A BETTER COMPARISON OF $\operatorname{cdh}$- AND $l\operatorname{dh}$-COHOMOLOGIES

Abstract

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$-linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

1 Introduction

Context—motives of singular schemes (with compact support). In [Voe00], Voevodsky constructed a triangulated category of motives $DM_{\operatorname{gm}}^{\operatorname{eff}}(k)$ using smooth schemes. In order to

• ∙ extend the motive functor $M:\mathsf{Sm}_{k}\rightarrow DM_{\operatorname{gm}}^{\operatorname{eff}}(k)$ from smooth $k$ -schemes to all separated finite type $k$ -schemes, and

• ∙ have access to a well-defined theory of motives with compact support $M^{c}:\mathsf{Sch}_{k}^{\operatorname{prop}}\rightarrow DM_{\operatorname{gm}}^{\operatorname{eff}}(k)$ ,

he proves a $\operatorname{cdh}$ -descent result. However, his proof only works in the presence of strong resolution of singularities (for example, in characteristic zero). In [Kel17], the resolution of singularities assumption was removed, at least if one works with $\mathbb{Z}[\frac{1}{p}]$ -coefficients, where $p$ is the exponential characteristic of the base field. The proof in [Kel17] has two main steps, namely [Kel17, Corollary 2.5.4] and [Kel17, Theorem 3.2.12]. This present article provides a shortcut to the first one. For more details about the strategy of [Kel17] we recommend consulting [Kel17, Chapter 1] and/or [Kel17, Chapter 4].

Main result. The main theorem of this article is:

Theorem 1. (cf. Theorem 30)

Suppose $S$ is a finite dimensional Noetherian separated scheme of positive characteristic $p\neq l$ , $\mathsf{Sch}_{S}$ the category of separated finite type $S$ -schemes, and $F$ a presheaf of $\mathbb{Z}_{(l)}$ -modules on $\mathsf{Sch}_{S}$ satisfying all of:

( $\operatorname{uh}$ -invariance)

$F(X)\cong F(Y)$ for any universal homeomorphism $Y\rightarrow X$ .

(Traces)

$F$ has covariant “trace” morphisms associated to finite flat surjective morphisms, see Definition 13.

(G1)

$F(R)\subseteq F(\operatorname{Frac}(R))$ for every finite rank Hensel valuation ring ¹   $R$ .

(G2)

$F(R)\rightarrow F(R/\mathfrak{p})$ is surjective for every finite rank Hensel valuation ring $R$ and prime $\mathfrak{p}\subset R$ .

Then the canonical comparison morphism is an isomorphism:

$$\begin{eqnarray}H_{\operatorname{cdh}}^{n}(S,F_{\operatorname{ cdh}})\stackrel{{\sim}}{\rightarrow }H_{l\operatorname{dh}}^{n}(S,F_{l\operatorname{dh}}).\end{eqnarray}$$

However, the main result of this article is its proof. The proof of [Kel17, Corollary 2.5.4] is a poorly structured collection of lemmas, which are difficult to arrange into some kind global narrative, and the hypotheses of [Kel17, Corollary 2.5.4] are an awkward list of very special properties that do not give much insight into why the cohomologies should agree.

The proof of Theorem 30 on the other hand, is short ²  and linear, and one can mostly explain how its hypotheses are used in the proof: $\mathbb{Z}_{(l)}$ -linearity and traces are to give descent for finite-flat-surjective-prime-to- $l$  morphisms (cf. Lemma 2), universal homeomorphism invariance is to correct non-Nagataness of Hensel valuation rings (cf. Lemma 3), and (G2) is to control $\text{}\underline{H}_{l\operatorname{dh}}^{1}F$ (cf. Equation (2) on page 204). Unfortunately, (G1) remains a little mysterious. It is easy to say how it is used in the proof (it produces inclusions $F(X)\subseteq F_{l\operatorname{dh}}(X)\subseteq \prod _{x\in X}F(x)$ , cf. Lemma 28, Proposition 29), but not why it should be a necessary ingredient. See also Remark 23 for more on this.

The class of presheaves covered by the hypotheses of Theorem 30 is reasonably large. Any cohomology theory representable in the motivic stable homotopy category is invariant under universal homeomorphisms (at least with $\mathbb{Z}[\frac{1}{p}]$ -coefficients), [EK18], [CD15, Lemma 3.15]. It is quite common for cohomology theories in algebraic geometry to have some kind of trace or transfer morphisms, see Example 14 for a list of examples. Many cohomology theories in algebraic geometry satisfy (G1), see also Gersten’s Conjecture for algebraic $K$ -theory [Qui73, Theorem 7.5.11], or Cousin complexes in the Bloch–Ogus–Gabber theorem [CTHK]. In particular, the cohomology theories representable in the motivic stable homotopy category that we are interested in satisfy (G1), [KM18, Remark 3.2], [Kel17, Theorem 3.3.1].

Condition (G2) seems newer. It is true for algebraic $K$ -theory, or more generally, for nilpotent invariant theories commuting with filtered colimits which satisfy “Milnor” excision, [KM18, Lemma 3.5]. The author is currently working with Elmanto, Hoyois, and Iwasa to prove “Milnor” excision for $SH$ with $\mathbb{Z}[\frac{1}{p}]$ -coefficients. See also Bhatt and Mathew’s newly minted arc-topology [BM18, Theorem 1.6].

The problem—non-Nagataness. One of the first things one might try when comparing the cohomology of a finer topology $\unicode[STIX]{x1D706}$ with a coarser one $\unicode[STIX]{x1D70E}$ is the change of topology spectral sequence

$$\begin{eqnarray}H_{\unicode[STIX]{x1D70E}}^{p}(X,(\text{}\underline{H}_{\unicode[STIX]{x1D706}}^{q}F)_{\unicode[STIX]{x1D70E}})\Rightarrow H_{\unicode[STIX]{x1D706}}^{p+q}(X,F_{\unicode[STIX]{x1D706}}).\end{eqnarray}$$

If one can show that $H_{\unicode[STIX]{x1D706}}^{p}(P,F_{\unicode[STIX]{x1D706}})=0$ (for $p>0$ and $F_{\unicode[STIX]{x1D706}}(P)=F(P)$ ) for schemes $P$ in a family inducing a conservative family of fiber functors of the $\unicode[STIX]{x1D70E}$ -topos, it follows that $(\text{}\underline{H}_{\unicode[STIX]{x1D706}}^{q}F)_{\unicode[STIX]{x1D70E}}=0$ (for $p>0$ and $F_{\unicode[STIX]{x1D70E}}=F_{\unicode[STIX]{x1D706}}$ ), the spectral sequence collapses, and one is done. If $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D706})=(\text{Zariski},\acute{\text{e}}\text{tale})$ then this would be to show that $H_{\operatorname{et}}^{p}(-,F_{\operatorname{et}})$ vanishes on all local rings. In our setting where $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D706})=(\operatorname{cdh},l\operatorname{dh})$ it amounts to showing that $H_{l\operatorname{dh}}^{p}(-,F_{l\operatorname{dh}})$ vanishes on finite rank Hensel valuation rings, (Appendix B).

To prove this vanishing, one would like to use a structure of trace morphisms on $F$ (as formalized in Definition 13) and the well-known fact that every $l\operatorname{dh}$ -covering of (the spectrum of) a Hensel valuation ring is refinable by a finite flat surjective morphism of degree prime to $l$ , via something like the following well-known lemma:

Lemma 2. (Lemma 15, [Kel17, Lemma 2.1.8])

Suppose that $F$ is a $\mathbb{Z}_{(l)}$ -linear presheaf with traces in the sense of Definition 13, and $f:Y\rightarrow X$ a finite flat surjective morphism of degree prime to $l$ . Then the complex

$$\begin{eqnarray}0\rightarrow F(X)\rightarrow F(Y)\rightarrow F(Y\times _{X}Y)\rightarrow F(Y\times _{X}Y\times _{X}Y)\rightarrow \cdots\end{eqnarray}$$

is exact.

The problem is that finite flat algebras over Hensel valuation rings are not necessarily products of Hensel valuation rings. We can try to return to Hensel valuation rings by normalizing (cf. Lemma 16), but then this takes us out of the category of finite algebras. To summarize:

Problem. Given a Hensel valuation ring $R$ and a finite faithfully flat $R$ -algebra $R\rightarrow A$ , in general, there is no $A$ -algebra $A^{\prime }$ such that $R\rightarrow A^{\prime }$ is finite faithfully flat and $A^{\prime }$ is a product of Hensel valuation rings.

The normalization $R\rightarrow \widetilde{A}$ is a product of Hensel valuation rings, and faithfully flat, but unless $\operatorname{Frac}(\widetilde{A})/\operatorname{Frac}(R)$ is finite separable and $R$ is discrete [Bou64, Chapter 6, Section 8, No. 5, Theorem 2, Corollary 1], the morphism $R\rightarrow \widetilde{A}$ is in general no longer finite, not even for a general discrete valuation ring $R$ (cf. Example 18).

The solution—pseudo-normalizations. The observation which rescues us is that $\mathbb{Z}[\frac{1}{p}]$ -motivic cohomology, and more generally the sheaves we are interested in, are invariant under universal homeomorphism. Since we can restrict our attention to finite rank Hensel valuation rings (cf. Corollary A.3), and all residue field extensions of a finite extension of valuation rings are finite, we do not have to normalize to catch all the information in the normalization that we need.

Lemma 3. (See Lemma 20)

Suppose $R$ is a finite rank Hensel valuation ring and $R\rightarrow A$ a finite faithfully flat algebra. Then there exists an $A$ -algebra $A^{\prime }$ such that $R\rightarrow A^{\prime }$ is finite faithfully flat and $\operatorname{Spec}(\widetilde{A^{\prime }})\rightarrow \operatorname{Spec}(A^{\prime })$ induces an isomorphism on underlying topological spaces and all residue fields.

