Deligne-Lurie completeness theorem

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## Idea

The statement that generalizes the [[Deligne completeness theorem]] from [[topos theory]] to [[(∞,1)-topos theory]].

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###### Theorem
**Deligne-Lurie completeness theorem**

A [[hypercomplete (∞,1)-topos]] which is a [[locally coherent (∞,1)-topos]] has [[enough points]].

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([[Spectral Schemes|Lurie SpecSchm, theorem 4.1]]).


Deligne-Mumford compactification

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## Idea

A standard [[compactification]] due to ([Deligne-Mumford 69](#DeligneMumford69)) of a [[moduli stack of curves]] by including curves with [[singularities]],in particular by including [[nodal curves]].

## Related concepts

* [[augmented Teichmüller space]]

* [[Deligne-Mumford stack]]

## References

The original article is

* {#DeligneMumford69} [[Pierre Deligne]], [[David Mumford]], _The irreducibility of the space of curves of given genus_, Publications Mathématiques de l'IHÉS (Paris) 36: 75–109 (1969) [numdam](http://www.numdam.org/item/PMIHES_1969__36__75_0)

Reviews include

* [pdf](http://www.math.harvard.edu/~kochs/DMC.pdf)

Discussion over the [[complex numbers]] showing that the DM-compactification here is equivalently the [[orbifold quotient]] of [[augmented Teichmüller space]] by the [[mapping class group]] is in

* {#HubbardKoch13} [[John Hubbard]], [[Sarah Koch]], _An analytic construction of the Deligne-Mumford compactification of the moduli space of curves_ ([arXiv:1301.0062](http://arxiv.org/abs/1301.0062))

[[!redirects Deligne-Mumford compactifications]]


Deligne-Mumford stack

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## Idea

A _Deligne-Mumford stack_ (after [Deligne-Mumford 69](#DeligneMumford69)) is the analogue in [[algebraic geometry]] of what in [[differential geometry]] is an [[orbifold]]: a [[quotient stack]] of a [[scheme]] over the [[étale site]] all whose [[automorphism group]]s are [[finite group]]s.

These are what originally were called [[algebraic stacks]]. The latter term nowadays often refers to the more general notion of [[Artin stacks]], where the [[automorphism groups]] ([[isotropy groups]]) are allowed to be more general [[algebraic groups]]. This case is the algebraic version of the general notion of [[geometric stack]].

## Definition

Given a [[scheme]] $S$. A $S$-[[stack]] $X$ (i.e under the [[Grothendieck construction]] a [[fibered category|category fibered]] in [[groupoid]]s over $(Aff/S)_{et}$ satisfying [[descent]]) is **Deligne-Mumford** when it has a [[representable morphism of stacks|representable]], separable and quasi-compact diagonal $\Delta: X\rightarrow X\times_S X$ and a covering $P:A\rightarrow X$ which is [[surjection|surjective]], representable and [[etale map|etale]], by an [[algebraic space]] $A$.

Deligne-Mumford stacks correspond to [[moduli space|moduli problems]] in which the objects being parametrized have finite [[automorphism group]]s. See also [[algebraic stack]].

### As a generalized scheme

From the perspective of [[derived algebraic geometry]]s a Deligne-Mumford stack is a special case of a [[generalized scheme]] (or $G$-scheme for $G$ a [[geometry (for structured (∞,1)-toposes)]]) as follows:

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###### Definition (etale geometry)

[[Structured Spaces|StSp Def 2.6.12]]

For $k$ a commutative [[ring]], let the **etale geometry** $G_{et}(k)$ be the [[geometry (for structured (∞,1)-toposes)]] defined as follows:

* the underlying [[(∞,1)-category]] is is ordinary [[category]]

  $$
    G_{et}(k) := (CRing_k^{fin})^{op}
  $$

  of finitely presented [[affine scheme]]s over $k$;

* a morphsim $f : Spec(A) \to Spec(B)$ is _admissible_ precisely if the corresponding morphism $f^* : B \to A$ of commutative $k$-[[algebra]]s is an [[etale map]];

* the [[Grothendieck topology]] on $G_{et}(k)$ is the restriction of the standard [[etale topology]].

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###### Theorem

[[Structured Spaces|SrSp Thm. 2.6.16]]

An [[(∞,1)-presheaf]] $F : CRing_k \to \infty Grpd$ is a Deligne-Mumford stack precisely if it is [[representable functor|representable]] by a $G_{et}(k)$-[[generalized scheme]] $(X,O_X)$ such that $X$ is [[n-localic (∞,1)-topos|1-localic]].

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## In moduli space theory

An important source of DM-stacks are [[moduli space|moduli]] problems, resulting often in [[moduli stack]]s (or their derived versions).

## Related concepts

* [[geometric stack]]

  * [[algebraic stack]]

    * **Deligne-Mumford stack**

      [[GIT-stable point]], [[geometric invariant theory]]

    * [[Artin stack]]

* [[Deligne-Mumford compactification]]

## References

The concept (cf. also *[[algebraic stack]]*) is due to:

* {#DeligneMumford69} [[Pierre Deligne]], [[David Mumford]], *The irreducibility of the space of curves of given genus*, Publications Mathématiques de l'IHÉS (Paris) **36**   (1969) 75-109 [[doi:10.1007/BF02684599](https://doi.org/10.1007/BF02684599), [numdam:PMIHES_1969__36__75_0](http://www.numdam.org/item?id=PMIHES_1969__36__75_0)]

Review in the general context of [[algebraic stacks]]:

* [[Angelo Vistoli]], _Grothendieck topologies, fibered categories and descent theory_ [[math.AG/0412512](http://arxiv.org/abs/math/0412512), [MR2223406](http://www.ams.org/mathscinet-getitem?mr=2223406)]  in: Fantechi et al. (eds.), _Fundamental algebraic geometry. Grothendieck's [[FGA explained]]_, Mathematical Surveys and Monographs __123__, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)]

and with emphases on the relation to [[orbifolds]]:

* {#Kresch09} [[Andrew Kresch]], _On the geometry of Deligne-Mumford stacks_ ([doi:10.5167/uzh-21342](https://doi.org/10.5167/uzh-21342), [pdf](https://www.zora.uzh.ch/id/eprint/21342/1/geodm.pdf)), in:  D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, M. Thaddeus (eds.) _Algebraic Geometry: Seattle 2005_, Proceedings of Symposia in Pure Mathematics 80, Providence, Rhode Island: American Mathematical Society 2009, 259-271 ([pspum-80-1](https://bookstore.ams.org/pspum-80-1))

Characterization of higher Deligne-Mumford stacks (see at *[[generalized scheme]]*):

* [[Jacob Lurie]], _Representability theorems_ [[pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XIV.pdf)]

* {#Carchedi13} [[David Carchedi]], _Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi_  [[arXiv:1312.2204](http://arxiv.org/abs/1312.2204)]

[[!redirects Deligne-Mumford stacks]]


delocalized equivariant cohomology

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## Idea

Let $G$ be a [[finite group]] and $X$ a suitable [[topological G-space]]. Then the _delocalized $G$-equivariant cohomology_ of $X$ ([Connes-Baum 89, p. 165](#ConnesBaum89)) is the [[ordinary cohomology]] of the [[quotient space]]

$$
  \left(
  \underset{
    h \in G
  }{\sqcup}
  X^{h}
  \right) / G
$$

of the [[disjoint union]] of [[fixed loci]], where the [[action]] of $g \in G$ is by

$$
  \left(h,\, x \in X^h  \right)
  \;\;
  \overset{g}{\mapsto}
  \;\;
  \left(g^{-1}\cdot h \cdot g ,\, x \cdot g \in X^{g^{-1} h g}  \right)
  \,.
$$

([Baum-Connes 89, above (7.3)](#ConnesBaum89))

## Properties

### Relation to Chen-Ruan cohomology
 {#RelationtoChenRuanCohomology}

The definition of delocalized equivariant cohomology coincides with that of [[Chen-Ruan cohomology]] for the [[global quotient orbifold]] ([[orbispace]]) $X \sslash G$. (Manifestly so, also highlighted in [Bunke-Spitzweck-Schick 06, Section 1.3](#BunkeSpitzweckSchick06))

### Relation to Bredon cohomology

In turn, [[Chen-Ruan cohomology]] of [[global quotient orbifolds]] (and hence, by the [above](#RelationtoChenRuanCohomology) delocalized equivariant cohomology) coincides with [[Bredon cohomology]] with [[coefficients]] in the [[rationalization|rationalized]] [[representation ring]]-functor $G/H \mapsto Rep(H)$ on the [[orbit category]].

([Mislin-Valette 03, Thm. 6.1](#MislinValette03), review in [Szabo-Valentino 07, Sec. 4.3](#SzaboValentino07)

### Relation to rationalized equivariant K-theory

In further turn, even-periodic [[Bredon cohomology]] with coefficients in the [[representation ring]]-functor is equivalent to [[rationalization|rationalized]] [[equivariant K-theory]]:

\linebreak

[[!include incarnations of rational equivariant topological K-theory -- table ]]

## Related concepts

* [[equivariant Chern character]]

## References

* {#ConnesBaum89} [[Paul Baum]], [[Alain Connes]], around (1.6) in: _Chern character for discrete groups_, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 ([doi:10.1016/B978-0-12-480440-1.50015-0](https://doi.org/10.1016/B978-0-12-480440-1.50015-0))

* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], Theorem 6.1 in: _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))

* {#BunkeSpitzweckSchick06} [[Ulrich Bunke]], [[Markus Spitzweck]], [[Thomas Schick]], _Inertia and delocalized twisted cohomology_, Homotopy, Homology and Applications, vol 10(1), pp 129-180 (2008) ([arXiv:math/0609576](https://arxiv.org/abs/math/0609576))

* {#SzaboValentino07} [[Richard Szabo]], [[Alessandro Valentino]], Section 4.3 of: _Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory_, Commun. Math. Phys. 294:647-702, 2010 ([arXiv:0710.2773](https://arxiv.org/abs/0710.2773))


delooping

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## Idea

The _delooping_ of an object $A$ is, if it exists, a uniquely [[pointed object]] $\mathbf{B} A$ such that $A$ is the [[loop space object]] of $\mathbf{B} A$:

$$
  A \simeq \Omega(\mathbf{B} A)
$$

In particular, if $A = G$ is a [[group]] then its delooping

* in the context [[Top]] is the [[classifying space]] $\mathcal{B}G$

* in the context [[∞-Grpd]] is the one-object groupoid $\mathbf{B}G$.

Under the [[homotopy hypothesis]] these two objects are identified: the [[geometric realization]] of the groupoid $\mathbf{B}G$ is the classifying space $\mathcal{B}G$:

$$
  |\mathbf{B}G| \simeq \mathcal{B}G
  \,.
$$

See [[looping and delooping]].

## Definition

Loop space objects are defined in any [[(∞,1)-category]] $\mathbf{C}$ with [[homotopy pullbacks]]: for $X$ any [[pointed object]] of $\mathbf{C}$ with point ${*} \to X$, its [[loop space object]] is [[generalized the|the]] [[homotopy pullback]] $\Omega X$ of this point along itself:

$$
  \array{
    \Omega X &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    {*} &\to& X
  }
  \,.
$$

Conversely, if $A$ is given and a [[homotopy pullback]] diagram

$$
  \array{
    A &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    {*} &\to& \mathbf{B}A
  }
$$

exists, with the point ${*} \to \mathbf{B} A$ being essentially unique, by the above $A$ has been realized as the [[loop space object]] of $\mathbf{B} A$

$$
  A = \Omega \mathbf{B} A
$$

and we say that $\mathbf{B} A$ is the **delooping** of $A$.

See the section delooping at [[groupoid object in an (∞,1)-category]] for more.

## Remarks

If $\mathbf{C}$ is even a [[stable (∞,1)-category]] then all deloopings exist and are then also denoted $\Sigma A$ and called the **[[suspension]]** of $A$.

## Characterization of deloopable objects

In section 6.1.3 of

* [[Jacob Lurie]], [[Higher Topos Theory]]

a definition of
[[groupoid object in an (infinity,1)-category]] $\mathbf{C}$ is given
as a homotopy simplicial object, i.e. a [[(infinity,1)-functor]]

$$
  C  : \Delta^{op} \to \mathbf{C}
$$
$$
  \cdots
  C_2 \stackrel{\to}\rightrightarrows
  C_1
  \rightrightarrows
 C_0
$$

satisfying certain conditions (prop. 6.1.2.6) which are such that if $C_0 = {*}$ is the [[point]] we have an internal _group_ in a homotopical sense, given by an object $C_1$ equipped with a coherently associative multiplication operation  $C_1 \times C_1 \to C_1$ generalizing that of [[Jim Stasheff|Stasheff]] [[H-space]] from the $(\infty,1)$-category [[Top]] to arbitrary $(\infty,1)$-categories.

Lurie calls the groupoid object $C$ an
_effective_ [[groupoid object in an (infinity,1)-category]] precisely if it arises as the delooping, in the above sense, of some object $\mathbf{B}C$.

One of the characterizing properties of an [[(infinity,1)-topos]] is that _every_ groupoid object in it is effective.

This is the analog of [[Jim Stasheff|Stasheff]]'s classical result about [[H-spaces]].

See the remark at the very end of section 6.1.2 in [[Higher Topos Theory|HTT]].

