A procdh topology
Shane Kelly and Shuji Saito
February 20, 2024
Abstract
In this article we propose a definition of a procdh topos. We show that it encodes
procdh excision, has bounded homotopy dimension and therefore is hypercomplete andadmits a conservative family of fibre functors. We also describe the local rings.
As an application, we show that nonconnective K-theory is the procdh sheafificationof connective K-theory, and that the motivic cohomology recently proposed by Elmanto
and Morrow is the procdh sheafification of Voevodsky’s motivic cohomology.

1Introduction

It has been known for many decades that blowups of smooth varieties in smooth centres giverise to a long exact sequence in algebraic K-theory (SGA71, Exp.VII), (TT90). As part ofhis work on the Bloch-Kato conjecture, Voevodsky encoded such blowup exact sequences ina Grothendieck topology—the cdh topology—giving access to the full arsenal of topos theory,(SV00).
For varieties which are not necessarily smooth, the picture has not been so complete. Algebraic K-theory fails to associate long exact sequences to blowups in general, as easy examples
show. The failure is in some sense a “quasi-coherent” part of algebraic K-theory. Indeed,
quasi-coherent cohomology also fails to have long exact sequences for blowups. On the other
hand, if one remembers infinitesimal information around the centre, one does get long exact sequences for quasi-coherent cohomology. This is described in Grothendieck’s theorem on formal
functions, (Gro61, Thm.4.1.5).
The analogous formal blowup sequences for algebraic K-theory have been studied for some
20 years now, (Wei01, §3-4), (KS02), (GH06), (GH11), (Mor18), and feature in Kerz, Strunk,
Tamme’s celebrated proof of Weibel’s conjecture, (KST18a); see (Mor16a) for a historical
overview. However, the absence of an associated topology has meant that the topos theoretic
techniques so skillfully employed by Voevodsky et al. were not available.
In this article we propose the following definition for a procdh topos.
Definition 1.1 (Definition 2.1). Let S be a qcqs scheme. The procdh topology on the category
SchS of S-schemes of finite presentation is generated by the following coverings.
1. Distinguished Nisnevich coverings: families of the formji

{U → X, V → X}such that i is a quasi-compact open immersion, j is an étale morphism, and j is an
isomorphism over X\U equipped with any (equivalently all) closed subscheme structure(s)of finite presentation.

12. Proabstract blowup squares: families of the form
{Zn → X}n∈N ⊔ {Y → X}
where Y → X is a proper morphism of finite presentation which is an isomorphism outsideof a closed subscheme Z0 ⊆ X of finite presentation, and Zn = Spec OX /IZn is the nth
infinitesimal thickening of Z0 .
A sign that this is a “correct” topology is that it captures procdh excision: presheaves withthe long exact sequences mentioned above are precisely the procdh sheaves.
Theorem 1.2 (Theorem 6.1, Corollary 7.11). Let S be a scheme and consider the following
conditions on a presheaf of spaces F ∈ PSh(SchS , S).1 .
1. Excision. For every distighished Nisnevich square {U → X, V → X} and every proabstract blowup square {Zn → X}n∈N ⊔ {Y → X} in SchS , Def.2.1, we have
∼

(1)

∼

(2)

F (X) → F (U ) ×F (U ×X V ) F (V ),F (X) → F (Y ) ×limn F (Zn ×X Y ) limn F (Zn ).
2. Čech descent. For every procdh covering family {Yλ → X}λ∈Λ we have∼

F (X) → lim F (Yn )
n

(3)

󰁔
where we write F (Yn ) for i∈I n F (Yi1 ×X · · · ×X Fin ). That is, in the terminology of(Lur09), F is a procdh sheaf.
3. Hyperdescent. For every X ∈ SchS and procdh hypercovering Y• → X we have∼

F (X) → lim Map(Yn , F ).
n

(4)

In the terminology of (Lur09), F is a hypercomplete procdh sheaf.
If S is qcqs then (Excision) ⇔ (Čech descent). If S is qcqs, has finite valuative dimension andNoetherian topological space, then (Čech descent) ⇔ (Hyperdescent).
Our procdh ∞-topoi have finite homotopy dimension, albeit not quite as optimal as one
could hope.
Theorem 1.3 (Theorem 7.9, Example 7.17). Let S be a qcqs scheme of finite valuative dimension d ≥ 0 with Noetherian underlying topological space. Then Shvprocdh (SchS , S) has homotopy
dimension ≤ 2d.
There exists a Noetherian scheme of Krull dimension one with procdh homotopy dimension
two.
The appearance of 2d instead of d is essentially because the topos PSh(N, S) has homotopy
dimension one. So every time we perform a limN , for example in Eq.(2) above, we potentially
increase the homotopy dimension by one. In a future article we consider a way around thisusing a site involving formal schemes.
Despite the unwanted factor of two, finite homotopy dimension is suﬃcient to imply thatthe topos is hypercomplete.
1 As usual, S := N Kan is the quasicategory associated to the simplicial category of Kan complexes, andPSh(SchS , S) is the quasicategory Fun(Schop
S , S). Before (Lur09), the category PSh(SchS , S) would have been
the category PSh(SchS , Set∆ ) of presheaves of simplicial sets with any model structure for which weak equivalences are objectwise weak equivalences. If one reads the statements using this latter interpretation, all limits
should be replaced with homotopy limits as described in (BK72) or (Hir03). Since homotopy limits calculatequasicategorical limits, the diﬀerence between the two points of view is superficial.

2

Corollary 1.4 (Corollary 7.11). Let S be a qcqs scheme of finite valuative dimension with
Noetherian underlying topological space. Then Shvprocdh (SchS , S) is hypercomplete.
Closely related to hypercompleteness is existence of enough points. Recall that a topos,
resp. ∞-topos T , is said to have enough points when the collection of all geometric morphisms
of topoi φ∗ : T → Set, resp. ∞-topoi φ∗ : T → S, detect isomorphisms, resp. equivalences.
Theorem 1.5 (Theorem 5.2, Corollary 7.15). Suppose S is a qcqs scheme with Noetherian
topological space of finite Krull dimension. Then both the classical topos Shvprocdh (SchS ), andthe ∞-topos Shvprocdh (SchS , S) have enough points.
We prove the classical version of Theorem 1.5 first and then upgrade it to the ∞-version.
Something worth noting is that since the generating covering families of the procdh topology are
not finite, the topos is not coherent, so we cannot simply apply Deligne’s completeness theorem.
We use the following proposition for the upgrade. For sites whose associated topos is coherent,
Deligne’s theorem combined with Prop.7.14 implies Lurie’s version, (Lur18, Thm.A.4.0.5).
Proposition 1.6 (Proposition 7.14). Let (C, τ ) be a small site admitting finite limits suchthat the topos Shv(C, Set) has enough points in the classical sense. Then the hypercompletion
Shv(C, S)∧ has enough points as an ∞-topos.
After all this talk of procdh points, the reader is surely asking for a characterisation of thelocal rings. Procdh local rings are essentially versions of cdh local rings—henselian valuation
rings—with some mild nilpotent thickening. Cf. (Kel24) for some philosophy motivating this.
Proposition 1.7 (Characterisation of procdh local rings). An aﬃne S-scheme Spec(R) → S
is procdh local, Def.3.2, if and only if it is of the formR∼
= O ×K A
with A a local ring of Krull dimension zero and O ⊆ K a henselian valuation ring of K=A/mA .
In other words, R ⊆ A is the set of elements whose residue mod mA lies in O ⊆ K = A/mA .
We also discuss how a general procdh local ring can often be written as a filtered colimit ofsmaller local rings, see Proposition 3.6 and Theorem 3.10.
In the latter sections, we apply the above theory to show the following topos-theoreticinterpretation of the Bass construction.
Theorem 1.8 (Theorem 8.1). For any Noetherian scheme X with dim(X) < ∞, there exists
a natural equivalence(aprocdh τ≥0 K)(X) ≃ K(X).
Here K(X) is the non-connective algebraic K-theory of X and τ≥0 K(X) the connective Ktheory.
The above equivalence is an analogue of the equivalence2(acdh τ≥0 )K(X) ≃ KH(X)(5)

with X still finite dimensional and Noetherian.
2 Admitting that KH satisfies cdh descent, (Hae04), (Cis13, Thm.3.9), a fast, clean, intuitive way to deduce
Eq.(5) is to prove that K≥0 (R) ∼= K(R) ∼= KH(R) for valuation rings, (KM21, Thm.1.3), cf.(KM21, Rem.3.4).
Equation (5) for finite dimensional Noetherian schemes is also proven in the landmark Kerz-Strunk-Tammepaper (KST18a, Thm.6.3) using derived algebraic geometry.

3

Using Theorem 8.1 we propose the following construction of a non-A1 -invariant motivicncohomology. For X smooth over a field F, let F ilmot
K(X) denote the motivic filtration on Ktheory whose associated spectral sequence is the Atiyah-Hirzebruch spectral sequence, (FS02),(Lev08),p−q
E2p,q = HM
(X, Z(−q)) ⇒ K−p−q (X).
mHere HM(X, Z(n)) = Rm ΓZar (X, Z(n)sm ) with Z(n)sm ∈ PSh(SmF , D(Z)) the graded piecesnof F ilmot K on the category SmF of smooth F-schemes. Note that Z(n)sm is identified with
Voevodsky’s A1 -invariant motivic complex and Bloch’s cycle complex.

Definition 1.9 (Definition 9.3). For integers n ≥ 0, we define the procdh local motivic complex
Z(n)procdh := aprocdh Lsm Z(n)sm ∈ Shvprocdh (Schqcqs, D(Z))),
F
as the procdh sheafification of the left Kan extension Lsm Z(n)sm of Z(n)sm along SmF → Schqcqs.
F
Here Schqcqs
is the category of qcqs F-schemes.
F
It follows essentially from the definition, together with a result of Bhatt-Lurie and the factthat the procdh site of a Noetherian scheme has finite cohomological dimension that we get an
Atiyah-Hirzebruch spectral sequence.
Theorem 1.10 (Theorem 9.5). For any Noetherian F-scheme󰀝 X with dim(X)󰀞 < ∞, there
exists a complete multiplicative decreasing N-indexed filtrationn
F ilprocdh
K(X)

on K(X)n∈N

and identifications

grnF ilprocdh K(X) ≃ Z(n)procdh (X)[2n].
Recently, in (EM23) Elmanto-Morrow have also proposed a non-A1 -invariant motivic cohomology for qcqs F-schemes. We will write Z(n)EM for these presheaves. They are constructed by modifying the cdh sheafification Z(n)cdh of the left Kan extension of Z(n)sm along
SmF → Schqcqs
by using Hodge-completed derived de Rham complexes in case F = Q and synF
tomic complexes in case F = Fp . The construction is motivated by trace methods in algebraic
K-theory using the cyclotomic trace map.
We conclude the article with the following comparison.
Corollary 1.11 (Corollary 9.9). Let F = Fp or Q and take a Noetherian F-scheme X. Then
there are equivalences
Z(n)procdh (X) ≃ Z(n)EM (X),functorial in X.

1.1

Future work

As mentioned above, a version of the material in this article for formal schemes is in progress.
We also have a version for derived schemes.

1.2

Notation and conventions

Throughout we write SchS for the category of S-schemes of finite presentation over a scheme S.
We write X gen for the set of generic points of a scheme X equipped with the topology induced
from the underlying topological space of X.
The first sections deal exclusively with presheaves of sets. When we pass to presheaves ofspaces we will write PSh(−, S), PSh(−, Spt), PSh(−, D(Z)), etc, and sometimes use PSh(−, Set)
if we want to emphasise that a presheaf takes values in discrete spaces.
4

1.3

Acknowledgements

We are very grateful to Matthew Morrow for many interesting questions and discussions about
the procdh topology and particularly for a very active scientific exchange without which thecomparison theorem would not have been possible.
We thank Marc Hoyois for answering the first author’s annoying emails about infinity categories.
We also thank Dustin Clausen and Ryomei Iwasa for their encouraging interest in the
project.
Much of the research in this article was conducted while the first author was Principal
Investigator of JSPS Grant-in-Aid for Early-Career Scientists, Project Number 19K14498 and
Bilateral Joint Research Projects (120213206). The second author is Principal Investigator of
JSPS Grant-in-Aid for Scientific Research (B), Project Number 20H01791.