Since our sheaves do not distinguish between $A^{\prime }$ and the (product of) valuation ring(s) $\widetilde{A^{\prime }}$ , we can use $A^{\prime }$ as though it were a valuation ring. To work with this lemma we introduce the notion of pseudo-normalization.

Definition 4. (See Definition 19)

Let $A$ be a ring and $A\rightarrow B$ an $A$ -algebra. Define a pseudo-integral closure ³  of $A$ in $B$ to be a finite sub- $A$ -algebra

$$\begin{eqnarray}A\rightarrow B^{\operatorname{pic}}\subseteq B^{\operatorname{ic}}\subseteq B\end{eqnarray}$$

of the integral closure $B^{\operatorname{ic}}$ of $A$ in $B$ such that $\operatorname{Spec}(B^{\operatorname{pic}})\rightarrow \operatorname{Spec}(B^{\operatorname{ic}})$ induces an isomorphism on topological spaces and residue fields. A pseudo-normalization of $A$ is a pseudo-integral closure of $A$ in its normalization

$$\begin{eqnarray}A\rightarrow \breve{A}\subseteq (A^{\operatorname{red}})^{{\sim}}.\end{eqnarray}$$

So the lemma above now becomes:

Lemma 5. (See Lemma 20)

Every finite faithfully flat algebra $A$ over a finite rank Hensel valuation ring admits a pseudo-normalization $A\rightarrow \breve{A}$ .

Outline. In Section 2 (resp. 3, resp. 4), we recall some well-known material on universal homeomorphisms (resp. presheaves with traces, resp. Hensel valuation rings). An interesting observation is that a morphism of schemes becomes an isomorphism of $l\operatorname{dh}$ -sheaves under Yoneda if and only if it is a universal homeomorphism, Corollary 9, Remark 10, (valid for any $l\neq p$ ). The $h$ -version of this statement is well-known and due to Voevodsky [Voe96] for excellent Noetherian schemes and Rydh [Ryd10] in general.

In Section 5, we introduce the notion of pseudo-integral closure and pseudo-normalization, and develop some basic properties.

Sections 6–8 (pages 199–205) contain our proof.

In Section 6, the condition (G1) appears, and we use it to show that for Hensel valuation rings $R$ we have $F(R)\cong F_{l\operatorname{dh}}(R)$ , Proposition 24.

In Section 7, we continue using (G1) to show that traces on $F$ induce traces on $F_{l\operatorname{dh}}$ , Proposition 29.

Section 8 (pages 203–205) contains our main theorem (Theorem 30) and the rest of its proof.

In Appendix A, we recall the definitions of the $\operatorname{cdh}$ - and $l\operatorname{dh}$ -topologies, and observe that for finite dimensional Noetherian schemes, the class of finite dimensional Hensel valuation rings induces a conservative family of fiber functors.

In Appendix B, we confirm that everything we need passes from finite type separated $S$ -schemes to all quasi-compact separated $S$ -schemes just as one would expect.

Convention. We work with a base scheme $S$ which will almost always be separated and Noetherian, and often of finite dimension. We write:

$\mathsf{Sch}_{S}$ for the category of separated finite type $S$ -schemes. Unless otherwise indicated, presheaf means a presheaf on $\mathsf{Sch}_{S}$ which is extended to the category

$\mathsf{SCH}_{S}$ of all quasi-compact separated $S$ -schemes by left Kan extension. That is $F(T)=\mathop{\varinjlim }\nolimits_{T\rightarrow X\rightarrow S}F(X)$ where the colimit is over factorizations through $X\in \mathsf{Sch}_{S}$ . Another way of saying this, (when $S$ is quasi-compact and quasi-separated), is that

presheaf means an additive functor $F:\mathsf{SCH}_{S}^{\operatorname{op}}\rightarrow \operatorname{Ab}$ that commutes with filtered colimits. For more on this see Appendix B.

We use

$h_{-}:C\rightarrow \mathsf{PreShv}(C);X\mapsto h_{X}$ for the Yoneda functor, and

$(-)_{\unicode[STIX]{x1D70F}}:\mathsf{PreShv}(C)\rightarrow \mathsf{Shv}_{\unicode[STIX]{x1D70F}}(C);F\mapsto F_{\unicode[STIX]{x1D70F}}$ for the sheafification functor. We write

$\text{}\underline{H}_{\unicode[STIX]{x1D70F}}^{n}F$ for the presheaf $H_{\unicode[STIX]{x1D70F}}^{n}(-,F_{\unicode[STIX]{x1D70F}})$ associated to a presheaf $F$ . This paper’s topologies (principally $\operatorname{cdh},l\operatorname{dh},\operatorname{fps}l^{\prime }$ ) are classically defined on $\mathsf{Sch}_{S}$ for $S$ Noetherian. We extend them to $\mathsf{SCH}_{S}$ in the canonical way (cf. Appendix B).

$Q(A)$ is the total ring of fractions of a ring $A$ . We will only ever apply this to reduced rings.

$l$ and $p$ are always distinct primes.

2 Universal homeomorphisms

In this section, we recall some basic material on universal homeomorphisms ( $\operatorname{uh}$ ), and show that $l\operatorname{dh}$ -sheafification preserves $\operatorname{uh}$ -invariances. For some background on the $l\operatorname{dh}$ -topology for Noetherian, and non-Noetherian schemes, we direct the reader to appendices A and B, respectively.

Definition 6. A morphism of schemes $f:Y\rightarrow X$ is a universal homeomorphism or $\operatorname{uh}$ if it satisfies the following equivalent conditions.

1. (1) For every $X$ -scheme $T\rightarrow X$ , the morphism $T\times _{X}Y\rightarrow T$ induces a homeomorphism on the underlying topological spaces.

2. (2) [Stacks, Tag 01S4, Tag 01S3] $f$ is a homeomorphism, and for every $y\in Y$ , the field extension $k(y)/k(f(y))$ is purely inseparable.

3. (3) [EGAIV4, Corollary 18.12.11] $f$ is integral, surjective and universally injective.

4. (4) [Stacks, Tag 01WM] $f$ is affine, surjective, and universally injective.

Note there are no finiteness conditions. There are universal homeomorphisms that are not of finite type, pertinently Example 18, and finite universal homeomorphisms that are not finitely presented. For example, $A=\mathop{\varinjlim }\nolimits_{n\rightarrow \infty }\mathbb{F}_{p}[x_{1},\ldots ,x_{n}]/\langle x_{i}x_{j}:i\neq j\rangle$ and the map $\unicode[STIX]{x1D719}:A\rightarrow A;x_{1}\mapsto x_{1}^{p}$ . To generate the kernel of $A[y]\rightarrow A;\sum a_{m}y^{m}\mapsto \sum \unicode[STIX]{x1D719}(a_{m})x_{1}^{m}$ , in addition to $y^{p}-x_{1}$ one needs all the $yx_{n}$ for $n>1$ .

Lemma 7. Suppose that $Y\rightarrow X$ is a universal homeomorphism, and ${\mathcal{A}}\subseteq {\mathcal{O}}_{Y}$ is a sub- ${\mathcal{O}}_{X}$ -algebra. Then both of $Y\rightarrow \text{}\underline{\operatorname{Spec}}({\mathcal{A}})\rightarrow X$ are universal homeomorphisms.

Proof. By Definition 6(3) it suffices to prove that $Y\rightarrow \text{}\underline{\operatorname{Spec}}({\mathcal{A}})$ is surjective. But this follows from integrality and injectivity of ${\mathcal{A}}\subseteq {\mathcal{O}}_{Y}$ , [Stacks, Tag 00GQ].◻

Recall that a morphism of schemes $Y\rightarrow X$ is completely decomposed if it induces a surjection $Y(K)\rightarrow X(K)$ of $K$ -points for every field $K$ .

Lemma 8. Let $S$ be a Noetherian separated scheme of positive characteristic $p$ . Every universal homeomorphism $Y\rightarrow X$ in $\mathsf{Sch}_{S}$ is refinable by a composition $Y_{1}\rightarrow Y_{0}\rightarrow X$ such that $Y_{0}\rightarrow X$ is proper and completely decomposed, and $Y_{1}\rightarrow Y_{0}$ is finite flat surjective and locally of degree a power of $p$ .

Proof. We reproduce the standard proof using Raynaud–Gruson flatification and induction on the dimension. The initial case is dimension $-1$ , that is, the empty scheme. Suppose that $Y\rightarrow X$ is a universal homeomorphism with $\dim X=n\geqslant 0$ , and suppose that the statement is true for a target scheme of dimension ${<}n$ . Replacing $X$ and $Y$ with their reductions, we can assume that $Y\rightarrow X$ is generically flat. In this case, by Raynaud–Gruson flatification [RG71, Theorem 5.2.2], there exists a nowhere dense closed subscheme $Z\subset X$ such that the strict transform $Y^{\prime }\rightarrow \operatorname{Bl}_{Z}X$ is globally flat. On the other hand, as $Y\rightarrow X$ is an integral morphism of finite type, it is finite. Therefore, $Y^{\prime }\rightarrow \operatorname{Bl}_{Z}X$ is flat and finite. Moreover, as $Y\rightarrow X$ is a universal homeomorphism, for each generic point $\unicode[STIX]{x1D709}\in X$ with corresponding point $\unicode[STIX]{x1D702}\in Y$ , the finite field extension $k(\unicode[STIX]{x1D702})/k(\unicode[STIX]{x1D709})$ is purely inseparable. In particular, as $Y^{\prime }$ and $\operatorname{Bl}_{Z}X$ are reduced, $Y^{\prime }\rightarrow \operatorname{Bl}_{Z}X$ is locally of degree a power of $p$ . Using the Noetherian inductive hypothesis to choose a refinement $W_{1}\rightarrow W_{0}\rightarrow Z$ of the universal homeomorphism $Y\times _{X}Z\rightarrow Z$ , we have produced the desired refinement $W_{1}\sqcup Y^{\prime }\rightarrow W_{0}\sqcup \operatorname{Bl}_{Z}X\rightarrow X$ of $Y\rightarrow X$ .◻

Corollary 9. Let $S$ be a Noetherian scheme of positive characteristic $p\neq l$ and $f:Y\rightarrow X$ a universal homeomorphism in $\mathsf{Sch}_{S}$ . Then the image $(h_{f})_{l\operatorname{dh}}\in \mathsf{Shv}_{l\operatorname{dh}}(\mathsf{Sch}_{S})$ of $f$ is an isomorphism.