## Examples

### Topological loop spaces

For $C = $ [[Top]] the [[(infinity,1)-category]]
of [[topological spaces]], a space is deloopable if it is an [[A-infinity-space]] and hence homotopy equivalent to a [[loop space]].

### Delooping of a group to a groupoid

Let $G$ be a [[group]] regarded as a [[discrete category|discrete groupoid]] in the [[(∞,1)-topos]] [[∞Grpd]] of [[∞-groupoids]].

Then $\mathbf{B} G$ exists and is, up to equivalence, the [[groupoid]]

* with a single object $\bullet$,

* with $Hom_{\mathbf{B} G}(\bullet, \bullet) = G$, or equivalently $Aut_{\mathbf{B}G}(\bullet) = G$,

* and with composition of morphisms in $\mathbf{B} G$ being given by the product operation in the group.

More informally but more suggestively we may write

$$
  \mathbf{B} G = \{ \bullet \stackrel{g}{\to} \bullet | g \in G\}
$$

or

$$
  \mathbf{B}G = \{
     \bullet \righttoleftarrow g \;|\; g \in G
  \}
$$

to emphasize that there is really only a single object.

Notice how the [[homotopy pullback]] works in this simple case:

the universal 2-cell $\eta$

$$
  \array{
    G &\to& {*}
    \\
    \downarrow &\Downarrow^{\eta}& \downarrow
    \\
    {*} &\to& \mathbf{B}G
  }
$$

filling this [[2-limit]] diagram is the [[natural transformation]] from the constant [[functor]]

$$
  G \to {*} \to \mathbf{B}G
$$

to itself, whose component map

$$
  \eta : Obj(G) \to Mor(\mathbf{B}G)
$$

is just the identity map, using that $Obj(G) = G$ and $Mor(\mathbf{B}G) = G$.

## Deloopings of higher categorical structures
{#DeloopingOfHigherCategoricalStructures}

There is also a notion of delooping which takes a pointed $(n, k+1)$-category $C$ to a pointed $(n+1, k)$-category $\mathbf{B} C$ in which $\mathbf{B} C$ has a single $0$-cell $\bullet$, and where $\hom(\bullet, \bullet) = C$. This is a tautological construction if one accepts the [[delooping hypothesis]], which views a $(n, k+1)$-category $C$ as a special type of $(n+k+1)$-category, namely a pointed $k$-connected $(n+k+1)$-category: by viewing such as *a fortiori* a pointed $(k-1)$-connected $(n+k+1)$-category, we get the delooping $\mathbf{B} C$.

This is just a generalization of the fact that a [[monoid]] $M$ gives rise to a one-object category (which we are denoting $\mathbf{B} M$). For an important example: a [[monoidal category]] $M$ has an associated **delooping [[bicategory]]** $\mathbf{B} M$, where

* $\mathbf{B} M$ has a single $0$-cell $\bullet$,

* the $1$-cells $\bullet \to \bullet$ of $\mathbf{B} M$ are named by objects of $M$, and the composite of $\bullet \stackrel{a}{\to} \bullet \stackrel{b}{\to} \bullet$ is $\bullet \stackrel{a \otimes b}{\to} \bullet$ (using the monoidal product $\otimes$ of $M$),

* the $2$-cells of $\mathbf{B} M$ are similarly named by morphisms of $M$; the vertical composition of $2$-cells in $\mathbf{B} M$ is given by composition of morphisms of $M$, and the horizontal composition of $2$-cells in $\mathbf{B} M$ is given by taking the monoidal product of the morphisms that name them in $M$.

Along similar lines, the delooping of a [[braided monoidal category]] produces a [[monoidal bicategory]], and delooping of *that* is a [[tricategory]] or (weak) $3$-category. See [[delooping hypothesis]] for more.

## Related concepts

* [[looping and delooping]]

* [[May recognition theorem]]

* [[loop space object]], [[free loop space object]],

  * **delooping**

  * [[loop space]], [[free loop space]], [[derived loop space]]

* [[suspension object]]

  * [[suspension]], [[reduced suspension]]

## References

Discussion in [[homotopy type theory]]:

* {#BCFR23} [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

* {#Wärn23} [[David Wärn]], _Eilenberg-MacLane spaces and stabilisation in homotopy type theory_ [[arXiv:2301.03685](https://arxiv.org/abs/2301.03685)]

[[!redirects deloopings]]
[[!redirects delooped group]]

[[!redirects delooped]]


delooping groupoid

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## Idea

For $G$ a [[group]], there is a [[pointed connected groupoid|pointed connected]] [[groupoid]] $(G \rightrightarrows \ast)$ with

* a single [[object]],

* [[morphisms]] the [[elements]] of $G$,

* [[composition]] the group operation in $G$.

This may be understood as the "[[delooping]]" of $G$ in the ambient [[(infinity,1)-topos]], and is, as such, naturally denoted $\mathbf{B}G$.

## Related concepts

* a [[skeletal groupoid]] is a [[disjoint union]] of delooping groupoids

* [[pair groupoid]]

* [[looping and delooping]]

* [[pointed connected groupoid]]

## References

For more see at

* [[Introduction to Topology -- 2]], *[Groupoids](https://ncatlab.org/nlab/show/Introduction+to+Topology+--+2#Groupoid)*

[[!redirects delooping groupoids]]


delooping hypothesis

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## Idea

The **delooping hypothesis** is one of the "guiding hypotheses of higher category theory."  Like the [[homotopy hypothesis]], it is generally accepted to be a "litmus test" that any suitable definition of [[n-category]] should satisfy.  It states that:

* [[k-tuply monoidal n-category|k-tuply monoidal n-categories]] can be identified with $(k-j)$-tuply monoidal, $(j-1)$-[[k-tuply connected n-category|simply connected]] $(n+j)$-categories, for any $0\le j \le k$.

The identification involves a degree shift: the $i$-morphisms of a $k$-tuply monoidal $n$-category become $(i+j)$-morphisms in the associated $(k-j)$-tuply monoidal $(n+j)$-category.

Here _$j$-(simply) connected_ means that any two parallel $i$-morphisms are [[equivalence|equivalent]] for $i \leq j$.  Also, _$0$-tuply monoidal_ is interpreted as meaning [[pointed object|pointed]].  We may also allow $n$ to be of the form [[(n,r)-category|(n,r)]] or $(\infty,r)$, with the usual conventions that $(n,r)+j=(n+j,r+j)$, $\infty+j=\infty$, and so on.  In particular, taking $j=k$ we have:

* $k$-tuply monoidal $n$-categories can be identified with pointed $(k-1)$-connected $(n+k)$-categories.

The $(n+j)$-category associated to a $k$-tuply monoidal $n$-category $C$ is called its **$j$-fold delooping** and sometimes written $B^j C$.  Conversely, any $k$-tuply monoidal $n$-category $C$ with a point $*\in C$ has a [[loop space object]] $\Omega C = C(*,*)$ which is a $(k+1)$-tuply monoidal $(n-1)$-category.

## Remarks

* Not infrequently the delooping hypothesis is used to supply a _definition_ of "$k$-tuply monoidal $n$-category."  See [[k-tuply monoidal n-category]] for an investigation in low dimensions.

* The delooping hypothesis is closely related to the [[stabilization hypothesis]].  In delooping language, the stabilization hypothesis says that once you have an $n$-category that can be delooped $n+2$ times, it can automatically be delooped infinitely many times.

* In low dimensions, the delooping hypothesis is a special case of the [[michaelshulman:exactness hypothesis|exactness hypothesis]].

## In homotopy theory

A "groupoidal" version of the delooping hypothesis may be stated as

* $k$-[[k-tuply groupal n-groupoid|tuply groupal]] $n$-groupoids can be identified with $(k-j)$-tuply groupal $(j-1)$-connected $(n+j)$-groupoids, for $0\le j\le k$.

Here "groupal" means "monoidal and such that all objects have inverses."  (This can actually be seen as a special case of the delooping hypothesis for $k$-tuply monoidal $(n,r)$-categories with $r$ set to $-1$.)

When $n=\infty$ the groupoidal delooping hypothesis can be interpreted (via the [[homotopy hypothesis]]) as a standard result of classical [[homotopy theory]]: "grouplike $E_k$-spaces" can be delooped $k$ times.  In particular, grouplike $A_\infty$-[[A-infinity-space|spaces]] can be delooped once, and grouplike $E_\infty$-[[E-infinity-space|spaces]] can be delooped infinitely many times (producing a [[spectrum]]).

Non-grouplike $A_\infty$-spaces can also be "delooped" in classical homotopy theory, but can only be recovered from their delooping up to [[group completion]].  This is because classical homotopy theory only works with $(\infty,0)$-categories, while the higher-categorical delooping of a non-grouplike $A_\infty$-space (that is, a monoidal $(\infty,0)$-category) should be an $(\infty,1)$-category, not an $(\infty,0)$-category.

## For $(\infty,n)$-categories

The delooping hypothesis also makes sense for $(\infty,n)$-categories rather than $n$-categories.  In this case a proof (using iterated $(\infty,1)$-categorical enrichment to define $(\infty,n)$-categories) is in

* [[David Gepner]], [[Rune Haugseng]], *Enriched ∞-categories via non-symmetric ∞-operads*, [arXiv](http://arxiv.org/abs/1312.3178)


delta lens

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## Idea ##
A delta lens (or d-lens) is a [[functor]] with additional structure specifying a suitable choice of lifts. Delta lenses can be understood in a number of ways:

* A morphism between [[categories]] which is both [[functor]] and [[cofunctor]];

* A generalization of split [[Grothendieck fibration|Grothendieck opfibrations]] where the chosen lifts need not be [[Cartesian morphism|opcartesian morphisms]];

* A [[vertical categorification|categorification]] of classical [[lens (in computer science)|lenses]] between [[sets]].

## Definition ##

A **delta lens** $(f, \varphi) : A \to B$ from a [[category]] $A$ to a category $B$ consists of a [[functor]] $f : A \to B$ together with a _lifting operation_ sending each pair $(a \in A, u : f a \to b \in B)$ to a [[morphism]] $\varphi(a, u) : a \to a'$ in $A$, where $a' = cod(\varphi(a, u))$, such that

* $\varphi$ defines a choice of lifts: $f \varphi(a, u) = u$,

* $\varphi$ preserves identity morphisms: $\varphi(a, 1_{f a}) = 1_{a}$,

* $\varphi$ preserves composition: $\varphi(a, v \circ u) = \varphi(a', v) \circ \varphi(a, u)$ where $a' = cod(\varphi(a, u))$.

\begin{centre}
\begin{tikzcd}
A
\arrow[d, "{(f, \varphi)}"']
& a
\arrow[d, phantom, "\vdots"]
\arrow[r, "{\varphi(a, u)}"]
& a'
\arrow[d, phantom, "\vdots"]
\\
B
& f a
\arrow[r, "u"]
& b = f a'
\end{tikzcd}
\end{centre}

## Properties ##

\begin{proposition}
A delta lens $(f, \varphi) : A \to B$ is equivalent to a [[functor]] $f \colon A \to B$ and a [[cofunctor]] $\varphi \colon A \nrightarrow B$ such that $f a = \varphi_{0} a$ for all objects $a \in A$, and $f \varphi_{1}(a \in A, u \colon fa \to b \in B) = u$.
\end{proposition}
\begin{proof}
See ([Ahman-Uustalu 2017, Section 6](#AhmanUustalu17)) and ([Clarke 2020](#Clarke20a)).
\end{proof}

## Examples ##

* A delta lens between [[indiscrete category|codiscrete categories]] is precisely a classical [[lens (in computer science)]]; see ([Johnson-Rosebrugh 2016, Proposition 4](#JohnsonRosebrugh16)).

* Every [[function]] $A \to B$ yields a delta lens $disc(A) \rightarrow disc(B)$ between [[discrete category|discrete categories]].

* Every [[discrete fibration|discrete opfibration]] is a delta lens whose lifting operation is determined by the unique [[Cartesian morphism|opcartesian lifts]]. Conversely, a delta lens is a discrete opfibration if the equation $\varphi(a, f w) = w$ holds for all morphisms $w \colon a \to a'$ in $A$.

* A split [[Grothendieck fibration|Grothendieck opfibration]] is a delta lens whose chosen lifts are [[Cartesian morphism|opcartesian morphisms]].

* Dually, a split [[Grothendieck fibration]] $A \to B$ is a delta lens $A^{op} \to B^{op}$.

* Every [[split epimorphism]] of [[monoid|monoids]] with a chosen [[section]] is a delta lens, when the monoids a considered as categories with a single object.

* More generally, every [[bijective on objects functor]] with a chosen [[section]] is a delta lens.