2Definition of a procdh topology

In this section we define a procdh topology.
Definition 2.1. Let S be a scheme. The procdh topology on SchS is generated by the following
coverings.
0. Zariski coverings.
1. Distinguished Nisnevich coverings: families of the formji

{U → X, V → X}such that i is a quasi-compact open immersion, j is an étale morphism, and j is an
isomorphism over X\U equipped with any (equivalently all) closed subscheme structure(s)of finite presentation.
2. Proabstract blowup squares: families of the form
{Zn → X}n∈N ⊔ {Y → X}
where Y → X is a proper morphism (of finite presentation) which is an isomorphism
outside of a closed subscheme Z0 ⊆ X of finite presentation, and Zn = Spec OX /IZn isthe nth infinitesimal thickening of Z0 .
Remark 2.2. More precisely, a family Y = {Yi → X}i∈I in SchS is a procdh covering if there
exists a rooted tree T of finite height and a functor T → SchS , such that the root is sent to X,
for each vertex v the family {Wc → Wv }c∈Child(v) is of one of the above forms, and for eachleaf l there exists a factorisation Wl → Yil → X for some il ∈ I.

3

Local rings

In this section we characterise procdh local rings, Prop.3.3. We also discuss how a general
procdh local ring can often be written as a filtered colimit of smaller local rings, Prop.3.6,
Theo.3.10. The latter is used in the proof of the comparison theorem, Corollary 9.9.
Definition 3.1. Recall that a fibre functor of a topos Shvτ (C) is a continuous morphism of
topoi φ∗ : Shvτ (C) ⇄ Set : φ∗ , or equivalently, a functor φ∗ : Shvτ (C) → Set which preserves
colimits and finite limits.
5For topologies τ on SchS for which every scheme is covered by aﬃne ones (e.g., anything
finer than the Zariski topology), there is a bijection between fibre functors of Shvτ (SchS ) andaﬃne S-schemes Spec(R) → S which are τ -local in the following sense.
Definition 3.2 (cf.(SGA72a, Thm.III.4.1), (SGA72a, I.8.10.14), (Gro66, Cor.8.13.2)). Let τ
be a topology on SchS such that every scheme is covered by aﬃne ones. An aﬃne S-scheme
Spec(R) → S is said to be τ -local if for every τ -covering {Yi → X}i∈I the morphism of sets󰁤hom(Spec(R), Yi ) → hom(Spec(R), X)(6)i∈I

is surjective.
Proposition 3.3 (Characterisation of procdh local rings). An aﬃne S-scheme Spec(R) → S
is procdh local, Def.3.2, if and only if it is of the formR∼
= O ×K A
with A a local ring of Krull dimension zero and O ⊆ K a henselian valuation ring of K=A/mA .
In other words, R ⊆ A is the set of elements whose residue mod mA lies in O.
Proof of Proposition 3.3. (⇒) The following composition of a Zariski covering and the procdh
covering󰀝󰀞󰀝󰀞OS [x, y]Spec n n⊔ Spec OS [x, xy ], Spec OS [ xy , y](7)〈x , y 〉 n∈Nshows that procdh local rings satisfy:(∗) ∀a, b ∈ R; we have a|b or b|a or a and b are both nilpotent.
It follows from this that Rred is a valuation ring, and in particular, R has a unique minimal
prime ideal n, which equals the set of nilpotents. All zero divisors are nilpotent by virtue ofthe procdh covering󰀝󰀞󰀝󰀞OS [x, y]OS [x, y]Spec n⊔ Spec(8)〈x , xy〉 n∈N〈y〉
S [x,y]of Spec O〈xy〉, so R → Rn , and therefore R → (R/n) ×k(n) Rn is injective. We claim that thelatter is also surjective. It follows from a diagram chase that n → nRn is surjective implies
R → (R/n) ×k(n) Rn is surjective.

0

󰈣n

󰈣R

󰈣 R/n

󰈣0

0

󰈃
󰈣 nRn

󰈃
󰈣 Rn

󰈃
󰈣 Rn /nRn󰈣0(9)

So suppose we have a ∈ n and s ∈ R \ n. We claim there is b ∈ n such that b/1 = a/s. Indeed,this follows from a ∈ n, s ∈ R \ n, and (∗).
So we have shown R → (R/n)×k(n) Rn is both injective and surjective. The Krull dimensionof Rn is zero because n is a minimal prime, and we have already observed that R/n = Rredis a valuation ring, so it suﬃces to show that R/n is henselian. But procdh local rings are
Nisnevich local rings, also known as henselian local rings, and quotients of henselian local rings
are henselian local rings.
(⇐). Suppose R = O×K A as in the statement. We want to show that (6) is an epimorphism
for all procdh coverings. Certainly it suﬃces to consider the generator coverings described in
the definition.
6

We immediately notice that R is henselian: O = Rred is henselian by assumption and∼− ⊗R (Rred ) induces an equivalence EtR → EtRred of categories of étale algebras, (Sta18, 039R).
So the desired lifting condition with respect to Nisnevich coverings is satisfied, cf.(Sta18, 04GG,
Item(7)).
Suppose we have a proabstract blowup square {Zn → X}n∈N ⊔ {Y → X} and a morphism
Spec(R) → X. If the generic point Spec(K) of Spec(R) doesn’t land in Z0 , then it lifts throughY because Y → X is an isomorphism over X \ Z0 . By the valuative criterion for properness,this lifting extends to a lifting of Spec(O). Since A is local the morphism Spec(K) → Y also
extends to Spec(A) → Y . These three morphisms factor through some open aﬃne of Y , sothey glue to give a lifting Spec(R) → Y → X since Spec(O ×K A) = Spec(O) ⊔Spec(K) Spec(A)is the categorical pushout in the category of aﬃne schemes.
On the other hand, if Spec(K) → X does factor through Z0 , then Spec(O) → X also
factors through Z0 . The morphism Spec(A) → X doesn’t necessarily factor through Z0 but the(finitely many) generators of IZ0 are sent inside mA = Nilpotents(A) so Spec(A) → X factors
through some Zn . Then we glue as in the previous case.
General procdh local rings can be quite large, but they are often filtered colimits of smaller
local rings. The rest of this section is devoted to such reductions and some consequences, forexample Corollary 5.3. We will use the following non-Nisnevich version of procdh local rings.
Definition 3.4. A prorh local ring are rings of the form O×K A with A a local ring of dimension
zero, and O ⊆ K a (not necessarily henselian) valuation ring of K = A/m.
One way to construct procdh local rings is to first build a prorh local ring, and then takethe henselisation.
We write Q(R) for the ring of total fractions. That is, Q(R) = R[S −1 ] where S is the set ofnonzero divisors.
Proposition 3.5. Suppose that R is a prorh local ring. That is, R = O ×K A with O a (notnecessarily henselian) valuation ring, A a dimension zero local ring, and K = Frac(O) ∼= A/m.
Let R → S be an étale morphism towards a local ring S.
Then S is prorh local, and the henselisation Rh is a procdh local ring. Moreover, in
this situation we have dim S ≤ dim R, length Q(R) = length Q(S) and dim Rh = dim R,
length Q(R) = length Q(Rh ).
Proof. The canonical morphism S → (S ⊗R O) ×(S⊗R K) (S ⊗R A) is an isomorphism because
R → S is flat, (Fer03, Thm.2.2(iv)). Since S is local, S ⊗R O = S ⊗R (Rred ) is also local,
and therefore a valuation ring of Krull dimension ≤ dim O, (Sta18, 0ASJ). It follows that theétale K-algebra S ⊗R K is actually a finite separable extension of K, and since (S ⊗R A)red =(S ⊗R (Ared ))red = (S ⊗R K)red , we find that S ⊗R A has exactly one prime ideal. Since
K → S⊗R K is a field extension and R → S is flat, for any composition series A ⊃ I0 ⊃ I1 ⊃ . . . ,the pullback S ⊗R A ⊃ S ⊗R I0 ⊃ S ⊗R I1 ⊃ . . . is a composition series for S ⊗R A. So
length A = length S ⊗R A.
In the case that R → S is a local homomorphism, Rred → Sred is also local, so it induces abijection of value groups of these valuation rings, (Sta18, 0ASF), and therefore a bijection of
posets of prime ideals. Consequently, dim R = dim Rred = dim Sred = dim S. Now the henselisation Rh is the colimit colim Sλ over étale algebras R → Sλ which are local homomorphisms of
local rings. We have just seen that all transition morphisms Sλ → Sµ induce homeomorphisms
on Spec, so we deduce that the prime ideals of Rh are in bijection with the prime ideals of R.
Therefore, dim Rh = dim R. As above, since K → Rh ⊗R K is a field extension and R → Rh
is flat, pullback preserves composition series so length A = length Rh ⊗R A.
7

It remains to show that Rh is procdh local. We have (Rh )red = (colim Sλ )red = colim Sλ,redis a filtered colimit of valuation rings and therefore a valuation ring. It follows that nh :=Nil(Rh ) is a unique minimal prime ideal of Rh , and also that Rh /nh is a henselian valuation
ring, since quotients of henselian rings are henselian. To conclude it suﬃces to show that
Rh → (Rh /nh ) ×k(nh ) (Rnhh ) is an isomorphism. This follows the fact that filtered colimits
commute with fibre products.
Proposition 3.6. If S is a Noetherian scheme (e.g., the spectrum of Z or a field) andSpec(R) → S an aﬃne S-scheme such that R is procdh local, then R is a filtered colimitof procdh local OS -algebras such that each length Q(Rλ ) is finite. If S has finite valuative
dimension, Def.7.1, then we can also assume each Rλ has finite Krull dimension.
Proof. We immediately observe that since R is local, we can assume S is aﬃne, say S =Spec(B0 ). Then the idea is that for any factorisation B0 → B → R, we can convert B into
a procdh local ring in a way which is functorial in B. If we write R as a filtered colimit
R = colim Bλ of B0 -algebras Bλ of finite presentation, we can use this fact to functoriallyconvert each Bλ into a procdh local ring Rλ , giving the expression R = colim Rλ . To conclude
we observe that the Rλ have the properties described in the statement if B0 is Noetherian,
resp. of finite valuative dimension.
Now we carry out this plan. Let B0 → Bλ → R be any factorisation, and consider the
preimage pλ ⊆ Bλ of the minimal prime n of R. The valuation ring R/n of R induces andfactors through a valuation ring Oλ = k(pλ ) ∩ (R/n) of k(pλ ), and the localisation Rn of R
factors through the localisation (Bλ )pλ . Form the epi-monic factorisation (Bλ )pλ →→ Aλ ↩→ Rn .
As nRn = Nil(Rn ), the preimage of this in Aλ also consists entirely of nilpotents. Since thisideal of nilpotents is the maximal ideal of Aλ , it follows that Aλ is a dimension zero local ring.
Glueing, we get a prorh local ring Pλ := Oλ ×k(pλ ) Aλ and a commutative diagram
Bλ ❖❖
❖❖❖
❖❖❖
󰈛
󰈃
(Bλ )pλ

❖❖❖
❖❖❖
❖󰈛

Pλ =

Bλ /pλ󰈃k(pλ )󰈣 Oλ ×k(p ) Aλ
λ
◗◗◗
◗◗◗
◗◗◗
󰈜󰈣
󰈃
󰈣 Aλ ◗
◗◗◗
◗◗◗
◗◗◗
◗󰈜

Oλ

󰈣R❏
❏❏❏
❏❏❏
󰈘
󰈣 R/n
󰈃
󰈣 Rn

󰈃
k(pλ )

❏❏
❏❏
❏❏
󰈘 󰈃
󰈣 k(n)

To conclude we take Rλ = Pλh to be the henselisation, which is a procdh local ring by Proposition 3.5.
One checks that the choice of pλ =preimage(n) ⊆ Bλ , and the constructions of k(pλ ),Oλ = k(pλ ) ∩ (R/n), (Bλ )pλ , Aλ =image((Bλ )pλ → Rn ), Pλ = Oλ ×k(pλ ) Aλ , and Rλ = Pλh arefunctorial in Bλ . Consider the system of factorisations B0 → Bλ → R with B0 → Bλ of finite
presentation. Note, colim Bλ = R. We get a factorisation
colim Bλ → colim Pλ → colim Rλ → R.
The composition is an isomorphism by construction. The maps Aλ → Rn and Oλ → R/nare injective by construction, so each Pλ = Oλ ×k(p) Aλ → R is injective, so colim Pλ → R
is injective, and therefore also an isomorphism. Finally, henselisation commutes with filteredcolimits of local rings with local transition homomorphisms, (Sta18, 07RP). So colim Rλ =colim Pλh = (colim Pλ )h = Rh = R.
8