Remark 10. In fact, the converse is also true: [Voe96, Theorem 3.2.9] and [Ryd10, Theorem 8.16] say that the image of $(h)_{\operatorname{h}}:\mathsf{Sch}_{S}\rightarrow \mathsf{Shv}_{\operatorname{h}}(\mathsf{Sch}_{S})$ is equivalent to the localization of $\mathsf{Sch}_{S}$ at the class $\operatorname{uh}$ . As $\operatorname{uh}$ is a right multiplicative system and satisfies the 2-out-of-6 property, it follows that a morphism in $\mathsf{Sch}_{S}$ is in $\operatorname{uh}$ if and only if it becomes an isomorphism in $\mathsf{Sch}_{\operatorname{h}}(\mathsf{Sch}_{S})$ , [KS06, 7.1.20]. Since $\mathsf{Sch}_{S}\rightarrow \mathsf{Shv}_{\operatorname{h}}(\mathsf{Sch}_{S})$ factors through $\mathsf{Shv}_{l\operatorname{dh}}(\mathsf{Sch}_{S})$ , the converse of Corollary 9 follows.

Proof. Surjectivity is a result of $f$ being (refinable by) an $l\operatorname{dh}$ -cover, Lemma 8. For injectivity, consider some $s,s^{\prime }\in (h_{Y})_{l\operatorname{dh}}(T)$ sent to the same element of $(h_{X})_{l\operatorname{dh}}(T)$ . Let $T^{\prime }\rightarrow T$ be an $l\operatorname{dh}$ -cover such that $s,s^{\prime }$ can be represented by some morphisms $t,t^{\prime }:T^{\prime }\rightarrow Y$ in $\mathsf{Sch}_{S}$ with $T^{\prime }$ reduced. Possibly refining $T^{\prime }$ , we can assume that $ft=ft^{\prime }$ . Let $\unicode[STIX]{x1D702}_{1},\ldots ,\unicode[STIX]{x1D702}_{n}$ be the generic points of $T^{\prime }$ . As $f:Y\rightarrow X$ is a universal homeomorphism, it is injective on the underlying topological space, and all residue field extensions are purely inseparable. Consequently, $\hom (\amalg \unicode[STIX]{x1D702}_{i},Y)\rightarrow \hom (\amalg \unicode[STIX]{x1D702}_{i},X)$ is injective, so $t|_{\amalg \unicode[STIX]{x1D702}_{i}}=t^{\prime }|_{\amalg \unicode[STIX]{x1D702}_{i}}$ . But $Y\rightarrow X$ is separated, and $T^{\prime }$ reduced, so $t=t^{\prime }$ .◻

Corollary 11. Let $S$ be a Noetherian scheme of positive characteristic $p\neq l$ . For any presheaf $F$ and $n\geqslant 0$ , the associated presheaf $(\text{}\underline{H}_{l\operatorname{dh}}^{n}F)(-)=H_{l\operatorname{dh}}^{n}(-,F_{l\operatorname{dh}})$ is $\operatorname{uh}$ -invariant.

Proof. Let $F_{l\operatorname{dh}}\rightarrow I^{\bullet }$ be an injective resolution in $\mathsf{Shv}_{l\operatorname{dh}}(\mathsf{Sch}_{S})$ . Then we have $(\text{}\underline{H}_{l\operatorname{dh}}^{n}F)(-)=H^{n}(\hom _{\mathsf{Shv}_{l\operatorname{dh}}}((h_{-})_{l\operatorname{dh}},I^{\bullet }))$ . But Corollary 9 says that $(h_{-})_{l\operatorname{dh}}$ sends universal homeomorphisms to isomorphisms. Hence, the same is true of $\text{}\underline{H}_{l\operatorname{dh}}^{n}F$ .◻

3 Traces and $\operatorname{fps}l^{\prime }$ -descent

This section contains material on presheaves with traces from [Kel17] included for the convenience of the reader.

Definition 12. We abbreviate finite flat surjective to $\operatorname{fps}$ , and finite flat surjective of degree prime to $l$ to $\operatorname{fps}l^{\prime }$ .

Definition 13. [Kel17, Definition 2.1.3]

Let ${\mathcal{S}}$ be a category of schemes admitting fiber products. A structure of traces on a presheaf $F:{\mathcal{S}}^{op}\rightarrow Ab$ is a morphism $\operatorname{Tr}_{f}:F(Y)\rightarrow F(X)$ for every $\operatorname{fps}$ morphism $f:Y\rightarrow X$ , satisfying the following axioms.

1. (1) (Add) We have $\operatorname{Tr}_{f\amalg f^{\prime }}=\operatorname{Tr}_{f}\oplus \operatorname{Tr}_{f^{\prime }}$ for every pair $Y\stackrel{f}{\rightarrow }X$ , $Y^{\prime }\stackrel{f^{\prime }}{\rightarrow }X^{\prime }$ of $\operatorname{fps}$ morphisms.

2. (2) (Fon) We have $\operatorname{Tr}_{f}\circ \operatorname{Tr}_{g}=\operatorname{Tr}_{f\circ g}$ for every composable pair $W\stackrel{g}{\rightarrow }Y\stackrel{f}{\rightarrow }X$ of $\operatorname{fps}$ morphisms.

3. (3) (CdB) We have $F(p)\circ \operatorname{Tr}_{f}=\operatorname{Tr}_{g}\circ F(q)$ for every $\operatorname{fps}$ morphism $f:Y\rightarrow X$ and every cartesian square

   

4. (4) (Deg) We have $\operatorname{Tr}_{f}\circ F(f)=d\cdot \operatorname{id}_{F(X)}$ for every $\operatorname{fps}$ morphism $f:Y\rightarrow X$ of constant degree $d$ .

A presheaf equipped with a structure of traces is called a presheaf with traces. A presheaf with traces taking values in the category of $R$ -modules for some ring $R$ , is called a presheaf of $R$ -modules with traces or an $R$ -linear presheaf with traces.

Example 14. (For these and more examples see [Kel17, Example 2.1.4])

1. (1) Every constant additive presheaf has a unique structure of traces determined by the axioms (Add) and (Deg).

2. (2) The trace and determinant equip $({\mathcal{O}},+)$ and $({\mathcal{O}}^{\ast },\ast )$ respectively with a structure of traces, [Kel17, Example 2.1.4(vii)].

3. (3) If $F$ is a presheaf with traces, then the associated Nisnevich sheaf $F_{\mathsf{Nis}}$ inherits a unique structure of traces compatible with the canonical morphism $F\rightarrow F_{\mathsf{Nis}}$ , [Kel17, Corollary 2.1.13]. This is essentially because a finite algebra over a Hensel local ring is a product of Hensel local rings.

4. (4) Pushforward of vector bundles would induce a structure of traces on higher $K$ -theory $K_{n}$ and homotopy invariant $K$ -theory $KH_{n}$ for every $n\in \mathbb{Z}$ , except (Deg) is only satisfied Zariski locally. The Nisnevich sheafifications of these sheaves $(K_{n})_{\mathsf{Nis}}$ and $(KH_{n})_{\mathsf{Nis}}$ have canonical structures of traces (cf. [Kel14, Proof of Lemma 3.1]).

Lemma 15. (cf. [Kel17, Lemma 2.1.8])

Suppose that $F$ is a $\mathbb{Z}_{(l)}$ -linear presheaf with traces, and $X_{\bullet }\rightarrow X_{-1}$ is a simplicial $X_{-1}$ -scheme such that each $X_{n+1}\rightarrow (\operatorname{cosk}_{n}X_{\bullet })_{n+1}$ is an $\operatorname{fps}l^{\prime }$ -morphism of constant degree, for example, $X_{n-1}=Y^{\times _{X}n}$ for some $\operatorname{fps}l^{\prime }$ -morphism of constant degree $Y\rightarrow X$ . Then the complex

$$\begin{eqnarray}0\rightarrow F(X_{-1})\rightarrow F(X_{0})\rightarrow F(X_{1})\rightarrow F(X_{2})\rightarrow \cdots\end{eqnarray}$$

is exact. Here the morphisms are $\sum (-1)^{i}d_{i}$ .

Consequently, $F$ is a $\operatorname{fps}l^{\prime }$ -sheaf, and we have both $\check{H}_{\operatorname{fps}l^{\prime }}^{n}(-,F)=0$ , and $H_{\operatorname{fps}l^{\prime }}^{n}(-,F_{\operatorname{fps}l^{\prime }})=0$ for $n>0$ .