## Related concepts

* [[lens (in computer science)]]

## References ##

The notion of delta lens was first defined in [[computer science]] as a generalization of the classical [[lens (in computer science)|lenses]] between sets:

* {#DiskinXiongCzarnecki11} Zinovy Diskin, Yingfei Xiong, Krzysztof Czarnecki, _From State- to Delta-Based Bidirectional Model Transformations: the Asymmetric Case_, Journal of Object Technology, 10, 2011 ([doi:10.5381/jot.2011.10.1.a6](http://dx.doi.org/10.5381/jot.2011.10.1.a6))

The connection between delta lenses and split [[Grothendieck fibration|Grothendieck opfibrations]] was first explored in the paper:

* {#JohnsonRosebrugh13} [[Michael Johnson]], [[Robert Rosebrugh]], _Delta Lenses and Opfibrations_, Electronic Communications of the EASST, 57, 2013 ([doi:10.14279/tuj.eceasst.57.875](http://dx.doi.org/10.14279/tuj.eceasst.57.875))

The following paper details the connection between delta lenses and the classical [[lens (in computer science)|lenses]]:

* {#JohnsonRosebrugh16} [[Michael Johnson]], [[Robert Rosebrugh]], _Unifying Set-Based, Delta-Based and Edit-Based Lenses_, CEUR Workshop Proceedings, 1571, 2016 ([pdf](http://ceur-ws.org/Vol-1571/paper_13.pdf))

The characterisation of delta lenses in terms of [[functors]] and [[cofunctors]] first appeared in the paper:

* {#AhmanUustalu17} [[Danel Ahman]], [[Tarmo Uustalu]], *Taking updates seriously*, Proceedings of the Sixth International Workshop on [Bidirectional Transformations (Bx 2017)](https://ceur-ws.org/Vol-1827/), CEUR Workshop Proceedings **1827** (2017) [[pdf](http://ceur-ws.org/Vol-1827/paper11.pdf)]

The notion of delta lens between [[internal category|internal categories]] as well as the link between [[cofunctor|cofunctors]], delta lenses, and split [[Grothendieck fibration|Grothendieck opfibrations]] is developed in the papers:

* {#Clarke20a} [[Bryce Clarke]], _Internal lenses as functors and cofunctors_, EPTCS, 323, 2020 ([doi:10.4204/EPTCS.323.13](http://dx.doi.org/10.4204/EPTCS.323.13))

* {#Clarke20b} [[Bryce Clarke]], _Internal split opfibrations and cofunctors_, Theory and Applications of Categories, 35, 2020 ([link](http://www.tac.mta.ca/tac/volumes/35/44/35-44abs.html))

[[!redirects delta lenses]]


delta-functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn}
###### Definition

Let $\mathcal{A}$, $\mathcal{B}$ be two [[abelian categories]].

A **homological $\delta$-functor** from $\mathcal{A}$ to $\mathcal{B}$ is for each $n \in \mathbb{N}$ a [[functor]]

$$
  T_n : \mathcal{A} \to \mathcal{B}
$$

equipped for each [[short exact sequence]] $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with a [[natural transformation]]

$$
  \delta_n : T_n(C) \to T_{n-1}(A)
$$

such that for each such short exact sequence there is, [[natural transformation|naturally]] a [[long exact sequence]]

$$
      \cdots T_{n+1}(C) \stackrel{\delta}{\to} T_n(A) \to T_n(B) \to T_n(C) \stackrel{\delta }{\to} T_{n-1}(A) \to \cdots
 \,.
$$

=--

## Examples

The archetypical example is the [[chain homology]] functor

$$
  H_\bullet(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}
$$

from the [[category of chain complexes]] of some [[abelian category]] (for $\mathbb{N}$-graded complexes).

The _universal_ example are (non-total) [[right derived functors]].

## Related concepts

* [[derived functor in homological algebra]]

## References

The notion is due to

* [[Alexander Grothendieck]], _[[Tohoku|Sur quelques points d'algèbre homologique]]_

A textbook account is for instance section 2.1 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

[[!redirects delta-functors]]
[[!redirects ∞-functor]]
[[!redirects ∞-functors]]

[[!redirects universal delta-functor]]
[[!redirects universal delta-functors]]
[[!redirects universal ∞-functor]]
[[!redirects universal ∞-functors]]

[[!redirects delta functor]]
[[!redirects delta functors]]
[[!redirects ? functor]]
[[!redirects ? functors]]


Delta-generated topological space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #DeltaGeneratedSpace}
###### Definition
**($\Delta$-Generated spaces)**

A **$\Delta$-generated space** (alias **numerically generated space**) ([Smith](#Smith), [Dugger 03](#Dugger03)) is a [[topological space]] $X$ whose topology is the [[weak topology|final topology]] induced by all [[continuous functions]] of the form $\Delta^n_{top} \to X$, hence those whose [[domain]] $\Delta^n_{top}$ is one of the standard [[topological simplices]], for $n \in \mathbb{N}$.

A [[morphism]] between $\Delta$-generated spaces is just a [[continuous function]], hence the [[category]] of $\Delta$-generated spaces is the [[full subcategory]] on these spaces inside all [[Top|TopologicalSpaces]],

=--

+-- {: .num_remark #AsColimitsOfTopologicalSimplices}
###### Remark
**(as colimits of topological simplices)**

Equivalently, the [[class]] of $\Delta$-generated spaces is the closure of the set of [[topological simplices]] $\Delta^n_{top}$ under [[small colimits]] in [[topological spaces]] (see at _[[Top]] -- [universal constructions](Top#UniversalConstructions)_).

=--

+-- {: .num_remark #AsEuclideanGeneratedSpaces}
###### Remark
**(as Euclidean-generated spaces)**

For each $n$ the [[topological simplex]] $\Delta^n$ is a [[retract]] of the ambient [[Euclidean space]]/[[Cartesian space]] $\mathbb{R}^n$ (as a [[inhabited|non-empty]] [[convex subset]] of a [[Euclidean space]] it is in fact an [[absolute retract]]). Hence the [[identity function]] on $\Delta^n$ factors as

$$
  id
  \;\colon\;
  \Delta^n_{top}
  \overset{\;\;\; i_n \;\;\;}{\hookrightarrow}
  \mathbb{R}^n
  \overset{\;\;\; p_n \;\;\;}{\longrightarrow}
  \Delta^n_{top}
  \,;
$$

and it follows that every [[continuous function]] $f$ with [[domain]] the [[topological simplex]] [[extension|extends]] as a [[continuous function]] to [[Euclidean space]]:

$$
  \array{
    \Delta^m_{top}
    &\overset{f}{\longrightarrow}&
    X
    \\
    \mathllap{{}^{i_n}}\big\downarrow & \nearrow _{\mathrlap{\exists}}
    \\
    \mathbb{R}^n
  }
$$

Therefore the condition that a topological space $X$ be $\Delta$-generated (Def. \ref{DeltaGeneratedSpace}) is equivalent to saying that its topology is [[final topology|final]] with respect to all continuous functions $\mathbb{R}^n \to X$ out of [[Euclidean spaces|Euclidean]]/[[Cartesian spaces]].

=--

+-- {: .num_remark #AsDTopologicalSpaces}
###### Remark
**(as D-topological spaces)**

By Prop. \ref{AdjunctionBetweenTopologicalSpacesAndDiffeologicalSpaces} below, the Euclidean-generated spaces and hence, by Remark \ref{AsEuclideanGeneratedSpaces}, the $\Delta$-generated spaces, are equivalently those that arise from equipping a [[diffeological space]] with its [[D-topology]], hence are, in this precise sense, the _D-topological spaces_. Luckily, "D-topological space" may also serve as an abbreviation for "Delta-generated topological space".

=--

## Properties

### Coreflection into all topological spaces
 {#CoreflectionIntoAllTopologicalSpaces}

[[!include adjunction between topological spaces and diffeological spaces]]

### Topological homotopy type and diffeological shape
 {#OnTopologicalHomotopyTypeAndDiffeologicalShape}

[[!include diffeological singular simplicial set -- definition]]

[[!include topological homotopy type is cohesive shape of continuous diffeology -- proposition]]

### Model category structure
 {#ModelCategoryStructure}

\begin{prop}
**([[model structure on Delta-generated topological spaces]])**
\linebreak
The category of $\Delta$-generated spaces carries the [[mathematical structure|structure]] of a [[cofibrantly generated model category]] with the same generating (acyclic) cofibrations as for the [[classical model structure on topological spaces]] and such that the [[coreflective subcategory|coreflection]] into all [[Top|TopologicalSpaces]] (Prop. \ref{AdjunctionBetweenTopologicalSpacesAndDiffeologicalSpaces}) is a [[Quillen equivalence]] to the [[classical model structure on topological spaces]].
\end{prop}

([Haraguchi 13, Theorem 3.3](#Haraguchi13))

This [[Quillen equivalence]] factors through the [[model structure on compactly generated topological spaces]] (e.g. [Gaucher 2007, p. 7](#Gaucher07)):

$$
  Top_{Qu}
  \underoverset
    { \underset{ k }{\longrightarrow} }
    { {\hookleftarrow} }
    { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; }
  k Top_{Qu}
  \underoverset
    { \underset{ D }{\longrightarrow} }
    { {\hookleftarrow} }
    { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; }
  D Top_{Qu}
  \,.
$$

### As a convenient category of topological spaces
 {#AsAConvenientCategoryOfTopologicalSpaces}

\begin{prop}\label{CartesianClosure}
  The [[category]] of [[Euclidean-generated spaces]]/$\Delta$-generated spaces (Def. \ref{DeltaGeneratedSpace}) is a [[Cartesian closed category]].

Explicitly, the [[internal hom]] $Maps_{DTop}$ is, equivalently:

* the image under $Cdfflg$  (eq:AdjointFunctorsBetweenTopSpAndDifflgSp) of the [[mapping space]] $Maps_{Top}(X,Y)$ with the [[compact-open topology]]:

  $$
    Maps_{DTop}(X,Y)
    \;\simeq\;
    Cdfflg
    \big(
      Maps_{Top}
      (
        X
        ,\,
        Y
      )
    \big)
    \,,
  $$

* the image under [[D-topology|$Dtplg$]]  (eq:AdjointFunctorsBetweenTopSpAndDifflgSp) of the [[internal hom]] $Maps_{Dfflg}$ formed in [[diffeological spaces]]:

  $$
    Maps_{DTop}(X,Y)
      \;\simeq\;
    Dtplg
    \big(
      Maps_{Dfflg}
      (
        X
        ,\,
        Y
      )
    \big)
    \,.
  $$

\end{prop}
The first statement is a special case of [Vogt 1971, Thm. 3.6](#Vogt71), as highlighted in [Gaucher 2007, Sec. 2](#Gaucher07).
The second statement is the image of [SYH 10, Prop. 4.7](#SYH10) under $Dtplg$ (using that $\nu = T \circ D$, in their notation from p. 4).

In fact, [SYH 10, Prop. 4.7](#SYH10) state something stronger, topologically characterizing $Maps_{Dfflg}(X,Y)$ even before applying $Dtplg$ to it. This stronger statement has a nice form when specialized to [[CW-complexes]]:

\begin{proposition}\label{CWComplexesAreDTopologicalSpaces}
  The [[category]] of [[Euclidean-generated spaces]]/$\Delta$-generated spaces (Def. \ref{DeltaGeneratedSpace}) contains all [[CW-complexes]].
\end{proposition}
([SYH 10, Cor. 4.4](#SYH10))

\begin{proposition}\label{DiffeologicalMapsOutOfCWComplexes}
**([[diffeological space|diffeological]] [[internal hom]] out of [[CW-complexes]])**
\linebreak

If $X$ is a [[CW-complex]],
regarded as an object in $DTopSp$ via Prop. \ref{CWComplexesAreDTopologicalSpaces},
then for every $A \,\in\, DTopSp$
their [[internal hom]] formed in [[diffeological spaces]]
is [[isomorphism|isomorphic]] to the image under $Cdffl$
(eq:AdjointFunctorsBetweenTopSpAndDifflgSp)
of the [[mapping space]] $Maps_{Top}$ with its [[compact open topology]]:
$$
  X \,\in\, CWComplex \hookrightarrow DTopSp
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  \underset{
    A \,\in\, DTopSp
  }{\forall}
  Maps_{Dfflg}
  \big(
    X
    ,\,
    A
  \big)
  \;\;
  \simeq
  Cdfflg
  \,
  Maps_{Top}
  \left(
    X
    ,\,
    A
  \right)
  \,.
$$
\end{proposition}
\begin{proof}
  This is the following combination of statements from [SYH 10](#SYH10):

1. Prop. 4.7 there says that, in general:

   $$
     Maps_{Dfflg}\big( X ,\, A\big)
     \;\simeq\;
     Cdfflgl
     \big(
       \mathbf{smap}(X,Y)
     \big)
   $$

   (where "$\mathbf{smap}$" is defined on their p. 6),

1. Prop. 4.3 there implies that, when $X$ is a CW-complex, as assumed here:

   $$
     Cdfflg
     \big(
       \mathbf{smap}(X,Y)
     \big)
     \;\simeq\;
     Cdfflg
     \big(
       Maps_{Top}(X,Y)
     \big)
     \,.
   $$

\end{proof}

In summary:

+-- {: .num_prop #EuclideanGeneratedSpacesAreConvenient}
###### Proposition
**([[Euclidean-generated spaces]] are [[convenient category of topological spaces|convenient]])**

The [[category]] of [[Euclidean-generated spaces]]/$\Delta$-generated spaces (Def. \ref{DeltaGeneratedSpace}) is
a [[convenient category of topological spaces]] in that:

* it is [[coreflective subcategory|coreflective]] in [[TopSp]] (by Prop. \ref{AdjunctionBetweenTopologicalSpacesAndDiffeologicalSpaces})

* it is [[complete category|complete]] and [[cocomplete category]]

(by [Vogt 1971](#Vogt71); [SYH 10, Prop. 3.4](#SYH10))

* it is [[locally presentable category|locally presentable]] ([FR 08, Cor. 3.7](#FajstrupRosicky08)),

* it contains all [[CW-complexes]] (Prop. \ref{CWComplexesAreDTopologicalSpaces}),

* it is [[cartesian closed]], with [[internal hom]] expressible as in Prop. \ref{CartesianClosure}.