If B0 is Noetherian, then each Bλ is also Noetherian, (Bλ )pλ is Noetherian, and the quotient Aλ = Q(Pλ ) is Noetherian, hence, of finite length. Included in Proposition 3.5 is that
length Q(Pλ ) = length Q(Rλ ).
If B0 has finite valuative dimension, then so does each Bλ and Bλ /pλ so dim Oλ = dim Pλis finite. By Proposition 3.5 dim Pλ = dim Rλ .
Sometimes finite length Q(R) is not strong enough, and we would like Nil(R) to be finitely
generated. This is not achievable for general procdh local rings.
Example 3.7. Let R be a procdh local ring such that n = Nil(R) is finitely generated. Then
R = Rn . That is, R is an local ring of Krull dimension zero.
As n is finitely generated, there is some n such that nn+1 = 0 and nn ∕= 0.3 Since R =(R/n) ×k(n) (Rn ) is defined by a Milnor square, the map R → Rn induces an identification of
non-unital rings n ∼= nRn , and consequently, an identification nn ∼= (nRn )n . Since nn+1 (andn+1nntherefore (nRn )) is zero, n (resp. (nRn ) ) is an R/n-module (resp. k(n)-vector space).
Moreover, since n is a finitely generated R-module, so is nn . Changing notation to O = R/n
and M = nn to make the consequences of this clearer, we have a finitely generated nonzero
module M over a valuation ring O such that M → M ⊗O Frac(O) is an isomorphism. Thisis only possible if O = Frac(O).4 Returning to the previous notation, we have shown that
R/n = k(n), and therefore R = (R/n) ×k(n) (Rn ) = Rn .
Now we start working towards Theorem 3.10 which says that we can get to finitely generated
nilradical if we relax the conditions on the nilpotents slighty. Theorem 3.10 is used in the proofof the comparison theorem, Corollary 9.9.
Lemma 3.8. Suppose R is a procdh local ring containing a field F such that Q(R)red /F isseparably generated. Then the canonical projection R → Rred admits a section in the categoryof F-algebras.
This essentially follows from Q(R)red /F being formally étale, and Q(R) → Q(R)red being anilpotent thickening, but we couldn’t find a reference with exactly the statement we wanted.
Proof. Since R = O ×K A (with O = Rred , K = Q(R)red , A = Q(R)), to get a factorisation
O 󰃚󰃚󰃄 R → O of the identity (in the category of F-algebras), it is equivalent to find a factorisation O 󰃚󰃚󰃄 A → K of the canonical inclusion O ⊆ K (in the category of F-algebras).
Furthermore, since K = O[(O \ {0})−1 ] and an element of A is a unit if and only if its image in
K is nonzero, giving such a factorisation is equivalent to giving a factorisation K 󰃚󰃚󰃄 A → K
of the identity (in the category of F-algebras). That is, it suﬃces to solve the lifting problem
K󰉃 󰉣
󰉲⑦

⑦

⑦

F1
⑦ 󰉃
F0

A󰉣

with F0 = F and F1 = K. We have assumed K/F is separably generated, so by assumption
there are elements {xλ ∈ K}λ∈Λ indexed by a well-ordered set Λ such that for every successor
λ < λ + 1 in Λ the element xλ+1 is transcendental or finite separable over the subfield K(xµ :3 We allow n = 0, in which case n0 = R.
4 For valuation rings we have torsion free ⇔ flat, for integral domains we have finitely generated flat ⇔ finitely
generated projective, and for local rings we have projective ⇔ free, so the isomorphism in question is isomorphic
to the canonical inclusion O ⊕r ⊆ Frac(O)⊕r for some r > 0.

9µ ≤ λ). So by transfinite induction, we can assume that the field extension F1 /F0 is generated
by a single element x ∈ F1 which is transcendental or finite separable.
Suppose that F1 = F0 (x) with x transcendental. Choosing any element of A in the preimageof x, we get an induced factorisation F0 [x] 󰃚󰃚󰃄 A → K. Since all non-zero polynomials f (x) ∈F0 [x] are non-zero in F0 (x) ⊆ K = A/m, they are sent to units in A, so we get a factorisation
F0 (x) 󰃚󰃚󰃄 A → K.
Now suppose x is finite separable with minimal polynomial f (X) ∈ F0 [X], so F1 ∼= F0 [X]/〈f (X)〉.
Choose a lift a ∈ A of x ∈ F1 ⊆ K and note that f (a) ∈ m since f (x) = 0. Since m = Nil(A),we have f (a)n = 0 for some n, so we get an induced commutative diagramK󰉃 󰉣

F0 [X]/〈f (X)〉
󰉃
φ

A󰉣

󰂵
󰂵󰂵

󰂵 F󰉃 1󰉫󰂵 󰂵
F0 [X]/〈f (X)n 〉 󰉣ιF0with unique diagonal since ι is étale, and ker(φ) is a nilpotent ideal, (Sta18, 02HM).
Lemma 3.9. Suppose F is any ring and R is any F-algebra such that R → Rred admits a
section in the category of F-algebras. Then we can write R as a filtered colimit R = colim Rλof sub-F-algebras Rλ ⊆ R such that (Rλ )red ∼= Rred but Nil(Rλ ) is finitely generated for all λ.
Proof. Given a set of elements S ⊆ Nil(R), write Rred [S] ⊆ R for the sub-Rred -module generatedmn
1 m2
by the monomials xm
i1 xi2 . . . xin for xi ∈ S. Since elements of Nil(R) are nilpotent, if S isfinite, Nil(Rred [S]) is a finite Rred -module. As Sλ ranges over all finite subsets of Nil(R), we
obtain R = colim Rλ with Rλ = Rred [Sλ ] as desired.
Theorem 3.10. Every procdh local ring over a perfect field F is a filtered colimit of F-algebras
Rλ (not necessarily procdh local rings) such that Nil(Rλ ) is finitely generated and (Rλ )red is a
finite rank henselian valuation ring.
Proof. We can assume our initial ring R has finite Krull dimension by Prop.3.6. The map R →Rred admits a section (as F-algebras) by Lem.3.8, so then the result follows from Lem.3.9.

4

Nisnevich-Riemann-Zariski spaces

In this section we consider a Nisnevich version RZ(XNis ) of the Riemann-Zariski space associated to a scheme X. These can be considered as small sites, cf.(ILO14, Exposé II). The main
result of this section is Corollary 4.13 which says that for each X ∈ SchS , the canonical comparison functor Shvprocdh (SchS ) → Shv(RZ(XNis )) preserves colimits and finite limits, at least
if S is a qcqs scheme with Noetherian topological space. This result will be used in Section 5 toshow that the topoi have enough points, and in Section 7 to show that the homotopy dimension
is finite.
Definition 4.1. By modification we will mean a morphism of schemes Y → X which isproper, of finite presentation, and an isomorphism over a dense qc open D ⊆ X. We write
ModX ⊆ SchX for the full subcategory of modifications.
Remark 4.2. If X is qcqs and Y ′ → Y , Y ′′ → Y are morphisms in ModX then Y ′ ×Y Y ′′ isagain in ModX . In particular, ModX admits finite limits, calculated in SchX , and is therefore
is filtered.

10

We do not ask modifications to be birational so that finite limits in ModX are more nicely
behaved. We can of course often refine any object in ModX by one which is birational to X,
Lem.4.14(1).
Definition 4.3. Let S be a qcqs scheme. For X ∈ SchS we define󰁝
RZ(XNis ) =YNis .
Y ∈ModX

Explicitly, RZ(XNis ) ⊆ Arr(SchX ) is the category whose objects are morphisms U → Y suchthat U ∈ YNis and Y ∈ ModX , and morphisms are commutative squares
U′

󰈣U

󰈃
Y′

󰈃
󰈣Y

We abbreviate U → Y to (U/Y ).
Remark 4.4. As it is a category of arrows in a category admitting finite limits, Arr(SchX )admits finite limits and they are calculated component wise: lim(Ai /Bi ) = (lim Ai / lim Bi ). If
each (Ai /Bi ) is in RZ(XNis ), then one checks that lim(Ai /Bi ) is again in RZ(XNis ).5 That is
RZ(XNis ) admits finite limits, and they are calculated termwise.
Definition 4.5. The category RZ(XNis ) is canonically equipped with the Grothendieck topology generated by:
1. families of the form{(Ui /Y ) → (U/Y )}i∈I(Nis)such that {Ui → U } is a Nisnevich covering, and
2. families of the form{(Y ′ ×Y U/Y ′ ) → (U/Y )}

(Car)

′

for morphisms Y → Y in ModX .
We will write Shv(RZ(XNis )) for the topos associated to the topology generated by coveringsof the form (Nis) and (Car).
Remark 4.6. Since the diagonal of a modification is again a modification, a presheaf (of sets)satisfies descent for all families (Car) if and only if it sends each (Y ′ ×Y U/Y ′ ) → (U/Y ) to an
isomorphism. Consequently,
Shvcar (RZ(XNis )) =lim

Y ∈ModX

PSh(YNis ),′where the limit is along pushforwards f∗ : PSh(YNis) → PSh(YNis ) for morphisms f : Y ′ → Y
in ModX . This implies

Shv(RZ(XNis )) =lim

Y ∈ModX

ShvNis (YNis ).

(10)

5 We observed lim B is in Modi
X in Remark 4.2. The fastest way to check that lim Ai → lim Bi is étale, is
probably to observe that it is formally étale and of finite presentation, (Sta18, 02HG, 00UR).

11

Remark 4.7. The same is true for presheaves of spaces (see Section 6 for conventions). Supposethat F satisfies descent for families of the form (Car). If Y ′ → Y in ModX is a closed immersion
then (Y ′ )×Y n = Y ′ so for any (U/Y ) in RZ(XNis ) we have
F (U/Y ) = lim F ((U ′ /Y ′ )×(U/Y ) (n+1) ) = lim F (U ′ /Y ′ ) = F (U ′ /Y ′ )
n

n

(11)

where U ′ = Y ′ ×Y U . It follows that (11) also holds for a general Y ′ → Y in ModX , since eachdiagonal Y ′ → (Y ′ )×Y n is a closed immersion in ModX . Conversely, if F sends families of theform (Car) to equivalences, then it clearly satisfies Čech descent for such families.
Proposition 4.8. Let X be a qcqs scheme and suppose F ∈ PSh(RZ(XNis )) has descent forthe coverings (Nis). Then the sheafification aF ∈ Shv(RZ(XNis )) satisfies
aF (U/Y ) = colimY ′ ∈(ModX )/Y F (Y ′ ×Y U/Y ′ ).

(12)

The same is true for presheaves of spaces.
Proof. First we show that the presheaf aF defined via Eq.(12) is a sheaf. By definition, apresheaf on RZ(XNis ) is a sheaf if and only if it has descent for coverings of the form (Nis)and (Car) in Definition 4.5. The presheaf aF in the statement certainly sends modifications
to isomorphisms, resp. equivalences, so it has descent for coverings of the form (Car) by
Remark 4.6 and Remark 4.7.
For Nisnevich coverings, we notice that a presheaf F has descent for coverings of the form
(Nis) if and only if the restriction to the small Nisnevich site YNis for each Y ∈ ModX has
Nisnevich descent if and only if it sends distinguished Nisnevich squares to cartesian squares.
If {U0 → U, U1 → U } is a distinguished Nisnevich square, and (U/Y ) ∈ RZ(XNis ) then for any
Y ′ → Y in ModX we have
F (Y ′ ×Y U/Y ′ ) = F (Y ′ ×Y U0 /Y ′ ) ×F (Y ′ ×Y U01 /Y ′ ) F (Y ′ ×Y U1 /Y ′ )where U01 = U0 ×U U1 by the assumption that F has descent for (Nis). Taking the colimit overY ′ and using the fact that filtered colimits commute with fibre products we findaF (U/Y ) = aF (U0 /Y ) ×aF (U01 /Y ) aF (U1 /Y ).
So aF is a sheaf. To conclude that a is the sheafification functor, it suﬃces to show that
if F is already a sheaf, then F → aF is an equivalence. But this is clear, since sheaves send
modifications to isomorphisms, Rem.4.6, resp. equivalences, Rem.4.7.
Definition 4.9. Let S be a qcqs scheme. For X ∈ SchS , we consider the canonical projection
functorρX : RZ(XNis ) → SchS ; (U/Y ) 󰀁→ U
and the functor induced by composition
PSh(SchS ) → PSh(RZ(XNis ));F 󰀁→ F ◦ ρX .