Proof. For each $0\leqslant i<j\leqslant n$ we have the commutative diagram

All morphisms are $\operatorname{fps}l^{\prime }$ -morphisms of constant degree, [Stacks, Tag 01GN ]. Setting, $D_{m}=\deg (X_{m}\rightarrow X_{-1})/\deg (X_{m-1}\rightarrow X_{-1})=\deg (X_{m}\stackrel{d_{i}}{\rightarrow }X_{m-1})$ , the composition $ba$ is of degree $D_{n+1}/D_{n}$ . Now, it follows from the above diagram, that

$$\begin{eqnarray}\displaystyle \frac{1}{D_{n+1}}Tr_{d_{i}}F(d_{j})\stackrel{(Fon)}{=}\frac{1}{D_{n+1}}Tr_{pr_{1}}Tr_{ba}F(ba)F(pr_{2}) & \stackrel{(\text{Deg})}{=} & \displaystyle \frac{1}{D_{n}}Tr_{pr_{1}}F(pr_{2})\nonumber\\ \displaystyle & \stackrel{(CdB)}{=} & \displaystyle \frac{1}{D_{n}}F(d_{j-1})Tr_{d_{i}}.\nonumber\end{eqnarray}$$

In particular, $(1/D_{\bullet })Tr_{d_{0}}$ is a chain homotopy between the zero morphism and the identity morphism of the chain complex in the statement.

For the second statement, notice that any $\operatorname{fps}l^{\prime }$ -morphism is refinable by an $\operatorname{fps}l^{\prime }$ -morphism of constant degree. Hence, the Čech cohomologies of these two classes of morphisms are the same, and moreover, they generate the same topology. Since the colimit over all hypercovers calculates sheaf cohomology, [SGA42, Theorem 7.4.1(2)], vanishing of sheaf cohomology also follows from exactness of the sequence.◻

4 Hensel valuation rings

Recall that an integral domain $R$ is a valuation ring if for all nonzero $a\in \operatorname{Frac}(R)$ , we have $a\in R$ or $a^{-1}\in R$ . Equivalently, the set of ideals of $R$ is totally ordered by inclusion, [Bou64, Chapitre VI, §1.2, Théorème 1].

A valuation ring $R$ is a Hensel valuation ring or hvr if it extends uniquely to every finite field extension, [EP05, §4.1]. That is, for every finite field extension $L/\operatorname{Frac}(R)$ , there exists a unique valuation ring $R^{\prime }\subseteq L$ such that $L=\operatorname{Frac}(R^{\prime })$ and $R=L\cap R^{\prime }$ . A valuation ring is a hvr if and only if it satisfies Hensel’s Lemma, [EP05, Theorem 4.1.3].

We will frequently use the following lemma. Recall that $Q(-)$ denotes the total ring of fractions.

Lemma 16. Let $R$ be a hvr, let $R\rightarrow A$ be a finite $R$ -algebra, and let $Q(A^{\operatorname{red}})\rightarrow L$ be a finite morphism with $L$ reduced. Then the integral closure $A^{\operatorname{ic}}$ of $A^{\operatorname{red}}$ in $L$ is a product of valuation rings. If $A$ and $L$ are integral domains the induced morphism $\operatorname{Spec}(A^{\operatorname{ic}})\rightarrow \operatorname{Spec}(A)$ is a homeomorphism.

Proof. The integral closure $A^{\operatorname{ic}}$ is the product of the integral closures of the images of $A^{\operatorname{red}}$ in the residue fields of $L$ , so we can assume $A$ is an integral domain and $L$ a field. Replacing $R$ with its image ⁴  in $A$ , we can assume $R\rightarrow A$ is injective. Note that since $A$ is finite, the integral closure of $A$ in $L$ is equal to the integral closure of $R$ in $L$ . Now the first claim follows from the facts that the integral closure of a valuation ring is the intersection of the extensions [EP05, Corollary 3.1.4], and since $R$ is Henselian, by definition there is a unique extension to $L$ . Now, $R\subseteq A\subseteq A^{\operatorname{ic}}$ are integral extensions of rings so $\operatorname{Spec}(A^{\operatorname{ic}})\rightarrow \operatorname{Spec}(A)\rightarrow \operatorname{Spec}(R)$ are surjective [Stacks, Tag 00GQ ]. Since $R\subseteq A\subseteq A^{\operatorname{ic}}$ are integral extensions, the incomparability property implies that $\operatorname{Spec}(A^{\operatorname{ic}})\rightarrow \operatorname{Spec}(A)\rightarrow \operatorname{Spec}(R)$ are injective. Finally, since the prime ideals of $A^{\operatorname{ic}}$ and $R$ are totally ordered, the bijection $\operatorname{Spec}(A^{\operatorname{ic}})\rightarrow \operatorname{Spec}(A)$ is a homeomorphism.◻

Lemma 17. Suppose that $R$ is a hvr of characteristic $p$ with fraction field $K$ , and suppose that $K\rightarrow K^{\prime }$ is a purely inseparable extension, and $R^{\prime }$ the integral closure of $R$ in $K^{\prime }$ . Then $\operatorname{Spec}(R^{\prime })\rightarrow \operatorname{Spec}(R)$ is a universal homeomorphism.

Proof. The extension $R\rightarrow R^{\prime }$ is integral by definition, and $\operatorname{Spec}(R^{\prime })\rightarrow \operatorname{Spec}(R)$ is surjective as it is dominant and satisfies the going up property. So it remains to see that it is injective, and each extension of residue fields is purely inseparable. As $R$ is Henselian, $R^{\prime }$ is a valuation ring, and so its poset of primes is totally ordered. By the incomparability property, it follows that there is exactly one prime of $R^{\prime }$ lying over any prime of $R$ . Let $\mathfrak{p}\subset R$ be a prime and $\mathfrak{p}^{\prime }\subset R^{\prime }$ the prime lying over it. Localizing at $\mathfrak{p}$ we can assume both are maximal ideals. Then a given $a\in k(\mathfrak{p}^{\prime })=R^{\prime }/\mathfrak{p}^{\prime }$ , lifts to some $b\in R^{\prime }$ , and since $K^{\prime }/K$ is purely inseparable, $b^{p^{i}}\in K$ for some $i$ . Now $R^{\prime }$ is the integral closure of $R$ in $K^{\prime }$ , so $b$ satisfies some monic $f(T)\in R[T]$ . But then $b^{p^{i}}$ satisfies the monic $f(T)^{p^{i}}$ . Since valuation rings are normal, it follows that $b^{p^{i}}\in R$ . Consequently, $a^{p^{i}}\in k(\mathfrak{p})=R/\mathfrak{p}$ . So $k(\mathfrak{p}^{\prime })/k(\mathfrak{p})$ is purely inseparable.◻

5 Pseudo-normalization

One of the obstacles to using valuation rings to study finite type morphisms of Noetherian schemes is non-Nagataness: Suppose $R$ is a hvr and $R\subset A$ a finite extension with $A$ an integral domain. Then the normalization $\widetilde{A}$ of $A$ is a valuation ring, Lemma 16. If $\operatorname{Frac}(A)/\operatorname{Frac}(R)$ is finite separable and $R$ is discrete then the morphism $R\rightarrow \widetilde{A}$ is also finite [Bou64, Chapter 6, Section 8, No. 5, Theorem 2, Corollary 1]. However, in general, the morphism $R\rightarrow \widetilde{A}$ may not be finite.

Example 18. [Bou64, Chapter 6, Section 8, Exercise 3b]

Let $k=\mathbb{F}_{p}(X_{n})_{n\in \mathbb{N}}$ , and equip $K=k(U,V)$ , with the (discrete) valuation induced by $\unicode[STIX]{x1D719}:k(U,V)\rightarrow k((U));V\mapsto \sum _{i=0}^{\infty }X_{i}^{p}U^{ip}$ . Let $K^{\prime }=K(V^{1/p})$ . Then, (exercise), the unique extension of valued fields $K^{\prime }/K$ induces an isomorphism on both value groups and residue fields. Consequently, the corresponding morphism $R\rightarrow R^{\prime }$ of discrete valuation rings is not finite, and so the normalization $R^{\prime }$ of $R[V^{1/p}]$ is not finite over $R[V^{1/p}]$ .

On the other hand, at least if the rank of $R$ is finite, if we take a large enough member $\breve{A}$ of the filtered poset of finitely generated sub- $A$ -algebras of $\widetilde{A}$ , the induced morphism $\operatorname{Spec}(\widetilde{A})\rightarrow \operatorname{Spec}(\breve{A})$ will be a universal homeomorphism (and even completely decomposed). Hence, as far as $\operatorname{uh}$ -sheaves (and even $\operatorname{cdh}$ -sheaves, cf. [HK18, Lemma 2.9]) are concerned, the normalization might as well be finite.

To formalize this phenomenon, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-hvr.

Convention. [EGAII, Corollary 6.3.8], [Stacks, Tag. 035N, 035P] . By normalization $\widetilde{A}$ of a ring $A$ with finitely many minimal primes, we mean the integral closure of its associated reduced ring $A^{\operatorname{red}}$ in the total ring of fractions of $Q(A^{\operatorname{red}})$ of $A^{\operatorname{red}}$ .

Note, that usually one only talks of normalizations of reduced rings.

Definition 19. Let $A$ be a ring and $A\rightarrow B$ an $A$ -algebra. Define a pseudo-integral closure of $A$ in $B$ to be a finite sub- $A$ -algebra $A\rightarrow B^{\operatorname{pic}}\subseteq B^{\operatorname{ic}}$ of the integral closure $B^{ic}\subseteq B$ of $A$ in $B$ such that $\operatorname{Spec}(B^{\operatorname{pic}})\rightarrow \operatorname{Spec}(B^{\operatorname{ic}})$ is a completely decomposed universal homeomorphism.