=--

Moreover, in further summary of the discussion further above, this [[convenient category of topological spaces]] is:

1. a [[full subcategory]] of the [[quasi-topos]] of [[diffeological spaces]] (see [there](diffeological+space#RelationToTopologicalSpaces)),

1. which is in turn a [[full subcategory]] of the [[cohesive topos]] of [[smooth sets]] (see [there](geometry+of+physics+--+smooth+sets#DiffeologicalSpacesAreTheConcreteSmoothSets));

1. which in turn is a [[full sub-(infinity,1)-category|full sub-$(\infty,1)$-category]] of the [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] of [[smooth infinity-groupoids|smooth $\infty$-groupoids]]

such that the canonical [[shape modality]] (the [[shape via cohesive path ∞-groupoid|smooth path ∞-groupoid construction]]) still sees the correct underlying [[homotopy type]] of topological spaces ([[schreiber:Proper Orbifold Cohomology|SS20, Ex. 3.18]], see also at [[model structure on Delta-generated topological spaces]]):

\begin{imagefromfile}
    "file_name": "TopologicalConvenienceBySmoothGroupoids210930.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [SS21](https://ncatlab.org/schreiber/show/Equivariant+principal+infinity-bundles)"
\end{imagefromfile}

## Related concepts

* [[Euclidean-generated ∞-groupoid]]

* [[compactly generated topological space]]

## References

### General

$\Delta$-generated spaces were originally proposed by [[Jeff Smith]] as a [[nice category of spaces]] for [[homotopy theory]].

* {#Smith} [[Jeff Smith]], _A really convenient category of topological spaces_ (unpublished and possibly non-existent, according to [Dugger 03, p. 4](#Dugger03))

* {#Dugger03} [[Daniel Dugger]], _Notes on Delta-generated spaces_, 2003 ([web](https://pages.uoregon.edu/ddugger/delta.html), [[DuggerDeltaGenerated.pdf:file]])

A [[proof]] that the category of $\Delta$-generated spaces is [[locally presentable category|locally presentable]] is in:

* {#FajstrupRosicky08} [[Lisbeth Fajstrup]], [[Jiří Rosický]], Thm. 3.6 of: _A convenient category for directed homotopy_, Theory and Applications of Categories, Vol. 21, 2008, No. 1, pp 7-20. ([arXiv:0708.3937](https://arxiv.org/abs/0708.3937), [tac:21-01](http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html))

See also at _[[directed homotopy theory]]_.

The [[model structure on Delta-generated topological spaces]] [[Quillen equivalence|Quillen equivalent]] to the [[classical model structure on topological spaces]] is due to

*  {#Haraguchi13} [[Tadayuki Haraguchi]], _On model structure for coreflective subcategories of a model category_, Math. J. Okayama Univ.57(2015), 79–84 ([arXiv:1304.3622](http://arxiv.org/abs/1304.3622), MR3289294, Zbl 1311.55027)

Discussion in the generality of subcategory-generated spaces, including [[compactly generated topological spaces]]:

* {#Gaucher07} [[Philippe Gaucher]], Section 2 of: *Homotopical interpretation of globular complex by multipointed d-space*, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 ([arXiv:0710.3553](https://arxiv.org/abs/0710.3553))

along the lines of

* {#Vogt71} [[Rainer M. Vogt]], *Convenient categories of topological spaces for homotopy theory*,  Arch. Math 22, 545–555 (1971) ([doi:10.1007/BF01222616](https://doi.org/10.1007/BF01222616))

* {#EscardoLawsonSimpson04} [[Martín Escardó]], [[Jimmie Lawson]], [[Alex Simpson]], Section 3 of: *Comparing Cartesian closed categories of (core) compactly generated spaces*, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 ([doi:10.1016/j.topol.2004.02.011](https://doi.org/10.1016/j.topol.2004.02.011))

Discussion about the [[q-model structure]], the [[m-model structure]], the [[h-model structure]] and the notion of $\Delta$-Hausdorff $\Delta$-generated space (a natural separation condition for $\Delta$-generated spaces) in:

* {#Gaucher21} [[Philippe Gaucher]], Section 2 and Appendix B of: *Left properness of flows*, Theory and Applications of Categories, **37** 19 (2021) 562-612  ([arXiv:1907.01454](https://arxiv.org/abs/1907.01454), [TAC:37-19](http://www.tac.mta.ca/tac/volumes/37/19/37-19abs.html))

### Relation to diffeological spaces

Relation to [[diffeological spaces]]:

* {#SYH10} [[Kazuhisa Shimakawa]], K. Yoshida, [[Tadayuki Haraguchi]], _Homology and cohomology via enriched bifunctors_, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 ([arXiv:1010.3336](https://arxiv.org/abs/1010.3336), [doi:10.2206/kyushujm.72.239]( https://doi.org/10.2206/kyushujm.72.239))

* {#CSW13} [[J. Daniel Christensen]], [[Gord Sinnamon]], [[Enxin Wu]], Sec. 3.2 of: _The $D$-topology for diffeological spaces_, Pacific Journal of Mathematics 272 (1), 87-110, 2014 ([arXiv:1302.2935](http://arxiv.org/abs/1302.2935), [doi:10.2140/pjm.2014.272.87](https://msp.org/pjm/2014/272-1/p04.xhtml))

[[!redirects Delta-generated topological spaces]]
[[!redirects Delta-generated space]]
[[!redirects Delta-generated spaces]]
[[!redirects Δ-generated topological space]]
[[!redirects Δ-generated topological spaces]]
[[!redirects Δ-generated space]]
[[!redirects Δ-generated spaces]]
[[!redirects numerically generated topological space]]
[[!redirects numerically generated topological spaces]]
[[!redirects numerically generated space]]
[[!redirects numerically generated spaces]]

[[!redirects Euclidean-generated space]]
[[!redirects Euclidean-generated spaces]]

[[!redirects Euclidean-generated topological space]]
[[!redirects Euclidean-generated topological spaces]]

[[!redirects D-topological space]]
[[!redirects D-topological spaces]]

[[!redirects ∞-generated space]]
[[!redirects ∞-generated spaces]]

[[!redirects ∞-generated topological space]]
[[!redirects ∞-generated topological spaces]]


[[!redirects lectures on p-divisible groups]]
This entry is about the text

* Michel Demazure, lectures on [[p-divisible group|p-divisible groups]] [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

## chapter I: schemes and formal schemes

### 1. $k$-functors

[[I.1, k-functors]]

### 2. affine $k$-schemes

[[I.1, k-functors|I.2, affine k-schemes]]

### 3. closed and open subfunctors; schemes

[[I.3, open- and closed subfunctors; schemes]]

### 4. the geometric point of view

[[I.4, the geometric point of view]]

### 5. finiteness conditions

[[I.5, finiteness conditions]]

### 6. the four definitions of formal schemes

[[I.6, the four definitions of formal schemes]]

### 7. operations on formal schemes

[[I.7. operations on formal schemes]]

### 8. constant- and étale schemes

[[I.8, constant- and étale schemes]], [[the fundamental theorem of Galois theory]], [[Grothendieck's Galois theory]]

### 9. the Frobenius morphism

[[I.9, the Frobenius morphism]]

### 10. Frobenius morphism and symmetric products

[[I.10, Frobenius morphism and symmetric products]]

## chapter II: group-schemes and formal group-schemes

### 1. group-functors

[[II.1, group -functors]], [[k-group-functor]], [[k-group]] (=[[k-group scheme]])

### 2. constant and étale k-groups

[[II.2, constant and étale k-groups]]

### 3. affine k-groups

[[II.3, affine k-groups]]

### 4. k-formal groups, Cartier duality
[[II.4, k-formal groups, Cartier duality]], [[Cartier duality]]

### 5. the Frobenius and the Verschiebung morphism

[[II.5, the Frobenius and the Verschiebung morphism]]

### 6. the category of affine k-groups

[[II.6, the category of affine k-groups]]

From now on ''$k$-group'' will mean by default ''commutative $k$-group'' and the field $k$ will be of characteristic $p\gt 0$. The case $p=0$ is rather trivial.

### 7. étale and connected formal k-groups

[[II.7, étale and connected formal k-groups]]

### 8. multiplicative affine groups

[[II.8, multiplicative affine groups]], [[diagonalizable group scheme]]

###  9. unipotent affine groups, decomposition of affine groups

[[II.9, unipotent affine groups, decomposition of affine groups]]

### 10. smooth formal groups

[[II.10, smooth formal groups]]

### 11. p-divisible formal groups

[[II.11, p-divisible formal groups]]

### 12 appendix

[[II.12, appendix]]

## chapter III: Witt groups and Dieudonné modules

### 1. the Artin-Hasse exponential series

[[III.1 the Artin-Hasse exponential series]]

### 2. the Witt rings over $\mathbb{Z}$

[[III.2, the Witt rings over Z]]

### 3. the Witt rings over $k$

[[III.3, the Witt rings over k]]

### 4. duality of finite Witt groups

[[III.4, duality of finite Witt groups]]

### 5. Dieudonné modules (affine unipotent groups)

[[III.5, Dieudonné modules (affine unipotent groups)]]

### 6. Dieudonné modules ($p$-torsion finite $k$-groups)

[[III.6, Dieudonné modules (p-torsion finite k-groups)]]

### 8. Dieudonne modules (p-divisible groups)

[[III.8, Dieudonné modules (p-divisible groups)]]

### 9. Dieudonné modules (connected formal groups of finite type)

[[III.9, Dieudonné modules (connected formal groups of finite type)]]

## chapter IV: classification of $p$ divisible groups

Unless otherwise stated let $k$ be a perfect field of prime characteristic.

We denote write $B(K):=Quot(W(k))$ for the [[quotient field]] of the [[Witt ring]] $W(k)$.

We extend the [[Frobenius morphism]] $x\mapsto x^{(p)}$ to an automorphism of $B(k)$. The set of fixed points of $x\mapsto x^{(p)}$ in $W(k)$ is $W(F_p)=\mathbb{Z}_p$. The set of fixed points of $x\mapsto x^{(p)}$ in $B(k)$ is $B(F_p)=\mathbb{Q}_p$.

### 1. isogenies

[[Demazure, lectures on p-divisible groups, IV.1, isogenies]]

### 2. the category of $F$-spaces

[[Demazure, lectures on p-divisible groups, IV.2, the category of F-spaces]]

### 3. the spaces $E^\lambda$, $\lambda \ge 0$

[[Demazure, lectures on p-divisible groups, IV.3, the spaces E^lambda, lambda \ge 0]]

### 4. classificaton of $F$-spaces over an algebraically closed field

[[Demazure, lectures on p-divisible groups, IV.4, classificaton of F-spaces over an algebraically closed field]]

### 5. slopes

[[Demazure, lectures on p-divisible groups, IV.5, slopes]]

### 6. the characteristic class of an endomorphism

[[Demazure, lectures on p-divisible groups, IV.6, the characteristic class of an endomorphism]]

### 7. specialization of $p$-divisible groups

[[Demazure, lectures on p-divisible groups, IV.7, specialization of p-divisible groups]]

### 8. some particular cases

[[Demazure, lectures on p-divisible groups, IV.8, some particular cases]]

## chapter V: $p$-adic cohomology of abelian varieties

### 1. abelian varieties, known facts

[[Demazure, lectures on p-divisible groups, V.1, abelian varieties, known facts]]

### 2. points of finite order and endomorphisms

[[Demazure, lectures on p-divisible groups, V.2, points of finite order and endomorphisms]]

### 3. structure of the $p$-divisible group $A(p)$

[[Demazure, lectures on p-divisible groups, V.3, structure of the p-divisible group A(p)]]

## Related nlab entries

[[relations of certain classes of group schemes]]


Demazure, lectures on p-divisible groups, I.1, k-functors

[[!redirects I.1, k-functors]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

+-- {: .num_defn}
###### Definition
(k-ring, k-functor,affine k-scheme)

For a ring $k$ the _category of $k$-rings_, denoted by $M_k:=k/Ring$ is defined to be the category of commutative associative $k$-algebras with unit which are rings. This is equivalently the category of pairs $(R,f:k\to R)$ where $R$ is a Ring and $f$ is a morphism of $k$-algebras.

The _category of $k$-functors_, denoted by $co Psh (M_k)$, is defined to be the category of covariant functors $M_k\to Set$.

The forgetful functor $O_k:R\to R$ sending a $k$-ring to its underlying set is called _affine line_.

For the full and faithful contravariant functor

$$Sp_k:\begin{cases}
M_k&\to& co Psh(M_k)
\\
A&\to& M_k(A,-)
\end{cases}$$

 $Sp_k A$ (and every isomorphic functor) is called an _affine $k$-scheme_. (In most modern texts one uses the notation ''$Spec$'' instead of ''$Sp$''.) $Sp_k$ restricts to an equivalence between the categories of $k$-rings and the category $Aff Sch_k$ of affine $k$-schemes. We think of this category as of $M_k^{op}$. The functor $Sp_k$ commutes with limits and skalar extension (see below). Consequently $Aff Sch_k$ is closed under limits and base change.

The affine line $O_k=M_k(k[t],-)$ is an affine $k$-scheme.