By composing this with the sheafification functor PSh(RZ(XNis )) → Shv(RZ(XNis )), we getρ∗X : Shvprocdh (SchS ) → Shv(RZ(XNis )).
Remark 4.10. Using Proposition 4.8 we have the following concrete description.
(ρ∗X F )(U/Y ) = colimY ′ ∈ModX F (Y ′ ×X U ).
12

(13)

Recall that a morphism of sites φ : (C, τ ) → (D, υ) is cocontinuous if for every U ∈ C
and covering family U = {Ui → φU }i∈I there is a covering family {Vi → Ui } such that
{φVi → φU }i∈I refines U , (SGA72a, Def.III.2.1), (Sta18, 00XJ).
Proposition 4.11. Let X be a qcqs scheme whose underlying topological space is Noetherian.
Then the morphism ρ is cocontinuous.
Proof. For (U/Y ) ∈ RZ(XNis ) and a procdh covering {Vi → U }i∈I in SchS , we want to finda covering {(Uj /Yj ) → (U/Y )}j∈J in RZ(XNis ), a function J → I; j 󰀁→ ij , and commutative
triangles
Uj ❴ ❴ ❴󰈣 Vij
❇❇
❇❇
❇❇
❇󰈔 󰈃
U.
Since procdh coverings are refined by finite length compositions of generator procdh coverings, using induction on the height of the tree in Remark 2.2, it suﬃces to prove the claim for
distinguished Nisnevich coverings and proabstract blowup squares.
For Nisnevich coverings the statement is obvious since for any (U/Y ) ∈ RZ(XNis ), a Nisnevich covering {Ui → U }i∈I of U gives rise to a Nisnevich covering {(Ui /Y ) → (U/Y )}i∈I of(U/Y ).
Consider (U/Y ) ∈ RZ(XNis ) and a proabstract blowup square U = {Zn → U }n∈N ⊔ {W →U }. Our task is to find a morphism Y ′ → Y in ModX and a Nisnevich covering {Vj → U ′ :=Y ′ ×Y U }j∈J such that for each j, we have a commutative diagram on the left for some n oron the right.
󰈣 Zn
󰈣W
Vj
Vj
(14)
󰈃
U′

󰈃
󰈣U

󰈃
U′

󰈃
󰈣U

󰈃
Y′

󰈃
󰈣Y

󰈃
Y′

󰈃
󰈣Y

Since U has finitely many generic points, by Lemma 4.17, we can assume that U is locallyirreducible. As such it suﬃces to treat the following two cases.
Case 1: U gen ⊆ Z0 . In this case, (Z0 )red = Ured . This means the (finitely many) generatorsof IZ0 are nilpotent, so Zn = U for some n. Hence, Y ′ = Y and the trivial covering {U → U }give a square on the left of (14).
Case 2: U gen ∩ Z0 = ∅. We will build a square as on the right of (14) with Vj = U ′ . By
the assumption U gen ∩ Z0 = ∅, the morphism W → U is an isomorphism over the genericpoints of U . These all lie over generic points of Y (because U → Y is étale) so W → U → Y
is generically flat. More precisely, letting T ⊂ Y be the closure of the image of Z0 in Y ,the morphism W → Y is flat over Y \T and T is nowhere dense in Y by the assumption
U gen ∩ Z0 = ∅. Write T = limλ Tλ for a cofiltered system of closed immersions Tλ → T of finite←−
presentation. Since Y gen is finite by the assumption, there exists λ such that Tλ is nowhere
dense.
By Raynaud-Gruson (RG71, Th.5.2.2), (Sta18, 081R), there is a blowup Y ′ → Y (projective,
but not necessarily of finite presentation) with a center contained in Tλ such that the strict
transform W ′ → Y ′ of W → Y is flat. Since U ′ := Y ′ ×Y U → Y ′ is étale, this implies thatW ′ → U ′ is flat, Lemma 4.15. So now we have a flat proper morphism which is genericallyan isomorphism. This implies it is globally an isomorphism, Lemma 4.16. So we obtain afactorisation U ′ ∼= W′ → W → U.
13

If Y ′ → Y is not of finite presentation, then we can at least write it as a filtered limit
Y = lim Yλ′ such that each Yλ′ → Y is in ModY and the transition morphisms are aﬃne,
Lemma 4.14. Setting Uλ′ = Yλ′ ×Y U we also have U ′ = lim Uλ′ and we can descend U ′ → W tosome Uλ′ , (Gro66, Prop.8.13.1). Noting that Yλ′ ∈ ModY implies Yλ′ ∈ ModX , we have found asquare on the right of (14).
′Counterexample 4.12. Let k be a field, I a set, and k[xI ] the polynomial ring in I-many
variables. Define RI := k[xI ]/〈xi xj : i ∕= j〉 and XI = Spec RI . So XI is I-many copies of theaﬃne line joined at the origin and (XI )gen is homeomorphic to I with the discrete topology.
Given a subset J ⊆ I there is an associated closed immersion XJ ⊆ XI defined by xi 󰀁→ 0 fori∈/ J which is finite presentation if and only if I \ J is finite.
′
For any two cofinite J, J ′ ⊆ I with J ∪ J ′ = I, the set {XJ → XI , XJ → XI } induces aproabstract blowup square. This cannot be refined by a modification (of finite presentation),so the functor RZ(XINis ) → SchXI is not cocontinuous.
Indeed, by Chevalley’s Theorem (Sta18, 054K), the image of any summand Y0 of a modification Y0 ∐ Y1 → XI is a closed subscheme of finite presentation. Since modifications are
isomorphisms over a dense open, this implies that for any modification Y → XI the scheme Y
is connected. Since it is an isomorphism over the generic points of X, it cannot factor throughany XJ → XI unless J = I.
Corollary 4.13. Let S be a qcqs scheme whose underlying topological space is Noetherian and
X ∈ SchS . The canonical functor ρ∗X from (13) preserves colimits and finite limits, (SGA72a,
III.2.3), (Sta18, 00XL).
Here are some lemmas that were used above.
Lemma 4.14. Suppose X is a qcqs scheme.
1. If the underlying topological space of X is Noetherian, then for every Y ∈ ModX there isa closed immersion Y ′ → Y in ModX such that (Y ′ )gen = X gen .
2. If Y → X is a proper morphism, not necessarily of finite presentation, which is an
isomorphism over a dense qc open D ⊆ X, then Y can be written as a cofiltered limit
Y = lim Yλ with Yλ ∈ ModX and whose transition morphisms are closed immersions
Yλ → Yµ which are isomorphisms over D.
Proof.
1. Let D ⊆ X be a dense qc open over which Y → X is an isomorphism. Let j : D →Y be the induced open immersion. Write I = ker(OY → j∗ OD ) as a filtered colimit I =colim Iλ of ideals Iλ of finite presentation, (DG71, Cor.6.9.15). Since Spec(OY /I) → Y
is an isomorphism over the open D ⊆ Y , there is some λ for which Yλ′ := Spec(OY /Iλ ) →Y is an isomorphism over D ⊆ Y , (Gro66, Thm.8.10.5(i)). Since X has Noetherian
topological space, so does Y , Lem.4.18, so up to changing λ, we can assume Yλ′ ∩ (Y gen \X gen ) = ∅, i.e., (Yλ′ )gen = X gen .
2. We can write Y → X as Y = lim Yλ with each Yλ → X proper and of finite presentation
and all transition morphisms closed immersions by (Sta18, 09ZR, 09ZQ). The isomorphism D ×X Y ∼= D induces closed immersions D → D ×X Yλ which are of finite presentation by (Sta18, 00F4(4)), and therefore defined by coherent sheaves of ideals, (Sta18,01TV). Since the D ×X Yλ are all quasi-compact, there is a λ for which D = D ×X Yµ forall µ ≥ λ.
Lemma 4.15. Suppose that W → U is any morphism of schemes, U → Y is étale and W → Y
is flat. Then W → U is also flat.
14

Proof. If U → Y is not already assumed to be separated, we can reduce to this case by replacing
U with an open aﬃne covering. Then the diagonal δ : U → U ×Y U is open (resp. closed)
because U → Y is unramified (resp. separated). Considering the cartesian squares
W
󰈃
U

δ′

δ

󰈣 U ×Y W󰈃󰈣 U ×Y U

pr2

pr2

󰈣W
󰈃
󰈣U

one sees that δ ′ : W → U ×Y W is also both open and closed. So we have U ×Y W = δ ′ (W )⊔W ′for some W ′ . Since W → U factors as the composition W → δ ′ (W ) ⊔ W ′ = U ×Y W → U of
the inclusion of a summand, and the pullback U ×Y W → U of the flat morphism W → Y , itis flat.
Lemma 4.16. Suppose that f : W → U is a flat, proper morphism of schemes, and D ⊆ U isa schematically dense open such that D ×U W → D is an isomorphism. Then W → U is an
isomorphism.
Proof. First note that f is faithfully flat: Since it is proper, it is closed, so it suﬃces to show
all generic points η of U are in the image. The inclusion of the localisation Uη → U is flat soUη ×U D → Uη is schematically dense. But Uη has a single point, so Uη ×U D is non-empty,
i.e., η ∈ D, so η ∈ f (W ).
Since W → U is faithfully flat, to show it is an isomorphism it suﬃces to show thatpr1 : W ×U W → W is an isomorphism, or equivalently, that the diagonal δ : W → W ×U W
pr1
is an isomorphism. The two morphisms W ×U W → W → U are flat, so D ×U W → W andD ×U W ×U W → W ×U W are schematically dense. But W → U is an isomorphism over Djδ
so D ∼= D ×U W ∼= D ×U W ×U W . So we have a factorisation D → W → W ×U W with bothj and δj schematically dense. It follows that δ is schematically dense. Since it is also a closed
immersion (f is proper, so separated) this implies δ is an isomorphism.
Lemma 4.17. Suppose Y is qcqs with Noetherian underlying space, let U → Y be an étalemorphism, and suppose U gen = η0 ⊔ η1 is a decomposition of the space of generic points of U
into clopens. Then there exists a cartesian squareU0′ ⊔ U1′

󰈣U

󰈃
Y′

󰈃
󰈣Y

such that
1. Y ′ → Y is a proper morphism of finite presentation which is an isomorphism over a dense
qc open of Y , and2. there are identifications η0 = (U0′ )gen and η1 = (U1′ )gen .
If Y is Noetherian, then we can take Y ′ → Y to be a blowup in a centre which is nowhere dense
in Y .
Proof. Choose any closed subschemes U0 ⊆ U and U1 ⊆ U of finite presentation whose genericschemes are η0 and η1 respectively (here we are using that U gen is finite).6 Since U0 ⊔ U1 →6 For example, let I

Z = ker(OU → OU gen ). Write IZ as a filtered union of ideals IZ,λ of finite presentation.
Z
Use the fact that U gen is a disjoint union of finitely many local schemes of dimension zero to deduce that theregen
is some Iλ with (Spec OU /Iλ ) ∩ UW = ∅.

15

U → Y is generically étale, there is some dense open D ⊆ Y over which it is étale, (Sta18,
07RP), and in particular, flat and of finite presentation. So we can apply Raynaud-Gruson
platification, (Sta18, 081R), to find a blowup Y ′ → Y (not necessarily of finite presentation)
which is an isomorphism over D ⊆ Y , and for which the strict transform U0′ ⊔ U1′ → Y ′ ofU0 ⊔ U1 → Y is flat. By Lemma 4.15 this implies U0′ ⊔ U1′ → Y ′ ×Y U is also flat. It isproper and an isomorphism over a dense open by construction, so it is in fact an isomorphism,
Lem.4.16. Now the second condition is satisfied, since U0′ → U factors through U0′ → U0 andthis latter is an isomorphism generically by construction.
The morphism Y ′ → Y might not be of finite presentation, but it is a cofiltered limit ofsome Yλ′ → Y which are proper, of finite presentation, and isomorphisms over dense qc opens′′
Dλ ⊆ Y , Lem.4.14. There is some λ for which we already have a decomposition Uλ′ = Uλ,0
⊔Uλ,1
′
′
′
′
where of course we have set Uλ = Yλ ×Y U and Uλ,i = Yλ ×Y Ui . Replacing Y with this Yλ′ ,both conditions are now satisfied.
Lemma 4.18. Suppose that X is a scheme with Noetherian topological space of finite dimension
and Y → X is a morphism of finite type. Then Y also has Noetherian topological space of finite
dimension.
Proof. We want to show that every descending chain
Z0 ⊇ Z1 ⊇ Z2 ⊇ . . .
of closed subspaces of Y stabilises. Since X has Noetherian topological space, it admits afinite open aﬃne covering {Spec(Ai )}ni=1 . It suﬃces to show that our chain stabilises in eachSpec(Ai ) ×X Y , that is, we can assume X is aﬃne.
Similarly, by definition, since Y → X is finite type it is quasi-compact, so we can also
assume Y is aﬃne, say Y = Spec(B). In particular, this means there is a closed immersion
Y → AnX for some n. So we can assume that Y = AnX . By induction we can assume n = 1.
Since X has Noetherian topological space, it has finitely many irreducible components. So
we can assume that X is integral.
Now we use induction on the dimension of X. If X is dimension -1 it is empty, and we aredone (the dimension zero case is also easy).
In general, suppose that
A1X ⊇ Z0 ⊇ Z1 ⊇ Z2 ⊇ . . .
is a decreasing chain of closed subschemes. I claim that it suﬃces to show that chains of finite
presentation closed subschemes stabilise. Indeed, if there exists a strictly decreasing chain ofclosed subsets, then we can manufacture a strictly decreasing chain of closed subsets of finite
presentation: For each i choose a prime pi ∈ Zi \ Zi+1 and an element fi+1 ∈ Ii+1 \ pi where
Ii+1 is the ideal corresponding to Zi+1 . Then setting Wi to be the closed corresponding to〈f1 , . . . , fi 〉, the chain W0 ⊇ W1 ⊇ W2 ⊇ . . . contains Z0 ⊇ Z1 ⊇ . . . as a subchain and also
has pi ∈ Wi \ Wi+1 , so it is strictly decreasing.
So we can assume Zn = 〈f1 , . . . , fn 〉 for some fi ∈ A[x] where A = Γ(X, OX ). Letting K be
the quotient field of A, the ring K[x] is Noetherian, so the corresponding chain of ideals (notjust closed subsets) in K[x] stabilises. Moreover, K[x] is a principal ideal domain, so there issome a ∈ A \ {0} and g(x) ∈ A[x] such that 〈 a1 g〉 = 〈f1 , . . . , fn 〉 for all n ≫is, we can󰁓0.1 That1
write each fi as fi = a1 g b1i hi for some hi ∈ A[x] and bi ∈ A \ {0}, and 1 =hkbi i ci i for someki ∈ A[x] and ci ∈ A \ {0} (for simplicity, we choose kj = 0 cj = 1 for j ≫ 0 so that the kj andcj are independent of n). If the a, bi , ci are all 1, then 〈g〉 = 〈f1 ,󰁓
. . . , fn 〉 holds
󰁓 in A[x], not justK[x]. That is, our chain stabilises. Indeed, fi = ghi and g = g( hi ki ) =fi ki .
16󰁔 󰁔
We now show how to reduce to the case all a, bi , ci are 1. Set ai = a bi ci . Since
Spec(A) has Noetherian topological space, the chain of opens Spec(A[ a10 ]) ⊆ Spec(A[ a01a1 ]) ⊆Spec(A[ a0 a11 a2 ]) ⊆ . . . stabilises. That is, there is some d ∈ A such that A[d−1 ] = A[ a0 a11...an ] forall n ≫ 0. Since d is nonzero and A is integral, Spec(A/〈d〉) has dimension strictly smaller thanSpec(A), so our chain of Zi ’s restricted to A1Spec(A/〈d〉) stabilises by the induction hypothesis,
and it suﬃces to show that it stabilises when restricted to A1Spec(A[d−1 ]) . By construction,a, bi , ci are all invertible in A[d−1 ]. So replacing A with A[d−1 ], we can assume they are all 1
by replacing g, hi , ki with a1 g, b1i hi and c1i ki .

