corresponding to the components of $X$, namely,
$E_{n}=\operatorname{Spec}k[x]/x^{n}\sqcup\operatorname{Spec}k[y]/y^{n}=:E_{n,x}\sqcup
E_{n,y}$.

To a sequence of abelian groups $\dots A_{2}\to A_{1}\to A_{0}$ we associated
the presheaf of abelian groups on $\mathrm{Sch}_{S}$,

$F:T\mapsto\left\\{\begin{array}[]{lr}A_{m},&m=\min\\{i\ |\ \exists\ T\to
E_{i,x}\to S\\}\\\ 0,&\forall i,\ \not\exists\ T\to E_{i,x}\to
S.\end{array}\right.$

For $τ=\operatorname{Nis},\mathrm{procdh}$, let $F_{τ}$ be the
$τ$-sheafifiaction of $F$ considered as a presheaf of chain complexes. So
$H^{i}(F_{τ}(S))=H_{τ}^{i}(S,F)$, etc. The only $V\in X_{\operatorname{Nis}}$
admitting an $S$-morphism $V\to E_{i,x}$ is the empty scheme, so
$F_{\operatorname{Nis}}(X)=0$ and by Proposition 7.24 we have
$F_{\mathrm{procdh}}(X)=F_{\operatorname{Nis}}(X)$, so (25) holds. We have
(26) for the same reason. The condition (27) holds because $E_{n}$ is a
disjoint union of procdh local schemes. More precisely we have

$F_{\mathrm{procdh}}(E_{n})=F_{\mathrm{procdh}}(E_{n,x})\times
F_{\mathrm{procdh}}(E_{n,y})$ $=F(E_{n,x})\times
F(E_{n,y})=A_{n}\times\\{0\\}.$

where the $0$ is because there are no factorisations $E_{n,y}\to E_{i,x}$ for
$n>1$. So choosing any sequence $(A_{n})_{n\in\mathbb{N}}$ with
${\lim}^{1}_{n\in\mathbb{N}}A_{n}\neq 0$ produces an $F$ with
$H^{2}_{\mathrm{procdh}}(S,F)\neq 0$.

######  Remark 7.26.

It seems possible to push the above technique further to at least show that
the cohomological dimension of certain surfaces is 4. For example, Gabber
proposed trying the surface $S\times S$, where $S$ is the scheme from Example
7.25.

##  8 Application to $K$-theory

In this section, we work with the category $\mathrm{Sch}^{\mathrm{qcqs}}$ of
qcqs schemes and the full subcategory $\mathrm{Sch}^{\mathrm{noe}}$ of
noetherian schemes. For
$\mathcal{C}=\mathrm{Spt},D(\mathbb{Z}),D(\mathbb{Q}),D(\mathbb{F}_{p})$, we
write $\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}},\mathcal{C})$ for the
$\infty$-category of presheaves on $\mathrm{Sch}^{\mathrm{qcqs}}$ with values
in $\mathcal{C}$ and write
$\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}^{\mathrm{qcqs}},\mathcal{C})$ for
the full subcategory of the procdh sheaves with the sheafification functor

$a_{\mathrm{procdh}}:\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}},\mathcal{C})\to\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}^{\mathrm{qcqs}},\mathcal{C}).$

######  Remark 8.1.

Cf. [BS13, Rem.4.1.2]. Sheafification on large sites is rare in the
literature—and for good reason, it can fail to exist. Indeed, the fpqc
topology on $\mathrm{Sch}^{\mathrm{qcqs}}$ does not admit a sheafification
functor, essentially because fpqc coverings can have arbitrarily large
cardinality, see Example 8.2. For the procdh topology (and many other
topologies), one way to proceed is as follows. Choose an uncountable strong
limit cardinal $κ$, and implicitly set $\mathrm{Sch}^{\mathrm{qcqs}}$ to be
the category $\mathrm{Sch}^{\mathrm{qcqs},κ}$ of qcqs schemes of cardinality
$<κ$. Since procdh coverings are refinable by covering families whose
morphisms are of finite presentation, for any second larger uncountable strong
limit cardinal $κ<κ^{\prime}$ the following two obvious squares commute.