A _[[k-ring of functions on a k-functor|function on a]]_ $k$-scheme $X$ is defined to be an object $f\in O(X):=co Psh (M_k)(X,O_k)$. $O(X)$ is a $k$-ring by component-wise addition and -multiplication.
=--

+-- {: .num_prop}
###### Proposition
There is an adjoint equivalence

$$(Sp\dashv O):Sch_{aff}\stackrel{O}{\to}Ring_k$$

of the categories of [[affine scheme|affine k-schemes]] and $k$-rings.
=--

+-- {: .num_remark}
###### Remark
The category of $k$-functors has limits.

The terminal object is $e:R\mapsto\{\varnothing\}$. Products and pullbacks are computed component-wise.
=--

+-- {: .num_remark}
###### Remark
For $\phi:k\to k^\prime$ the ''base change'' functor $(-)\otimes_k k^\prime:co Psh(M_k)\to co Psh(M_{k^\prime})$ induced by $(-)\circ \phi:M_k\to M_{k^\prime}$ given by postcompositions with $\phi$ is called _skalar extension_.
=--


Demazure, lectures on p-divisible groups, I.10, Frobenius morphism and symmetric products

[[!redirects I.10, Frobenius morphism and symmetric products]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)
We give here another characterization of the [[Frobenius morphism]] in terms of symmetric products.

Let $p$ be a prime number, let $k$ be a field of [[characteristic]] $p$, let $V$ be a $k$-vector space, let $\otimes^p V$ denote the $p$-fold tensor power of $V$, let $TS^p V$ denote the subspace of symmetric tensors. Then we have the symmetrization operator

$$s_V:
\begin{cases}
\otimes^p V\to TS^p V
\\
a_1\otimes\cdots\otimes a_n\mapsto \Sigma_{\sigma\in S_p}a_{\sigma(1)}\otimes\cdots\otimes a_{\sigma(n)}
\end{cases}$$

end the linear map

$$\alpha_V:
\begin{cases}
TS^p V\to\otimes^p V
\\
a\otimes \lambda\mapsto\lambda(a\otimes\cdots\otimes a)
\end{cases}$$

then the map $V^{(p)}\stackrel{\alpha_V}{\to}TS^p V\to TS^p V/s(\otimes^p V)$ is bijective and we define $\lambda_V:TS^p V\to V^{(p)}$ by

$$\lambda_V\circ s=0$$

and

$$\lambda_V \circ \alpha_V= id$$

If $A$ is a $k$-ring we have that $TS^p A$ is a $k$-ring and $\lambda_A$ is a $k$-ring morphism.

If $X=Sp_k A$ is a ring spectrum we abbreviate $S^p X=S^p_k X:=Sp_k (TS^p A)$ and the following diagram is commutative.

$$\array{
X
&\stackrel{F_X}{\to}&
X^{(p)}
\\
\downarrow&&\downarrow
\\
X^p
&\stackrel{can}{\to}& S^p X
}$$


Demazure, lectures on p-divisible groups, I.3, open- and closed subfunctors; schemes

[[!redirects I.3, open- and closed subfunctors; schemes]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

For a $k$-functor $X\in coPsh(M_k)$ and $E\subseteq O(X)=M_k(X,O_k)$ a set of functions on $X$, Definition in [[k-ring]], we define

$$V(E)(R):=\{x\in X(R) | f\in E, f(x)=0\}$$

and

$$D(E)(R):=\{x\in X(R)|f\in E, \text{the} f(x) \text{generate the unit ideal of} R\}$$

(The ''unit ideal of $R$'' is just $R$ itself.) For a transformation $u:Y\to X$ of $k$-functors and $Z\subseteq X$ a subfunctor we define

$$u^{-1}(Z)(R):=\{y\in Y(R)|u(y)\in Z(R)\}$$

A subfunctor $Y\subseteq X$ is called _open subfunctor_ resp. _closed subfunctor_ if for every transformation $u:T\to X$ we have $u^{-1}(Y)$ is of the form $V(E)$ resp. $D(E)$.

+-- {: .num_defn}
###### Definition

A $k$-functor $X$ is called a $k$_-scheme_ if the following two conditions hold:

1. ($X$ is a sheaf for the Zarisky Grothendieck topology on $M_k^{op}$) For all $k$-rings and all families $(f_i)_i$ such that $R=\coprod_i R f_i$ we have: if for all $x_i\in R[f_i^{-1}]$ such that the images of $x_i$ and $x_j$ coincide in $X(R[f_i^{-1} f_j^{-1}])$ there is a unique $x\in X(R)$ mapping to the $x_i$.

1. ($X$ has a cover of Zarisky open immersions of affine $k$-schemes) There exists a family $(U_i)_i$ of open affine subfunctors of $X$ such that for all fields $K\in M_k$ we have that $X(K)=\coprod_i U_i(K)$.

=--

+-- {: .num_remark}
###### Remark
The category of $k$-schemes is closed under limits, forming open- and closed subfunctors and skalar extension.

=--


Demazure, lectures on p-divisible groups, I.5, finiteness conditions

[[!redirects I.5, finiteness conditions]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Let $k$ be a field.

An affine $k$-scheme $X$ is called finite if $O(X)$ is a finite dimensional vector space.


Demazure, lectures on p-divisible groups, I.6, the four definitions of formal schemes

[[!redirects I.6, the four definitions of formal schemes]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

+-- {: .num_defn}
###### Definition and Remark

Let $k$ be a field. Let $Mf_k$ denote the category of finite dimensional $k$-rings.

1. A $k$-scheme is called a $k$_-formal scheme_ if it is is equivalent to a codirected colimit of finite (affine) $k$-schemes.

1. A $k$-scheme is a $k$-formal scheme if it is presented by a profinite $k$-ring or -equivalently- by a $k$-ring which is the limit of discrete quotients which are finite $k$-rings. If $A$ is such a topological $k$-ring $Spf_k(A)(R)$ denotes the set of continous morphisms from $A$ to the topological discrete ring $R$. We have $Spf_k$ is a contravariant equivalence between the category of profinite $k$-rings and the category $fSch_k$ of formal $k$-schemes.

1. Instead of defining $fSch_k$ as the opposite of $Mf_k$ define it covariantly on the category of finite dimensional $k$-corings.

1. A formal $k$-scheme is precisely a left exact (commutig with finite limits) functor $X:Mf_k\to Set$.

The inclusion $Mf_k\hookrightarrow M_k$ induces a functor

$${}^\hat\; :Sch_k\to fSch_k$$

called _completion functor_.
=--
## 3

### [[k-coalgebra]] and [[k-coring]]

A *$k$-coalgebra* is a $k$-[[vector space]] $C$ equipped with a $k$-linear map $\Delta:C\to C\otimes_k C$.

A $k$-coalgebra $C$ is called a *$k$-coring* if $\Delta$ is

1. coassociative in that $(\Delta\otimes 1_C)\circ \Delta=(1_C\otimes \Delta)\circ \Delta$

1. cocommutative in that the image of $\Delta$ consists only of symmetric tensors.

1. has a counit $\epsilon$ to $\Delta$ satisfying $\Delta\circ (1_C\otimes \epsilon)=\Delta\circ (\epsilon\otimes 1_C)=1_C$.

### [[spectrum of a k-coring]]

Let $A$ and $R$ be two finite $k$-rings, let $A^*$ denote the [[k-coalgebra|dual k-coring]] of $A$.

Linear maps $A\to R$ correspond bijectively to elements of the tensor product $A^*\otimes R$. The $k$-linear maps $\Delta_{A^*}$ and $\epsilon_{A^*}$ extends to $R$-linear maps

$$A^*\otimes R\to (A^*\otimes R)\otimes_R(A^*\otimes R)$$

and

$$A^*\otimes R\to R$$

denoted by $\Delta$ and $\epsilon$.

+--{: .num_lemma}
###### Lemma and Definition

1. A $k$-linear map $A\to R$ associated to $u\in A^*\otimes R$ is a ring morphism iff $\Delta u= u\otimes u$ and $\epsilon u=1$.

1. There is a functorial isomorphism

$$Sp(A(R)=\{u\in A^*\otimes R|\Delta u=u\otimes u, \epsilon u =1\}$$

and the *$k$-formal spectrum of the coring $C$* is defined by

$$Sp^* C(R)=\{u\in C\otimes R|\Delta u =u\otimes u, \epsilon u =1\}$$

in particular $Sp^*: M_k^{op}\to coPsh(M_k)$ is a covariant functor from the category of $k$-corings to the category of $k$-formal functors.
=--

## References

* Michel Demazure, lectures on p-divisible groups [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)


Demazure, lectures on p-divisible groups, I.7. operations on formal schemes

[[!redirects I.7. operations on formal schemes]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

A finite limit of $k$-formal schemes is $k$-formal. The category of $k$-formal schemes has arbitrary products.

A formal $k$-scheme is called _local $k$-scheme_ if it is the formal spectrum of a [[local ring]] (=has a unique maximal ideal).

Any formal $k$-scheme is a coproduct of local $k$-schemes ([Demazure, p.15](#Demp)).


Demazure, lectures on p-divisible groups, I.8, constant- and étale schemes

[[!redirects I.8, constant- and étale schemes]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

$Sch_k$ is [[copower|copowered (= tensored)]] over $Set$. We define the _constant $k$-scheme_ on a set $E$ by

$$E_k:=E\otimes Sp_k k=\coprod_{e\in E}Sp_k k$$

For a scheme $X$ we compute $M_k(E_k,E)=Set(Sp_k k,X)^E=X(k)^E=Set(E,X(k))$ and see that there is an adjunction

$$((-)_k\dashv (-)(k)):Sch_k\to Set$$

A _constant formal scheme_ is defined to be a completion of constant scheme. The completion functor induces an equivalence between the category of constant schemes and the category of constant formal schemes.

An _étale $k$-scheme_ is defined to be a directed colimit of $k$-spectra $Sp_k k^\prime$ of finite separable field-extensions $k^\prime$ of $k$.

An _étale formal $k$-scheme_ is defined to be a directed colimit of formal $k$-spectra $Spf_k k^\prime$ of finite separable field-extensions $k^\prime$ of $k$.

+-- {: .num_remark}
###### Remark
Let $X$ be a $k$-scheme or a formal $k$-scheme. Then the following statements are equivalent:

1. $X$ is étale.

1. $X\otimes_k cl(k)$ is constant.

1. $X\otimes_k k_s$ is constant.

where $cl(k)$ denotes an algebraic closure of $k$, $k_s$ denotes the subextension of $cl(k)$ consisting of all separable elements of $cl(k)$ and $\otimes_k$ denotes skalar extension.
=--

+-- {: .num_prop}
###### Proposition
Let $X$ be a $k$-formal scheme (resp. a [[locally algebraic scheme]]) then $X$ is [[etale scheme|étale]] iff the [[Frobenius morphism]] $F_X:X\to X^{(p)}$is a monomorphism (resp. an isomorphism).
=--

+-- {: .num_theorem}
###### Theorem
([[the fundamental theorem of Galois theory|fundamental theorem of Galois theory]])

The functor

$$\begin{cases}
Sch_{et}\to Gal(k_s / k)- Mod
\\
X\mapsto X(k_s)
\end{cases}$$

from [[etale scheme|étale schemes]] to the category of [[Galois module|Galois modules]] $Gal(k_s/s)-Mod$ is an equivalence of categories.
=--

+-- {: .num_remark}
###### Remark
The completion functor

$$\begin{cases}
Sch_{et}\to fSch_{et}
\\
X\mapsto \hat X
\end{cases}$$
 is an equivalence of categories.
=--

[[!redirects étale scheme]]

[[!redirects etale scheme]]


Demazure, lectures on p-divisible groups, I.9, the Frobenius morphism

[[!redirects I.9, the Frobenius morphism]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

## Reminder

Let $s:R\to S$ be a morphism of rings. Then we have an [[adjunction]]

$$(s^*\dashv s_*):S.Mod\stackrel{s_*}{\to} R.Mod$$

from the category of $S$-[[module|modules]] to that of $R$-modules where

$$s^*:A\mapsto A\otimes_s S$$
is called *scalar extension* and $s_*$ is called *scalar restriction*.

## Idea

(Frobenius recognizes [[p-torsion]])

## Definition

Let $p$ be a prime number, let $k$ be a field of characteristic $p$. For a $k$-ring $A$ we define

$$f_A:
\begin{cases}
A\to A
\\
x\mapsto x^p
\end{cases}$$

The $k$-ring obtained from $A$ by scalar restriction along $f_k:k\to k$ is denoted by $A_{f}$.

The $k$-ring obtained from $A$ by scalar extension along $f_k:k\to k$ is denoted by $A^{(p)}:=A\otimes_{k,f} k$.

There are $k$-ring morphisms $f_A: A\to A_f$ and $F_A:\begin{cases}
A^{(p)}\to A
\\
x\otimes \lambda\mapsto x^p \lambda
\end{cases}$.

For a $k$-functor $X$ we define $X^{(p)}:=X\otimes_{k,f_k} k$ which satisfies $X^{(p)}(R)=X(R_f)$. The _Frobenius morphism_ for $X$ is the transformation of $k$-functors defined by

$$F_X:
\begin{cases}
X\to X^{(p)}
\\
X(f_R):X(R)\to X(R_f)
\end{cases}$$

If $X$ is a $k$-scheme $X^{(p)}$ is a $k$-scheme, too.

Since the completion functor ${}^\hat\;:Sch_k\to fSch_k$ commutes with the above constructions the Frobenius morphism can be defined for [[formal scheme|formal k-schemes]], too.