5Conservativity of the fibre functors

Recall that a topos is said to have enough points when a morphism f is an isomorphism if andonly if φ(f ) is an isomorphism for all fibre functors φ, Def.3.1, (SGA72a, Exposé IV, Déf.6.4.1).
Equivalently, (SGA72a, Exposé IV, Prop.6.5(a)), a topos of the form Shvτ (C) has enoughpoints when a family {Yi → X}i∈I in C is a covering family if and only if ⊔i∈I φ(Yi ) → φ(X)is surjective for all fibre functors φ.7
Deligne’s completeness theorem says that if C is an essentially small category with fibre
products, and every τ -covering is refinable by a finite one, then Shvτ (C) has enough points,(SGA72b, Prop.VI.9.0) or (Joh77, Thm.7.44, 7.17).
Example 5.1. If X is a qcqs scheme then Shv(RZ(XNis )) has enough points.
Theorem 5.2. Suppose S is a qcqs scheme with Noetherian topological space of finite Krull
dimension. Then the topos Shvprocdh (SchS ) has enough points.
Proof. We want to show that if Y = {Yi → Y }i∈I is a family of morphisms in SchS suchthat the morphism of sets ⊔i φ(Yi ) → φ(Y ) is surjective for every fibre functor φ, then Y isa procdh-covering. We work by induction on the Krull dimension of Y , the base case being
Y = ∅ with dim Y = −1, where it is clearly true. The inductive hypothesis is:(Hyp.) If dim X < dim Y and ⊔φXi → φX is surjective for every fibre
functor φ, then {Xi → X}i∈I is a procdh covering.

The functor ρ∗Y : Shvprocdh (SchS ) → Shv RZ(YNis ) from (13) preserves colimits and finite limits,
Corollary 4.13. So by composition, every fibre functor φ of Shv RZ(YNis ) induces a fibre functor
ρ∗φY
Shvprocdh (SchS )→RZ(YNis ) → Set.

By assumption, our family Y is sent to a surjection of sets under each such fibre functor φ ◦ ρ∗Y .
Since RZ(YNis ) has enough points, Exam.5.1, it follows that ρ∗Y Y is a surjective family of sheaves
in Shv(RZ(YNis )). This means that, locally, we can lift the section idY of (ρ∗Y Y )((Y /Y )) =homSchS (ρ((Y /Y )), Y ) = homSchS (Y, Y ). Explicitly, this means that there exists a covering{(Uj /Y ′ ) → (Y /Y )}j∈J such that the family {Uj → Y ′ → Y }j∈J refines Y. Since Y ′ → Y is a
modification, there is a nowhere dense closed subscheme of finite presentation Z0 → Y outsideof which Y ′ → Y is an isomorphism. Since Y has finite Krull dimension and Z0 → Y is nowhere
dense, dim Z0 < dim Y . Since fibre functors commute with finite limits, for each fibre functor
φ, the morphism of sets ⊔φ(Zn ×Y Yi ) = ⊔φ(Zn ) ×φ(Y ) φ(Yi ) → φ(Zn ) is the pullback of the
7 Here, we have used the same symbol for an object X of C and the sheafification of the presheaf hom(−, X)
it represents.

17

surjective morphism of sets ⊔φ(Yi ) → φ(Y ), and therefore is surjective. So by the induction
hypothesis, for n ∈ N the families {Zn ×Y Yi → Zn }i∈I are procdh coverings. Then the family{Zn ×Y Yi → Zn → Y }n∈N,i∈I ⊔ {Uj → Y ′ → Y }j∈J

(15)

is the composition of the procdh covering {Zn → Y }n∈N ⊔ {Y ′ → Y }, the Nisnevich covering{Uj → Y ′ }j∈J and the procdh coverings {Zn ×Y Yi → Zn }j∈Jn , and each morphism in Eq.(15)factors through some Yi → Y .
Corollary 5.3. Suppose S is a Noetherian scheme of finite Krull dimension. Then procdh local
S-rings R with length Q(R) and dim R finite induce a conservative family of fibre functors of
Shvprocdh (SchS ).
Proof. Apply Proposition 3.6.

6

ExcisionIn this section we discuss the equivalence of procdh excision and Čech descent.
Sheaves of spaces. Since (Lur09), the default has become to define a sheaf of spaces as apresheaf of spaces satisfying Čech descent.8
Theorem 6.1. Let S be a scheme and consider the following conditions on a presheaf of spaces
F ∈ PSh(SchS , S).
1. Excision. For every distighished Nisnevich square {U → X, V → X} and every proabstract blowup square {Zn → X}n∈N ⊔ {Y → X} in SchS , Def.2.1, we have
∼

(16)

∼

(17)

F (X) → F (U ) ×F (U ×X V ) F (V ),F (X) → F (Y ) ×lim F (Zn ×X Y ) lim F (Zn ).
2. Čech descent. For every procdh covering family {Yλ → X}λ∈Λ we have∼

F (X) → lim F (Yn )
n

(18)

󰁔
where we write F (Yn ) for i∈I n F (Yi1 ×X · · · ×X Fin ). That is, in the terminology of(Lur09), F is a procdh sheaf.
If S is qcqs then (Excision) ⇔ (Čech descent).
Proof. The proofs from (AHW17, Thm.3.2.5) work verbatim. See (KS24, §6) for complete
proofs in our setting.

7

Homotopy dimension

7.1

Valuative dimension

Definition 7.1. Recall that the valuative dimension of a scheme is the supremum of the ranksof all valuation rings of residue fields of generic points of X centred on X.
󰀏󰀝󰀞󰀏 ∃ x ∈ X gen ; R is a valuation ring of k(x)dimv (X) = sup dim R 󰀏󰀏.
∃ Spec(R) → X compatible with R ⊆ k(x)8 As opposed to the hyperdescent used in work of Artin, Brown, Deligne, Friedlander, Gersten, Jardine, Joyal,
Mazur, Morel, Thomason, Verdier, Voevodsky, and many others in work dating back at least to the 70’s.

18The valuative dimension is of interest to us because it controls the size of RZ(X). Indeed,
the R appearing in the definition of valuative dimension are precisely the reductions of the local
rings of the locally ringed topological space limY ∈ModX Y .
The following basic properties of valuative dimension follow directly from well-known facts
about valuation rings.
Lemma 7.2. Let X be a scheme.
1. If Y ⊆ X is a nowhere dense subscheme and dimv (Y ) is finite, thendimv Y 󰃥 dimv X.
2. If f : Y → X is a proper morphism inducing a bijection of sets Y gen = X gen andisomorphisms of residue fields k(y) ∼= k(f (y)) for all y ∈ Y gen thendimv Y = dimv X.
3. If X has finite valuative dimension then so does every X-scheme of finite type.
Proof. The statements already appear in (EHIK21, Prop.2.3.2). For more reasonable proofssee (KS24, Lem.7.2).

7.2

Homotopy dimension∼

Recall that for n ≥ −2 one says that a space K ∈ S is n-truncated if Map(Dn+2 , K) →Map(S n+1 , K), where Dn+2 is the (n + 2)-disc, S n+1 is its boundary, the (n + 1)-sphere, so
S 0 = ∗⊔∗ and S −1 := ∅, (Lur09, Lem.5.5.6.17). The notion of n-truncatedness is extended to a
general ∞-category T , such as T = PSh(C, S), by declaring an object F ∈ T to be n-truncated
if the space Map(G, F ) is n-truncated for all objects G of T . If T is presentable9 then the
inclusion T≤n ⊆ T of the full subcategory T≤n of n-truncated objects is ω-accessible and theinclusion admits a left adjoint, (Lur09, Prop.5.5.6.18),(−)≤n : T → T≤n .
Suppose T, T ′ are ∞-categories admitting finite limits and ρ : T ′ → T is a functor. If ρ
preserves finite limits, then it preserves n-truncated objects, cf.(Lur09, Prop.5.5.6.16), and we
obtain the commutative square on the right hand side of the following diagram.
T
(−)≤n

󰈃

λ

󰈣 T′

λ≤n

󰈃

T󰉃 󰉣

(−)≤n

ρ

inc.

T󰉃 ′

(19)

inc.
ρ≤n

′
T≤n ❴ ❴ ❴󰈣 T≤n′T≤n 󰉣❴ ❴ ❴ T≤n′
If ρ and the inclusions T≤n ⊆ T , T≤n⊆ T ′ all admit left adjoints λ, (−)≤n , (−)≤n respectively,
then an adjunction argument shows that the functor λ≤n := (−)≤n ◦ λ ◦ inc. produces thecommutative square on the left hand side of Eq.(19) above. In fact, since the inc. are fully
faithful, λ≤n is a left adjoint to ρ≤n . Note that inc. and (−)≤n preserve final objects. So if λalso preserves the final object, we have

λ≤n (∗) = (−)≤n ◦ λ ◦ inc.(∗) = ∗.
Therefore, for F ∈ T we have F≤n ∼= ∗ implies λ(F )≤n ∼= ∗.
9 This means that T admits all small colimits and is of the form T = Ind(T ′ ) for some small category T ′ .

19Example 7.3.
1. If Φ : Shvτ (C, S) → S is any fibre functor10 and F ∈ Shvτ (C, S), we have F≤n ∼=∗⇒Φ(F )≤n ∼= ∗.
2. If X ∈ C is any object and F ∈ Shv(C, S), we have F≤n ∼= ∗ ⇒ (F |X )≤n ∼= ∗, where(−)|X : Shv(C, S) → Shv(C/X , S) is the restriction functor with C/X equipped with theinduced topology: coverings in C/X are precisely those families which are sent to coverings
in C; the projection C/X → C is a continuous and cocontinuous morphism of sites.
3. If X is a qcqs scheme with Noetherian topological space and F ∈ Shvprocdh (SchS ) wehave F≤n ∼= ∗ ⇒ (ρ∗X F )≤n ∼= ∗ in Shv(RZ(XNis ), S), cf. Corollary 4.13.
As a right adjoint, global sections Map(∗, −) does not preserve connectivity in general.
Homotopy dimension describes how badly this fails.
Definition 7.4 ((Lur09, Prop.6.5.1.12, Def.7.2.1.1)). One says the ∞-topos Shvτ (C, S) has
homotopy dimension ≤ d if for every sheaf F ∈ Shvτ (C, S) we haveF≤d−1 ∼
= ∗ implies Map(∗, F )≤−1 ∼= ∗.

