perfectoid field. Let $X$ be a $K$-adic space locally of finite type. Suppose $X$ is smooth. Then, up to replacing $X$ by an affinoid cover we can find an étale morphism

$$ X \to \mathbb{T}_n = \mathrm{Spa}(K\langle T_1^{\pm 1}, \dots, T_n^{\pm 1} \rangle, \mathcal{O}_K\langle T_1^{\pm 1}, \dots, T_n^{\pm 1} \rangle) $$
that is an open rational subset inside a finite étale morphisme toward an open rational subset of $\mathbb{T}_n$. Let $\mathbb{T}_n^\infty = \lim_{\leftarrow k \ge 0} \mathbb{T}_n$ where the transition map between level $k$ and $l$ with $l \le k$ is the Kummer map $t_i \mapsto t_i^{\frac{k-l}{l}}$, $1 \le i \le n$. This is the affinoid perfectoid space

$$ \mathbb{T}_n^\infty = \mathrm{Spa}(K\langle T_1^{\pm 1/p^\infty}, \dots, T_n^{\pm 1} \rangle, \mathcal{O}_K\langle T_1^{\pm 1/p^\infty}, \dots, T_n^{\pm 1} \rangle). $$

Then, using the purity theorem,

$$ X \times_{\mathbb{T}_n} \mathbb{T}_n^\infty = \lim_{\underset{k \ge 0}{\leftarrow}} (X \times_{\mathbb{T}_n, t \mapsto t^{p^k}} \mathbb{T}_n) $$

is a perfectoid space pro-étale over $X$.

In another situation Faltings used the following elementary result.

**Proposition 4.1.5 (Faltings).** — Let $R$ be an integral normal $p$-torsion free noetherian $\mathbb{Z}_{(p)}$-algebra. Fix an algebraic closure $\bar{K}$ of $K = \mathrm{Frac}(R)$ and let $L|K$ inside $\bar{K}$ be the union of all finite degree extensions $K'$ of $K$ such that $\overline{R[\frac{1}{p}]}^{K'}$ is an étale $R[\frac{1}{p}]$-algebra. This means that if $\bar{x} : \mathrm{Spec}(\bar{K}) \to \mathrm{Spec}(R[\frac{1}{p}])$,

$$ \mathrm{Gal}(L|K) = \pi_1(\mathrm{Spec}(R[\frac{1}{p}]), \bar{x}). $$

Let $\bar{R}$ be the integrale closure of $R$ in $L$. Then, the $p$-adic completion $\widehat{\bar{R}}$ is integral perfectoid and thus $\widehat{\bar{R}}[\frac{1}{p}]$ is perfectoid with $\widehat{\bar{R}}[\frac{1}{p}]^\circ$ almost equal to $\widehat{\bar{R}}$.

*Proof.* — It it clearly $p$-adic without $p$-torsion. One can take $p^{1/p}$, a $p$-root of $p$ in the fixed algebraic closure of $\mathrm{Frac}(R)$, as a pseudo-uniformizer. Since $\bar{R}$ is integrally closed in $\overline{R[\frac{1}{p}]}$,

$$ \mathrm{Frob} : \overline{R}/p^{1/p} \to \overline{R}/p $$

is injective. For the surjectivity of the preceding Frobenius, let $a \in \bar{R}$. Then the polynomial

$$ P(X) = X^p + pX - a $$

satisfies $(P(X), P'(X)) = \overline{R[\frac{1}{p}]}[X]$. Thus, $\overline{R}[X]/(P(X))$ is a finite $\bar{R}$-algebra that is étale outside $p$ and we can find $x \in \bar{R}$ such that $P(x) = 0$. $\square$

### 4.1.2. The universal cover of a $p$-divisible group.

**Proposition 4.1.6.** — Let $C|\mathbb{Q}_p$ be an algebraically closed complete non-archimedean field. Let $\mathscr{G}$ be a $p$-divisible formal group over $\mathcal{O}_C$. Let $\mathscr{G}_\eta$ be its generic fiber as an adic space. Then,

$$ \lim_{\underset{\times p}{\leftarrow}} \mathscr{G}_\eta $$

is perfectoid with tilting isomorphic to

$$ \lim_{\underset{\mathrm{Frob}}{\leftarrow}} \mathscr{G}_{k_C} \times_{k_C} \mathrm{Spa}(C^b). $$

Let us explain the meaning of this result. Here $\mathscr{G}$ is a formal Lie group,

$$ \mathscr{G} \simeq \mathrm{Spf}(\mathcal{O}_C[\![x_1, \dots, x_d]\!]) $$

for some integer $d$, the dimension of our formal group. Then,

$$ \mathscr{G}_{\eta} \simeq \dot{\mathbb{B}}_{C}^{d}, $$

the $d$-dimensional open ball as an adic space over $\mathrm{Spa}(C)$. Multiplication by $p$,

$$ \mathscr{G} \xrightarrow{\times p} \mathscr{G} $$
is finite locally free of rank $p^h$ where $h$ is by definition the height of $\mathcal{G}$. This is in fact an fppf torosor under the finite locally free group scheme $\mathcal{G}[p]$ over $\text{Spec}(\mathcal{O}_C)$. In generic fiber,

$$ \mathcal{G}_{\eta} \xrightarrow{\times p} \mathcal{G}_{\eta} $$

is finite étale, a torsor under the finite étale group $\mathcal{G}[p]_\eta$.

The formal group $\mathcal{G}_{k_C}$ is isomorphic to $\text{Spf}(k_C[[x_1, \dots, x_d]])$ and

$$ \lim_{\xleftarrow{\text{Frob}}} \mathcal{G}_{k_C} \simeq \text{Spf}(k_C[[x_1^{1/p}, \dots, x_d^{1/p}]]) $$

is a formal $\mathbb{Q}_p$-vector space. Now, in the statement we have in fact previously fixed a section of the projection $\mathcal{O}_C/p \to k_C$ so that $k_C \hookrightarrow C^b$. Then,

$$ \lim_{\xleftarrow{\text{Frob}}} \mathcal{G}_{k_C} \times_{k_C} \text{Spa}(C^b) \simeq \mathring{\mathbb{B}}_{C^b}^{d, 1/p^\infty}, $$

a perfectoid open ball with a structure of $\mathbb{Q}_p$-vector space.

## 4.2. Definition and elementary properties

Recall that the category of affinoid perfectoid spaces admits cofiltered limits.

**Definition 4.2.1.** — Let $f: X \to Y$ be a morphism of perfectoid spaces.

1. $f$ is affinoid pro-étale if $X$ and $Y$ are affinoid perfectoid and $Y \to X$ is a cofiltered limit of étale affinoid perfectoid spaces over $X$.

2. $f$ is pro-étale if locally on $X$ and $Y$ it is affinoid pro-étale.

Let us begin with some examples.

**Example 4.2.2.** — For $X$ perfectoid and $x \in X$, $\text{Spa}(K(x), K(x)^+) = X_x \to X$ is pro-étale, $X_x = \lim_{\leftarrow\limits_{U \ni x}} U$ with $U$ going through the set of affinoid neighborhoods of $x$.

**Example 4.2.3.** — If $X = \text{Spa}(A, A^+)$ and $I$ is an ideal of $A$ then $V(I) \hookrightarrow X$ is affinoid pro-étale: any Zariski closed subset of $X$ is affinoid pro-étale inside $X$ !

**Example 4.2.4.** — If $X$ is a perfectoid space and $P$ a locally profinite set then $X \times_P X \to X$ is pro-étale. If $P = \lim_{\leftarrow\limits_i} P_i$ with $P_i$ finite and the limit is cofiltered then $X \times_P X = \lim_{\leftarrow\limits_i} (X \times P_i)$.

**Example 4.2.5.** — Let $X$ be a qc qs perfectoid space and $P \to \pi_0(X)$ be a continuous map of profinite spaces. Then, $X \times_{\pi_0(X)} P \to X$ is pro-étale. For example, if $P = \{c\}$ where $c \in \pi_0(X)$ and $P = \{c\} \hookrightarrow \pi_0(X)$ is the evident inclusion then one finds that any connected component of $X$ is pro-étale inside $X$. In general, $\pi_0(X \times_{\pi_0(X)} P) = P$.

In particular, let us note that if $Y \to X$ is a morphism of qc qs perfectoid spaces, there is a canonical factorization

$$ Y \xrightarrow{\text{iso. on connected comp.}} X \times_{\pi_0(X)} \pi_0(Y) \xrightarrow{\text{pro-\'etale}} X. $$

**Definition 4.2.6.** — A family $(Y_i \xrightarrow{f_i} X)_{i \in I}$ of pro-étale morphisms toward the perfectoid space $X$ is a pro-étale covering if for any quasi-compact open subset $U$ of $X$ there exists $I' \subset I$ finite and for each $i \in I'$ a quasi-compact open subset $V_i \subset V_i$ such that

$$ \bigcup_{i \in I'} f_i(V_i) = U. $$
**Example 4.2.7.** — 1. For any perfectoid space $X$, the family of pro-étale morphisms

$$ (\mathrm{Spec}(K(x), K(x)^+) \to X)_{x \in X} $$

is surjective but not a covering in general.

2. For a perfectoid field $K$, if $\mathbb{B}_K^{1,1/p^\infty}$ is the closed perfectoid ball over $K$, the pro-étale morphism

$$ \{0\} \amalg \mathbb{B}_K^{1,1/p^\infty} \setminus \{0\} \to \mathbb{B}_K^{1,1/p^\infty} $$

is surjective but not a covering.

Although Point (1) of the preceding example is not a pro-étale cover, there is a way to “amalgamate” all $\mathrm{Spa}(K(x), K(x)^+)$, $x \in X$, in a pro-étale cover of $X$ when $X$ is affinoid perfectoid. This is the following example.

**Example 4.2.8.** — If $X = \mathrm{Spa}(A, A^+)$ is affinoid perfectoid, $\varpi$ is a pseudo-uniformizer of $A$, $R^+ = \coprod_{x \in X} K(x)^+$ and $R = R^+ [\frac{1}{\varpi}]$ then $\mathrm{Spa}(R, R^+) \to \mathrm{Spa}(A, A^+)$ is a pro-étale cover. In fact, one has

$$ \mathrm{Spa}(R, R^+) = \lim_{\substack{|X|=\coprod_{i \in I} Z_i \\ (U_i)_{i \in I} \\ U_i \text{ nbd. of } Z_i}} \coprod_{i \in I} U_i \to X. $$

## 4.3. Affinoid pro-étale morphisms

**Proposition 4.3.1.** — Let $X_\infty = \lim_{\leftarrow i \in I} X_i$ be a cofiltered limit of affinoid perfectoid spaces.

1. There is an equivalence

$$ 2 - \lim_{\underset{i \in I}{\longrightarrow}} \{\text{finite étale}/X_i\} \xrightarrow{\sim} \{\text{finite étale}/X_\infty\}. $$

2. There is an equivalence

$$ 2 - \lim_{\underset{i \in I}{\longrightarrow}} \{qc \ \mathrm{qs \ étale} / X_i\} \xrightarrow{\sim} \{qc \ \mathrm{qs \ étale} / X_\infty\}. $$

**Proposition 4.3.2.** — Let $X$ be an affinoid perfectoid space. The functor

$$ \lim_{\leftarrow} : \mathrm{Pro}(\text{affinoid perfectoid, étale}/X) \to \{\text{affinoid pro-étale}/X\} $$

is an equivalence of categories.

## 4.4. Totally disconnected perfectoid spaces

**Definition 4.4.1.** — A perfectoid space is totally disconnected if it is $qc \ \mathrm{qs}$ and $|X|$ is totally disconnected.

Thus, a perfectoid space is totally disconnected if and only if for each connected component $C$ of $X$ there exists a closed point $c \in X$ such that

$$ C = \mathrm{Spa}(K(c), K(c)^+) \hookrightarrow X. $$

One can think of a totally disconnected perfectoid spaces as an “amalgamation” of $\mathrm{Spa}(K(c), K(c)^+)$ when $c$ varies in a profinite set, its set of connected components.