### In terms of symmetric products

We give here another characterization of the [[Frobenius morphism]] in terms of symmetric products.

Let $p$ be a prime number, let $k$ be a field of [[characteristic]] $p$, let $V$ be a $k$-vector space, let $\otimes^p V$ denote the $p$-fold tensor power of $V$, let $TS^p V$ denote the subspace of symmetric tensors. Then we have the symmetrization operator

$$s_V:
\begin{cases}
\otimes^p V\to TS^p V
\\
a_1\otimes\cdots\otimes a_n\mapsto \Sigma_{\sigma\in S_p}a_{\sigma(1)}\otimes\cdots\otimes a_{\sigma(n)}
\end{cases}$$

end the linear map

$$\alpha_V:
\begin{cases}
TS^p V\to\otimes^p V
\\
a\otimes \lambda\mapsto\lambda(a\otimes\cdots\otimes a)
\end{cases}$$

then the map $V^{(p)}\stackrel{\alpha_V}{\to}TS^p V\to TS^p V/s(\otimes^p V)$ is bijective and we define $\lambda_V:TS^p V\to V^{(p)}$ by

$$\lambda_V\circ s=0$$

and

$$\lambda_V \circ \alpha_V= id$$

If $A$ is a $k$-ring we have that $TS^p A$ is a $k$-ring and $\lambda_A$ is a $k$-ring morphism.

If $X=Sp_k A$ is a ring spectrum we abbreviate $S^p X=S^p_k X:=Sp_k (TS^p A)$ and the following diagram is commutative.

$$\array{
X
&\stackrel{F_X}{\to}&
X^{(p)}
\\
\downarrow&&\downarrow
\\
X^p
&\stackrel{can}{\to}& S^p X
}$$

## Properties

Note that the Frobenius $F_p$ is an endomorphism of a field $R$ only if the characteristic of $R$ is $p$. In this case it is automatically a monomorphism, since field homomorphisms always are.

However if we pass from rings to schemes, in general it is not true that Frobenius is a monomorphism. The following proposition gives necessary and sufficient conditions for the Frobenius to be a monomorphism in case of [[formal scheme|formal schemes]].

+-- {: .num_prop}
###### Proposition
Let $X$ be a $k$-formal scheme (resp. a [[locally algebraic scheme]]) then $X$ is [[etale scheme|étale]] iff the [[Frobenius morphism]] $F_X:X\to X^{(p)}$is a monomorphism (resp. an isomorphism).
=--

## Examples

If $X=Sp_k A$ is a $k$-ring spectrum we have $X^{(p)}=Sp_k A^{(p)}$ and $F_X=Sp_k F_A$.

If $k=\mathbb{F}$ is a finite field we have $X^{(p)}=X$ however $F_X$ will not equal $id_X$ in general.

If $k\hookrightarrow k^\prime$ is a field extension we have $F_{X\otimes_k k^\prime}=F_X\otimes_k k^\prime$.

[[!redirects Frobenius morphism]]


Demazure, lectures on p-divisible groups, II.1, group -functors

[[!redirects II.1, group -functors]]
[[!redirects k-group]]

[[!redirects k-group-functor]]

[[!redirects k-group scheme]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

A _group functor_ is a [[group object]] in a functor category.

A *$k$-group functor* is a [[k-functor]] which is a group functor.

A group functor which is a scheme is called *$k$-group scheme* or just *$k$-group*. The category of $k$-groups is denoted by $Gr_k$.

## Terminology

The terminology of group-functors is mainly used in the theory of [[algebraic group|algebraic groups]] in algebraic geometry. If $G$ is a group functor the multiplication morphism $\pi:G\times G\to G$ is called _group law on_ $G$.

## References

* Michel Demazure, lectures on p-divisible groups [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups), chapter II


Demazure, lectures on p-divisible groups, II.10, smooth formal groups

[[!redirects II.10, smooth formal groups]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

+-- {: .num_defn}
###### Definition
A not necesarily commutative connected formal group $G=\Sp f A$ is called *[[smooth formal k-group]] if $A$ is a power-series algebra $k[ [X_1,...X_n] ]$ in $n$ variables

The coproduct $\Delta:A\to A\hat \otimes A$ is given by a set af formal power series $\Phi(X,Y)=(\Phi_i(X_1,..., X_n, Y_1,..., Y_n)), i_1,...,n$ satisfying the axioms (Ass),(Un) and (Com). Such a set $\{\Phi_i\}$ is called a *Dieudonné group law*.
=--

+-- {: .num_theorem}
###### Theorem
Let $G=Sp f A$ be a (not necessarily commutative) connected formal group of finite type.
1.If $p=0$ then $G$ is smooth.

1. If $p\not =0$ then the following conditions are equivalent

1. $G$ is smooth

1. $A\otimes_k k^{p-1}$ is reduced.

1. $F_G$ is an epimorphism.
=--

The previous theorem can be strengthened:

+-- {: .num_theorem}
###### Theorem
(Cartier)

Let $p=0$, let $G=Sp^* C$ be a connected (not necessarily commutative) formal $k$-group (realized as the [[spectrum of a coring|formal spectrum of a k-coring]] $C$).

1. $C$ is the universal enveloping algebra of the Lie algebra $\mathcal{g}$ of $G$.

1. The category of connected formal $k$-groups is equivalent to the category of all Lie algebras over $k$.

1. If $\mathcal{g}$ is finite dimensional then $G$ is smooth.

1. If $G$ is commutative $\mathcal{g}$ is abelian.

1. $G\simeq (\alpha^\circ)^{(I)}$; by duality any unipotent (commutative) $k$-group is a power of the additive group.
=--

+-- {: .num_theorem}
###### Theorem
(Dieudonné-Cartier-Gabriel)
Let $p\gt 0$, let $k$ be a [[perfect field]] of [[characteristic]] $p$ let $G=Sp^* C$ be a connected (not necessarily commutative) connected formal $k$[[k-group of finite type|-group of finite type]], let $H$ be a subgroup of $G$, let $G/H:=Spf A$ (this quit ion has not been defined in these lectures).

Then $A$ is of the form

$$k [ [ X_1,\dots,X_n] ][Y_1,\dots,Y_d]/(Y_1^{p^{r_1}},\dots,Y_d^{p^{r_d}})$$

This applies for instance to $A=\hat O_{G,e}$, for an [[algebraic group]] $G$.

=--

+-- {: .num_cor}
###### Corollary
Let $p\ge 0$, let $G$ be a [[connected formal group]] (=local formal group) of [[formal group of finite type|finite type]]. Then

1. If $k$ is prefect, there exists a unique exact sequence of connected groups

$$0\to G_{red}\to G\to G/G_{red}\to 0$$

with $G_{red}$ smooth and $G/G_{red}$ [[infinitesimal k-group|infinitesimal]].

2. For large $r$, the group $G/ker F^r_G$ is smooth.
=--

+-- {: .num_cor}
###### Corollary
Let $G$ be a connected formal group of finite type, let $n:=dim G$. Then $rk(co her F^i_G)$ is bounded and

$$rk(ker F^i_G)=p^{n i}\cdot rk(coker F^i_G)$$
=--

+-- {: .num_cor}
###### Corollary
1. Let $0\to G^\prime\to G\to G^n\to 0$ be an exact sequence of connected formal groups. Then $dim(G)=dim(G^\prime)+dim(G^n)$.

2. If $f:G^\prime \to G$ is a morphism of connected formal groups, with $G$ smooth and $dim G=dim G^\prime$, then $f$ is an epimorphism iff $ker f$ is finite.
=--


Demazure, lectures on p-divisible groups, II.11, p-divisible formal groups

[[!redirects II.11, p-divisible formal groups]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Let $k$ be a field of prime characteristic $p\gt 0$.

## Definition

+-- {: .num_defn}
###### Definition
($p$-divisible group)

A commutative formal $k$-group $G$ is called *[[p-divisible group|p-divisible formal k-group]]* or *Barsotti-Tate group* if it satisfies the following properties:

: (pdg1) $p\cdot id_G\to G$ is an epimorphism.

: (pdg2) $G$ is a $p$-torsion group in that $G=\cup_j ker(p^j \cdot id_G)$

: (pdg3) $ker(p\cdot id_G)$ is finite.

We have $rk(ker \,p \cdot id_G)=p^h$, $h\in \mathbb{N}$. This $h$ is called the *height $ht(G)$ of $G$*.
=--

+-- {: .num_rem}
###### Definition
(alternative definition of $p$-divisible group)

 Let

$$G_1\stackrel{i_1}{\to}G_2\stackrel{i_2}{\to}G_3\stackrel{i_3}{\to}\cdots$$

be a codirected diagram of [[finite k-group|finite k-groups]] such that

1. $rk(G_j)=p^{h j}$, $h$ a fixed integer,

1. all sequences $0\stackrel{}{\to}G_j\stackrel{i_j}{\to}G_{j+1}\stackrel{p^j}{\to}G^{j+1}$ are exact.

Then $colim_n G_n$ is a $p$-divisible group of height $h$ and $ker(p^n id_G:G\to G)\simeq G_n$.
=--

+-- {: .num_remark}
###### Remark

If $G$ is a p-divisible group in the sense of the first definition, from (pdg1) follows $rk(ker p^j \cdot id_G)0p^{j\cdot ht (G)}$. Since $rk$ is multiplicative

$$0\to ker \,p^j\hookrightarrow ker \,p^{j+k}\stackrel{p^j}{\to}ker\, p^k\to 0$$

is exact.
=--

+-- {: .num_defn}
###### Definition
([[Serre dual]] of a $p$-divisible group)

Let $G$ be a $p$-divisible group $G$. The *Serre dual* $G^\prime$ of $G$ is defined by: let $G_j:=ker(p^j id_G)$ and let $p_j:G_{j+1}\to G_j$ is the map induced by $p id_G$. Then we define

$$G_j^\prime:=D(G_j)$$

$$i_j^\prime:=D(p_j):G^\prime_j\to G^\prime_{j+1}$$

$$G^\prime:=colim_{j^\prime}G_j^\prime$$

This is a $p$-divisible formal group with $ht(G^\prime)=ht(G)$ and we have $p_j^\prime=D(i_j)$ and $(G^\prime)^\prime\simeq G$.

=--

## Examples

+-- {: .num_example}
###### Example
Let $\mathbb{Z}_p$ denote the [[p-adic number|ring of p-adic integers]], let $\mathbb{Q}_p$ denote the [[p-adic number|field of p-adic numbers]]. The constant formal group $(\mathbb{Q}_p /\mathbb{Z}_p)_k$ is a $p$-divisible group of height $1$.

Conversely any $p$-divisible group of height $h$ is isomorphic to $(\mathbb{Q}_p /\mathbb{Z}_p)^h_k$.

=--

+-- {: .num_example}
###### Example
Let $A$ be a commutative [[algebraic group|algebraic k-group]], such that $p id_G:A\to A$ is an epimorphism. Then

1. $ker(p\cdot id_A)$ is finite.

1. $A(p):=\cup_j ker(p^j id_A)$ is a $p$-divisible group containing $\hat A^\circ=\cup_j ker(F^j G)$.

1. If $A=\mu_k$ we have $A(p)=\cup_j p^j \mu_k=(\mathbb{Q}_p /\mathbb{Z}_p)^\prime_k$.

1. If $A$ is an abelian variety of dimension $g$ $p id_A$ is an epimorphism with $rk(her p id_G)=p^{2g}$ and consequently $A(p)$ is a $p$-divisible group of height $2g$. This example is further described in chapter [[V, p-adic cohomology of abelian varieties]], particularly in [[V.3, structure of the p-divisible group A(p)]].
=--

+-- {: .num_prop}
###### Proposition
Let $G$ be a $k$-formal group. Then $G$ is $p$-divisible iff the following conditions hold:

1. $\pi_\circ(G)(\overline k)\simeq (\mathbb{Q}_p /\mathbb{Z}_p)^r$, $r$ finite.

1. $G^\circ$ is of finite type, smooth, and $ker(V:G^{\circ (p)}\to G^\circ)$ is finite.
=--

+-- {: .num_example}
###### Example
Let $A$ be an algebraic unipotent $k$-group, then $\hat A^\circ$ is never $p$-divisible unless $A$ is finite.
=--

+-- {: .num_remark}
###### Remark
Let $G$ be $p$-divisible. Then we have $height(G)=dim(G)+dim(G^\prime)$
=--

+-- {: .num_prop}
###### Proposition
Let $G$ be a connected, finite type, smooth formal group. There exist two subgroups $H$,K\subseteq G$ with $H$ is $p$-divisible, $p^n K = 0$ for large $n$, $H\cap K$ is finite, and $G=H+ K$.
=--


Demazure, lectures on p-divisible groups, II.2, constant and étale k-groups

[[!redirects II.2, constant and étale k-groups]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Let $k$ be a field.

+-- {: .num_defn}
###### Definition

If $E$ is a group then the [[constant scheme]] $E_k$ on $E$ carries a [[group scheme|group structure]]. Such an $E_k$ is called constant $k$-group.

=--

+-- {: .num_defn}
###### Definition and Remark

Let $k_s$ denote the field extension of $k$ consisting of separable elements. Let $\Pi:=Gal(k_s/k)$ denote the [[Galois group]] of this extension. Then the functor

$$(-)(k_s):\begin{cases}
et Sch_k
&\to&
Gal(k_s/k)-Set
\\
X
&\to&
X(k_s)
\end{cases}$$

is an equivalence between the category of étale $k$-schemes and that of sets equipped with a group action of$Gal(k_s/k)$.