A pseudo-normalization of a ring $A$ with finitely many minimal primes is a pseudo-integral closure of $A$ in its normalization $A\rightarrow \breve{A}\subseteq (A^{\operatorname{red}})^{{\sim}}$ . We will write $\mathsf{PseIntClo}(B/A)$ and $\mathsf{PseNor}(A)$ for the set of pseudo-integral closures, and pseudo-normalizations respectively.

A pseudo-hvr is an integral domain $A$ such that $\widetilde{A}$ is an hvr, and $A$ is a pseudo-normalization of itself, $A=\breve{A}$ . That is, $\operatorname{Spec}(\widetilde{A})\rightarrow \operatorname{Spec}(A)$ is a completely decomposed universal homeomorphism.

Even if pseudo-integral closures and pseudo-normalizations exist they are certainly not unique in general. If the normalization, respectively, integral closure is finite, then it is the final pseudo-normalization, respectively, pseudo-integral closure.

The following lemma contains the basic facts we need about pseudo-normalizations $\breve{A}$ , at least for finite rings $A$ over a finite rank hvr $R$ : (3) they exist, (2), (4) they are (ind-)functorial for dominant morphisms, and (5) they preserve flat morphisms.

Lemma 20. Let $R$ be a finite rank hvr.

1. (1) Let $A\rightarrow B$ be a ring homomorphism. If $B^{\operatorname{pic}}\subseteq B^{\operatorname{ic}}$ is a pseudo-integral closure, then any finitely generated subalgebra $B^{\operatorname{pic}}\subseteq B^{\prime }\subseteq B^{\operatorname{ic}}$ is also a pseudo-integral closure.

2. (2) If the collection of pseudo-integral closures of a ring homomorphism $A\rightarrow B$ is nonempty, then it is a filtered poset and $B^{\operatorname{ic}}=\bigcup _{B^{\operatorname{pic}}\in \mathsf{PseIntClo}(B/A)}B^{\operatorname{pic}}$ .

3. (3) If $A$ is a finite $R$ -algebra, and $Q(A^{\operatorname{red}})\rightarrow K$ finite with $K$ reduced, the poset $\mathsf{PseIntClo}(K/A)$ of pseudo-integral closures is nonempty. In particular, the conclusion of part (2) holds.

4. (4) If $\unicode[STIX]{x1D719}:A\rightarrow B$ is a morphism of finite $R$ -algebras such that $\operatorname{Spec}(\unicode[STIX]{x1D719})$ sends generic points to generic points, and

   
   a square of finite morphisms of reduced rings, then for any $A^{\operatorname{pic}}\in \mathsf{PseIntClo}(K/A)$ there is a $B^{\operatorname{pic}}\in \mathsf{PseIntClo}(L/B)$ compatible with the above square. In other words, the canonical morphism of integral closures $A^{\operatorname{ic}}\rightarrow B^{\operatorname{ic}}$ of $A\rightarrow B$ in $K\rightarrow L$ induces a morphism of ind-objects
   

5. (5) If $\unicode[STIX]{x1D719}:A\rightarrow B$ is a finite flat morphism of finite $R$ -algebras, then there exists an inclusion $\breve{A}\subseteq \breve{B}$ of pseudo-normalizations compatible with $\unicode[STIX]{x1D719}$ , which is flat.

Proof.

1. (1) Since $B^{\operatorname{ic}}\supseteq B^{\operatorname{pic}}$ is integral so is $B^{\operatorname{ic}}\supseteq B^{\prime }$ , and consequently, $\operatorname{Spec}(B^{\operatorname{ic}})\rightarrow \operatorname{Spec}(B^{\prime })$ is surjective, [Stacks, Tag 00GQ ]. But $\operatorname{Spec}(B^{\operatorname{ic}})\rightarrow \operatorname{Spec}(B^{\operatorname{pic}})$ is a homeomorphism, so $\operatorname{Spec}(B^{\operatorname{ic}})\rightarrow \operatorname{Spec}(B^{\prime })$ is also injective. Finally, for any point $x\in \operatorname{Spec}(B^{\operatorname{ic}})$ with images $y,z$ in $\operatorname{Spec}(B^{\prime }),\operatorname{Spec}(B^{\operatorname{pic}})$ , The isomorphism $k(x)\cong k(z)$ implies an isomorphism $k(x)\cong k(y)$ .

2. (2) Follows from part (1).

3. (3) Let $A^{\operatorname{ic}}$ be the integral closure of $A^{\operatorname{red}}$ in $K$ . Since $A^{\operatorname{red}}$ is reduced with finitely many minimal primes, $Q(A^{\operatorname{red}})$ is a finite product of fields, [Stacks, Tag 02LV], so $K$ is also a finite product of fields. Hence $A^{\operatorname{ic}}$ is the product of the integral closures in the residue fields of $K$ , and therefore it suffices to consider the case $A$ and $K$ are both integral domains. Replacing $R$ with its image in $A$ , we can also assume $R\rightarrow A$ is injective. Now $R\subseteq A^{\operatorname{ic}}$ is an extension of valuation rings, Lemma 16. As $R\subseteq A^{\operatorname{ic}}$ is generically finite, for every prime $\mathfrak{p}\subset R$ , the extension $k(\mathfrak{p})\subset k(\mathfrak{q})$ is finite, where $\mathfrak{q}\subset A^{\operatorname{ic}}$ is the prime lying over $\mathfrak{p}$ , [Stacks, Tag 0ASH]. For each prime of $R$ , choose a set of generators of the corresponding finite field extension, lift them to $A$ , and let $A^{\operatorname{pic}}$ be the sub- $R$ -algebra that they generate. Certainly, $A^{\operatorname{pic}}$ is a finite $R$ -algebra. The morphism $A^{\operatorname{pic}}\subseteq A^{\operatorname{ic}}$ is an integral ring extension so $\operatorname{Spec}(A^{\operatorname{ic}})\rightarrow \operatorname{Spec}(A^{\operatorname{pic}})$ is surjective, [Stacks, Tag 00GQ ]. As $R$ is Henselian, $\operatorname{Spec}(A^{\operatorname{ic}})\rightarrow \operatorname{Spec}(R)$ is a homeomorphism, and we conclude that $\operatorname{Spec}(A^{\operatorname{ic}})\rightarrow \operatorname{Spec}(A^{\operatorname{pic}})$ is also a homeomorphism. By construction it induces isomorphisms on all field extensions. Hence, it is a completely decomposed universal homeomorphism. So $\mathsf{PseIntClo}(K/A)$ is nonempty.

4. (4) This follows from Part (3)—Existence, and Part (1)—Closure under subextension: Choose any $B_{0}^{\operatorname{pic}}\in \mathsf{PseIntClo}(B)$ and take $B^{\operatorname{pic}}$ to be the sub- $B$ -algebra of $L$ generated by the image of $A^{\operatorname{pic}}$ and $B_{0}^{\operatorname{pic}}$ .

5. (5) We will construct the following diagram. Note all morphisms except possibly the ones with source $A$ and $B$ are inclusions.

   

   Given some $\breve{A_{1}}\rightarrow \breve{B_{1}}$ extending $A\rightarrow B$ , consider the sub- $(A^{\operatorname{red}})^{{\sim}}$ -algebra $\breve{B}_{1}\subseteq B^{\prime }\subseteq (B^{\operatorname{red}})^{{\sim}}$ generated by the image of $\breve{B_{1}}$ . As $(B^{\operatorname{red}})^{{\sim}}$ is $(A^{\operatorname{red}})^{{\sim}}$ -torsion-free, so is $B^{\prime }$ , and as $(A^{\operatorname{red}})^{{\sim}}$ is a (product of) valuation rings, Lemma 16, we deduce that $B^{\prime }$ is $(A^{\operatorname{red}})^{{\sim}}$ -flat. As $\breve{A}_{1}\rightarrow \breve{B}_{1}$ is finite type, $(A^{\operatorname{red}})^{{\sim}}\rightarrow B^{\prime }$ is also finite type. Since $(A^{\operatorname{red}})^{{\sim}}$ is a product of local rings, the finitely generated flat module $B^{\prime }$ is a free module (Noetherianness is not needed, [Mat89, Theorem 7.10]). As $(A^{\operatorname{red}})^{{\sim}}=\bigcup _{\mathsf{PseNor}(A)}\breve{A}$ , we have ⁵   $B^{\prime }\cong (A^{\operatorname{red}})^{{\sim}}\otimes _{\breve{A_{2}}}\breve{B_{2}}$ for some finite free $\breve{A_{2}}\rightarrow \breve{B_{2}}$ with $\breve{A_{2}}$ in $\mathsf{PseNor}(A)$ (the ring $\breve{B_{2}}$ is not necessarily in $\mathsf{PseNor}(B)$ yet). Since $(\bigcup _{\breve{A}\supseteq \breve{A_{2}}\in \mathsf{PseNor}(A)}\breve{A})\otimes _{\breve{A_{2}}}\breve{B_{2}}\cong B^{\prime }\supseteq \breve{B}_{1}$ , tensoring with some large enough $\breve{A}_{3}\supseteq \breve{A}_{2}$ in $\mathsf{PseNor}(A)$ , we get our $\breve{B}_{3}\cong \breve{A}_{3}\otimes _{\breve{A}_{2}}\breve{B}_{2}\supseteq \breve{B}_{1}\supseteq \breve{B}_{0}$ , with $\breve{A}_{3}\rightarrow \breve{B}_{3}$ in $\mathsf{PseNor}(\unicode[STIX]{x1D719})$ . ◻

Lemma 21. Let $R$ be a hvr of characteristic $p\neq l$ . Then any $\operatorname{fps}l^{\prime }$ -morphism $R\rightarrow A$ is corefinable by the composition of a generically étale $\operatorname{fps}l^{\prime }$ -morphism $R\rightarrow A^{\prime }$ and a $\operatorname{uh}$ -morphism $A^{\prime }\rightarrow A^{\prime \prime }$ such that $R\rightarrow A^{\prime \prime }$ is also $\operatorname{fps}l^{\prime }$ .