**Lemma 4.4.2.** — Any totally disconnected perfectoid space is affinoid perfectoid.

*Proof.* — Let $X$ be totally disconnected. We can fix a finite open covering of $X$ by affinoid perfectoid spaces $X = \cup_i U_i$. It splits, $X = \coprod_i V_i$ with $V_i \subset U_i$. Since $V_i$ is open/closed in $U_i$ affinoid perfectoid, it is affinoid perfectoid. □
The following result is inspired by the following remark. Let $X$ be a $qc$ $qs$ perfectoid space. If $Y$ is a $qc$ $qs$ perfectoid space together with a morphism $Y \to X$ then the image of $|Y|$ in $|X|$ is a pro-constructible generalizing subset. In general this subset does not have a structure of perfectoid space. Nevertheless suppose $X$ is totally disconnected perfectoid space with $\pi_0(X)$ finite. One has

$$X = \coprod_{\substack{c \in X \\ \text{closed point}}} \text{Spa}(K(c), K(c)^+) .$$

Let $Z \subset |X|$ be a pro-constructible generalizing subset. For any $c \in X$ a closed point one has $\text{Spa}(K(c), K(c)^+) \cap Z$ is pro-constructible generalizing. It thus has a closed point $z$ (see Corollary 1.3.9). Being stable under generalizations we deduce that

$$\text{Spa}(K(c), K(c)^+) \cap Z = X_z = \text{Spa}(K(z), K(z)^+)$$

with $K(z) = K(c)$. We thus have

$$Z \cap \text{Spa}(K(c), K(c)^+) = \{x \in \text{Spa}(K(c), K(c)^+) \mid \forall f \in K(c)^+, |f(x)| \le 1\}.$$

We have $\mathcal{O}(X) = \coprod_{c \text{ closed pt.}} K(c)$ and we deduce that there exists a subset $S \subset \mathcal{O}(X)$ such that

$$Z = \{x \in X \mid \forall f \in S, |f(x)| \le 1\}.$$

In particular this is an intersection of Laurent domains and it has a structure of a totally disconnected perfectoid space pro-étale inside $X$. In fact, this extends from the case when the set of connected components is finite to any profinite set.

**Theorem 4.4.3.** — Let $X$ be a totally disconnected perfectoid space and $Z \subset |X|$ pro-constructible generalizing. Then, $Z$ is represented by a totally disconnected perfectoid space such that the morphism $Z \hookrightarrow X$ is pro-étale.

*Proof.* — We prove that

$$Z = \varinjlim_{\substack{S \subset \mathcal{O}(X) \text{ finite} \\ Z \subset \{|f| \le 1| \mid f \in S\}}} \{x \in X \mid \forall f \in S, |f(x)| \le 1\}.$$

Let $x \in X \setminus Z$. We are going to construct $f \in \mathcal{O}(X)$ satisfying $|f(x)| > 1$ and for all $z \in Z$, $|f(z)| \le 1$. This will prove the theorem.

For this let $C$ be the connected component of $x$. One has $C = \text{Spa}(K(c), K(c)^+)$ where $c$ is the unique closed point of $C$. Let us look at $C \cap Z$. Suppose first it is empty. Then,

$$\bigcap_{U \supset C} U \cap Z = \emptyset,$$

where $U$ is an open/closed neighborhood of $C$, and by compacity of the constructible topology there exists such a $U$ satisfying $U \cap Z = \emptyset$. The function $f \in \mathcal{O}(X)$ whose value on $U$ is $\varpi^{-1}$ and 1 on $X \setminus U$ satisfies $|f(x)| > 1$ and $\forall z \in Z$, $|f(z)| \le 1$.

Suppose now that $C \cap Z \neq \emptyset$. Since $Z$ is pro-constructible, $C \cap Z$ is pro-constructible in $C$ and thus has a closed point $z$. Since $Z$ is generalizing and $C$ is a chain, we have

$$C \cap Z = \text{Spa}(K(c), K(z)^+)$$

with $K(c) = K(z)$. We thus have

$$C \cap Z = \bigcap_{f \in K(z)^+} \{y \in C \mid |f(y)| \le 1\}.$$

Since $x \in C \setminus C \cap Z$, there exists $f \in K(z)^+$ such that $|f(x)| > 1$. The local ring $\mathcal{O}_{X,c}$ is the colimit of $\mathcal{O}_X(U)$ where $U$ goes through the set of open/closed neighborhoods of $C$. There thus exists $U$
an open/closed neighborhood of $C$ and $g \in \mathcal{O}_X(U)$ satisfying $|g(x)| > 1$ and for all $y \in C \cap Z$, $|g(y)| \le 1$. Let

$$V = \{y \in U \mid |g(y)| \le 1\},$$

an open quasi-compact subset of $U$. One has

$$\bigcap_{\substack{C \subset U' \subset U \\ U' \text{ open/closed}}} U' \cap (Z \cap (U \setminus V)) = \emptyset.$$

By compacity of the constructible topology we can thus suppose, up to shrinking $U$, that for all $y \in Z \cap U$, $|g(y)| \le 1$. We can now define $h \in \mathcal{O}(X)$ be setting $h|_U = g$ and $h|_{X \setminus U} = 0$. It satisfies $|h(x)| > 1$ and for all $y \in Z$, $|h(y)| \le 1$. $\square$

Thus, if $f : Y \to X$ is a morphism of qcqs perfectoid spaces with $X$ totally disconnected, there is a canonical factorization

$$Y \xrightarrow{\text{surjective}} \underbrace{Z}_{\text{aff. perf.}} \xrightarrow{\text{pro-\'etale immersion}} X.$$

**Proposition 4.4.4.** — For any qcqs perfectoid space $X$, there exists a pro-étale open surjective morphism $\tilde{X} \to X$ with $\tilde{X}$ a totally disconnected perfectoid space.

## 4.5. Strictly totally disconnected perfectoid spaces

**Proposition 4.5.1.** — For $X$ a perfectoid space the following are equivalent:

1. $X$ is strictly totally disconnected

2. $X$ is totally disconnected and for any $x \in X$, $K(x)$ is algebraically closed.

*Proof.* — (1) ⇒ (2) The only thing to verify is that the residue fields are algebraically closed. We use the equivalence of Corollary 2.7.6. Since $X$ is totally disconnected, the residue field of $x \in X$ is the same as the one of $c$, the unique closed point of the connected component $C$ of $x$. Now,

$$2 - \lim_{\substack{U \subset C \\ \text{open/closed}}} \{\text{\'etale finite}/U\} \xrightarrow{\sim} \{\text{\'etale finite}/K(c)\}.$$

But now is $V \to U$ with $U$ open and closed containing $C$, $V \to U$ extends to an étale cover $V \coprod X \setminus U \to X$ of $X$. This étale cover has a section and we deduce the result.

(2) ⇒ (1) Let $Y \to X$ be an étale cover. Using the quasi-compacity of $X$ and the definition of an étale morphism, it can be refined to a cover $\coprod_{i \in I} V_i \to X$ with $I$ finite and $V_i \to U_i$ étale finite where $U_i \subset X$ is open. The topological cover $\coprod_{i \in I} U_i \to X$ splits and we can thus suppose that $Y \to X$ is étale finite. We can now use the equivalence of Corollary 2.7.6 to deduce that any point of $X$ has a neighborhood $U$ such that $Y \times_X U \to U$ splits. Thus, over a topological cover of $X$ the covering splits and we deduce the result by splitting this last topological cover. $\square$

## 4.6. Quasi-pro-étale morphisms

**4.6.1. Scholze’s Berkley example.** — We will prove the following result explicitly. Let $K$ be a perfectoid field of characteristic different from 2.

**Proposition 4.6.1.** — Let $\mathbb{B} = \text{Spa}(K\langle T^{1/p^\infty} \rangle, \mathcal{O}_K\langle T^{1/p^\infty} \rangle)$ be the one dimensional closed perfectoid ball over $K$. Consider the Kummer morphism $\mathbb{B} \to \mathbb{B}$ defined by $T^{1/p^k} \mapsto T^{2/p^k}$ for all $k \ge 0$. There exists an explicit pro-étale covering $\tilde{\mathbb{B}} \to \mathbb{B}$ such that the pullback of the Kummer map via this pro-étale cover is pro-étale.
*Proof.* — Fix an absolute value $|\cdot|$ on $K$ defining its topology and a radius $\rho \in ]0, 1[ \cap |K|$. We note

$$f : \mathbb{B} \to \mathbb{B}$$

our Kummer map. It is étale finite outside the origin $V(T)$. For $n \ge 1$ let

$$U_n = \mathbb{B}(\rho^n) \amalg \left( \prod_{i=0}^{n-1} \mathcal{C}(\rho^{i+1}, \rho^i) \right)$$

where if $\rho = |\varpi|$, $\mathbb{B}(\rho^n) = \{x \in \mathbb{B} \mid |T(x)| \le |\varpi(x)|^n\}$ is the closed ball with radius $\rho^n$ and

$$\mathcal{C}(\rho^{i+1}, \rho^i) = \{x \in \mathbb{B} \mid |\varpi(x)|^{i+1} \le |T(x)| \le |\varpi(x)|^i\}$$

is the annulus with radii $[\rho^{i+1}, \rho^i]$. This forms a projective system of covers of $\mathbb{B}$ and we set

$$U_{\infty} = \lim_{n \ge 0} U_n \to \mathbb{B}$$

that is a pro-étale cover of $\mathbb{B}$. Let

$$V_n = \mathbb{B}(\rho^{n/2}) \amalg \left( \prod_{i=0}^{n-1} \mathcal{C}(\rho^{\frac{i+1}{2}}, \rho^{\frac{i}{2}}) \right)$$

and

$$V_{\infty} = \lim_{n \ge 0} V_n.$$

There is then a cartesian diagram

<figure><img src="image_38.png" /></figure>

We are going to prove that $V_\infty \to U_\infty$ is pro-étale. For this note

$$W_n = \mathbb{B}(\rho^n) \amalg \left( \prod_{i=0}^{n-1} \mathcal{C}(\rho^{\frac{i+1}{2}}, \rho^{\frac{i}{2}}) \right).$$

Define the following transitions morphisms

$$\alpha_{n+1} : W_{n+1} \to W_n$$

using the formulas:

* $\alpha_{n+1}|_{\mathbb{B}(\rho^{n+1})} : \mathbb{B}(\rho^{n+1}) \hookrightarrow \mathbb{B}(\rho^n)$ followed by the evident inclusion of $\mathbb{B}(\rho^n)$ in $W_n$

* $\alpha_{n+1}|_{\mathcal{C}(\rho^{\frac{n+1}{2}}, \rho^{\frac{n}{2}})} : \mathcal{C}(\rho^{\frac{n+1}{2}}, \rho^{\frac{n}{2}}) \xrightarrow{z \to z^2} \mathcal{C}(\rho^{n+1}, \rho^n) \hookrightarrow \mathbb{B}(\rho^n)$ followed by the evident inclusion.

* $\alpha_{n+1}|_{\mathcal{C}(\rho^{\frac{i+1}{2}}, \rho^{\frac{i}{2}})}$ is the evident inclusion of $\mathcal{C}(\rho^{\frac{i+1}{2}}, \rho^{\frac{i}{2}})$ in $W_n$ if $0 \le i \le n-1$.

Define now

$$W_{\infty} = \lim_{n \ge 0} W_n.$$

There is an étale morphism of projective systems

$$g_n : W_n \to U_n$$

defined by:

* $g_n|_{\mathbb{B}(\rho^n)}$ is the evident inclusion in $U_n$

* $g_{n|\mathcal{C}(\rho^{\frac{i+1}{2}}, \rho^{\frac{i}{2}})} : \mathcal{C}(\rho^{\frac{i+1}{2}}, \rho^{\frac{i}{2}}) \xrightarrow{z \to z^2} \mathcal{C}(\rho^i, \rho^{i+1})$ followed by the evident inclusion inside $U_n$, for $0 \le i \le n-1$.
This defines a pro-étale morphism

$$g : W_{\infty} \to U_{\infty}.$$

Define now a morphism of projective systems

$$h_n : V_n \to W_n$$

using the formulas:

* $h_{n|\mathbb{B}(\rho^{\frac{n}{2}})} : \mathbb{B}(\rho^{\frac{n}{2}}) \xrightarrow{z \mapsto z^2} \mathbb{B}(\rho^n)$ followed by the evident inclusion in $W_n$

* $h_{n|C(\rho^{\frac{i+2}{2}}, \rho^{\frac{i}{2}})}$ is the evident inclusion in $W_n$ for $0 \le i \le n-1$.

The following diagram commutes for all $n$

<figure><img src="image_39.png" /></figure>

and we thus have a diagram

<figure><img src="image_40.png" /></figure>

We now say that $h_{\infty} : V_{\infty} \to W_{\infty}$ is an isomorphism. For this we apply Proposition 3.12.1. In fact, this is an evident isomorphism outside the origin of $\mathbb{B}$ since for any radius $R \in ]0, 1[ \cap ]K|$

$$V_n \times_{\mathbb{B}} \mathbb{B}(R) \to W_n \times_{\mathbb{B}} \mathbb{B}(R)$$

is an isomorphism for $n \gg 0$. Moreover, computing the fibers at the origin using Proposition 2.11.2 we find

$$V_{\infty} \times_{\mathbb{B}} \{0\} = \varinjlim_{n \ge 0} \mathbb{B}(\rho^{n/2}) = \text{Spa}(K, \mathcal{O}_K)$$

and

$$W_{\infty} \times_{\mathbb{B}} \{0\} = \varinjlim_{n \ge 0} \mathbb{B}(\rho^n) = \text{Spa}(K, \mathcal{O}_K).$$

This is thus an isomorphism over the origin of the ball $\mathbb{B}$. We thus have a diagram

<figure><img src="image_41.png" /></figure>

and $V_{\infty} \to U_{\infty}$ is pro-étale.