Moreover this gives a functor

$$(-)(k_s):\begin{cases}
et Gr_k
&\to&
Gal(k_s/k)-Mod
\\
X
&\to&
X(k_s)
\end{cases}$$

which is an equivalence between the category of étale $k$[[group scheme|-groups]] and that of groups equipped with a group action of $Gal(k_s/k)$. Note that commutative $Gal(k_s/k)$-groups are called Galois modules, too.

A $k$-group scheme is étale iff its coefficient extension $X\otimes_k k_s$ along $k\hookrightarrow k_s$ is a constant k-scheme.

=--

* Michel Demazure, lectures on p-divisible groups [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)


Demazure, lectures on p-divisible groups, II.3, affine k-groups

[[!redirects II.3, affine k-groups]]
[[!redirects affine k-group]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Let $A\in M_k$, let $\Delta:A\to A\otimes_k A$.

Then the $k$-group $(Sp_k A, Sp_k \Delta_k)$ is called *affine k-group*.

A *$k$-biring* is defined to be a $k$-module which is a $k$-ring and a $k$-coring such that $A\to A\otimes A$ is a $k$-ring morphism and $A\otimes A\to A$ is a $k$-coring morphism.

## Examples

+-- {: .num_example}
###### Example

The additive $k$-group $\alpha_k$ assigns to a $k$-ring its underlying additive group. We have $O(\alpha_k)=K[t]$ since by the Yoneda lemma we have $O(\alpha_k)=hom(\alpha_k, O_k)=\alpha_k(k[t])$

=--

+-- {: .num_example}
###### Example

The multiplicative $k$-group $\mu_k$ assigns to a $k$-ring the multiplicative group of its invertible elements. We have $O(\mu_k)=K[tt^{-1}]$.

=--

+-- {: .num_example}
###### Example

There is a group homomorphism

$$n:=n_{\mu_k}:\begin{cases}
\mu_k&\to& \mu_k
\\
x&\to &x^n
\end{cases}$$

its kernel we denote by $n\mu_k:=ker(n)$. We have $n\mu_k(R)=\{x\in R|x^n=1\}$ and $O(n\mu_k)=k[t]/(t^n -1)$. If $k$ is a field and $n$ is not $0$ in $k$ the $k$-group $n\mu_k$ is [[etale scheme|étale]] since $t^n-1$ is a separable polynomial. $n\mu_k(k_s)$ is the Galois module of the $n$-th root of unity.
=--

+-- {: .num_example}
###### Example

Let $k$ be a field of prime [[characteristic]] $p$, let $r$ be an integer. Then there is a $k$-group morphism.

$$p^r:=p^r_{\alpha_k}:\begin{cases}
\alpha_k&\to& \alpha_k
\\
x&\to &x^{p^r}
\end{cases}$$

We have $ker(p^r\alpha_k)(R)=\{x\in R | x^{p^r}=0\}$ and $O(ker(p^r \alpha_k))=K[t]/t^{p^r}$. For any field $K$ we have $ker(p^r_{\alpha_k})(K)=\{0\}$.

=--


Demazure, lectures on p-divisible groups, II.4, k-formal groups, Cartier duality

[[!redirects II.4, k-formal groups, Cartier duality]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

A _$k$-formal group_ is a $k$-group whose underlying $k$-functor is a $k$-formal functor.

The previous constructions in chapter II carry over to $k$-formal groups.

+-- {: .num_defn}
###### Definition
Let $G$ be a commutative $k$-group functor. Then the *[[Cartier duality|Cartier dual]]* $D(G)$ of $G$ is defined by

$$D(G)(R)=Gr_R(G\otimes_k R,\mu_R)$$

Moreover we have

$$hom(G,D(H))=hom(H,D(G))=hom(G\times H,\mu_k)$$
=--


Demazure, lectures on p-divisible groups, II.5, the Frobenius and the Verschiebung morphism

[[!redirects II.5, the Frobenius and the Verschiebung morphism]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Let $k$ be a field with prime characteristic $p$.

The [[Frobenius morphism]] $F_G: G\to G^{(p)}$ commutes with finite products and hence if $G$ is a [[k-group-functor]] $G^{(p)}$ is a $k$-group functor, too, and $F_G$ is a $k$-group morphism.

We abbreviate $F_G^n:G\to G^{(p^n)}$.

The same is true for $k$-formal groups.

Let $G$ be a commutative affine $k$-group. Then for the [[Cartier duality|Cartier dual]] $D(-)$ we have

$$D(G^{(p)})=D(G)^{(p)}$$

By Cartier duality we obtain the [[Verschiebung morphism]] $V_G:G^{(p)}\to G$ for which holds $\hat D(V_G)=F_{\hat D(G)}$. We abbreviate $V_G^n:G^{(p^n)}\to G$.

Let $f:G\to H$ be a morphism of commutative affine $k$-groups. The the following diagram is commutative

$$\array{
G^{(p)}
&\stackrel{V_G}{\to}&
G
&\stackrel{F_G}{\to}&
G^{(p)}
\\
\downarrow^{f^{(p)}}&&\downarrow^f&&\downarrow^{f^{(p)}}
\\
F^{(p)}
&\stackrel{V_H}{\to}&
H
&\stackrel{F_H}{\to}&
H^{(p)}
}$$

Moreover we have

$$V_G\circ F_G=p id_G$$

and

$$F_G\circ V_G=p id_{G^{(p)}}$$

## Examples

$V_{\mu_k}$ is the identity and $V_{\alpha_k}$ is zero.

This follows since $F$ is an epimorphism and $p id_{\mu_k}=F_{\mu_k}$ and $p id_{\alpha_k}=0$


Demazure, lectures on p-divisible groups, II.6, the category of affine k-groups

[[!redirects II.6, the category of affine k-groups]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

+-- {: .num_theorem}
###### Theorem
The category $AC_k$ of affine commutative $k$-groups is an [[abelian category]].
=--

As such it has in particular kernels and cokernels.

Remark: A category is abelian if it is [[Ab-enriched category|Ab-enriched]]( i.e. enriched over the category $AB$ of abelian groups) and has finite limits and finite colimits and every monomorphism is a kernel and every epimorphism is a cokernel.

+-- {: .num_theorem}
###### Theorem
Let $f:G\to H$ be a morphism in $AC_k$..

1. The following conditions are equivalent: $f$ is a monomorphism, $O(f)$ is surjective (i.e. $G$ is a closed subgroup of $H$), $f$ is a kernel.

1. The following conditions are equivalent: $f$ is a epimorphism, $O(f)$ is injective, $O(f):O(H)\to O(G)$ exhibits $O(G)$ as a faithful flat $O(H)$ module,  $f$ is a cokernel.
=--

+-- {: .num_cor}
###### Corollary
If $k\hookrightarrow k^\prime$ is a field extension skalar extension $G\mapsto G\otimes_k k^\prime$ is an exact functor.
=--
+-- {: .num_cor}
###### Corollary

=--
+-- {: .num_cor}
###### Corollary

=--
+-- {: .num_cor}
###### Corollary

=--
+-- {: .num_cor}
###### Corollary

=--
+-- {: .num_cor}
###### Corollary

=--

+-- {: .num_theorem}
###### Theorem
1. The category $AC_k$ satisfies the axiom (AB5): it has directed limits and the directed limit of an epimorphism is an epimorphism.

1. The [[artinian object|artinian objects]] of $AC_k$ are [[algebraic group|algebraic groups]]. Any object of $AC_k$ is the directed limit of its algebraic quotients.

1. By Cartier duality, the dual statements hold for the category of com-
mutative $k$-formal-groups.
=--


Demazure, lectures on p-divisible groups, II.7, étale and connected formal k-groups

[[!redirects II.7, étale and connected formal k-groups]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

[[étale group scheme|Étale]] affine (resp. étale formal) $k$-groups are equivalent to finite (resp. all) Galois modules by

$$E\to E\otimes_{k_s}k^\prime=\coprod_{K/k\;\text{separable}}E(k)$$

$G$ is etale iff $ker F_G=e$. This implies that $F$ is an isomorphism.

+-- {: .num_defn}
###### Definition

A $k$-formal group $G=Spf A$ is called  *[[local formal group scheme|local]]* (or *connected*) if the following two equivalent conditions hold:

1. $A$ is local

1. $G(k)=\{0\}$ for any field $K$.

=--

A morphism from a connected group to an étale group is zero.

+-- {: .num_prop}
###### Proposition
Let $G$ be a $k$-formal group.

1. Then there is an exact sequence

$$0\to G^\circ\to G\to \pi_\circ(G)\to 0$$

where $G^\circ$ is connected and $\pi_\circ(G)$ is étale. If $R\in Mf_k$ is a finite dimensional $k$-ring and $n_0$ is the [[nilradical]] (i.e. -if $R$ is commutative- the set of all nipotent elements) of $R$ then

$$G^\circ(R)=ker(G(R)\to G(R/n_0))$$

If $p$ is not $0$, then

$$G^\circ= lim_n \; ker(F\,G^{(p^n)})$$

If $k\to k^\prime$ is a [[field extension]] then $(G\otimes_k k^\prime)^\circ =G^\circ\otimes_k k^\prime$, and $\pi_0(G\otimes_k k^\prime)=\pi_0(G)\otimes_k k^\prime$.

2. If $k$ is perfect, there is a unique isomorphism $G=G^\circ\times \pi_0(G)$
=--

+-- {: .num_defn}
###### Definition

An affine $k$-group $G$ is called *infinitesimal* if it one of the followig conditions is satisfied:

1. $G$ is finite and local.

1. $G$  is algebraic and $G(cl(k))=e$ (the terminal $k$-group)
=--

+-- {: .num_cor}
###### Corollary
A finite group is an extension of an étale group by an infinitesimal group.

This extension splits if $k$ is perfect.
=--

+-- {: .num_defn}
###### Definition
A (not necessarily commutative) connected formal $k$-group $G=Spf A$ is said to be *of finite type* if $A$ is noetherian. The *dimension of $G$* is defined to be the [[Krull dimension]] of $A$.
=--


Demazure, lectures on p-divisible groups, II.8, multiplicative affine groups

[[!redirects II.8, multiplicative affine groups]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

+-- {: .num_remark}
###### Remark
Let $G$ be a $k$-group-functor. Then the following conditions are equivalent:

1. $G$ is the [[Cartier dual]] of a constant group.

1. $G$ is an affine $k$-group and the $k$-ring $O(G)$ is generated by the morphisms $G\to \mu_k$ (these are called *[[character|characters]] of $G$*).
=--

+-- {: .num_defn}
###### Definition
A $k$-group satisfying the conditions of the previous remark is called *[[diagonalizable group scheme|diagnalizable k-group]]*.
=--

+-- {: .num_theorem}
###### Theorem
Let $G$ be a $k$-group. Then the following conditions are equivalent:

1. $G\otimes_k k_s$ is diagonalizable.

1. $G\otimes_k K$ is diagonalizable for a field $K\in M_k$.

1. $G$ is the [[Cartier duality|Cartier dual]] of an étale $k$-group.

1. $\hat D(G)$ is an [[étale group scheme|étale]] $k$-formal group.

1. $Gr_k(G,\alpha_k)=0$

1. (If $p\neq 0)$, $V_G$ is an epimorphism

1. (If $p\neq0)$, $V_G$ is an isomorphism
=--

+-- {: .num_defn}
###### Definition and Remark
1. A $k$-group satisfying the conditions of the previous theorem is called *[[multiplicative group scheme|multiplicative k-group]]*.

1. Multiplicative $k$-groups correspond by duality to étale formal $k$-groups.

1. The category $ACm_k$ of multiplicative $k$-groups forma a subcategory of the category $AC_k$ of affine commutative $k$-groups which is stable under forming subgroups, quatients, extensions (the set of these properties says that the subcategory is [[thick subcategory|thick]]) and limits.

1. $ACm_k$ is (contravariant) equivalent to the category of [[Galois module|Galois modules]]: To $G$ corresponds the Galois module $\hat D(G\otimes_k k_s)(k_s)=Gr_{k_s}(G\otimes_k k_s,\mu_{k_s})$.

1. If $E$ is an étale $k$-group, then $D(E)$ is multiplicative and $\hat D(D(E))=E$. And we have $D(D(E))=E$. The duality is hence given by $E\to D(E)$ , $G\to D(G)$ without reference to formal groups.
=--


Demazure, lectures on p-divisible groups, II.9, unipotent affine groups, decomposition of affine groups

[[!redirects II.9, unipotent affine groups, decomposition of affine groups]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

+-- {: .num_theorem}
###### Theorem
Let $G$ be an [[affine scheme|affine]] [[group scheme|k-group]]. Then the following conditions are equivalent.

1. The [[completion]] of the [[Cartier duality|Cartier dual]] $\hat D(G)$ of $G$ is a connected formal group.

1. Any [[multiplicative group scheme|multiplicative subgroup]] of $G$ is zero.

1. For any subgroup $H$ of $G$ with $H\neq 0$ we have $Gr_k(H,\alpha_k)\neq 0$.

1. Any algebraic quotient of $G$ is an extension of subgroups of $\alpha_k$.

1. (If $p\neq 0)$, $\cap Im V^n_G =e$.
=--

+-- {: .num_defn}
###### Definition and Remark
1. A group satisfying the conditions of the previous theorem is called *[[unipotent k-group]]*.