Proof. First consider the case when $A$ is an integral domain. Consider the separable closure $L$ of $\operatorname{Frac}(R)$ in $\operatorname{Frac}(A)$ , and choose pseudo-integral closures $A^{\prime },A^{\prime \prime }$ of $R,A$ in $L,\operatorname{Frac}(A)$ respectively, Lemma 20(4).

By definition, $A^{\prime }\rightarrow \widetilde{A^{\prime }}$ and $A^{\prime \prime }\rightarrow \widetilde{A^{\prime \prime }}$ induce completely decomposed $\operatorname{uh}$ -morphisms, and $\widetilde{A^{\prime }}\rightarrow \widetilde{A^{\prime \prime }}$ induces a universal homeomorphism by Lemma 17. Hence $A^{\prime }\rightarrow A^{\prime \prime }$ is a universal homeomorphism.

So now it suffices to show that every $\operatorname{fps}l^{\prime }$ -morphism $R\rightarrow A$ is corefinable by an $\operatorname{fps}l^{\prime }$ -morphism $R\rightarrow A^{\prime \prime }$ with $A^{\prime \prime }$ an integral domain. Suppose first that $R$ is a field $K=R$ . Then one of the residue fields of $A$ is of degree prime to $l$ : Indeed, write $A=\prod A_{\mathfrak{p}_{i}}$ as the product of its local rings. As $l\nmid \dim _{K}A$ we have $l\nmid \dim _{K}A_{\mathfrak{p}_{i}}$ for some $i$ . As $\dim _{K}A_{\mathfrak{p}_{i}}=\sum _{j}\dim _{K}\mathfrak{p}_{i}^{j}/\mathfrak{p}_{i}^{j+1}$ and each $\mathfrak{p}_{i}^{j}/\mathfrak{p}_{i}^{j+1}$ is a $k(\mathfrak{p}_{j})$ -vector space, we see that $l\nmid [k(\mathfrak{p}_{i}):K]$ . For a general $R$ , the previous case applied to $\operatorname{Frac}(R)\rightarrow \operatorname{Frac}(R)\otimes _{R}A$ produces a minimal prime $\mathfrak{p}$ of $A$ such that $l\nmid [A_{\mathfrak{p}}^{\operatorname{red}}:\operatorname{Frac}(R)]$ . But then since flat  $=$  torsion-free over valuation rings, $R\rightarrow A/\mathfrak{p}$ is $\operatorname{fps}l^{\prime }$ .◻

Lemma 22. If $R$ is a finite rank hvr and $R\rightarrow A$ a finite $R$ -algebra, then every $l\operatorname{dh}$ -cover of $\operatorname{Spec}(A)$ is refinable by a finite one.

Proof. Since every $l\operatorname{dh}$ -covering is refinable by the composition of an $\operatorname{fps}l^{\prime }$ -covering and a $\operatorname{cdh}$ -covering (see the proof of B.5(4)) it suffices to prove the statement for $\operatorname{cdh}$ -coverings. Replacing $A$ with $\prod _{\substack{ \mathfrak{p}\subset A \\ \mathfrak{p}\text{ prime}}}(A/\mathfrak{p})^{\smallsmile }$ we can assume that $A$ is a pseudo-hvr. That is, $\widetilde{A}$ is a hvr and $\operatorname{Spec}(\widetilde{A})\rightarrow \operatorname{Spec}(A)$ is a completely decomposed universal homeomorphism. Let $\operatorname{Spec}(B)\rightarrow \operatorname{Spec}(A)$ be the $\operatorname{cdh}$ -covering in question, which we assume is affine, and since our $\operatorname{cdh}$ -topology is pulled back from a Noetherian scheme (cf. Proposition B.4), we also assume its of finite presentation $B\cong A[x_{1},\ldots ,x_{n}]/\langle f_{1},\ldots ,f_{m}\rangle$ . Every $\operatorname{cdh}$ -covering of a hvr admits a section, so there is a factorization $A\rightarrow B\rightarrow \widetilde{A}$ . As $A\rightarrow \widetilde{A}$ is an integral extension, the images of the $x_{i}$ in $\widetilde{A}$ satisfy some monic polynomials $g_{i}(T)\in A[T]$ . Then the composition $A\rightarrow B\rightarrow B^{\prime }:=A[x_{1},\ldots ,x_{n}]\langle f_{1},\ldots ,f_{m},g_{1}(x_{1}),\ldots ,g_{n}(x_{n})\rangle$ is a finite morphism, and its $\operatorname{Spec}$ is completely decomposed because $\operatorname{Spec}$ of the composition $A\rightarrow B^{\prime }\rightarrow \widetilde{A}$ is completely decomposed.◻

6 $l\operatorname{dh}$ -descent for pseudo-hvrs

From this point we start working with the following “Gestern” condition. A presheaf $F$ satisfies (G1) if:

1. (G1) For every hvr $R$ , the canonical morphism $F(R)\rightarrow F(\operatorname{Frac}(R))$ is a monomorphism.

Remark 23. We find it disappointing that we do not know a proof avoiding this condition, as its not really clear heuristically why it should be involved in passing traces from $F$ to $F_{l\operatorname{dh}}$ .

Voevodsky shows in [Voe00b, Corollary 4.18] that for a homotopy invariant presheaf with transfers $F$ and a smooth semilocal $k$ -scheme $X$ , the morphism $F(X)\rightarrow F(\unicode[STIX]{x1D702})$ to the generic scheme $\unicode[STIX]{x1D702}$ is a monomorphism. So, at least in the homotopy invariant setting over a field, traces imply a version of (G1). However, in our setting we do not have this: if $S$ is a Noetherian base scheme of dimension ${>}0$ , $s$ a nongeneric point, and $F$ the constant additive sheaf of some nontrivial abelian group $A$ , then $F(-\times _{S}s)$ is clearly $\operatorname{uh}$ -invariant, and has traces by Example 14(1), however, will not satisfy (G1).

Proposition 24. Suppose that $F$ is a $\operatorname{uh}$ -invariant $\mathbb{Z}_{(l)}$ -linear presheaf with traces satisfying (G1). Then for any pseudo-hvr $R$ of positive characteristic $p\neq l$ , we have $F(R)=F_{l\operatorname{dh}}(R)$ .

Proof. Since $F$ is $\operatorname{uh}$ -invariant, so is $F_{l\operatorname{dh}}$ , Corollary 11, so we can assume $R$ is a hvr.

Injectivity: $s\in \ker (F(R)\rightarrow F_{l\operatorname{dh}}(R))$ if and only if there is some $l\operatorname{dh}$ -covering $f:Y\rightarrow \operatorname{Spec}(R)$ such that $F(f)s=0$ . Since $R$ is a hvr we can assume that $f$ is an $\operatorname{fps}l^{\prime }$ -morphism, Proposition B.5(4). But then $s\stackrel{(\text{Deg})}{=}(1/\deg f)\operatorname{Tr}_{f}F(f)s=0$ .

Surjectivity: For every $s\in F_{l\operatorname{dh}}(R)$ there is an $l\operatorname{dh}$ -covering $f:Y\rightarrow \operatorname{Spec}(R)=:X$ such that $s|_{Y}\in \operatorname{im}(F(Y)\rightarrow F_{l\operatorname{dh}}(Y))$ . Since $R$ is a hvr we can assume that $f$ is an $\operatorname{fps}l^{\prime }$ -morphism, Proposition B.5(4). In fact, we can assume $f$ factors as $Z_{0}\stackrel{f_{1}}{\rightarrow }Y_{0}\stackrel{f_{0}}{\rightarrow }X$ with $f_{1}$ a $\operatorname{uh}$ -morphism, and $f_{0}$ a generically étale $\operatorname{fps}l^{\prime }$ -morphism, Lemma 21. Since $F$ is $\operatorname{uh}$ -invariant, so is $F_{l\operatorname{dh}}$ , Corollary 11, so we can forget $f_{1}$ and just work with $f_{0}$ . Since $f_{0}$ is generically étale, $Y_{0}\times _{X}Y_{0}$ is generically reduced. Choose $Y_{1}=(Y_{0}\times _{X}Y_{0})^{\smallsmile }$ such that the composition $\unicode[STIX]{x1D70B}_{1}:Y_{1}\rightarrow Y_{0}\times _{X}Y_{0}\stackrel{pr_{1}}{\rightarrow }Y_{0}$ is still flat, Lemma 20(5), and therefore $\operatorname{fps}l^{\prime }$ . We claim that

$$\begin{eqnarray}0\rightarrow F(X)\stackrel{F(f_{0})}{\longrightarrow }F(Y_{0})\stackrel{F(\unicode[STIX]{x1D70B}_{1})-F(\unicode[STIX]{x1D70B}_{2})}{\longrightarrow }F(Y_{1})\end{eqnarray}$$

is exact where $\unicode[STIX]{x1D70B}_{2}$ is the composition $Y_{1}\rightarrow Y_{0}\times _{X}Y_{0}\stackrel{pr_{2}}{\rightarrow }Y_{0}$ . Indeed, by $\operatorname{id}\stackrel{(\text{Deg})}{=}(1/\deg f_{0})\operatorname{Tr}_{f_{0}}F(f_{0})$ it is exact at $F(X)$ . For exactness at $F(Y_{0})$ , we claim that $\operatorname{Tr}_{\unicode[STIX]{x1D70B}_{1}}F(\unicode[STIX]{x1D70B}_{2})=F(f_{0})\operatorname{Tr}_{f_{0}}$ . Indeed, since $Y_{1}$ is $\operatorname{Spec}$ of a (product of) pseudo-hvrs, and $F$ is $\operatorname{uh}$ -invariant, by (G1) it suffices to check this after pulling back to the generic points of $Y_{1}$ . But by (CdB), it suffices to show $\operatorname{Tr}_{\unicode[STIX]{x1D70B}_{1}}F(\unicode[STIX]{x1D70B}_{2})=F(f_{0})\operatorname{Tr}_{f_{0}}$ holds generically, that is, over the generic point $\unicode[STIX]{x1D702}$ of $X$ . But