<figure><img src="image_42.png" /></figure>

### 4.6.2. Structure over a strictly totally disconnected space. —

**Theorem 4.6.2.** — Let $f : Y \to X$ be a separated morphism ofqcqs perfectoid spaces. Suppose

1. $X$ is strictly totally disconnected,

2. The fibers of $f$ at maximal points are profinite i.e. for any maximal point $x$ of $X$, $x = \text{Spa}(C, \mathcal{O}_C)$, there exists a profinite set $P_x$ such that

$$Y \times_X \text{Spa}(C, \mathcal{O}_C) \simeq \text{Spa}(C, \mathcal{O}_C) \times_{\underline{P}_x}.$$

Then, $f$ is affinoid pro-étale and $Y$ is strictly totally disconnected.
*Proof.* — We use the factorization

$$Y \to X \times_{\pi_0(X)} \pi_0(Y) \to X.$$

Using this we can suppose that $\pi_0(Y) \xrightarrow{\sim} \pi_0(X)$. Let $Z = \text{Im}(|f|) \subset |X|$, a pro-constructible generalizing subset that is represented by an affinoid perfectoid space pro-étale over $X$, see Theorem 4.4.3. Let us note, by abuse of notation, this perfectoid space $Z$. We are going to prove that $Y \xrightarrow{\sim} Z$. For this, according to Proposition 3.12.1, we have to prove that for any affinoid perfectoid field $(K, K^+)$ the map $Y(K, K^+) \to X(K, K^+)$ is injective. The valuation criterion of separatedness implies it is sufficient to prove that $Y(K, K^\circ) \to X(K, K^\circ)$ is injective. Up to replacing $X$ by a connected component $C$ and $Y$ by $Y \times_X C$, we can suppose that $X$ and thus $Y$ is connected. We have now, $X = \text{Spa}(C, C^+)$.

Let now $y_1, y_2 \in Y$ be two maximal points that have the same image $x = \text{Spa}(C, \mathcal{O}_C)$, the unique maximal point of $X$, in $X$. We have $y_1, y_2 \in P_x$ a profinite set. Suppose $y_1 \neq y_2$. We can thus find a decomposition $f^{-1}(x) = U_1 \sqcup U_2$ in open/closed subsets $U_1$ and $U_2$ of the profinite set $f^{-1}(x)$ such that $y_1 \in U_1$ and $y_2 \in U_2$. Since $U_1$ and $U_2$ are closed in the pro-constructible subset $f^{-1}(x)$ of $Y$, they are pro-constructible in $Y$. Let us look at $\bar{U}_1$ and $\bar{U}_2$ their closure in $Y$ that are thus the set of their specializations. Since $Y \to X = \text{Spa}(C, C^+)$ is separated, any point of $Y$ has a unique generalization in $f^{-1}(x)$. We deduce that $Y = \bar{U}_1 \sqcup \bar{U}_2$ which contradicts the connectedness of $Y$. $\square$

**Definition 4.6.3.** — A morphism $Y \to X$ of perfectoid spaces is quasi-pro-étale if there exists a pro-étale cover $X' \to X$ such that $Y \times_X X' \to X'$ is pro-étale.

**Corollary 4.6.4.** — For $f : Y \to X$ a morphism of perfectoid spaces the following are equivalent:

1. $f$ is quasi-pro-étale

2. There exists an open cover $Y = \cup_{i \in I} V_i$, such that $f|_{V_i} : V_i \to X$ has profinite geometric fibers at all maximal points of the target.

3. For any strictly totally disconnected perfectoid space $X'$ with a morphism $X' \to X$, $Y \times_X X' \to X'$ is pro-étale.

## 4.7. The structure pro-étale sheaf

**Proposition 4.7.1.** — The presheaf $\mathcal{O} : X \mapsto \mathcal{O}(X)$ on the big pro-étale perfectoid site is a sheaf.

*Proof.* — Let us fix a base affinoid perfectoid space $X$ and $\varpi \in \mathcal{O}(X)$ a pseudo-uniformizer. One has $\mathcal{O} = \mathcal{O}^+ [\frac{1}{\varpi}]$ and $\mathcal{O}^+ = \varinjlim_{n \geq 1} \mathcal{O}^+ / \varpi^n$ as presheaves on affinoid perfectoid spaces/X. It thus suffices to prove that $\mathcal{O}^+ / \varpi$ is a pro-étale sheaf. But if $\nu : X_{\text{proét}} \to X_{\text{ét}}$ then

$$\mathcal{O}_{X_{\text{proét}}}^{+} / \varpi = \nu^{-1}(\mathcal{O}_{X_{\text{ét}}}^{+} / \varpi).$$

$\square$
# CHAPTER 5

## THE v-TOPOLOGY

### 5.1. Definition and basic properties

### 5.2. The v-sheaf O

**Theorem 5.2.1.** — The correspondence $X \mapsto \mathcal{O}(X)$ is a sheaf for the v-topology on perfectoid spaces.

*Proof.* — Since for any perfectoid space $X$, $\mathcal{O}(X) \subset \coprod_{x \in X} K(x)$ (use Theorem 2.1.16), this is a separated presheaf. Let $Y \to X$ be a v-cover. We want to prove that $\mathcal{O}(X)$ surjects to $\ker(\mathcal{O}(Y) \to \mathcal{O}(Y \times_X Y))$. Using the separatedness, we can suppose that $X$ is affinoid perfectoid. Using the quasi-compacity assumption in the definition of a v-cover, one can suppose, up to replacing $Y$ by a finite disjoint union of affinoid perfectoid opensubsets of $Y$, that $Y$ is affinoid perfectoid too. Using the sheaf property for the pro-étale topology we can suppose, up to replacing $X$ by a totally disconnected pro-étale cover $\widetilde{X} \to X$ of $X$ and $Y$ by a totally disconnected pro-étale cover of $Y \times_X \widetilde{X}$, that $Y$ and $X$ are totally disconnected. The result is then a consequence of Proposition 5.2.2. □

**Proposition 5.2.2.** — Let $X = \text{Spa}(A, A^+)$ be a totally disconnected perfectoid space and $Y \to X$ with $Y = \text{Spa}(B, B^+)$ affinoid perfectoid. Then

1. $B^+/\varpi$ is a flat $A^+/\varpi$-algebra.

2. If $Y \to X$ is surjective then $B^+/\varpi$ is a faithfully flat $A^+/\varpi$-algebra.

*Proof.* — Let $\pi : |X| \to \pi_0(X)$ and note $f : Y \to X$. Consider the sheaf of $\mathcal{O}_X^+/\varpi$-algebras $(f_*\mathcal{O}_Y^+)/\varpi$ on $|X|$. We push it forward to $\pi_0(X)$ and look at

$$ \mathcal{F} := \pi_* (f_* (\mathcal{O}_Y^+)/\varpi) $$

as a sheaf of $\mathcal{A} := \pi_*(\mathcal{O}_X^+/\varpi)$-algebras on the profinite set $\pi_0(X)$. For $c \in \pi_0(X)$, if $x \in X$ is the closed point of the connected component $\pi^{-1}(c)$,

$$ \mathcal{A}_c = \mathcal{O}_{X,x}^+/\varpi. $$

One has $R^1\pi_* = 0$ since $X$ is totally disconnected and thus

$$ \mathcal{F} = ((\pi f)_* \mathcal{O}_Y^+)/\varpi. $$

We deduce that

$$ \mathcal{F}_c = ((\pi f)_* \mathcal{O}_Y^+)_c / \varpi. $$

Since $((\pi f)_* \mathcal{O}_Y^+)_c$ has no $\varpi$-torsion and $\mathcal{O}_{X,x}^+$ is a valuation ring, $\mathcal{F}_c$ is a flat $\mathcal{A}_c$-module. Now, if

$$ 0 \to M' \to M \to M'' \to 0 $$
is an exact sequence of $\Gamma(\pi_0(X), \mathcal{A})$-modules, it defines an exact sequence of $\mathcal{A}$-modules

$$0 \to M' \otimes \mathcal{A} \to M \otimes \mathcal{A} \to M'' \otimes \mathcal{A} \to 0.$$

If we apply $- \otimes_{\mathcal{A}} \mathcal{F}$ to this sequence the obtained sequence

$$0 \to M' \otimes \mathcal{F} \to M \otimes \mathcal{F} \to M'' \otimes \mathcal{F} \to 0$$

is exact since it is exact stalkwise thanks to the preceding flatness result. Taking global sections we obtain that the sequence

$$0 \to M' \otimes \Gamma(\pi_0(X), \mathcal{F}) \to M \otimes \Gamma(\pi_0(X), \mathcal{F}) \to M'' \otimes \Gamma(\pi_0(X), \mathcal{F}) \to 0$$

is exact and thus $\Gamma(\pi_0(X), \mathcal{F}) = B^+/\varpi$ is a flat $\Gamma(\pi_0(X), \mathcal{A}) = A^+/\varpi$-modules.

The faithful flatness result is obtained using the surjectivity of the specialization map, see Proposition 2.13.6. $\square$

## 5.3. Subcanonicity

## 5.4. A descent result
# CHAPTER 6

## DIAMONDS

### 6.1. Definition and first properties

In the following we note $\text{Perf}_{\mathbb{F}_p}$ for the category of $\mathbb{F}_p$-perfectoid spaces.

**Definition 6.1.1.** — A pro-étale sheaf $X$ on $\text{Perf}_{\mathbb{F}_p}$ is a diamond if there exists a perfectoid space $\tilde{X}$ and an equivalence relation $R \subset \tilde{X} \times \tilde{X}$ represented by a perfectoid space such that

1. both morphisms $R \to \tilde{X}$ are pro-étale i.e. $R$ is a pro-étale equivalence relation,

2. $X \simeq \tilde{X}/R$ as pro-étale sheaves.

### 6.2. $\text{Spa}(\mathbb{Q}_p)^{\diamond}$

We note $\text{Perf}_{\mathbb{Q}_p}$ for the category of perfectoid spaces over $\text{Spa}(\mathbb{Q}_p)$.

**Definition 6.2.1.** — We note $\text{Spa}(\mathbb{Q}_p)^{\diamond}$ the functor on $\text{Perf}_{\mathbb{F}_p}$ defined by the formula

$$ \text{Spa}(\mathbb{Q}_p)^{\diamond}(S) = \{ (S^{\sharp}, \iota) \mid S^{\sharp} \in \text{Perf}_{\mathbb{Q}_p}, \iota : S \xrightarrow{\sim} S^{\sharp, \flat} \} / \sim. $$

One of the first basic results is the following.

**Proposition 6.2.2.** — The functor $\text{Spa}(\mathbb{Q}_p)^{\diamond}$ is a v-sheaf.

*Proof.* — We prove more generally that $\text{Spa}(\mathbb{Z}_p)^{\diamond}$ is a v-sheaf. We have to prove that the functor on $\mathbb{F}_p$-affinoid perfectoid rings

$$ (A, A^+) \to \mathcal{D}_1(A, A^+)/W(A^+)^\times $$

is a v-sheaf. We can rewrite this functor as

$$ (A, A^+) \to \varinjlim_{\varpi \in A^{\circ\circ \cap A^\times}} ([\varpi]W(A^+) + p)/1 + [\varpi]W(A^+). $$

where $\varpi \le \varpi'$ if $\varpi\varpi'^{-1} \in A^+$. Now the result is an easy consequence of the following two facts:

* $\mathcal{O}^+$ and thus $(A, A^+) \to \varinjlim_{\varpi \in A^{\circ\circ \cap A^\times}} p + [\varpi]W(A^+)$ is a v-sheaf.