1. Unipotent groups correspond by duality to connected formal $k$-groups.

1. The category $ACu_k$ of affine commutative unipotent groups form a thick subcategory of $AC_k$ which is stable under limits.
=--

The following theorem is the dual to the theorem of the previos chapter.

+-- {: .num_theorem}
###### Theorem
1. An affine $k$ group is in a unique way an extension of a unipotent group by a multiplicative group.

1. This extension splits if $k$ is perfect.

1. If $k$ is perfect any finite group is uniquely the product of four subgroups which are respectively [[étale group scheme|étale]] multiplicative, étale unipotent, [[infinitesimal group scheme|infinitesimal]] multiplicative and infinitesimal unipotent.

1. The category $F_k$ of finite commutative $k$-groups splits as a product of four subcategories: $Fem_k$, $Feu_k$, $Fim_k$, $Fiu_k$.

1. The categories $Feu_k$ and $Fim_k$ are dual to each other.

1. The categories $Fem_k$ and $Fiu_k$ are selfdual.
=--

+-- {: .num_prop}
###### Proposition
1. Let $p = 0$, then Then $F_k=Fem_k$.

1. Let $p\neq 0$, let $k$ be algebraically closed. Then any commutative finite $k$-group is an extension of copies of $p \alpha_k$, $p \mu_k$ and $(\mathbb{Z}/r\mathbb{Z})_k$ where $r$ is prime.
=--

+-- {: .num_cor}
###### Corollary
If $m$ is a prime and $G$is a finite commutative $k$-group, then$m^\alpha id_G=0$ for large $\alpha$ iff $rk(G)$ is a power of $m$.
=--


Demazure, lectures on p-divisible groups, III.1 the Artin-Hasse exponential series

[[!redirects III.1 the Artin-Hasse exponential series]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Let $p$ be a fixed prime number.

+-- {: .num_defn}
###### Definition
Let $k$ be a ring. Then the assignation

$$\Lambda_k:\begin{cases}
M_k&\to&R[[t]]
\\
R&\to&1+tR[[t]]
\end{cases}$$

sending a [[k-ring]] to the multiplicative group of formal power series with coefficients in $R$ and constant term $1$ is an [[affine k-group]]. As a [[k-functor]] $\Lambda_k$ is equivalent to $O_k^\mathbb{N}$.
=--

+-- {: .num_remark}
###### Remark
1. There is an exact sequence $0\to \Lambda_k^{(n+1)}\to \Lambda_k^{(n)}\to \alpha_k\to 0$

1. We have $\Lambda_k=lim_n \Lambda_k/\Lambda_k^{(n+1)}$.

1. Each $\Lambda_k /\Lambda_k^{(n+1)}$ is is an $n$-fold extension of the additive group $\alpha_k$.

1. If $k$ is a field then $\Lambda_k$ is a unipotent group.
=--

+-- {: .num_remark}
###### Remark
For $F=(1-t+\cdots )\in \Lambda(k)$ there is an isomorphism of $k$-schemes

$$\phi_F:\begin{cases}
O_k^{\mathbb{N}_{\gt 0}}\to \Lambda_k
\\
(a_n)\mapsto \Pi_n F(a_n t^n)
\end{cases}$$

If $k=\mathbb{Q}$ and $F(t)=exp(-t)$ we have $F(at)F(bt)=F(A(a+b)t)$ and $\phi_F$ is an isomorphism $\phi_F:\alpha_k^{\mathbb{N}_{\gt 0}}\to \Lambda_k$.

If $k$ is a field with characteristic $p$ it is not possible an $F$ such that $F(at)F(bt)=F(ct)$. However there is always a formula $F(at)F(bt)=\Pi_{i\gt 0} F(\lambda_i(a,b)t^i$ where $\lambda_i(X,Y)\in k[X, Y]$. This is verified by [[Möbius inversion]].
=--

+-- {: .num_remark}
###### Remark
([[Möbius inversion]])
Let $\mu$ be the Möbius function. Then [[Möbius inversion]] gives
$$exp(-t)=\Pi_n (1-t^n)^{\mu(n)/n}$$
=--

+-- {: .num_defn}
###### Definition and Remark
The *Artin-Hasse exponential* is defined by the morphism

$$E:\begin{cases}
O^\mathbb{N}_{\mathbb{Z}_{(p)}}\to \Lambda_{\mathbb{Z}_{(p)}}
\\
((a_n),t)\mapsto \Pi_{n\ge 0}F(a_n t^{p^n})
\end{cases}$$

We have $E((a_i),t)E((b_i),t)=E((S_i(a_0,\dots,a_i,b_0,\dots,b_i),t)$ where $s_i\in \mathbb{Z}_{(p)}[X_0,\dots, X_i,Y_0,\dots, Y_i]$

$P\in \Lambda(R)$, $R\in M_{\mathbb{Z}_{(p)}}$ can be uniquely written as

$$P(t)=\Pi_{(n,p)=1} E((a_n),t^n)$$

where $(a_n)\in R^\mathbb{N}$
=--

+-- {: .num_prop}
###### Proposition
The $\mathbb{Z}_{(p)}$-group $\Lambda_{\mathbb{Z}_{(p)}}$ is isomorphic to the n/(n,p)=1-power of the subgroup image of $E$.

By base-change a similar statement applies to $\Lambda_{\mathbb{F}_p}$. This shows that the Artin-Hasse exponential plays over $\mathbb{F}_p$ a similar role as the usual exponential over $\mathbb{Q}$.
=--


Demazure, lectures on p-divisible groups, III.2, the Witt rings over Z

[[!redirects III.2, the Witt rings over Z]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

The [[Artin-Hasse exponential series]] $E$ can be written as

$$E((a_0,\cdots),t)=exp(-\Sigma_{n\ge 0}t^{p^n}\Phi_n /p^n)$$

with $\Phi_n(a_0,\cdots)=a_0^{p^n} + p a_1^{p^{n-1}}+\dots p^n a_n$

The $S_i$ in the multiplication formula $E((a_i),t)E((b_i),t)=E((S_i(a_0,\dots,a_i,b_0,\dots,b_i),t)$ where $s_i\in \mathbb{Z}_{(p)}[X_0,\dots, X_i,Y_0,\dots, Y_i]$ of the Artin-Hasse exponential series can also be written as $\Phi_n(a_0,\cdots,a_n)+\Phi_n(d_0,\dots,b_n)=\Phi_n(S_0,\dots,S_n)$

+-- {: .num_lemma}
###### Lemma
We have $S_n\in\mathbb{Z}[X_0,\dots,X_n]$
=--

+-- {: .num_theorem}
###### Definition and Theorem
There exists a unique commutative group law $W$ on the $\mathbb{Z}$-group scheme $W:=O_\mathbb{Z}^\mathbb{N}$ called **$\mathbb{Z}$-group of Witt-vectors of finite length relative to $p$** satisfying the following properties:

1. $E:O_\mathbb{Z}^\mathbb{N}\otimes_\mathbb{Z} \mathbb{Z}_{(p)}\to \Lambda_{\mathbb{Z}_{(p)}}$ is a morphism of $k$-groups.

1. Each $\Phi_n:O_\mathbb{Z}^\mathbb{N}\to \alpha_\mathbb{Z}$  is a morphism of $k$-groups.

1. $(a_n)+(b_n)=(S_n(a_0,\dots, a_n,b_0,\dots,b_n))$

For $w=(a_n)\in W(R)=R^\mathbb{N}$, $a_n$ is called the *$n$-th component of $w$* and $\Phi_n(w)$ is called the *$n$-th phantom component of $w$*.  The phantom components of $w$ define a group isomorphism

$$W\otimes_\mathbb{Z}\mathbb{Z}[p^{-1}]\to \alpha^\mathbb{N}_{\mathbb{Z}[p^{-1}]}$$
=--

+-- {: .num_remark}
###### Remark and Definition
There is an endomorphism of the group of Witt vectors

$$T:\begin{cases}
W\to W
\\
(a_o,\dots,a_n,\dots)\mapsto(0,a_o,\dots,a_n,\dots)
\end{cases}$$
called translation.

We define the group $W_n$ of Witt vectors of length $n$ by $W_n(R):=co ker\, T^n(R)$ or equivalently by the exact sequence

$$\label{W_n}0\to W\stackrel{T^n}{\to}W\stackrel{R_n}{\to}W_n\to 0$$

and we have

1. $(a_0,a_1,\dots)=(a_0,\dots,a_{n-1},0,\dots)\overset{\bullet}{+} T^n(a_n,a_{n+1},\dots)$

1. The underlying scheme of $W_n$ is $O_k^n$, the projection morphism $W/to W_n$ is $(a_0,\dots,)\mapsto(a_0,\dots,a_{n-1}$.

1. The group law on $W_n$ is $(a_0,\dots,a_{n-1})\overset{\bullet}{+}(b_0,\dots,b_{n-1})=(S_0(a_0,b_0),\dots,S_{n-1}(a_0,\dots,a_{n-1},b_0,\dots,b_{n-1}))$

1. In particular $W_1=\alpha$

The [[snake lemma]] gives from diagram (1) translation morphisms $T:W_n\to W_{n+1}$ such that $T(a_0,\dots,a_{n-1})=(0,a_0,\dots,a_{n-1})$, projection morphisms $R_W_{n+1}\to W_n$ such that $R(a_0,\dots,a_n)=(a_0,\dots,a_{n-1})$ and exact sequence

$$0\to W_m\stackrel{T^n}{\to}W_{n+m}\stackrel{R^m}{\to}W_n\to 0$$

and the projections $W\to W_n$ give rise to an isomorphism

$$W\simeq lim_n W_n$$
=--

+-- {: .num_lemma}
###### Definition and Lemma
Let $\iota:\begin{cases}O_\mathbb{Z}\to W\\a\to(a,0,\dots)\end{cases}$ be an inclusion.

Then we have $\Phi_n \iota(a)=a^{p^n}$ and$E(\iota(a),t)=F(at)$).
=--

+-- {: .num_theorem}
###### Theorem and Definition

There is a unique ring-structure on the $\mathbb{Z}$-group $W$ such that either of the two following conditions is satisfied:

1. Each $\Phi_n:W\to O_\mathbb{Z}$ is a ring homomorphism.

1. $\iota(ab)=\iota(a)\iota(b)$, $a,b\in R\in M_\mathbb{Z}$.

The $\mathbb{Z}$-ring $W$ is called **Witt-Ring**, each $W_n$ is a quotient ring of $W$. The canonical morphisms $R_n:W\to W_n$ and $R:W_{n+1}\to W_n$ are ring homomorphisms (but not $T$).
=--


Demazure, lectures on p-divisible groups, III.3, the Witt rings over k

[[!redirects III.3, the Witt rings over k]]
This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

Let $k$ be a field of prime characteristic $p$. Let $W$ denote the [[Witt ring over Z]]

+-- {: .num_defn}
###### Definition
The **Witt ring over $k$** denoted by is defined by the coefficient extension $W_k:=W\otimes_\mathbb{Z} k$. and $W_{nk}:=W_n\otimes_\mathbb{Z}k$

The [[Witt ring over Z|phantom-components]] $W_k\to \alpha_k$ reduce now to $(a_i)_{0\le i\le n}\mapsto a_0^{p^n}$.

Since $W_k=W_{F_p}\otimes_{F_p}k$ we can identify $W_k^{(p)}$, see Definition [[Frobenius morphism]],  and $W_k$ and the [[Frobenius morphism]] becomes the endomorphism

$$F:\begin{cases}W_k\to W_k\\(a_0,\dots,a_n,\dots)\mapsto (a_0^p,\dots,a_n^p,\dots)\end{cases}$$

This is a ring morphism since since $F$ commutes with products. Similar statements are true for $W_{nk}$ and the affine $k$-group $\Lambda_k$ defined in [[Artin-Hasse exponential series]].
=--

+-- {: .num_prop}
###### Proposition
1. The [[Verschiebung morphism]] of $\Lambda_k$ is given by $\phi(t)\to \phi(t^p)$.

1. The Verschiebung morphism of $K_k$ is the [[Witt ring over Z|translation]] $T$.

1. The Verschiebung morphism of $W_nk$ is $R \cdot T=T\cdot R$.

1. If $x,y\in W_k(R)$, $R\in M_k$, then $V(F x\cdot y)=x\cdot V y$.
=--

+-- {: .num_cor}
###### Corollary

=--

+-- {: .num_cor}
###### Corollary

=--

+-- {: .num_cor}
###### Corollary
Let $k$ be perfect. Then

1. $W(k)$ is a discrete valuation ring.

1. $W(k)$ is complete.

1. $W(k)/p W(k)=k$
=--

+-- {: .num_prop}
###### Proposition
(Witt)
Let $k$ be perfect, let $A$ be compete, noetherian local ring with residue field $k$. Let $\pi:A\to k$ be the canonical projection. There exists a unique ring morphism

$$u:W(k)\to A$$

which is compatible with the projections $W(k)\to k$ and $\pi:A\to k$.

If moreover $A$ is a discrete valuation ring with $p\cdot 1_A\not 0$, then $A$ is a free finite $W(k)$-module of rank $[A/p : A]$.

In particular if $pA=A$, then $u$ is an isomorphism.
=--