is cartesian, so the claim follows from (CdB). Hence, the above sequence is exact at $F(Y_{0})$ since if $F(\unicode[STIX]{x1D70B}_{2})s^{\prime }=F(\unicode[STIX]{x1D70B}_{1})s^{\prime }$ , then $s^{\prime }\stackrel{\text{(Deg)}}{=}(1/\deg \unicode[STIX]{x1D70B}_{1})\operatorname{Tr}_{\unicode[STIX]{x1D70B}_{1}}F(\unicode[STIX]{x1D70B}_{1})s^{\prime }\stackrel{\text{cycle}}{=}(1/\deg \unicode[STIX]{x1D70B}_{1})\operatorname{Tr}_{\unicode[STIX]{x1D70B}_{1}}F(\unicode[STIX]{x1D70B}_{2})s^{\prime }\stackrel{\text{(&#x201C;CdB&#x201D;)}}{=}(1/\deg \unicode[STIX]{x1D70B}_{1})F(f_{0})\operatorname{Tr}_{f_{0}}s^{\prime }$ .

So we have established that the top row in the following diagram is exact. Injectivity for pseudo-hvrs says the right vertical morphism is injective. Hence, by diagram chase, we find a preimage of $s$ in $F(X)$ .

7 Traces on $F_{l\operatorname{dh}}$

In this section, we show that for a nice presheaf with traces $F$ , the associated $l\operatorname{dh}$ -sheaf also has traces. We essentially transplant the method of [Kel12, Theorem 3.5.5], which becomes much easier in the context of this paper. For another approach to putting trace morphisms on $F_{l\operatorname{dh}}$ (with more restrictive hypotheses than we use here) see [Kel17].

Recall that in [Kel12, § 3.5] we defined

$$\begin{eqnarray}F_{\operatorname{cdd}}(X)=\mathop{\prod }_{x\in X}F(x)\end{eqnarray}$$

for $F$ a presheaf and $X$ a scheme.

Remark 25. One sees directly that $F_{\operatorname{cdd}}$ is a $\operatorname{cdh}$ -sheaf. More specifically, for any completely decomposed morphism $Y\rightarrow X$ , the sequence

$$\begin{eqnarray}0\rightarrow F_{\operatorname{cdd}}(X)\rightarrow F_{\operatorname{cdd}}(Y)\rightarrow F_{\operatorname{cdd}}(Y\times _{X}Y)\end{eqnarray}$$

is exact (showing this by hand using a splitting of $\amalg _{y\in Y}y\rightarrow \amalg _{x\in X}x$ is an easy exercise).

What takes a little bit more work is the following theorem.

Proposition 26. [Kel12, Theorem 3.5.5, Lemma 3.3.6(2), Definition 3.3.4, Proposition 3.5.7]

Let $F$ be a $\operatorname{uh}$ -invariant $\mathbb{Z}_{(l)}$ -linear presheaf with traces, where $l\neq \operatorname{char}s$ for all points $s\in S$ of the base scheme. Then there is a unique structure of traces on $F_{\operatorname{cdd}}$ such that $F\rightarrow F_{\operatorname{cdd}}$ is a morphism of presheaves with traces. Moreover, the trace morphisms on $F_{\operatorname{cdd}}$ satisfy:

(Tr)

If $A$ is a pseudo-hvr, $\unicode[STIX]{x1D719}:A\rightarrow B$ an $\operatorname{fps}$ morphism, $\mathfrak{p}_{1},\ldots ,\mathfrak{p}_{n}$ are the minimal ideals of $B$ , and $\unicode[STIX]{x1D702}_{i}:B\rightarrow (B/\mathfrak{p}_{i})^{\smallsmile }$ the canonical morphisms, then

where $m_{i}=\operatorname{length}~B_{\mathfrak{p}_{i}}$ , and of course, $(B/\mathfrak{p}_{i})^{\smallsmile }$ are chosen to be flat over $A$ , (or some $\operatorname{cduh}$ -extension of it, and we implicitly use $\operatorname{uh}$ -invariance), Lemma 20(5).

Remark 27. We do not need the following description, but in case the reader is interested, we recall that the trace morphisms on $F_{\operatorname{cdd}}$ are defined as follows. Given an $\operatorname{fps}$ morphism $f:Y\rightarrow X$ , and $y\in Y$ , the $y$ th component of the trace morphism $F_{\operatorname{cdd}}(Y)=\prod _{y\in Y}F(y)\rightarrow \prod _{x\in X}F(x)=F_{\operatorname{cdd}}(X)$ is given by $(\operatorname{length}\,{\mathcal{O}}_{x\times _{X}Y,y})\operatorname{Tr}_{f|_{y/x}}$ .

Lemma 28. Suppose that $F$ is a $\operatorname{uh}$ -invariant $\mathbb{Z}_{(l)}$ -linear presheaf with traces satisfying (G1), and $R$ a finite rank hvr of positive characteristic $p\neq l$ . Then on the category of finite $R$ -algebras, there is a factorization $F\rightarrow F_{l\operatorname{dh}}\stackrel{\unicode[STIX]{x1D704}}{{\hookrightarrow}}F_{\operatorname{cdd}}$ with $\unicode[STIX]{x1D704}$ a monomorphism.

Proof. On hvrs, and in particular on fields, $F\cong F_{l\operatorname{dh}}$ , Proposition 24, so $F_{\operatorname{cdd}}\cong (F_{l\operatorname{dh}})_{\operatorname{cdd}}$ . Our injection is $\unicode[STIX]{x1D704}:F_{l\operatorname{dh}}\rightarrow (F_{l\operatorname{dh}})_{\operatorname{cdd}}\stackrel{{\sim}}{\leftarrow }F_{\operatorname{cdd}}$ .

Let $A$ be a finite $R$ -algebra. Spec of the morphism $A\rightarrow \prod _{\substack{ \mathfrak{p}\subset A \\ \mathfrak{p}~\text{prime}}}(A/\mathfrak{p})^{\smallsmile }$ is a $\operatorname{cdh}$ -covering. Since $F_{l\operatorname{dh}}$ is $\operatorname{uh}$ -invariant, Corollary 11, each of the morphisms $F_{l\operatorname{dh}}((A/\mathfrak{p})^{\smallsmile })\rightarrow F_{l\operatorname{dh}}((A/\mathfrak{p})^{{\sim}})$ is an isomorphism. But $F\cong F_{l\operatorname{dh}}$ on the hvrs $(A/\mathfrak{p})^{{\sim}}$ and $k(\mathfrak{p})$ , Proposition 29, and $F$ satisfies (G1), so each $F_{l\operatorname{dh}}((A/\mathfrak{p})^{{\sim}})\rightarrow F_{l\operatorname{dh}}(k(\mathfrak{p}))$ is injective. Hence, $F_{l\operatorname{dh}}(A)\rightarrow \prod _{\substack{ \mathfrak{p}\subset A \\ \mathfrak{p}~\text{prime}}}F_{l\operatorname{dh}}(k(\mathfrak{p}))=(F_{l\operatorname{dh}})_{\operatorname{cdd}}(A)\cong F_{\operatorname{cdd}}(A)$ is injective.◻

Proposition 29. Suppose that $F$ is a $\operatorname{uh}$ -invariant $\mathbb{Z}_{(l)}$ -linear presheaf with traces satisfying (G1), and $R$ is a finite rank hvr of positive characteristic $p\neq l$ . Then on the category of finite $R$ -algebras, the trace morphisms on $F_{\operatorname{cdd}}$ descend to the subpresheaf $F_{l\operatorname{dh}}$ . In particular, $F_{l\operatorname{dh}}$ has a structure of traces on the category of finite $R$ -algebras.

Proof. Explicitly, we want to show that for every $\operatorname{fps}$ -morphism of $R$ -algebras $\unicode[STIX]{x1D719}:A\rightarrow B$ , the trace morphism $\operatorname{Tr}_{\unicode[STIX]{x1D719}}:F_{\operatorname{cdd}}(B)\rightarrow F_{\operatorname{cdd}}(A)$ sends the image of $F_{l\operatorname{dh}}(B){\hookrightarrow}F_{\operatorname{cdd}}(B)$ inside the image of $F_{l\operatorname{dh}}(A){\hookrightarrow}F_{\operatorname{cdd}}(A)$ . First note that if $A\rightarrow A^{\prime }$ is a morphism whose $\operatorname{Spec}$ is a completely decomposed proper ( $=$ finite) morphism, then a diagram chase using the short exact sequences, Remark 25, associated to $A\rightarrow A^{\prime }\rightrightarrows A^{\prime }\otimes _{A}A^{\prime }$ by $F_{l\operatorname{dh}}{\hookrightarrow}F_{\operatorname{cdd}}$ shows that $F_{l\operatorname{dh}}(A)=F_{l\operatorname{dh}}(A^{\prime })\cap F_{\operatorname{cdd}}(A)$ . So replacing $A$ with $A^{\prime }=\prod _{\substack{ \mathfrak{p}\subset A \\ \mathfrak{p}~\text{prime}}}(A/\mathfrak{p})^{\smallsmile }$ and using (Add) and (CdB), we can assume that $A$ is a pseudo-hvr. Now, using the morphism $B\rightarrow \prod _{\substack{ \mathfrak{q}\subset B \\ \mathfrak{q}~\text{prime}}}(B/\mathfrak{q})^{\smallsmile }$ and the property (Tr) stated in Proposition 26, we can assume that $B$ is also a pseudo-hvr. So now $A\rightarrow B$ is a $\operatorname{fps}$ -morphism between pseudo-hvrs. But then $F(A)\stackrel{{\sim}}{\rightarrow }F_{l\operatorname{dh}}(A)$ , $F(B)\stackrel{{\sim}}{\rightarrow }F_{l\operatorname{dh}}(B)$ are isomorphisms, Proposition 24. So the result follows from the fact that $F\rightarrow F_{\operatorname{cdd}}$ is a morphism of presheaves with traces, Proposition 26.◻

8 Comparison of $\operatorname{cdh}$ - and $l\operatorname{dh}$ -descent

Theorem 30. Suppose $S$ is a finite dimensional Noetherian separated scheme of positive characteristic $p\neq l$ and $F$ is a $\operatorname{uh}$ -invariant $\mathbb{Z}_{(l)}$ -linear presheaf with traces satisfying both conditions:

(G1)

$F(R)\rightarrow F(\operatorname{Frac}(R))$ is injective for every finite rank hvr $R$ .