* If $(R, R^+) \text{ is an } \mathbb{F}_p\text{-perfectoid ring then}$

$$ \varinjlim_{\varpi \in A^{\circ\circ \cap A^\times}} H_v^1(\text{Spa}(R, R^+), 1 + [\varpi]W(\mathcal{O}^+)) = 0. $$

This last fact is an easy consequence of the almost vanishing of $H_v^1(\text{Spa}(R, R^+), \mathcal{O}^+)$ using some standard dévissages. $\square$

**Proposition 6.2.3.** — The v-sheaf $\text{Spa}(\mathbb{Q}_p)^{\diamond}$ is a diamond.
*Proof.* — The tilting equivalence defines a morphism of $v$-sheaves

$$ \mathrm{Spa}(\mathbb{C}_p^b) \to \mathrm{Spa}(\mathbb{Q}_p)^{\diamond}. $$

This is an epimorphism of pro-étale sheaves. In fact, if $X^\sharp$ is an untilt over $\mathbb{Q}_p$ of $X \in \mathrm{Perf}_{\mathbb{F}_p}$, for any $K|\mathbb{Q}_p$ of finite degree contained in $\overline{\mathbb{Q}}_p$, according to the purity theorem (Theorem 3.11.1), $X^\sharp \otimes_{\mathbb{Q}_p} K$ is perfectoid with $X^\sharp \otimes_{\mathbb{Q}_p} K \to X$ is étale finite. Over the pro-étale cover

$$ (X^\sharp \hat{\otimes}_{\mathbb{Q}_p} \mathbb{C}_p)^b = \lim_{K|\mathbb{Q}_p} (X^\sharp \otimes_{\mathbb{Q}_p} K)^b \to X $$

there is a morphism to $\mathrm{Spa}(\mathbb{C}_p^b)$. We have thus proven the pro-étale surjectivity assertion. It remains to compute the diagram

$$ \mathrm{Spa}(\mathbb{C}_p)^b \times_{\mathrm{Spa}(\mathbb{Q}_p)^\circ} \mathrm{Spa}(\mathbb{C}_p^b) \implies \mathrm{Spa}(\mathbb{C}_p^b). $$

This is equivalent to computing

$$ \mathrm{Spa}(\mathbb{C}_p) \times_{\mathrm{Spa}(\mathbb{Q}_p)} \mathrm{Spa}(\mathbb{C}_p) \implies \mathrm{Spa}(\mathbb{C}_p). $$

where the fiber product is taken in the category of functors on $\mathrm{Perf}_{\mathbb{Q}_p}$. But as functors on $\mathrm{Perf}_{\mathbb{Q}_p}$, $\mathrm{Spa}(\mathbb{C}_p) = \lim_{K|\mathbb{Q}_p} \mathrm{Spa}(K)$ where as before $K|\mathbb{Q}_p$ is finite degree. We then have

$$ \mathrm{Spa}(\mathbb{C}_p) \times_{\mathrm{Spa}(\mathbb{Q}_p)} \mathrm{Spa}(\mathbb{C}_p) = \lim_{K|\mathbb{Q}_p} \mathrm{Spa}(\mathbb{C}_p) \times_{\mathrm{Spa}(\mathbb{Q}_p)} \mathrm{Spa}(K). $$

We deduce that

$$ \mathrm{Spa}(\mathbb{C}_p) \times_{\mathrm{Spa}(\mathbb{Q}_p)} \mathrm{Spa}(\mathbb{C}_p) = \mathrm{Spa}(\mathbb{C}_p) \times \overline{\mathrm{Gal}(\overline{\mathbb{Q}}_p|\mathbb{Q}_p)} $$

and thus our equivalence relation is pro-étale with

$$ \mathrm{Spa}(\mathbb{Q}_p)^{\diamond} = \mathrm{Spa}(\mathbb{C}_p^b)/\overline{\mathrm{Gal}(\overline{\mathbb{Q}}_p|\mathbb{Q}_p)}. \quad \square $$

The proof gives formulas like

$$ \begin{aligned} \mathrm{Spa}(\mathbb{Q}_p)^{\diamond} &= \mathrm{Spa}(\mathbb{C}_p^b)/\overline{\mathrm{Gal}(\overline{\mathbb{Q}}_p|\mathbb{Q}_p)} \\ &= \mathrm{Spa}(\overline{\widehat{\mathbb{Q}}_p(\zeta_{p^\infty})}^b)/\overline{\mathbb{Z}_p^\times} \\ &\qquad \widehat{\mathbb{F}_p((T^{1/p^\infty}))} \end{aligned} $$

where if $\varepsilon = (\zeta_{p^n})_{n \ge 0} \in \overline{\mathbb{Q}_p(\zeta_{p^\infty})}^b$ then $T = \varepsilon + 1$. Here the action of $\overline{\mathbb{Z}_p^\times}$ is given by the usual formula

$$ T^a = \sum_{k \ge 0} \binom{a}{k} (T-1)^k $$

for $a \in \mathbb{Z}_p^\times$.

**Remark 6.2.4.** — The $v$-sheaf $\mathrm{Spa}(\mathbb{Z}_p)^\diamond$ is not a diamond. This still has the geometric structure of a spatial $v$-sheaf and one can do some geometry with it. For example (????) one can prove that $\mathrm{Spa}(\mathbb{Z}_p)^\diamond \to *$ is $\ell$-cohomologically smooth for $\ell \ne p$.

**Proposition 6.2.5.** — There is an equivalence of categories

$$ \mathrm{Perf}_{\mathbb{Q}_p} \xrightarrow{\sim} \mathrm{Perf}_{\mathbb{F}_p} / \mathrm{Spa}(\mathbb{Q}_p)^{\diamond}. $$

This induces an isomorphism of the $v$, resp. pro-étale, resp. étale, topoi

$$ \overline{\mathrm{Perf}_{\mathbb{Q}_p}} \xrightarrow{\sim} \overline{\mathrm{Perf}_{\mathbb{F}_p}} / \mathrm{Spa}(\mathbb{Q}_p)^{\diamond}. $$
## 6.3. The diamond associated to a $\mathbb{Q}_p$-adic space

**6.3.1. Background on the Shilov boundary.** — Let $K$ be a complete non-archimedean field. Let $A$ be a $K$-affinoid algebra (by this we mean “à la Tate” in the sense that it is topologically of finite type. Let

$$ \tilde{A} = A^{\circ}/A^{\circ\circ} $$

that is a reduced $k_K$-algebra of finite type ([10, Corollary 3-Section 6.3.4]). Let $\|\cdot\|_{\infty}: A \to \mathbb{R}_+$ be the spectral norm. One has

$$ A^{\circ} = \{f \in A \mid \|f\|_{\infty} \le 1\}. $$

Let us look at the specialization map

$$ \mathrm{sp}: \mathcal{M}(A) \to \mathrm{Spec}(\tilde{A}). $$

This is surjective (Proposition 2.13.6). Moreover, for each minimum prime ideal $\mathfrak{p}$ of $\tilde{A}$,

$$ \mathrm{sp}^{-1}(\mathfrak{p}) = \{\xi_{\mathfrak{p}}\} $$

is one point.

## 6.4. The main result

**Theorem 6.4.1.** — 1. Let $X$ be a $\mathrm{Spa}(\mathbb{Q}_p)$-adic space. The functor

$$ X^{\circ}: \mathrm{Perf}_{\mathbb{F}_p} \to \mathrm{Sets} $$

$$ S \mapsto \{ (S^{\sharp}, \iota, f) \mid S^{\sharp} \in \mathrm{Perf}_{\mathbb{Q}_p}, \iota: S \xrightarrow{\sim} S^{\sharp, \flat}, f: S^{\sharp} \to X \} / \sim $$

is represented by a diamond.

2. If $K|\mathbb{Q}_p$ is a complete non-archimedean field, this defines a fully faithful functor

$$ (-)^{\circ}: \{ \text{locally of finite type normal adic spaces} / \mathrm{Spa}(K) \} \hookrightarrow \{ \text{diamonds} \} / \mathrm{Spa}(K)^{\circ}. $$

*Proof.* — Everything is easily reduced to the case $X = \mathrm{Spa}(A, A^+)$ with $(A, A^+)$ a complete sheafy affinoid Tate ring. Let $(B, B^+)$ be the completion of $(A, A^+)$ with respect to the spectral norm associated to a choice of a pseudo-uniformizer and some number in $]0, 1[$ as in Theorem 2.1.16. There is a morphism $(A, A^+) \to (B, B^+)$ inducing an isomorphism of functors on $\mathrm{Perf}_{\mathbb{F}_p}$, $\mathrm{Spa}(B, B^+)^{\circ} \xrightarrow{\sim} \mathrm{Spa}(A, A^+)^{\circ}$. We can thus suppose that $A$ is uniform. We can then apply Proposition 4.1.1 to deduce the existence of a filtered inductive system $(A_i, A_i^+)$$_{i \in I}$ of uniform affinoid finite étale $(A, A^+)$-algebras such that

$$ (A_{\infty}, A_{\infty}^{+}) = \widehat{\lim}_{i \in I} (A_i, A_i^{+}) $$

is perfectoid and for all indices $i \ge j$, $|\mathrm{Spa}(A_i, A_i^+)| \to |\mathrm{Spa}(A_j, A_j^+)|$ is surjective. Let

$$ X_{\infty} = \mathrm{Spa}(A_{\infty}, A_{\infty}^{+}) \in \mathrm{Perf}_{\mathbb{Q}_p}. $$

There is a morphism

$$ (10) \qquad X_{\infty}^{\flat} \to X^{\diamond}. $$

It is easily deduced from the fact that $\mathcal{O}$ is a $v$-sheaf on $\mathrm{Perf}_{\mathbb{Q}_p}$, see Theorem 5.2.1, that the functor $S \mapsto X(S)$ is a $v$-sheaf on $\mathrm{Perf}_{\mathbb{Q}_p}$. Proposition 6.2.2 coupled with the preceding fact implies that $X^{\diamond}$ is a $v$-sheaf on $\mathrm{Perf}_{\mathbb{F}_p}$. Let us prove that the morphism (10) is an epimorphism of pro-étale sheaves. $\square$
## 6.5. Diamonds as v-sheaves

Recall the following result by Gabber ([51, Proposition 0APL]): every algebraic space is an fpqc sheaf. This result has the following analog.

**Theorem 6.5.1.** — *Diamonds are v-sheaves.*

## 6.6. Subdiamonds

The following result says that a lot of objects are diamonds.

**Proposition 6.6.1.** — Let $Y$ be a diamond and $X \subset Y$ a sub-v-sheaf. Then $X$ is a diamond.

*Proof.* — Let $\tilde{Y}$ be a perfectoid space and $R \subset \tilde{Y} \times \tilde{Y}$ be a pro-étale equivalence relation such that

$$Y \simeq \tilde{Y}/R$$

as pro-étale sheaves.

□
# CHAPTER 7

## ARTIN'S CRITERION FOR SPATIAL DIAMONDS
<figure><img src="image_43.png" /></figure>
# APPENDIX A

## THE COTANGENT COMPLEX AND DEFORMATION THEORY

### A.1. Background on homotopical algebra

#### A.1.1. Simplicial objects. —

**A.1.1.1. Definition and basic properties.** — Simplicial sets have been introduced, mainly under the influence of Daniel Kan ([5]), as a combinatorial model for the homotopy type of topological spaces.

We note $\Delta$ for the category of finite ordered sets. For $n \ge 0$ we note

$$[n] = \{0 < 1 < \cdots < n\} \in \mathrm{Ob}\,\Delta.$$

Let $\mathcal{C}$ be a category. By definition a simplicial object $X$ in $\mathcal{C}$ is a functor

$$X : \Delta^{\mathrm{op}} \to \mathcal{C}.$$

For such an $X$ and $n \ge 0$ we note

$$X_n$$

for the value of the functor on $[n]$. When $\mathcal{C}$ is the category of sets we call $X_n$ the set of $n$-simplices.

**Definition A.1.1.** — Let $n \ge 1$. For $0 \le i \le n$ the inclusion $[n-1] \to [n]$ obtained by “missing $i$” gives rise to what we call a *face morphism*

$$d_{n,i} : X_n \to X_{n-1}$$

For $0 \le i \le n-1$, the surjection $[n] \to [n-1]$ obtained by “identifying $i$ and $i+1$” gives rise to a *degeneracy morphism*

$$s_{n-1,i} : X_{n-1} \to X_n$$

We send to Remark A.1.6 for the terminology face and degeneracy morphism. Those satisfy the relations

1. $d_i d_j = d_{j-1} d_i$ if $i < j$

2. $d_i s_j = s_{j-1} d_i$ if $i < j$

3. $d_i s_j = \mathrm{Id}$ if $i = j$ or $i = j+1$

4. $d_i s_j = s_j d_{i-1}$ if $i > j + 1$

5. $s_i s_j = s_{j+1} s_i$ if $i \le j$.

Reciprocally, a collection of objects $(X_n)_{n \ge 0}$ together with the face and degeneracy morphisms satisfying the preceding relations defines a simplicial object. We note

$$s\mathcal{C}$$
for the category of simplicial objects in $\mathcal{C}$.