Note that the latter condition is equivalent to Map(∗, F )≤−1 ∕= ∅.
Remark 7.5. More generally, if Shvτ (C, S) has homotopy dimension ≤ d then we haveF≤d+n ∼
= ∗ implies Map(∗, F )≤n ∼=∗for all n ≥ −1, (Lur09, Def.7.2.1.6, Lem.7.2.1.7).
Remark 7.6. Even more generally, one says that Shvτ (C, S) is locally of finite homotopy dimension if for every X ∈ C there is some dX < ∞ such that F≤dX +n ∼= ∗ implies Map(X, F )≤n ∼=∗ for all n ≥ −1.
By the canonical adjunction Shvτ (C, S) ⇄ Shvτ (C/X , S) (cf. Example 7.3(2)), this isequivalent to asking that each Shvτ (C/X , S) has finite homotopy dimension.
Example 7.7. Consider the category N = {0 → 1 → 2 → . . . }. An object of PSh(N, S)
is a diagram · · · → K(2) → K(1) → K(0) and the global sections functor is {K(n)}n∈N 󰀁→
limn∈N K(n). If (K(n))≤0 ∼= ∗, that is, π0 K(n) ∼= ∗ for all n, then it follows from the short
exact sequences of pointed sets, (BK72, §7.4),∗ → lim 1 π1 K(n) → π0 lim K(n) → lim π0 K(n) → ∗
n∈N

n∈N

n∈N

(20)

∼ ∗. So the topos PSh(N, S) has homotopy dimension ≤ 1. The sequencethat (lim K(n))≤−1 =of non-empty discrete spaces K(n) = N≥n shows that the homotopy dimension is ∕≤ 0 sinceK(n)≤−1 = ∗ for all n but hom(∗, {K(n)}n∈N )≤−1 = ∅.
Example 7.8 ((CM21, Cor.3.11, Thm.3.18)). If Cλ is a filtered system of finitary11 excisive12
sites with colimit C, then Clausen and Mathew show that Shv(C, S) has homotopy dimension≤ d if all Shv(Cλ , S) do. Using this they show that for any qcqs algebraic space whose underlying
topological space has Krull dimension ≤ d, the ∞-topos Shv(XNis , S) has homotopy dimension≤ d.
It also follows from this that if X is a qcqs scheme of valuative dimension d then RZ(XNis )has homotopy dimension ≤ d.
10 As in the case of sets, Φ is a fibre functor if it preserves all colimits and finite limits, cf.(Lur09,

Rem.6.3.1.2,
Cor.5.5.2.9, Thm.6.1.0.6).
11 A site is finitary if it has finite limits and every covering family is refineable by a finite one.
12 A site is excisive if for all U ∈ C, the functor F 󰀁→ Map(U, F ) commutes with filtered colimits.

20

Theorem 7.9. Let S be a qcqs scheme of finite valuative dimension d ≥ 0 with Noetherian
underlying topological space. Then Shvprocdh (SchS , S) has homotopy dimension ≤ 2d.
Proof. The proof is by induction on the valuative dimension of S. Suppose F ∈ Shvprocdh (SchS , S)
has F≤2d−1 = ∗. We want to show that F (S)≤−1 ∼= ∗.
Since S has Noetherian topological space,ρ∗ = ρ∗S : Shvprocdh (SchS , S) → Shv(RZ(SNis ), S)
is a left adjoint of a morphism of ∞-topoi, Cor.4.13, so (ρ∗ F )≤2d−1 ∼= ∗, Exam.7.3. Since thehomotopy dimension of Shv(RZ(SNis ), S) is ≤ d, Exam.7.8, we have (ρ∗ F )(S)≤−1 ∼= ∗, that is,
the space (ρ∗ F )(S) is non-empty. Concretely, one can calculate (ρ∗ F )(S) as colimY ∈ModS F (Y ),Remark 4.10, so we can find a modification Y → S such that F (Y ) is non-empty.
If d = 0, we have Y = S and we are done with this step.
If d > 0, up to refining Y we can assume that Y gen = S gen , Lem.4.14. In particular,
there exists a nowhere dense non-empty closed subscheme of finite presentation Z0 ⊆ S suchthat Y → S is an isomorphism over S\Z0 and E0 := Z0 ×S Y is also a nowhere dense closed
subscheme of finite presentation. Note we now have 0 ≤ dimv Z0 ≤ d − 1 and similar for
E0 , Lem.7.2. We continue to have (F |SchZn )≤2d−1 ∼= ∗ and (F |SchEn )≤2d−1 ∼= ∗, Exam.7.3.
By the induction hypothesis Shvprocdh (SchZn , S) and Shvprocdh (SchEn , S) all have homotopy
dimension ≤ 2d − 2, so F (Zn )≤1 , F (En )≤1 ∼= ∗ for all n, Rem.7.5. We have seen that PSh(N, S)
has homotopy dimension ≤ 1, Exam.7.7 so (limn∈N F (Zn ))≤0 ∼= ∗ and (limn∈N F (En ))≤0 ∼= ∗.
∼Combining this with F (Y )≤−1 = ∗, cartesianness of the square, Thm.6.1,
F (S)󰈣 F (Y )󰈃limn∈N F (Zn )󰈃󰈣 limn∈N F (En )implies that F (S)≤−1 ∼= ∗. Indeed, both limn∈N F (Zn ) and limn∈N F (En ) are non-empty
connected and F (Y ) is non-empty, so the pullback is also non-empty.
Corollary 7.10. Let S be a qcqs scheme of finite valuative dimension d ≥ 0 with Noetherian
underlying topological space. For any sheaf of abelian groups F ∈ Shvprocdh (SchS , Ab) we haven
Hprocdh(S, F ) = 0;n > 2d.

Proof. This is (Lur09, Cor.7.2.2.30). Cf. also (Lur09, Def.7.2.2.14, Rem.7.2.2.17).
Corollary 7.11. Let S be a qcqs scheme of finite valuative dimension with Noetherian underlying topological space. Then Shvprocdh (SchS , S) is Postnikov complete,13 and hypercomplete.14
Proof. Lemma 7.2(3) and Lemma 4.18 say that each X ∈ SchS also have finite valuative dimension with Noetherian underlying topological space so by Theorem 7.9 the ∞-topos Shvprocdh (SchS )is locally of finite homotopy dimension, hence, Postnikov comlete, (Lur09, Prop.7.2.1.10), andhypercomplete, (Lur09, Cor.7.2.1.12).
13 Postnikov complete means that for every object F we have F = limn τ≤n F .
means that for every hypercovering U∗ → X and object F we have Map(X, F ) =limn∈∆ Map(Un , F ), (Lur09, pg.667).
14 Hypercomplete

21

7.3

Points of ∞-topoi

As in the 1-topos case, one says that an ∞-topos Shv(C, S) has enough points when a morphism
f is an equivalence if and only φ∗ f is an equivalence for every geometric morphism of ∞-topoi
φ∗ : Shv(C, S) ⇄ S : φ∗ , (Lur09, Rem.6.5.4.7). Unlike the 1-topos case, in general, detecting
equivalences is not equivalent to detecting coverings unless Shv(C, S) is hypercomplete, (Lur18,
Prop.A.4.2.1).15
The purpose of this subsection is to prove Proposition 7.14 which says that if the 1-topos ofa site has enough points and the ∞-topos is hypercomplete, then the ∞-topos also has enoughpoints.
Remark 7.12. This should be compared to Lurie’s result (Lur18, Thm.A.4.0.5), which saysthat any ∞-topos which is locally coherent and hypercomplete has enough points. We cannot
apply this to the procdh topos because it is not locally coherent.
Conversely, combining Deligne’s 1-categorical completeness theorem with our Proposition 7.14
recovers (Lur18, Thm.A.4.0.5).
We apply Proposition 7.14 to the procdh topos in Corollary 7.15.
Lemma 7.13. Let (C, τ ) be a site admitting finite limits. Every fibre functor φ∗Set of the 1-topos
induces a fibre functor φ∗S of the ∞-topos fitting into a commutative square Eq.(21). The sameis true with φ∗Set and φ∗S reversed.
Shv(C, S)
φ∗
S

(−)≤0

󰈣 Shv(C, Set)

(21)

φ∗
Set

󰈃
S

(−)≤0

󰈃
󰈣 Set

Proof. Since C admits finite limits, in both the 1-category and ∞-category setting there is a
canonical correspondence between fibre functors φ∗? of PSh(C,󰁕 ?) and pro-objects P of C, givenby (P : Λ → C) 󰀁→ (F 󰀁→ colimλ∈Λ F (Pλ )) and φ∗ 󰀁→ (P : C F → C), (Lur09, Cor.5.3.5.4).
So, for φ∗Set : Shv(C, Set) → Set as in the lemma, we automatically get the extended diagramon the left.
PSh(C, S)
φ∗
PSh

(−)≤0

󰈣 PSh(C, Set)

PSh(C, S)

󰈃
󰈃
(−)≤0
󰈣 Shv(C, Set)
Shv(C, S)
󰈛

S

(−)≤0

(−)≤0

󰈣 PSh(C, Set)

󰈃
󰈃
(−)≤0
󰈣 Shv(C, Set)
Shv(C, S)∗φ∗S󰈃 φSet
󰈣 Set

󰈃
S

(−)≤0

(22)
φ∗
PSh

󰈣 Set 󰉬Since Shv(C, S) is the topological localisation of PSh(C, S) generated by covering sieves, (Lur09,
Def.5.5.4.5, Def.6.2.1.4) and φ∗PSh preserves colimits, to show that φ∗PSh factors through Shv(C, S),
it suﬃces to show it sends covering sieves to equivalences. Any sieve16 in PSh(C, S) is sentby φ∗PSh to an inclusion of connected components A → A ⊔ B in S. Such an inclusion is anequivalence if and only if it induces an isomorphism on connected components, i.e., if it is an
isomorphism after τ≤0 . So the result follows from the fact that covering sieves are sent toisomorphisms in Set by the outside square in Eq.(22).
15 This is unsurprising as the definition of sieve in (Lur09) basically encodes the notion of Čech descent.
16 Or more generally, any monomorphism, i.e., any morphism F → G such that F → F ×

22

G F is an equivalence.

Conversely, by the same argument, any φ∗S gives rise to a φ∗PSh as in the diagram on theright. Again, this factors through Shv(C, Set) if and only if it sends covering sieves to isomorphisms, but this is guaranteed by the fact that PSh(C, S) → Shv(C, S) sends covering sieves toequivalences.
Proposition 7.14. Let (C, τ ) be a small site admitting finite limits such that the 1-topos
Shv(C, Set) has enough points as a 1-topos. Then the hypercompletion Shv(C, S)∧ has enoughpoints as an ∞-topos.
Proof. We have the following facts.
1. PSh(C, S)[S −1 ] ∼= (LS PSh(C, Set∆ ))◦ for any set of morphisms S in PSh(C, Set∆ ).
That is, the localisation (in the sense of (Lur09, Def.5.2.7.2)) of the ∞-category PSh(C, S)
at the image of S is the ∞-category associated to the Bousfield localisation of the simplicial
model category PSh(C, Set∆ ) (with either the injective or projective model structures).
2. Shv(C, S) is the localisation of PSh(C, S) at the class of Čech hypercoverings.
3. Shv(C, S)∧ is the localisation of PSh(C, S) at the class HR of all hypercoverings, (Lur09,
Thm.6.5.3.12).
4. the weak equivalences in LHR PSh(C, Set∆ ) are the morphisms such that each of theinduced maps
π0 E → π0 B

πn (E|C/X , b) → πn (B|C/X , b),

X ∈ C, b ∈ B(X)are isomorphisms of sheaves, (Jar87, Prop.2.8), (DHI04).
Putting these facts together, the result holds almost by definition.
Corollary 7.15. Let S be a qcqs scheme of finite valuative dimension d ≥ 0 with Noetherian
underlying topological space. Then Shvprocdh (SchS , S) has enough points as an ∞-topos.
Proof. The 1-topos Shvprocdh (SchS , Set) has enough points, Thm.5.2, and the ∞-topos Shvprocdh (SchS , S)is hypercomplete, Cor.7.11, so it has enough points by Proposition 7.14.

7.4Counterexample

We give an example that suggests the bound on the homotopy dimension in Theorem 7.9 isstrict.
Proposition 7.16. Let S be scheme with finitely many generic points and such that for allhs ∈ S the henselisation OS,sis procdh local (e.g., a smooth curve over a field, or a procdh local
scheme). Suppose F ∈ ShvNis (SchS , ∗) is a Nisnevich sheaf of sets / abelian groups / chain
complexes of abelian groups / spaces. Then F (S) = Fprocdh (S).
Proof. Since F and Fprocdh are both Nisnevich sheaves, and the (small) Nisnevich ∞-topos is
hypercomplete, it suﬃces to show that for s ∈ S we have φ∗Nis,s F = φ∗Nis,s Fprocdh where φ∗Nis,s G
means colims→V →S G(V ) and the colimit is over all factorisations with V → S étale of finitehpresentation. By assumption OS,sis procdh local. Note that {s → V → S : V ∈ SNis } is ahpro-object of SchS and Spec(OS,s) = lims→V →S V so the functor φ∗Nis,s is the fibre functor ofV ∈SNish
Shvprocdh (SchS ) associated to the procdh local ring OS,s. Hence, φ∗Nis,s F = φ∗Nis,s Fprocdh .