(G2)

$F(R)\rightarrow F(R/\mathfrak{p})$ is surjective for every finite rank hvr $R$ and prime $\mathfrak{p}$ of codimension one.

Then the canonical comparison morphism is an isomorphism:

$$\begin{eqnarray}H_{\operatorname{cdh}}^{n}(S,F_{\operatorname{ cdh}})\stackrel{{\sim}}{\rightarrow }H_{l\operatorname{dh}}^{n}(S,F_{l\operatorname{dh}}).\end{eqnarray}$$

Proof. By the change of topology spectral sequence

$$\begin{eqnarray}H_{\operatorname{cdh}}^{i}(S,(\text{}\underline{H}_{l\operatorname{dh}}^{j}F)_{\operatorname{ cdh}})\Rightarrow H_{l\operatorname{dh}}^{i+j}(S,F_{l\operatorname{dh}})\end{eqnarray}$$

it suffices to show that $F_{\operatorname{cdh}}=F_{l\operatorname{dh}}$ , and $(\text{}\underline{H}_{l\operatorname{dh}}^{j}F)_{\operatorname{cdh}}=0$ for $j>0$ . Since finite rank hvrs form a conservative family of fiber functors for the $\operatorname{cdh}$ -site $\mathsf{Sch}_{S}$ , Corollary A.3, and cohomology commutes with filtered limits of schemes with affine transition morphisms, Proposition B.5, it suffices to show that for every finite rank hvr $R$ we have $F(R)=F_{l\operatorname{dh}}(R)$ , and $H_{l\operatorname{dh}}^{j}(R,F_{l\operatorname{dh}})=0$ for $j>0$ .

It was already shown in Proposition 24 that we have $F(R)=F_{l\operatorname{dh}}(R)$ . We now show that for any finite $R$ -algebra $A$ , we have

(1) $$\begin{eqnarray}H_{l\operatorname{dh}}^{j}(A,F_{l\operatorname{dh}})=0\end{eqnarray}$$

for $j>0$ . We work by induction on $(\dim \operatorname{Spec}(A),j)$ where $\mathbb{N}\times \mathbb{N}_{{>}0}$ has the lexicographical ordering. Explicitly, we suppose that (1) is true for $\dim \operatorname{Spec}(A)<\dim \operatorname{Spec}(R)$ , and all $0<j$ and suppose also that (1) is true when $\dim \operatorname{Spec}(A)=\dim \operatorname{Spec}(R)$ and $0<j<J$ . We will show that it is true for $\dim \operatorname{Spec}(A)=\dim \operatorname{Spec}(R)$ and $j=J$ .

Here is a plan of what we will prove, where $A^{\prime }=A^{\operatorname{pic}}$ is a pseudo-integral closure of $A^{\operatorname{red}}$ in $(Q(R)\otimes _{R}A)^{\operatorname{red}}$ , and $R^{\prime }=\widetilde{A^{\prime }}$ :

Step $\unicode[STIX]{x1D6FC}$ . Let $A^{\prime }=A^{\operatorname{pic}}$ be a pseudo-integral closure of $A^{\operatorname{red}}$ inside $(Q(R)\otimes _{R}A)^{\operatorname{red}}$ , let $\mathfrak{p}\subset R$ be the prime of height one (if $\dim \operatorname{Spec}(R)=0$ set $\mathfrak{p}$ to be the unit ideal $\mathfrak{p}=R$ ), and consider the blowup sequence, Proposition B.5(5),

(2) $$\begin{eqnarray}\displaystyle \cdots & \rightarrow & \displaystyle H_{l\operatorname{dh}}^{j-1}(A^{\prime }/\mathfrak{p},F_{l\operatorname{dh}})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{l\operatorname{dh}}^{j}(A,F_{l\operatorname{dh}})\rightarrow H_{l\operatorname{dh}}^{j}(A^{\prime },F_{l\operatorname{dh}})\oplus H_{l\operatorname{dh}}^{j}(A/\mathfrak{p},F_{l\operatorname{dh}})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{l\operatorname{dh}}^{j}(A^{\prime }/\mathfrak{p},F_{l\operatorname{dh}})\rightarrow \cdots\end{eqnarray}$$

By induction on $\dim \operatorname{Spec}(A)$ , (1) is true for $A/\mathfrak{p}$ and $A^{\prime }/\mathfrak{p}$ , so we obtain an isomorphism

(3) $$\begin{eqnarray}H_{l\operatorname{dh}}^{J}(A,F_{l\operatorname{dh}})\cong H_{l\operatorname{dh}}^{J}(A^{\prime },F_{l\operatorname{dh}}).\end{eqnarray}$$

Here, (G2) and $\operatorname{uh}$ -invariance of $F_{l\operatorname{dh}}$ , Corollary 11, is used in the case $J=1$ to obtain surjectivity of the morphism $F_{l\operatorname{dh}}(A^{\prime })\rightarrow F_{l\operatorname{dh}}(A^{\prime }/\mathfrak{p})$ .

Step $\unicode[STIX]{x1D6FD}$ . Note $\operatorname{Spec}$ of $A^{\prime }\rightarrow R^{\prime }:=\widetilde{A^{\prime }}$ is a $\operatorname{uh}$ . Since $H_{l\operatorname{dh}}^{J}(-,F_{l\operatorname{dh}})$ is $\operatorname{uh}$ -invariant, Corollary 11, we are reduced to showing that $H_{l\operatorname{dh}}^{J}(R^{\prime },F_{l\operatorname{dh}})$ vanishes.

Step $\unicode[STIX]{x1D6FE}$ . Consider the Čech spectral sequence

(4) $$\begin{eqnarray}E_{2}^{i,j}=\check{H}_{l\operatorname{dh}}^{i}(R^{\prime },\text{}\underline{H}_{l\operatorname{dh}}^{j}F)\Rightarrow H_{l\operatorname{dh}}^{i+j}(R^{\prime },F_{l\operatorname{dh}}).\end{eqnarray}$$

Note that the $i=0,j>0$ part vanishes automatically, since $\check{H}_{\unicode[STIX]{x1D70F}}^{0}(-,\text{}\underline{H}_{\unicode[STIX]{x1D70F}}^{j}F)=0$ vanishes for any topology $\unicode[STIX]{x1D70F}$ and $j>0$ . In particular,

(5) $$\begin{eqnarray}E_{2}^{0,J}=\check{H}_{l\operatorname{dh}}^{0}(R^{\prime },\text{}\underline{H}_{l\operatorname{dh}}^{J}F)\cong 0.\end{eqnarray}$$

Since every $l\operatorname{dh}$ -covering of $R^{\prime }$ is refinable by a finite one, Lemma 22, and $\text{}\underline{H}_{l\operatorname{dh}}^{j}F(-)=H_{l\operatorname{dh}}^{j}(-,F_{l\operatorname{dh}})$ vanishes on finite $R^{\prime }$ -algebras for $0<j<J$ by induction, it follows that

(6) $$\begin{eqnarray}E_{2}^{J-j,j}=\check{H}_{l\operatorname{dh}}^{J-j}(R^{\prime },\text{}\underline{H}_{l\operatorname{dh}}^{j}F)\cong 0;\quad \text{for }0<j<J.\end{eqnarray}$$

The vanishing so far, (5) and (6), with the spectral sequence (4) shows that

(7) $$\begin{eqnarray}E_{2}^{J,0}=\check{H}_{l\operatorname{dh}}^{J}(R^{\prime },F_{l\operatorname{dh}})\cong H_{l\operatorname{dh}}^{J}(R^{\prime },F_{l\operatorname{dh}}).\end{eqnarray}$$

Step $\unicode[STIX]{x1D6FF}$ . Since $R^{\prime }$ is a product of hvrs, every $l\operatorname{dh}$ -covering is refinable by an ${\operatorname{fps}l^{\prime }}^{\prime }$ -covering, Proposition B.5(4). So it suffices to show that the isomorphic groups

(8) $$\begin{eqnarray}\check{H}_{l\operatorname{dh}}^{J}(R^{\prime },F_{l\operatorname{dh}})\cong \check{H}_{\operatorname{fps}l^{\prime }}^{J}(R^{\prime },F_{l\operatorname{dh}})\end{eqnarray}$$

are zero.

Step $\unicode[STIX]{x1D716}$ . Since $F_{l\operatorname{dh}}$ has a structure of traces, Proposition 29, this vanishing follows directly from $\mathbb{Z}_{(l)}$ -linearity and the structure of traces, Lemma 15. ◻