**Example A.1.2.** — The truncated object $X_1 \xrightarrow{\leftarrow s_0} \xrightarrow{\leftarrow d_1} X_0$ is defined by the relations $d_1 s_0 = \text{Id} = d_0 s_0$.

**Example A.1.3.** — The functor $S$ that sends a finite ordered set $E$ to the topological space $\{(\lambda_e)_{e \in E} \in \mathbb{R}_+^E | \sum_e \lambda_e = 1\}$ is a simplicial topological space with $S_n$ the standard simplex. The face maps are projections to faces of the simplex and the degeneracy maps are inclusions of the faces.

**Example A.1.4.** — Let $X$ be a topological space. We can compose the preceding functor $\Delta \to \text{Top}$ with the functor $\mathcal{C}(-, X)$ of continuous functions toward $X$. We obtain the singular set

$$ \mathrm{Sing}(X) \in s \text{ Sets} $$

whose $n$-simplices, $\mathrm{Sing}_n X$, are continuous maps from the standard simplex $\{(t_0, \dots, t_n) \in \mathbb{R}_+^n | \sum_i t_i = 1\}$ to $X$. This defines a functor

$$ \mathrm{Top} \to s \text{ Sets.} $$

**A.1.1.2. Reconstructing a simplicial set from its simplices.** — We are going to see that one can reconstruct a simplicial set as a gluing of its $n$-simplices when $n$ varies in a canonical way.

**Definition A.1.5.** — For any integer $n \ge 0$ define

$$ \Delta_n \in s \text{ Sets} $$

as $\mathrm{Hom}(-, [n])$.

Yoneda lemma shows that for any simplicial set $X$,

$$ \mathrm{Hom}(\Delta_n, X) = X_n $$

For $n \ge 1$ and $0 \le i \le n - 1$ there is a morphism

$$ \sigma_{n-1,i} : \Delta_n \to \Delta_{n-1} $$

that corresponds to the projection $[n] \to [n - 1]$ identifying $i$ and $i + 1$. The degeneracy map $s_{n-1,i} : X_{n-1} \to X_n$ is then given by

$$ (11) $$

$$ s_{n-1,i} = \sigma_{n-1,i}^* $$

that is to say it sends the $n - 1$-simplex $\Delta_{n-1} \xrightarrow{u} X$ to

$$ \Delta_n \xrightarrow{\sigma_{n-1,i} \text{ projection}} \Delta_{n-1} \xrightarrow{u} X. $$

There is a morphism for $0 \le i \le n$,

$$ \delta_{n,i} : \Delta_{n-1} \to \Delta_n $$

that corresponds to the injection $[n - 1] \to [n]$ missing $i$. One then has

$$ (12) $$

$$ d_{n,i} = \delta_{n,i}^* $$

that is to say it sends the $n$-simplex $\Delta_n \xrightarrow{u} X$ to

$$ \Delta_{n-1} \xrightarrow{\delta_{n,i}} \Delta_n \xrightarrow{u} X. $$
**Remark A.1.6.** — Equations (11) and (12) show that, in the case of simplicial sets, one can interpret the face and degeneracy maps as “real” face and degeneracy maps after interpreting $X_n$ as a set of “n-simplices”. The strength of the simplicial objects approach is that one can use the intuition of algebraic topology and simplicial sets in a more general context of simplicial objects in any category where this interpretation does not exist but the intuition remains.

Let $\mathcal{S}_X$ be the category whose objects are couples $(E, x)$ where $E \in \text{Ob}\Delta$ and $x \in X(E)$, and

$$ \operatorname{Hom}((E, x), (E', x')) = \{f : E \to E' \mid f^*x' = x\}. $$

This is equivalent to the category of couples $(n, x)$ where $n \in \mathbb{N}$, $x \in X_n$ and

$$ \operatorname{Hom}((n, x), (n', x')) = \{f : [n] \to [n'] \mid f^*x' = x\}. $$

We note $\Delta_n \to X$ for an object of this last category. Its morphisms are thus commutative diagrams

<figure><img src="image_44.png" /></figure>

via the identification $\operatorname{Hom}(\Delta_n, \Delta_m) = \operatorname{Hom}([n], [m])$. There is a functor

$$ F : \mathcal{S}_X \to s \text{Sets} $$

$$ (E, x) \mapsto \operatorname{Hom}(-, E) $$

and a morphism of functors, where $X$ is the constant functor with values $X$,

$$ F \implies X $$

$$ F(E, x) \mapsto x. $$

This induces an isomorphism of simplicial sets

$$ \lim_{\mathcal{S}_X} F \xrightarrow{\sim} X. $$

We note this as

$$ \lim_{\substack{\Delta_n \xrightarrow{\sim} X \\ \Delta_n \to X}} $$

This gives a meaning to the fact that $X$ is obtained by gluing

$$ \coprod_{X_n} \Delta_n $$

when $n$ varies using the face and degeneracy maps, like a CW complex together with its cell decomposition. More precisely, $X$ is the quotient of

$$ \coprod_{n \ge 0} \coprod_{X_n} \Delta_n $$

by the equivalence relation in the category of simplicial sets generated by

$$ (x, \delta_{n,i}(a)) \sim (d_{n,i}(x), a) \text{ for } x \in X_n \text{ and } a \in \Delta_{n-1} $$

and

$$ (x, \sigma_{n,i}(a)) \sim (s_{n,i}(x), a) \text{ for } x \in X_n \text{ and } a \in \Delta_{n+1}. $$
**A.1.1.3.** *Geometric realization of a simplicial set.* — There is a geometric realization functor

$$|-| : \text{sets} \longrightarrow \text{Top}$$

$$X \mapsto \varinjlim_{\Delta_n \to X} |\Delta_n|$$

where $|\Delta_n|$ is the usual simplex in $\mathbb{R}_+^{n+1}$. This is a left adjoint functor to the singular set functor

$$\mathrm{Hom}(|X|, T) = \mathrm{Hom}(X, \mathrm{Sing}(T))$$

**Example A.1.7.** — 1. The degeneracy map

$$|\sigma_{n-1,i}| : |\Delta_n| \longrightarrow |\Delta_{n-1}|$$

of Equation (12) is given by, for $(\lambda_0, \dots, \lambda_n) \in \mathbb{R}_+^n$ satisfying $\sum_{i=0}^n \lambda_i = 1$, $(\lambda_0, \dots, \lambda_n) \mapsto (\lambda_0, \dots, \lambda_{i-1}, \lambda_i + \lambda_{i+1}, \lambda_{i+2}, \dots, \lambda_n)$.

2. The face map

$$|\delta_{n,i}| : |\Delta_{n-1}| \longrightarrow |\Delta_n|$$

of Equation (11) is given by

$$(\lambda_0, \dots, \lambda_{n-1}) \mapsto (\lambda_0, \dots, \lambda_{i-1}, 0, \lambda_i, \dots, \lambda_{n-1}).$$

The topological space $|X|$ is naturally a CW complex. More precisely, let

$$X_n^{\mathrm{ndeg}} = X_n \setminus \bigcup_{i=0}^{n-1} \mathrm{Im}(s_{n-1,i})$$

be the set of “non-degenerate” simplices. There is a quotient map

$$\coprod_{n \ge 0} \coprod_{X_n} |\Delta_n| \longrightarrow |X|.$$

For $k \ge 0$ let

$$|X|_k = \mathrm{Im} \left( \coprod_{n \le k} \coprod_{X_n} |\Delta_n| \longrightarrow |X| \right).$$

This is the $k$-skeleton of $|X|$ and

$$|X|_k \setminus |X|_{k-1} = \coprod_{X_k^{\mathrm{ndeg}}} |\Delta_k|$$

where

$$|\Delta_k| = \{ (\lambda_0, \dots, \lambda_k) \in \mathbb{R}_{>0}^k \mid \sum_{i=0}^k \lambda_i = 1 \}.$$

**A.1.1.4.** *Augmented simplicial objects.* —

**A.1.2. Homotopies.** —

**A.1.2.1.** *Two equivalent formulations.* — Let $X$ and $Y$ be simplicial objects in a category $\mathcal{C}$. Let

$$f, g : X \longrightarrow Y$$

be two morphisms in $\mathcal{C}$.

**Definition A.1.8.** — An homotopy between $f$ and $g$ is the data of morphisms $(h_{n,i})_{n \ge 0, 0 \le i \le n}$, where $h_{n,i} : X_n \to Y_{n+1}$, satisfying

* $d_0h_0 = f_n$

* $d_{n+1}h_n = g_n$
$$
\bullet \ d_i h_j = \begin{cases} h_{j-1} d_i & \text{if } i < j \\ d_i h_{i-1} & \text{if } i = j \\ h_j d_{i-1} & \text{if } i > j + 1 \end{cases}
$$
$$
\bullet \ s_i h_j = \begin{cases} h_j s_{i-1} & \text{if } i > j \\ h_{j+1} s_i & \text{if } i \le j \end{cases}
$$

This definition is justified by the following results whose proof are elementary computation (as
usual in homotopical algebra, a lot of notions come from natural notions issued from simplicial
sets and classical algebraic topology, see Remark A.1.6).

**Proposition A.1.9 ([4, Proposition 3.1].)** — For $h = (h_{n,i})_{0\le i\le n}$ an homotopy between $f$ and $g$ define $(r_{n,i})_{0\le i\le n+1}$ where $r_{n,i} : X_n \to Y_n$ is defined by the formula

$$
r_{n,0} = f_n, \quad r_{n,n+1} = g_n, \quad r_{n,i} = d_{n+1,i} h_{n,i} \text{ if } 1 \le i \le n.
$$

The correspondence $h \mapsto r$ induces a bijection between homotopies between $f$ and $g$ and collections of maps $r_{n,i} : X_n \to Y_n, 0 \le i \le n+1$ satisfying

$$
\bullet \ r_{n,0} = f_n, \ r_{n,n+1} = g_n
$$

$$
\bullet \ d_{n,i} r_{n,j} = \begin{cases} r_{n-1,j-1} d_{n,i} & \text{if } i < j \\ r_{n-1,j} d_{n,i} & \text{if } i \ge j \end{cases}
$$

$$
\bullet \ s_{n,i} r_{n,j} = \begin{cases} r_{n+1,j+1} s_{n,i} & \text{if } i < j \\ r_{n+1,j} s_{n,i} & \text{if } i \ge j. \end{cases}
$$

The inverse to this bijection if given by the formula

$h_{n,i} = r_{n+1,i} s_{n,i}.$

The preceding proposition allows us to recast the definition of an homotopy in its classical
framework of algebraic topology.

**Lemma A.1.10.** — Suppose that *C* is the category of Sets. Then, an homotopy between *f* and *g* is the same as a morphism

$h : X \times \Delta_1 \to Y$

such that the following diagram

<figure><img src="image_45.png" /></figure>

commutes.

**A.1.2.2.** *The Kan extension property.* —

**A.1.3. Homotopy groups.** —

**A.1.3.1. Connected components.** — Let $X$ be a simplicial set. Consider the relation on $X_0$ generated by the relation

$x \sim y \Longleftrightarrow \exists z \in X_1, d_1(z) = x \text{ and } d_0(z) = y.$

One has $x \sim x$ and we still denote $\sim$ for the equivalence relation generated by the preceding. We then define

$$
\pi_0(X) = X_0 / \sim = \text{coeq} \ (X_1 \overset{h}{\longrightarrow} X_0).
$$
**A.1.3.2. Homotopy groups.** — Let $n \ge 1$. Define the simplicial set

$$ \partial\Delta_n = \bigcup_{0 \le i \le n} \delta_{n,i}(\Delta_{n-1}) \subset \Delta_n. $$

One has

$$ (\partial\Delta_n)_m = \{f : [m] \to [n] \mid f \text{ is not surjective}\}. $$

Its geometric realization is

$$ |\partial\Delta_n| = \{ (\lambda_0, \dots, \lambda_n) \in \mathbb{R}_+^n \mid \sum_{i=0}^n \lambda_i = 1 \text{ and } \exists i, \lambda_i = 0 \}. $$