23

Example 7.17. The idea is to find a scheme S of valuative dimension 1, a proabstract blowup
square󰈣X“colim”En
(23)
n∈N

󰈃
“colim”Zn
n∈N

󰈃
󰈣S

and a sheaf of abelian groups F such thati
Hprocdh(X, F ) = 0 for all i(24)i
Hprocdh(Zn , F ) = 0 for all ii
Hprocdh (En , F ) = 0 i > 0.

(25)
(26)

Then the corresponding long exact sequence isi· · · → 0 → Hprocdh(S, F ) → Ri−1 lim F (En ) → 0 → . . . .
n∈N

2
So if we can also arrange that the lim1n∈N is non-zero, we have non-zero Hprocdh(S, F ) giving acounterexample to the statement that Shvprocdh (SchS , S) has homotopy dimension ≤ 1.
Our example is two aﬃne lines joined at the origin, S = A1 ⊔{0} A1 , with normalisation
X = A1k ⊔ A1k and associated proabstract blowup square“colim”En󰈣X󰈃
“colim”Zn

󰈃
󰈣S

n∈N

n∈N

(27)

where Zn is the nth thickening of the origin and En = Zn ×S X. Note that En decomposes
into two components corresponding to the components of X, namely, En = Spec k[x]/xn ⊔Spec k[y]/y n =: En,x ⊔ En,y .
To a sequence of abelian groups . . . A2 → A1 → A0 we associated the presheaf of abeliangroups on SchS ,󰀝Am , m = min{i | ∃ T → Ei,x → S}
F : T 󰀁→0,∀i, ∕ ∃ T → Ei,x → S.

For τ = Nis, procdh, let Fτ be the τ -sheafifiaction of F considered as a presheaf of chain
complexes. So H i (Fτ (S)) = Hτi (S, F ), etc. The only V ∈ XNis admitting an S-morphism
V → Ei,x is the empty scheme, so FNis (X) = 0 and by Proposition 7.16 we have Fprocdh (X) =FNis (X), so (24) holds. We have (25) for the same reason. The condition (26) holds because
En is a disjoint union of procdh local schemes. More precisely we have
Fprocdh (En ) = Fprocdh (En,x ) × Fprocdh (En,y )= F (En,x ) × F (En,y ) = An × {0}.

where the 0 is because there are no factorisations En,y → Ei,x for n > 1. So choosing any
2
sequence (An )n∈N with lim1n∈N An ∕= 0 produces an F with Hprocdh(S, F ) ∕= 0.
Remark 7.18. It seems possible to push the above technique further to at least show thatthe cohomological dimension of certain surfaces is 4. For example, Gabber proposed trying thesurface S × S, where S is the scheme from Example 7.17.
24

8Application to K-theory

In this section, we work with the category Schqcqs of qcqs schemes and the full subcategory
Schnoe of noetherian schemes. For C = Spt, D(Z), D(Q), D(Fp ), we write PSh(Schqcqs , C) for
the ∞-category of presheaves on Schqcqs with values in C and write Shvprocdh (Schqcqs , C) for
the full subcategory of the procdh sheaves with the sheafification functor17
aprocdh : PSh(Schqcqs , C) → Shvprocdh (Schqcqs , C).

8.1

Algebraic K-theory

By (KST18b), non-connective K-theory satisfies procdh excision for Noetherian schemes. Consequently, by Theorem 6.1 it defines a sheaf K ∈ Shvprocdh (Schnoe , Spt). We have the following
topos-theoretic interpretation of the Bass construction.
Theorem 8.1. For X ∈ Schnoe with dim(X) < ∞, there exists a natural equivalence(aprocdh τ≥0 K)(X) ≃ K(X).
Proof. For X as above, we have the descent spectral sequencesp󰁨 −q ) ⇒ K−p−q (X),E2p,q = Hprocdh(X, Kp󰁨 −q ) ⇒ π−p−q (aprocdh τ≥0 K(X)),
E2p,q = Hprocdh(X, τ≥0 K

󰁨 i is the procdh sheafification of the presheaf Ki = πi K of abelian groups on SchX ,where K
󰁨i = K󰁨 i for i ≥ 0 and τ≥0 K
󰁨 i = 0 for i < 0. By Corollary 7.10, the spectral sequencesand τ≥0 K
󰁨 i = 0 for i < 0. By Theorem 5.2 the procdh
are bounded. So, it suﬃces to show that K
1-topos has enough points, so it is enough to show that φ∗ Ki = 0 for all i < 0 and fibre
functors φ∗ : Shvprocdh (SchX ) → Set. Since K-theory commutes with filtered colimits ofrings, we have φ∗ Ki = Ki (R) where R = colim Rλ for (Spec(Rλ ) → X)λ∈Λ the proobject
corresponding to φ∗ . Let N ⊂ R be the ideal of nilpotent elements. Then, for i < 0, we get
Ki (R) = Ki (R/N) = 0, where the first equality follow from the nil-invariance of the negative
K-theory and the last equality follows from (KM21, Th.1.3) since R/N is a valuation ring. This
completes the proof.
Remark 8.2. The proof of Theorem 8.1 shows the following. Note that we use Corollary 5.3
to reduce to the smaller class of those procdh local rings R with length Q(R) and dim R finite.
Proposition 8.3. Take F = Fp or Q, set C = Spt or D(Z), and let E ∈ PSh(Schqcqs, C) be aF
presheaf. Consider the following conditions.
(Desc) For each Noetherian F-scheme X the restriction E|SchX is a procdh sheaf.
(Fin) E is finitary, in the sense that it preserves filtered colimits of F-algebras.
(BB)≥N For each procdh local ring R with length Q(R) and dim R finite, the procdh stalk E(R) is
homologically bounded below N . That is, πi E(R) = 0 for i < N .
If E satisfies (Desc), (Fin), and (BB)N then for every Noetherian F-scheme X there is a natural
equivalence
aprocdh (F≥N )(X) ≃ F (X).
(28)
We will produce a number of presheaves that satisfy the conditions of Proposition 8.3 using
the following definition.
17 For existence of sheafification for certain large sites (including ours) see (BS13, Rem.4.1.2).

25

Definition 8.4. Let F and C be as in Proposition 8.3. For E ∈ PSh(Schqcqs, C), defineF
Nil E := fib(E → acdh E) ∈ PSh(Schqcqs
, C),
F

(29)

where acdh : PSh(Schqcqs, C) → Shvcdh (Schqcqs, C) is the cdh sheafification functor.
F
F

8.2

Negative cyclic homology
1

Recall that for k → R a morphism of commutative rings, one defines HC− (R/k) = HH(R/k)hS
as the homotopy fixed points of the Hochsdhild homology and for qcqs k-schemes using Zariski
descent. In (Ant19, Thm.1.1), Antieau defines a functorial complete decreasing multiplicative
Z-indexed filtration,18󰀝󰀞−n
F ilHKR HC (X/k)
on HC− (X/k)
(30)
n∈Z

and natural equivalences≥n󰁦 X/k [2n].
grnF ilHKR HC− (X/k) ≃ LΩ

(31)

󰁦 X/k ≃ ∧n LX/k [−n]grn LΩ(32)

󰁦 X/Q is the Hodge-completed derived de Rham complex equipped with the Hodge
Here, LΩ󰀋 ≥n 󰀌
󰁦 X/kfiltration LΩ.19 The graded pieces of this filtration are computed byn∈N

where LX/k ∈ PSh(Schqcqs, D(k))) is the cotangent complex, (Bha12, Construction 4.1).
k0
By (EM23, Lem.4.5), the cofibre cofib(F ilHKR
HC− (−/Q) → HC− (−/Q)) is a cdh sheaf on∼
qcqs
0
SchQ , so the canonical morphism Nil F ilHKR
HC− (−/Q) → Nil HC− (−/Q) is an equivalenceof presheaves. Therefore, applying Nil to (30) produces a complete and exhaustive N-indexedfiltration
󰀝󰀞n
F ilHKRNil HC− (−/Q)
on Nil HC− (−/Q)
(33)
n∈N

with identifications≥n󰁦 −/Q [2n].
grnF ilHKR Nil HC− (−/Q) ≃ Nil LΩ

(34)

≥n

󰁦 −/Q ∈ PSh(Schqcqs , D(Q)) satisfy (Desc), (Fin), andLemma 8.5. The presheaves Nil LΩ
Q(BB)≥−n .

Proof.
noe(Desc) By (Mor16b, Thm.2.10), the ∧n L−/Q ∈ PSh(Schqcqs
Q , D(Q)) are procdh sheaves on SchQ .
󰁦 −/Q = limn LΩ<n are also procdh sheaves. From
Hence, the LΩ<n , and their limit LΩ
−/Q

−/Q

≥n

󰁦 −/Q = fib(LΩ󰁦 −/Q → LΩ<n ) are procdh sheaves. Since
this we deduce that the fibres LΩ
−/Q
≥n

≥n

≥n

󰁦 −/Q = fib(LΩ󰁦 −/Q → acdh LΩ󰁦 −/Q ) are
all cdh sheaves are procdh sheaves, the fibres Nil LΩprocdh sheaves.

18 It is not necessarily exhaustive in general, but is exhaustive if X/k is L
X/k has Tor-amplitude contained in[0, 1].
19 That is, LΩ󰁦 X/k = lim LΩ<n where for aﬃnes X, the complex LΩ<n is the totalisation of the simplicial
X/k
X/k
←−n
chain complex [r] 󰀁→ σ <n Ω∗P /A for P• → B a polynomial A-algebra resolution of B and where σ <n refers torthe stupid truncation in cohomological degrees < n.

26󰁦 −/Q is a procdh sheaf. In fact, it is a cdh sheaf, (EM23, Lem.4.5),(Fin) Above we saw that LΩ󰁦
so Nil LΩ−/Q = 0 leading to an equivalence≥n󰁦 −/Q ∼
Nil LΩ= Nil LΩ<n−/Q [−1].

(BB)≥−n(35)i
So it suﬃces to show finitarity of Nil LΩ<n−/Q . The presheaves ∧ L−/Q are finitary and<nLΩ−/Q admits a finite filtration whose graded quotients are shifts of ∧i L−/Q , so it follows
that the LΩ<n−/Q are finitary. The cdh sheafification functor preserves finitary sheaves so<nthe acdh LΩ<n−/Q , and therefore Nil LΩ−/Q are finitary.
By Eq.(35) it suﬃces to show that LΩ<nR/Q is supported in cohomological degree ≤ n − 1
for all R as in Proposition 8.3. We have<n<n(acdh LΩ<n−/Q )(R) = (acdh LΩ−/Q )(R/N) = LΩ(R/N)/Q ,(36)

where the first (resp. second) equality holds since any finitary cdh sheaf is nil-invariant(resp. a valuation ring is a point of the cdh topos). Since LΩ<nA/Q for a local Q-algebra A
is supported in degrees ≤ n − 1, we are reduced to showing the surjectivity of the mapn−1
H n−1 (LΩ<n(LΩ<nR/Q ) → H
(R/N)/Q ).

n−1
This holds since the map is identified with Ωn−1
R/Q → Ω(R/N)/Q .

Proposition 8.6. Let X be a Noetherian Q-scheme of finite Krull dimension. Then Nil HC− (−/Q) ∈PSh(SchX , D(Q)) satisfy (Desc), (Fin), and (BB)≥0 . Therefore
aprocdh (Nil HC− (−/Q))≥0 (X) ≃ Nil HC− (X/Q).
Proof. The property (Desc) follows from the corresponding property for the graded pieces≥n󰁦 −/Q of the filtration (33). For (Fin) and (BB)≥0 consider the spectral sequence induced
Nil LΩ
by (33)≥−j󰁦 X/Q ) ⇒ H i+j Nil HC− (X/Q).
E i,j = H i−j (Nil LΩ
2
qcqs
If X ∈ SchQ has finite valuative dimension d with Noetherian underlying topological spaces,≥−j󰁦 −/Q from Lemma 8.5 imply E i,j = 0 for i − j > 2d − j,Corollary 7.10 and (BB)≥j for Nil LΩ
2which implies that the spectral sequence is bounded. Hence, (Fin) and (BB)≥0 for HC− (−/Q)≥−j󰁦 −/Q , Lemma 8.5.
follows from (Fin) and (BB)≥j for Nil LΩ

8.3

Integral topological cyclic homology

For X ∈ Schqcqs
write TC(X) for the integral topological cyclic homology of X. By (BMS19),
Fp
for any qcqs Fp -scheme X, there exists a functorial complete decreasing N-indexed filtration20
󰀝󰀞n
F ilBMS TC(X)
on TC(X)(37)
n∈Nwith associated graded quotients
grnF ilBMS TC(X) ≃ Z(n)syn (X)[2n]for a natural object Z(n)syn ∈ PSh(SchFp , D(Zp )) called the syntomic complex.
20 (BMS19) treats quasi-syntomic rings and it is extended to all p-complete rings in (AMMN22).