Let $X$ be a simplicial set and fix a base point $x \in X_0$. For any $n$, the unique map $[n] \to [0]$ defines a map $X_0 \to X_n$ that is a composite of degeneracy maps. The sphere

$$ S^n = \Delta_n / \partial\Delta_n $$

is pointed too since $(S^n)_0$ has only one element. Let us note 0 this point. Let us now look at the morphisms of pointed simplicial sets

$$ \operatorname{Hom}((S^n, 0), (X, x)). $$

This is identified with

$$ \{y \in X_n \mid d_{n,0}(y) = \dots = d_{n,n}(y) = x_{n-1}\} $$

where $x_{n-1} \in X_{n-1}$ is obtained from $x$ via $[n-1] \to [0]$. We define a relation $\sim$ on $\operatorname{Hom}((S^n, 0), (X, x))$ in the following way. For two morphisms

$$ f, g : (S^n, 0) \to (X, x) $$

we say that $f \sim g$ if there exists a morphism

$$ h : S^n \times \Delta_1 \to X $$

such that

<figure><img src="image_46.png" /></figure>

and

$$ h|_{0 \times \Delta_1} = x. $$

**A.1.4. The Dold-Kan correspondence.** —

**A.1.5. Simplicial rings.** —

**A.1.6. Bisimplicial sets and the Eilenberg-Zilber theorem.** —

**A.1.7. The Hurewicz theorem.** —
## A.2. Adjoint functors, comonads and the Bar construction

**A.2.1. Construction of the simplicial object.** — Let $C \xrightarrow{F} \mathcal{D}$ be a couple of adjoint functors where $F$ is left adjoint to $G$. Let

$$u : F \circ G \implies \mathrm{Id}_{\mathcal{D}}$$

$$v : \mathrm{Id}_{\mathcal{C}} \implies G \circ F$$

be the natural transformations associated with the adjunction. Let us note

$$T = F \circ G$$

and

$$\eta : T \xrightarrow{u} \mathrm{Id}_{\mathcal{D}}$$

$$\mu : T \xrightarrow{FvG} T^2.$$

The triple $(T, \eta, \mu)$ is what is called a *comonad*, the comonad property being the relations

$$T\mu \circ \mu = \mu T \circ \mu : T \implies T^3$$

$$T\eta \circ \mu = \eta T \circ \mu = \mathrm{Id}_T : T \implies T.$$

**Remark A.2.1.** — If we have a diagram of functors between categories

$$C_1 \xrightarrow{I} C_2 \underset{K}{\overset{J}{\biguplus}} \underset{u}{\bigdownarrow} C_3 \xrightarrow{L} C_4$$

with $u$ a natural transformation, we note

$$LuI : LJI \implies LKI$$

the natural transformation defined by the following rule: for $X \in C_1$,

$$(LuI)_X = L(u_{I(X)}) : L(J(I(X))) \to L(K(I(X))).$$

Starting from such a comonad we can construct a functor

$$\mathcal{D} \to \{\text{augmented simplicial objects in } \mathcal{D}\}$$

via the following rule. Let $X$ be an object of $\mathcal{D}$. For $n \ge 0$ set

$$X_n = T^{n+1}(X)$$

with the face and degeneracy maps

$$d_{n,i} = T^i \eta T^{n+1-i} : X_n \to X_{n-1}$$

$$s_{n,i} = T^i \mu T^{n+1-i} : X_n \to X_{n+1}.$$

This is augmented toward $X$ via

$$X_{\bullet} \xrightarrow{\eta} X.$$

The simplicial identities are deduced from the comonad relations. In low degree this is the following diagram

$$T^3(X) \xrightarrow{\substack{T^2 \eta \\ \leftarrow T \mu \\ \rightarrow T \eta T}} T^2(X) \xrightarrow{\substack{T \eta \\ \leftarrow \mu \\ \rightarrow \eta T}} T(X) \xrightarrow{\eta} X.$$

This is the so-called *Bar construction*.
**A.2.2. Contractibility and acyclicity.** — One then has the following result: the augmented simplicial object

$$G(X_{\bullet}) \to G(X)$$

is contractible (see [4] for the different notions of contractible simplicial objects). In fact, the augmented simplicial complex $G(X_{\bullet}) \to G(X)$ comes with an extra map

$$t_n : G(X_n) \to G(X_{n+1})$$

(where we note $X_{-1} = X$ and $t_{-1} : G(X) \to G(X_0)$) defined by the formula

$$G(X_n) \xrightarrow{v} GFG(X_n) = G(X_{n+1}).$$

Let $G(X)_{\text{cons}}$ be now the constant simplicial object with value $G(X)$ and face/degeneracy maps given by the identity. There are then two maps of simplicial objects

$$G(X_{\bullet}) \xrightarrow{a} G(X)_{\text{cons}}$$

where $G(X_{\bullet}) \to G(X)_{\text{cons}}$ is given by the augmentation and the other one by

$$G(X) \xrightarrow{t_{n-1} \circ \cdots \circ t_{-1}} G(X_n).$$

In fact, one verifies immediately that for $n \ge -1$,

$$(13) \qquad \begin{cases} d_{n+1,0} \circ t_n = \text{Id} \\ d_{n+1,i} \circ t_n = t_{n-1} \circ d_{n,i-1} \quad \text{for } 0 < i \le n+1 \end{cases}$$

which implies in particular that the preceding map is a morphism of simplicial objects. Recall now the following definition.

**Definition A.2.2.** — Let $f, g : X \to Y$ be two maps of simplicial objects in a category. An homotopy between $f$ and $g$ is the data of maps $(h_{n,i})_{0 \le i \le n}$, where $h_{n,i} : X_n \to Y_{n+1}$, satisfying

* $d_0 h_0 = f_n$

* $d_{n+1} h_n = g_n$

* $d_i h_j = \begin{cases} h_{j-1} d_i & \text{if } i < j \\ d_i h_{i-1} & \text{if } i = j \\ h_j d_{i-1} & \text{if } i > j + 1 \end{cases}$

* $s_i h_j = \begin{cases} h_j s_{i-1} & \text{if } i > j \\ h_{j+1} s_i & \text{if } i \le j \end{cases}$

For simplicial objects in the category of sets or in an abelian category this is the same as a morphism $X \times \Delta_1 \to Y$ giving $f$ via the inclusion $X \times \{0\} \hookrightarrow X \times \Delta_1$ and $g$ via the inclusion $X \times \{1\} \hookrightarrow X \times \Delta_1$. This is in fact deduced from the following proposition whose proof is a simple computation.

**Proposition A.2.3** ([4, Proposition 3.1]). — For $h = (h_{n,i})_{0 \le i \le n}$ an homotopy between $f$ and $g$ define $(r_{n,i})_{0 \le i \le n+1}$ where $r_{n,i} : X_n \to Y_n$ is defined by the formula

$$r_{n,0} = f_n, \quad r_{n,n+1} = g_n, \quad r_{n,i} = d_{n+1,i} h_{n,i} \text{ if } 1 \le i \le n.$$

The correspondence $h \mapsto r$ induces a bijection between homotopies between $f$ and $g$ and collections of maps $r_{n,i} : X_n \to Y_n, 0 \le i \le n+1$ satisfying

* $r_{n,0} = f_n, r_{n,n+1} = g_n$

* $d_{n,i} r_{n,j} = \begin{cases} r_{n-1,j-1} d_{n,i} & \text{if } i < j \\ r_{n-1,j} d_{n,i} & \text{if } i \ge j \end{cases}$

* $s_{n,i} r_{n,j} = \begin{cases} r_{n+1,j+1} s_{n,i} & \text{if } i < j \\ r_{n+1,j} s_{n,i} & \text{if } i \ge j. \end{cases}$
The inverse to this bijection if given by the formula

$$h_{n,i} = r_{n+1,i} s_{n,i}.$$

If the ambient category is abelian such an homotopy induces an homotopy of the associated
morphisms between the (unnormalized) chain complexes:

$$\tilde{h} := \sum_{i=0}^{n} (-1)^i h_{n,i} : X_n \to Y_{n+1}$$

satisfies

$$\tilde{h}\partial + \partial\tilde{h} = f - g$$

where $\partial_n = \sum_{i=0}^n (-1)^i d_{n,i}$ is the usual boundary map.

**Remark A.2.4.** — One has to be careful that in general the relation “f is homotpic to g” is not an equivalence relation. This is the case if Y is a Kan simplicial set but not in general.

**Remark A.2.5.** — Both definitions of an homotopy, via h or r, are useful. The point of view of h as a collection of morphisms $X_n \to Y_{n+1}$ gives rise to the homotopy of the associated chain complexes if the category is abelian. The point of view of r as a collection of morphisms $X_n \to Y_n$ is typically useful for the proof of Lemma A.2.13.

One then has the following result.

**Proposition A.2.6.** — One has $a \circ b = \mathrm{Id}$ and $b \circ a \sim \mathrm{Id}$ (homotopic to the identity).

*Proof.* — The equality $a \circ b = \mathrm{Id}$ is immediate. Define now for $0 \le i \le n$, $h_{n,i} : G(X_n) \to G(X_{n+1})$ as the composite

$$G(X_n) \xrightarrow{d_{n,0}} G(X_{n-1}) \xrightarrow{d_{n-1,0}} \dots \xrightarrow{d_{n-i+1,0}} G(X_{n-i}) \xrightarrow{t_{n-i}} G(X_{n-i+1}) \xrightarrow{t_{n-i+1}} \dots \xrightarrow{t_n} G(X_{n+1})$$

that is to say

$$h_{n,i} = t_n \circ t_{n-1} \circ \dots \circ t_{n-i} \circ d_{n-i+1,0} \circ \dots \circ d_{n-1,0} \circ d_{n,0}.$$

One verifies using Equation (13) that this satisfies the axioms of Definition A.2.2 of an homotopy between the identity and $b \circ a$. $\square$

As a particular case, if $\mathcal{C}$ is an abelian category then the chain complex associated with $G(X_\bullet)$ is homotopy equivalent to the zero complex. For $n \ge 0$, if

$$\partial_n = \sum_{i=0}^{n} (-1)^i d_{n,i} : G(X_n) \to G(X_{n-1})$$

one has, using Equation (13),

$$\partial_{n+1}t_n + t_{n-1}\partial_n = \mathrm{Id}.$$

This allows us to construct plenty of canonical resolutions when the functor $G$ is conservative.
This is summarized in the following proposition.

**Proposition A.2.7 (Bar resolution).** — Let $\mathcal{C} \stackrel{F}{\underset{G}{\rightleftarrows}} \mathcal{D}$ be a couple of adjoint functors where $F$ is left adjoint to $G$. Let $T = FG : \mathcal{D} \to \mathcal{D}$.

1. There is a functor

$$\mathcal{D} \to \{\text{augmented toward } \mathrm{Id}_{\mathcal{D}} \text{ simplicial objects in } \mathcal{D}\}$$

$$X \mapsto [T^{\bullet+1}(X) \to X].$$
2. The composite of this functor with $G$ is an homotopy equivalence

$$G(T^{\bullet+1}(X)) \to G(X)_{\text{cons}}$$

where $G(X)_{\text{cons}}$ is the constant simplicial object with value $G(X)$.

Sometimes we use the same formalism as before but in the opposite categories. In this case the preceding structures are called monads. Let us give some examples.

**Example A.2.8.** — If $X$ is a topos and $f: U \to e$ is a covering of the final object of $X$ the couple of adjoint functors $(f^*, f_*)$ defines for any $\mathcal{F}$ an abelian group in $X$ an augmented cosimplicial complex $\mathcal{F} \to \mathcal{F}_\bullet$ with $\mathcal{F}_n = (f_*f^*)^{n+1}\mathcal{F}$. After applying $f^*$ to this augmented cosimplicial abelian groups in $X$ it becomes contractible. Since $f$ is a cover this implies that the chain complex associated to $\mathcal{F} \to \mathcal{F}_\bullet$ is a resolution of $\mathcal{F}$. This gives rise to the Čech cohomology spectral sequence associated to the cover $U \to e$.

**Example A.2.9.** — Using the notations of the preceding example, the couple of adjoint functors $(f_1, f^*)$ gives rise to a resolution $(f_1f^*)^{\bullet+1}\mathcal{F} \to \mathcal{F}$. When $X$ is the topos associated to a topological space and $U$ is associated to an open cover of this topological space, this gives rise to the compactly supported Čech cohomology spectral sequence.

**Example A.2.10.** — If $X$ is a topological space and $f: \coprod_{x \in X} \{x\} \to X$ then the couple of adjoint functors $(f^*, f_*)$ gives rise to the Godement resolution.

**Example A.2.11.** — If $A \to B$ is a morphism of rings the couple of adjoint functors $(B \otimes_A -, \text{Res}_{B/A})$ gives rise for any $A$-module $M$ to an augmented cosimplicial $A$-module $M \to B^{\otimes(\bullet+1)} \otimes_A M$. If $A \to B$ is faithfully flat this becomes contractible after applying $B \otimes_A -$ and is thus acyclic as a chain complex. This is the standard Čech resolution of $M$ associated to the fpqc cover $\text{Spec}(B) \to \text{Spec}(A)$.

**Example A.2.12.** — With the notations of the preceding example, for any $B$-module $M$ there is an augmented simplicial $B$-module $B^{\otimes(\bullet+1)} \otimes_A M \to M$. Since the associated restriction of scalars functor from $B$ to $A$ is conservative, this is a resolution. For example, for $M = B$, this is the usual Bar resolution

$$\cdots B \otimes_A B \otimes_A B \xrightarrow{\partial} B \otimes_A B \xrightarrow{\partial} B$$

with $\partial(b_1 \otimes b_2 \otimes b_3) = b_1b_2 \otimes b_3 - b_1 \otimes b_2b_3$ and $\partial(b_1 \otimes b_2) = b_1b_2$ that gives rise to the standard resolution in group cohomology when we take $A = \mathbb{Z} \to \mathbb{Z}[G] = B$.

We will need the following lemma later.

**Lemma A.2.13.** — Let $f: \text{Id}_D \Rightarrow \text{Id}_D$ be an endomorphism of the identity functor. Then the two maps

$$ \begin{array}{ccc} T^{\bullet+1}(X) & \longrightarrow & X \\ \underset{f T^{\bullet+1}}{\downarrow} \underset{T^{\bullet+1}f}{\big|} & & \downarrow f \\ T^{\bullet+1}(X) & \longrightarrow & X \end{array} $$

are homotopic maps of simplicial complexes.

*Proof.* — The fact that those are morphisms of simplicial objects is easy. We use the definition of an homotopy of Proposition A.2.3. It suffices in fact to set

$$r_{n,i} = T^i f T^{n+1-i}$$

to obtain the result.

□
## A.3. The cotangent complex

**A.3.1. General construction.** — Let $A$ be a ring. We consider the two adjoint functors

$$A\text{-algebras } \xrightarrow{\text{forget}}_{A[-]} \text{Sets}$$

where for a set $X$, $A[X]$ is the symmetric algebra on the free $A$-module $A^{(X)}$ which is a polynomial algebra. We then have a canonical polynomial resolution for any $A$-algebra $B$,

$$\dots \mathrel{\overbrace{\hspace{2.5cm}}^{\hspace{2.5cm}} \hspace{2.5cm}} \mathfrak{A}[\mathfrak{A}[B]] \mathrel{\overbrace{\hspace{2.5cm}}^{\hspace{2.5cm}} \hspace{2.5cm}} \mathfrak{A}[B] \xrightarrow{d_1} \mathfrak{A}[B] \longrightarrow B$$

where for $b_1, \dots, b_n \in B$ distinct and $P \in A[X_1, \dots, X_n]$ one has

$$d_0([P([b_1], \dots, [b_n])]) = P([b_1], \dots, [b_n])$$

$$d_1([P([b_1], \dots, [b_n])]) = [P(b_1, \dots, b_n)].$$

and the degeneracy map $s_0 : A[B] \to A[A[B]]$ is given by

$$s_0([b]) = [[b]].$$

This is a simplicial $A$-algebra $P_{B/A, \bullet}$ satisfying:

* for each $n \ge 0$, $P_{B/A, n}$ is a polynomial $A$-algebra on a set $X_n$,

* one has $\pi_0(P_{B/A, \bullet}) = B$ and $\pi_i(P_{B/A, \bullet}) = 0$ for $i > 0$ by applying point (2) of Proposition A.2.7 that says that the augmented simplicial set $P_{B/A, \bullet} \to B$ is contractible,

* for each $n$ and $0 \le i \le n$, the degeneracy morphism $s_{n,i}$ satisfies $s_{n,i}(X_n) \subset X_{n+1}$.

**Definition A.3.1.** — The cotangent complex $\mathbb{L}_{B/A}$ of $A \to B$ is the chain complex of $B$-modules associated to the simplicial $B$-module

$$\Omega_{P_{B/A, \bullet}/A}^{1} \otimes_{P_{B/A, \bullet}} B.$$

We will often see it as an object of $D(B)$, the derived category of $B$-modules, but nevertheless let us remark it is well defined canonically as a complex. It is concentrated in $\le 0$ degrees. The natural augmentation $\mathbb{L}_{B/A} \to \Omega_{B/A}^{1}$ induces an isomorphism

$$H^0(\mathbb{L}_{B/A}) \xrightarrow{\sim} \Omega_{B/A}^{1}.$$

**A.3.2. Definition as a derived functor.** — Let us begin with a definition.

**Definition A.3.2.** — Let $\mathcal{B}_{\bullet}$ be a simplicial $A$-algebra augmented toward the $A$-algebra $B$

$$\mathcal{B}_{\bullet} \to B.$$

We say that $\mathcal{B}_{\bullet}$ is a polynomial resolution of $B$ is

* $\pi_n(\mathcal{B}_{\bullet}) = 0$ for $n > 0$,

* $\pi_0(\mathcal{B}_{\bullet}) \xrightarrow{\sim} B$,

* for all $n \ge 0$, $\mathcal{B}_n$ is a polynomial $A$-algebra on a $X_n \subset \mathcal{B}_n$ i.e. $A[X_n] \xrightarrow{\sim} \mathcal{B}_n$,

* the degeneracy maps $s_{n,i} : \mathcal{B}_n \to \mathcal{B}_{n+1}$, $0 \le i \le n$, satisfy $s_{n,i}(X_n) \subset X_{n+1}$,

Typically, $P_{B/A, \bullet} \to B$ is a polynomial resolution of $B$. We now have the following deep result by Quillen.

**Theorem A.3.3 (Quillen [45, Chapter II-Section 4], [31, Theorem 4.17])**

The category of simplicial $A$-algebras has the structure of a model category in the sense of Quillen where a morphism $f : \mathcal{B}_{\bullet} \to \mathcal{C}_{\bullet}$ is

1. a weak equivalence if for all $n \ge 0$, $\pi_n(\mathcal{B}_{\bullet}) \xrightarrow{\sim} \pi_n(\mathcal{C}_{\bullet})$,

2. a fibration if the induced map $\mathcal{B}_{\bullet} \to \pi_0(\mathcal{B}_{\bullet}) \times_{\pi_0(\mathcal{C}_{\bullet})} \mathcal{C}_{\bullet}$ is surjective.
Moreover, if $\mathcal{B}_{\bullet} \to B$ is a polynomial resolution then $\mathcal{B}_{\bullet} \to B$ is a cofibration.

We can now see the cotangent complex as derived functor i.e. a Kan extension. In fact, we have

$$ \lim_{\mathcal{B}_{\bullet} \to B} \Omega_{\mathcal{B}_{\bullet}/A} \otimes_{\mathcal{B}_{\bullet}} B \xrightarrow{\sim} \mathbb{L}_{B/A} \in D(B) $$

where $\mathcal{B}_{\bullet} \to B$ goes through the set of resolutions of $B$ by simplicial $A$-algebras and where the colimit is essentially constant. Here by a resolution we mean $\pi_i(\mathcal{B}_{\bullet}) = 0$ if $i > 0$ and $\pi_0(\mathcal{B}_{\bullet}) \xrightarrow{\sim} B$. The limit becomes constant as soon as $\mathcal{B}_{\bullet} \to B$ is a polynomial resolution, in which case

$$ \Omega_{\mathcal{B}_{\bullet}/A} \otimes_{\mathcal{B}_{\bullet}} B \xrightarrow{\sim} \mathbb{L}_{B/A} \in D(B). $$

### A.4. Basic properties of the cotangent complex

**A.4.1. The fundamental triangle.** — For $A \to B \to C$ morphisms of rings we obtain the usual exact sequence of $C$-modules

$$ \Omega_{B/A}^{1} \otimes_{B} C \to \Omega_{C/A}^{1} \to \Omega_{C/B}^{1} \to 0. $$

We are going to see that this extends to a long exact sequence and even an exact triangle in $D(C)$.

**Proposition A.4.1.** — For $A \to B \to C$ morphisms of rings there is an exact triangle in the triangulated category $D(C)$

<figure><img src="image_47.png" /></figure>

*Proof.* — Consider the polynomial resolution

$$ \mathcal{B}_{\bullet} = P_{B/A, \bullet} \to B. $$

Via the projection to $B$ and the morphism $B \to C$, there is a morphism of simplicial rings $\mathcal{B}_{\bullet} \to C$ where $C$ is seen as a constant simplicial ring. Consider now the bi-simplicial ring

$$ P_{C/\mathcal{B}_{\bullet}, \bullet}. $$

Let us form its diagonal

$$ \mathcal{C}_{\bullet} = \Delta P_{C/\mathcal{B}_{\bullet}, \bullet}. $$

It is augmented toward $C$ and this is again a polynomial resolution of $C$ thanks to [32, Proposition 1.9-Chapter IV] (since we work with simplicial abelian groups, one can use alternatively the generalized Eilenberg-Zilber theorem, [32, Theorem 2.5-Chapter IV]). We now have a diagram of simplicial rings

<figure><img src="image_48.png" /></figure>

where $\mathcal{B}_{\bullet}$ is a polynomial resolution of $B$ as an $A$-algebra and for each $n$, $\mathcal{C}_n$ is a polynomial $\mathcal{B}_n$-algebra. From this we deduce an exact sequence of simplicial $\mathcal{C}_{\bullet}$-modules

$$ 0 \to \Omega_{\mathcal{B}_{\bullet}/A}^{1} \otimes_{\mathcal{B}_{\bullet}} \mathcal{C}_{\bullet} \to \Omega_{\mathcal{C}_{\bullet}/A}^{1} \to \Omega_{\mathcal{C}_{\bullet}/\mathcal{B}_{\bullet}}^{1} \to 0. $$
To any derivation $D \in \text{Der}_A(B, M)$ one associated the automorphism $\text{Id} + D$ of the extension $E$. Let us note

$$ \text{Exalg}_A(B, M) = \pi_0(\text{Exalg}_A(B, M)) $$

the set of automorphism classes of such extensions. The groupoid $\text{Exalg}_A(B, M)$ is functorial in $B$ and $M$. If $f: B' \to B$ is a morphism of $A$-algebras the pull back via this morphism induces a functor

$$ f^*: \text{Exalg}_A(B, M) \to \text{Exalg}_A(B', M) $$

where the pullback by $f$ is defined by the diagram

$$ \begin{tikzcd} 0 \arrow[r] \arrow[d] & M \arrow[r] \arrow[d] & E' \arrow[r] \arrow[d] & B' \arrow[r] \arrow[d] & 0 \\ 0 \arrow[r] & M \arrow[r] & E \arrow[r] \arrow[d] & B \arrow[r] \arrow[d] & 0 \end{tikzcd} $$

i.e. $E' = E \times_B B'$. For $u: M \to M'$ a morphism of $B$-modules, it induces a functor

$$ u_*: \text{Exalg}_A(B, M) \to \text{Exalg}_A(B, M') $$

where the pushforward by $u$ is defined

$$ \begin{tikzcd} 0 \arrow[r] \arrow[d, "u"] & M \arrow[r] \arrow[d] & E \arrow[r] \arrow[d] & B \arrow[r] \arrow[d] & 0 \\ 0 \arrow[r] & M' \arrow[r] & E' \arrow[r] \arrow[d] & B \arrow[r] \arrow[d] & 0 \end{tikzcd} $$

i.e. $E' = E \coprod_M M'$. Both operations commute: there is a natural isomorphism of functors $f^*u_* \xrightarrow{\sim} u_*f^*$. We now use the language of Picard groupoids ([1, Exposé XVIII-Section 1.4]). The preceding two fonctorialities defines a (strictly commutative) Picard groupoid structure on $\text{Exalg}_A(B, M)$

$$ \text{Exalg}_A(B, M) \times \text{Exalg}_A(B, M) \xrightarrow{+} \text{Exalg}_A(B, M) $$

via the following rule

$$ \xi + \xi' = \Sigma_* \Delta^*(\xi \times \xi') $$

where

* if $\xi$ is the extension $0 \to M \to E \to B \to 0$ and $\xi'$ is the extension $0 \to M \to E' \to B \to 0$, $\xi \times \xi'$ is the extension

$$ 0 \to M \times M \to E \times E' \to B \times B \to 0, $$

* $\Sigma: M \times M \to M$ is the morphism $(m_1, m_2) \mapsto m_1 + m_2$,

* $\Delta: B \to B \times B$ is the diagonal morphism $b \mapsto (b, b)$.