27

Lemma 8.7. The presheaf Nil Z(n)syn ∈ PSh(Schqcqs
Fp , Spt) satisfies (Desc), (Fin) and (BB)≥−n .
Proof.
(Desc) This property for Nil Z(n)syn follows from the fact that Z(n)syn is a procdh sheaf on
SchnoeFp by (EM23, Th.8.6) and Theorem 6.1.
(Fin) The argument is taken from the proof of (EM23, Th.4.24(4)). First, we claim Nil Z(n)syn [ p1 ] =•0. Using the fact that F ilBMS
TC from (37) naturally splits after inverting p, the claim
1
is reduced to Nil TC[ p ] = 0. To prove this, we use the following pullback square in
PSh(Schqcqs , Spt)tr
󰈣 TC ,
K(38)
󰈃󰈃
trcdh 󰈣KH
acdh TC(BB)≥−nwhere KH is the homotopy K-theory and tr is the cyclotomic trace ((BHM93), (DGM13)),and trcdh is induced by tr via the equivalence KH ≃ acdh K, (KST18b, Th.6.3), (Cis13),(KM21).
The pullback square follows from the latter equivalence and the fact that the fiber of tr is a
cdh sheaf by (LT19, Th. A.3). By (38), Nil TC[ p1 ] = 0 follows from fib(K → KH)[ p1 ] = 0
by (TT90, Th. 9.6). Thus, it suﬃces to show that Z(n)syn (−)/p is finitary noting that the
cdh sheafification of a finitary presheaf is finitary. By (EM23, Lem.4.16), Z(n)syn (−)/p
admits a finite increasing filtration whose graded pieces are some shifts of ∧i LΩ−/Fp withi ≤ n, so the desired assertion follows from the finitarity of ∧i LΩ−/Fp .
By the finitarity and Theorem 3.10, it suﬃces to show that Nil Z(n)syn (R) ∈ D(Zp ))≤nfor any henselian local ring R such that the ideal N ⊂ R of nilpotent elements is finitely
generated and R/N is a valuation ring. Similarly to (36), we haveacdh Z(n)syn (R) = acdh Z(n)syn (R/N) = Z(n)syn (R/N).
Note that Z(n)syn and acdh Z(n)syn are not finitary in general so the first equality requires
the assumption that N is finitely generated. Hence, (BB)≥−n follows from (AMMN22,
Th.5.2) noting (R, N) is a henselian pair. This completes the proof of the claim.

Proposition 8.8. The presheaf Nil TC ∈ PSh(Schqcqs
Fp , Spt) satisfies (Desc), (Fin), and (BB)≥0 .
Consequently, for any Noetherian Fp -scheme X with dim(X) < ∞ we haveaprocdh (Nil TC)≥0 (X) ≃ Nil TC(X).
Proof. The desired properties follow from Lemma 8.7 by the same argument deducing Proposition 8.6 from Lemma 8.5: The property (Desc) follows from (Desc) for Nil Z(n)syn by thefiltration (37). The properties (Fin) and (BB)≥0 for Nil TC follow from (Fin) and (BB)≥j for
Nil Zp (−j)syn by using the spectral sequenceE2i,j = H i−j (Nil Zp (−j)syn (X)) ⇒ π−i−j Nil TC(X),arising from the filtration (37). Note that the spectral sequence is bounded since Corollary 7.10
and (BB)≥j for Nil Zp (−j)syn from Lemma 8.7 imply E2i,j = 0 for i − j > 2d − j.

28

9Procdh motivic complex

In this section, we address the following conjecture.
Conjecture 9.1 (Beilinson, (Beı̆87), cf. also (EM23, Introduction)). For any reasonable
scheme X, there is a natural spectral sequence:p−q
E2p,q = HM
(X, Z(−q)) ⇒ K−p−q (X)(39)

iwhere K∗ (X) is the non-connective algebraic K-theory of X, and HM(X, Z(n)) is the motivic
cohomology of X yet to be defined.

When X is smooth over a field k, an answer was given by the following theorem.
Theorem 9.2 ((FS02) and (Lev08)). Let Smk denote the category of smooth schemes separated
of finite type over󰀝
󰀞 a field k. There exists a complete multiplicative decreasing N-indexed filtrationn
F ilmotK(X)

on K(X) and identifications of spectran∈NgrnF ilmot K(X) ≃ Z(n)sm (X)[2n](40)functorial in X ∈ Smk .

Above, Z(n)sm (X)[2n] is a chain complex of abelian groups regarded as a spectrum via theEilenberg-Maclane functor D(Z) → Spt. As a complex of abelian groups, it can be defined as
Z(n)sm (X) = C ∗ (Ztr (G∧nm ))(X)[−n]for X ∈ Smk ,(41)

1
where C ∗ (Ztr (G∧qm ))[−q] is Voevodsky’s A -invariant motivic complex defined in (SV00). Thisis strictly functorial in Smk in the sense that it defines a functor between the 1-category Smk
and the 1-category of chain complexes of abelian groups. Scheme-wise this is shown to be quasiisomorphic to Bloch’s cycle complex, (Blo86), in (Voe02, Cor.2). Of course, another approachis to just take Eq.(40) as the definition for Z(n)sm (X)[2n].

In what follows, we write F = Q or Fp and continue to let Schqcqs
be the category of qcqs
F
schemes over F and Schnoe
be its full subcategory of noetherian schemes. Recently, ElmantoF
Morrow (EM23) extended Theorem 9.2 to all X ∈ Schqcqs
by using instead of Z(n)sm a newFmotivic complex
Z(n)EM ∈ PSh(Schqcqs, D(Z))).
F
They construct Z(n)EM by modifying the cdh sheafification Z(n)cdh of the left Kan extensionof Z(n)sm along SmF → Schqcqs
by using Hodge-completed derived de Rham complexes in caseF
F = Q and syntomic complexes in case F = Fp . The construction is motivated by trace methods
in algebraic K-theory using the cyclotomic trace map tr from (38). The purpose of this section
is to give a diﬀerent approach to Conjecture 9.1 by using our procdh topology.
Definition 9.3. For integers n ≥ 0, we define the procdh-local motivic complex
Z(n)procdh := aprocdh Lsm Z(n)sm ∈ Shvprocdh (Schqcqs, DZ)),
F
as the procdh sheafification of the left Kan extension Lsm Z(n)sm of Z(n)sm along SmF → Schqcqs.
FRemark 9.4. Note that there have been various constructions of motivic cohomology onsmooth schemes over Dedekind domains, (Gei04), (Lev01), (Spi18), (CD19), etc., and one
could apply the same construction, i.e., procdh sheafification of left Kan extension, to any ofthese.
29

Theorem 9.5. For X ∈ Schnoe< ∞, there exists a complete multiplicative21
F󰀝 with dim(X) 󰀞decreasing N-indexed filtrationn
F ilprocdh
K(X)

on K(X) and identificationsn∈N

grnF ilprocdh K(X) ≃ Z(n)procdh (X)[2n].
Proof. By Bhatt-Lurie (see (EHK+ 20, Ex. A.0.6)), there is a natural equivalenceK≥0 ≃ Lsm (K|SmF ),where the right hand side is the left Kan extension of K|SmF along SmF → Schqcqs. UsingF
Theorem 8.1, we obtain a filtration on K by procdh sheafifying the left Kan extension along•
SmF → SchF of F ilmot
K|SmF from Theorem 9.2. The identification of the graded pieces is clear.
Completeness follows from the boundedness described in Eq.(44) below.
iiRemark 9.6. By definition, Eq.(41), we have HZar
(Z(n)sm ) = 0 for i > n, where HZar
(Z(n)sm )smis the Zariski cohomology sheaf of Z(n) . It implies that for any local k-algebra A, we have

H i ((Lsm Z(n)sm )(A)) = 0

for i > n.

(42)

Thus, Corollary 7.10 implies that for X ∈ Schqcqs
of finite valuative dimension d with NoetheF
rian underlying topological space, we have
H i (Z(n)procdh (X)) = 0

for i > 2d + n.

(43)

In particular, when X is noetherian, we have for each n ∈ N
nπ−i F ilprocdh
K(X) = 0;i > 2d − n(44)

so the induced spectral sequence (39) withi
HM
(X, Z(n)) := H i (Z(n)procdh (X))

is bounded.
Now, it is a natural question if two constructions Z(n)EM and Z(n)procdh coincide.
Theorem 9.7. Assume given Z(n) ∈ PSh(Schqcqs, D(Z))) and consider the following condiF
tions.
(a) There is a natural comparison map
ψ : Z(n)sm → Z(n)|SmF
in PSh(SmF , D(Z))), whose induced map φ : Lsm Z(n)sm → Z(n) has the properties that
φ(R) is an equivalence for all procdh local rings R with length Q(R) and dim R finite.
(b) Z(n) is finitary. That is,
Z(n)(lim Pλ ) = colimZ(n)(Pλ )λλfor any cofiltered system (Pλ )λ∈Λ in Schqcqs
with aﬃne transition morphisms.
F
21 For compatibility of the monoidal structures with left Kan extension see (EM23, §2.3). Sheafification is also
compatible since it is defined using filtered colimits, see (Lur09, Prop.6.2.2.7) which works for a very generalclass of coeﬃcient categories.

30

(c) Z(n) is a procdh sheaf on Noetherian schemes. That is,Z(n) ∈ Shvprocdh (Schnoe
F , D(Z))).
If (a), (b), and (c) are satisfied then, φ induces an equivalence Z(n)procdh (X) ≃ Z(n)(X) for
any X ∈ Schnoe
F .
Proof. For procdh sheaves of chain complexes E ∈ Shvprocdh (Schnoe
F , D(Z)), we have a descent
spectral sequencepq
E2p,q = Hprocdh(X, HprocdhE) ⇒ H p+q E(X),qwhich converges by finiteness of cohomological dimension, Cor.7.10. Here Hprocdh
E is theprocdh sheafification of the presheaf of abelian groups X 󰀁→ π−q E(X). So since Z(n) andZ(n)procdh are procdh sheaves, Assumption (c), it suﬃces to show that the morphismqq
HprocdhZ(n)procdh → HprocdhZ(n)of procdh cohomology sheaves of abelian groups is an isomorphism on Noetherian schemes.
Since the 1-topos Shvprocdh (SchX ) has enough points, Thm.5.2, and cohomology commuteswith filtered colimits, and both Z(n) and Lsm Z(n)sm are finitary, Assumption (b), it suﬃces toshow that for all procdh local rings Spec(R) → X with R as in Assumption (a), the comparison
H q Lsm Z(n)sm (R) → H q Z(n)(R)
is an isomorphism for all q ∈ Z. This is precisely Assumption (a).
Theorem 9.8 (Elmanto-Morrow). The presheaf Z(n)EM satisfies the conditions (a), (b), (c)of Theorem 9.7.
Proof. Condition (b), i.e., finitary-ness, is (EM23, Th.4.10(5), Th.4.24(4)) and Condition (c),i.e., procdh descent, is (EM23, Th.8.2).
We prove (a). Existence of the morphism Z(n)sm → Z(n)|SmF is contained in (EM23,
Th.1.1(9)), although we don’t need the full force of their theorem since we are only askingfor existence, not the stronger fact that the morphism is an equivalence. Now we want toshow that the induced map Lsm Z(n)sm → Z(n) is an equivalence on procdh local rings R withlength Q(R) and dim R finite.
By (EM23, Th.7.7), it suﬃces to prove Z(n)EM (R) is supported in cohomological degrees≤ n for any procdh local ring R. By (EM23, Th.4.10(2), Th.4.24.(2)), there are fiber sequences(cf. Eq.(29))≥n󰁦 R/Q → Z(n)EM (R) → Z(n)cdh (R) if F = Q,
Nil LΩ
Nil Z(n)syn (R) → Z(n)EM (R) → Z(n)cdh (R) if F = Fp ,≥n

󰁦 R/Q and Nil Z(n)syn (R) are supported inwhere Z(n)cdh = acdh Lsm Z(n)sm . Note that Nil LΩdegrees ≤ n by Lemma 8.5 and Lemma 8.7. Noting that Z(n)sm is finitary, the same argument
as Lemma 8.7 gives
Z(n)cdh (R) = Z(n)cdh (R/N) = (Lsm Z(n)sm )(R/N).
So, Z(n)cdh (R) is supported in degrees ≤ n by Eq.(42), which proves the desired assertion.
Corollary 9.9. There is an equivalence Z(n)procdh ≃ Z(n)EM in PSh(Schnoe
F , D(Z))).
31

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