This is moreover equipped with a “B-linear structure”. This means that for $\lambda \in B$, there is a multiplication by $\lambda$ functor

$$ \mathfrak{m}_{\lambda}: \text{Exalg}_A(B, M) \to \text{Exalg}_A(B, M) $$

that is simply the pushforward via $M \xrightarrow{\times\lambda} M$. It is equipped with compatibilities: natural isomorphisms

$$ c_{\lambda, \mu}: \mathfrak{m}_{\lambda} \circ \mathfrak{m}_{\mu} \xrightarrow{\sim} \mathfrak{m}_{\lambda\mu} $$

and

$$ a_{\lambda}: \mathfrak{m}_{\lambda} \circ + \xrightarrow{\sim} + \circ (\mathfrak{m}_{\lambda} \times \mathfrak{m}_{\lambda}). $$

All those natural transformations satisfy some evident compatibility relations (that are left to the reader) that equip $\text{Exalg}_A(B, M)$ with what we call a B-linear Picard groupoid structure.
**A.5.2. Expression in terms of the cotangent complex.** — Now, there is a correspondence

$$[M^{-1} \to M^0] \to gp(M^{-1} \to M^0)$$

that associated to a length 2 complex of $B$-modules a $B$-linear Picard groupoid. The objects of $gp(M^{-1} \xrightarrow{\partial} M^0)$ are $M^0$ and

$$\mathrm{Hom}(m, m') = \{n \in M^{-1} \mid m' = \partial(n) + m\}.$$

The composition rule $\mathrm{Hom}(m, m') \times \mathrm{Hom}(m', m'') \to \mathrm{Hom}(m, m'')$ is given by the addition law of $M^{-1}$. The $B$-linear groupoid structure is simply given by the $B$-module structure of $M^0$.

This correspondence $gp(-)$

• transforms morphisms of complexes into $B$-linear functors,

• sends homotopies between morphisms of complexes to $B$-linear natural transformations of $B$-linear functors,

• is such that a morphism of complexes induces an equivalence if and only if it is a quasi-isomorphism.

**Proposition A.5.1.** — There is an equivalence of $B$-linear Picard groupoids

$$gp(\tau_{\le 1} R \mathrm{Hom}_B(\mathbb{L}_{B/A}, M)[1]) \xrightarrow{\sim} \underline{\mathrm{Exalg}}_A(B, M).$$

*Proof.* — Consider the surjection

$$\pi : A[B] \to B$$

and note $I$ its kernel. This defines a length 2-complex of $B$-modules

$$I/I^2 \xrightarrow{\delta} \Omega_{A[B]/A}^1 \otimes_{A[B]} B$$

where for $x \in I$, $\delta(x \mod I^2) = dx \otimes 1$. Let us define a $B$-linear functor

$$gp(\mathrm{Hom}_B(\Omega_{A[B]/A}^1 \otimes_{A[B]} B, M) \to \mathrm{Hom}_B(I/I^2, M)) \to \underline{\mathrm{Exalg}}_A(B, M).$$

For this let us look at the extension

$$(14) \qquad 0 \to I/I^2 \to A[B]/I^2 \to B \to 0.$$

Pushforward of this extension via a morphism $I/I^2 \to M$ defines a map

$$\mathrm{Ob}(gp(\mathrm{Hom}_B(\Omega_{A[B]/A}^1 \otimes_{A[B]} B, M) \to \mathrm{Hom}_B(I/I^2, M))) \to \mathrm{Ob}(\underline{\mathrm{Exalg}}_A(B, M)).$$

This map extends to a functor in the following way. The Equation (14) fits into a bigger diagram

<figure><img src="image_49.png" /></figure>

where for $x \in A[B]$, $f(x \mod I^2) = dx \otimes 1$. If $\varepsilon$ is the upper extension of $B$ by $I/I^2$, $f$ induces an isomorphism of extensions of $B$ by $\Omega_{A[B]/A}^1 \otimes_{A[B]} B$,

$$0 \xrightarrow{\sim} \delta_* \varepsilon$$

where $0$ is the trivial extension, de dual numbers $B[I/I^2]$. Now, if $u : I/I^2 \to M$ and $v : \Omega_{A[B]/A}^1 \otimes_{A[B]} B \to M$, we have a canonically defined isomorphism

$$0 \xrightarrow{\sim} v_* \delta_* \varepsilon \xrightarrow{\sim} (v\delta)_* \varepsilon.$$

Thus, for $u : I/I^2 \to M$, there is an isomorphism

$$u_* \varepsilon \xrightarrow{\sim} u_* \varepsilon + 0 \xrightarrow{\sim} u_* \varepsilon + v_* \delta_* \varepsilon \xrightarrow{\sim} (u + v\delta)_* \varepsilon.$$
This defines our functor

$$gp(\operatorname{Hom}_B(\Omega_{A[B]/A}^1 \otimes_{A[B]} B, M) \to \operatorname{Hom}_B(I/I^2, M)) \to \underline{\operatorname{Exalg}}_A(B, M).$$

For the essential surjectivity, given an extension of $B$ by $M$, $\alpha$, the choice of a set-theoretical section of $E \to B$ defines a morphism $g$ as in the following diagram

<figure><img src="image_50.png" /></figure>

Now, $g_I$ defines a $B$-linear morphism $u = (g_I \mod I^2) : I/I^2 \to M$. One verifies immediately that $g$ modulo $I^2$ defines an isomorphism.

$$u_*\varepsilon \xrightarrow{\sim} \alpha.$$

For the full faithfulness we have to verify the two following points:

1. $\ker(\operatorname{Hom}_B(\Omega_{A[B]/A}^1 \otimes_{A[B]} B, M) \to \operatorname{Hom}_B(I/I^2, M)) \xrightarrow{\sim} \operatorname{Aut}(\mathbf{0})$,

2. $u_*\varepsilon \simeq \mathbf{0}$ if and only if $u$ factorizes through $\delta$.

Point (1) is an immediate consequence of the usual exact sequence

$$I/I^2 \xrightarrow{\delta} \Omega_{A[B]/A}^1 \otimes_{A[B]} B \to \Omega_{B/A}^1 \to 0.$$

For point (2), this is simply that in the diagram

<figure><img src="image_51.png" /></figure>

there exists an $A$-derivation $A[B]/I^2 \to M$ making the diagram commute if and only if there exists a $B$-linear map making the following diagram commute

<figure><img src="image_52.png" /></figure>

It remains to prove that

$$[\operatorname{Hom}_B(\Omega_{A[B]/A}^1 \otimes_{A[B]} B, M) \to \operatorname{Hom}_B(I/I^2, M)] \simeq \tau_{\le 1} \operatorname{Hom}_B(\mathbb{L}_{B/A}, M)$$

in $D(B)$. For this let us look at the truncated simplicial augmented $A$-algebra

$$P_1 \xrightarrow{\partial_0} P_0 \xrightarrow{\pi} B.$$

Since it is a resolution, $\operatorname{Im}(\partial_0 - \partial_1) = I$. The $B$-linear morphism

$$\Omega_{P_1/A}^1 \otimes_{P_1} B \to \Omega_{P_0/A}^1 \otimes_{P_0} B$$

sends $dx \otimes 1$ to $d\partial_0 x \otimes 1 - d\partial_1 x \otimes 1 = d(\partial_0 - \partial_1)(x) \otimes 1$. Moreover, $x \mapsto (\partial_0 - \partial_1)(x) \mod I^2$ is an $A$-derivation from $P_1$ to $I/I^2$; this is a consequence of the formula

$$\forall x, y \in P_1, \quad \partial_0(x)(\partial_0(y) - \partial_1(y)) + \partial_1(y)(\partial_0(x) - \partial_1(x)) = \partial_0(xy) - \partial_1(xy).$$
From this we deduce a morphism of complexes

<figure><img src="image_53.png" /></figure>

In fact, one verifies by an explicit computation that the composite $\mathbb{L}_{B/A}^{-2} \to \mathbb{L}_{B/A}^{-1} \to I/I^2$ is zero.
We thus have a surjective $B$-linear morphism

$$ \nu : \mathbb{L}_{B/A}^{-1} / \mathrm{Im} \, \partial \to I/I^2. $$

This is in fact an isomorphism. We are going to construct an explicit inverse $\mu$. For this let $x \in I$.
Write $x = \partial_0(y) - \partial_1(y)$ with $y \in P_1$ and set

$$ \mu(x) = dy \otimes 1 \in \mathbb{L}^{-1} / \mathrm{Im} \, \partial. $$

This is well defined since if $x = \partial_0(y') - \partial_1(y')$ then $y - y' \in \ker \partial$ and thus $y - y' = \partial z$ with $z \in P_2$ (where here $\partial = \partial_0 - \partial_1 + \partial_2$ is the usual boundary map from $P_2$ to $P_1$). We thus have $dy \otimes 1 - dy' \otimes 1 \in \mathrm{Im} \, \partial$. Let $s : P_0 \to P_1$ be the degeneracy morphism. Recall that we have $\partial_0 s = \mathrm{Id}$ and $\partial_1 s = \mathrm{Id}$. Let now $x \in P_0$, $y \in I$ and write $y = \partial(z)$ with $z \in P_1$. We have

$$ \mu(xy) = d(s(x)z) \otimes 1 \mod \mathrm{Im} \, \partial. $$

But now,

$$ \begin{aligned} d(s(x)z) &= s(x)dz \otimes 1 + zd\left( \underbrace{s(x)}_{\text{and thus } \in \mathrm{Im} \, \partial} \right) \otimes 1 \\ &\equiv dz \otimes \pi(x) \mod \mathrm{Im} \, \partial. \end{aligned} $$

We thus have a factorization as a $B$-linear morphism

$$ \mu : I/I^2 \to \mathbb{L}^{-1} / \mathrm{Im} \, \partial. $$

It is immediate that $\mu$ and $\nu$ are inverse to each other. At the end we have constructed a quasi-isomorphism

$$ \tau_{\ge -1} \mathbb{L}_{B/A} \xrightarrow{\sim} [I/I^2 \xrightarrow{\delta} \Omega_{A[B]/A}^1 \otimes_{A[B]} B]. \quad \square $$

**Corollary A.5.2.** — There is a canonical isomorphism of $B$-modules for any $B$-module $M$,

$$ \mathrm{Ext}_B^1(\mathbb{L}_{B/A}, M) \xrightarrow{\sim} \mathrm{Exalg}_A(B, M). $$

## A.6. Deformation theory

We now have all the tools to do some deformation theory.

**Theorem A.6.1.** — Let $A \to B$ be a morphism of rings. Let $A' \twoheadrightarrow A$ be a surjection with square zero kernel $I$. Suppose given a $B$-module $N$ and a morphism of $A$-modules $I \to N$. Consider the category of couples $(B', u)$

<figure><img src="image_54.png" /></figure>

where $B'$ is a ring, $B' \to B$ is a surjection with square zero kernel $N$ and $u$ is a morphism of rings.

1. This category is a groupoid.

2. There is an obstruction in $\mathrm{Ext}_B^2(\mathbb{L}_{B/A}, N)$ so that this groupoid is non-empty.
3. If this is non-empty then its isomorphism classes, the $\pi_0$ of this groupoid, is a torsor under $\mathrm{Ext}_B^1(\mathbb{L}_{B/A}, N)$.

4. The automorphism group of any object is identified with $\mathrm{Ext}_B^0(\mathbb{L}_{B/A}, N)$.

*Proof.* — We only verify point (3), the other assertions being more elementary. The extension $\varepsilon$, $0 \to I \to A' \to A \to 0$, has its isomorphism class that is an element

$$[\varepsilon] \in \mathrm{Ext}_A^1(\mathbb{L}_{A/Z}, I).$$

The pushforward $v_*\varepsilon$ of $\varepsilon$ by $v: I \to N$ has an isomorphism class that is

$$v_*[\varepsilon] \in \mathrm{Ext}_A^1(\mathbb{L}_{A/Z}, N).$$

Our groupoid is non-empty if and only if

$$v_*[\varepsilon] \in \mathrm{Im} \left( \mathrm{Ext}_B^1(\mathbb{L}_{B/Z}, N) \to \underbrace{\mathrm{Ext}_A^1(\mathbb{L}_{A/Z}, N)}_{\mathrm{Ext}_B^1(\mathbb{L}_{A/Z} \otimes_A^{\mathbb{L}} B, N)} \right).$$

We now use the exact triangle

<figure><img src="image_55.png" /></figure>

to deduce that the obstruction we are looking for is

$$\partial(v_*[\varepsilon])$$

where

$$\mathrm{Ext}_B^1(\mathbb{L}_{B/Z}, N) \to \mathrm{Ext}_B^1(\mathbb{L}_{A/Z} \otimes_A^{\mathbb{L}} B, N) \xrightarrow{\partial} \mathrm{Ext}_B^2(\mathbb{L}_{B/A}, N).$$

This prove point (3). □

**Corollary A.6.2.** — If $A' \to A$ is a surjection with square zero kernel $I$ and $B$ is a flat $A$-algebra there is an obstruction in

$$\mathrm{Ext}_B^2(\mathbb{L}_{B/A}, I \otimes_A B)$$

to lift $B$ to a flat $A'$-algebra.
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