$\textstyle{\mathrm{PSh}(\mathrm{Sch}^{κ^{\prime},\mathrm{qcqs}},\mathcal{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$restriction$\scriptstyle{a_{\mathrm{procdh}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}^{κ^{\prime},\mathrm{qcqs}},\mathcal{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$inclusionrestriction$\textstyle{\mathrm{PSh}(\mathrm{Sch}^{κ,\mathrm{qcqs}},\mathcal{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{\mathrm{procdh}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}^{κ,\mathrm{qcqs}},\mathcal{C})}$inclusion

######  Example 8.2.

Let ${\operatorname{Field}}_{\mathbb{Q}}$ be the category of all field
extensions of $\mathbb{Q}$ and consider the presheaf $F$ that sends an
extension $K/\mathbb{Q}$ to the the smallest ordinal $ω$ of the same
cardinality as $K$, so $|F(K)|=|K|$. Note that for any inclusion $K\subseteq
L$ with $|K|=|L|$ we have _equality_ $F(K)=F(L)$. Suppose that $F$ has a
sheafification $F_{\operatorname{fpqc}}$ for the topology on
${\operatorname{Field}}_{\mathbb{Q}}$ induced by the fpqc topology, and set
$κ=|F_{\operatorname{fpqc}}(\mathbb{Q})|$. Choose an extension $K/\mathbb{Q}$
with $κ<|K|$. Let $R=im(\hom(K,-)\to\hom(\mathbb{Q},-))$ be the sieve
generated by $K$ so $R(L)$ is $\ast$ or $\varnothing$ according to whether
there exists a morphism $K\to L$ or not. All transitions $F(K)\to
F(K^{\prime})$ are inclusions, so $F\to F_{\operatorname{fpqc}}$ is an
inclusion of presheaves. This implies that
$\hom(R,F)\to\hom(R,F_{\operatorname{fpqc}})$ is also injective. But we have
$\hom(R,F)=\lim_{L\in R}F(L)=F(K)=|K|$, and therefore
$κ<|K|=|\hom(R,F)|\leq|\hom(R,F_{\operatorname{fpqc}})|$. Since
$κ=F_{\operatorname{fpqc}}(\mathbb{Q})=\hom(\hom(\mathbb{Q},-),F_{\operatorname{fpqc}})$,
we cannot have
$\hom(R,F_{\operatorname{fpqc}})=\hom(\hom(\mathbb{Q},-),F_{\operatorname{fpqc}})$,
contradicting the assumption that $F_{\operatorname{fpqc}}$ is an fpqc sheaf.
Note that this can be turned into an example on $\mathrm{Sch}^{\mathrm{qcqs}}$
by left Kan extending along
${\operatorname{Field}}_{\mathbb{Q}}^{op}\to\mathrm{Sch}^{\mathrm{qcqs}}$.

###  8.1 Algebraic $K$-theory

By [KST18b], non-connective $K$-theory satisfies procdh excision for
Noetherian schemes. Consequently, by Proposition 6.3 it defines a sheaf
$K\in\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}^{\mathrm{noe}},\mathrm{Spt})$.
We have the following topos-theoretic interpretation of the Bass construction.

######  Theorem 8.3.

For $X\in\mathrm{Sch}^{\mathrm{noe}}$ with $\dim(X)<\infty$, there exists a
natural equivalence

$(a_{\mathrm{procdh}}τ_{\geq 0}K)(X)\simeq K(X).$

###### Proof.

For $X$ as above, we have the descent spectral sequences

$E_{2}^{p,q}=H^{p}_{\mathrm{procdh}}(X,\widetilde{K}_{-q})\Rightarrow
K_{-p-q}(X),$ $E_{2}^{p,q}=H^{p}_{\mathrm{procdh}}(X,\tau_{\geq
0}\widetilde{K}_{-q})\Rightarrow\pi_{-p-q}(a_{\mathrm{procdh}}τ_{\geq
0}K(X)),$

where $\widetilde{K}_{i}$ is the procdh sheafification of the presheaf
$K_{i}=\pi_{i}K$ of abelian groups on $\mathrm{Sch}_{X}$, and $\tau_{\geq
0}\widetilde{K}_{i}=\widetilde{K}_{i}$ for $i\geq 0$ and $\tau_{\geq
0}\widetilde{K}_{i}=0$ for $i<0$. By Corollary 7.11, the spectral sequences
are bounded. So, it suffices to show that $\widetilde{K}_{i}=0$ for $i<0$. By
Theorem 5.2 the procdh 1-topos has enough points, so it is enough to show that
$\phi^{*}K_{i}=0$ for all $i<0$ and fibre functors
$\phi^{*}:\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}_{X})\to\mathrm{Set}$.
Since $K$-theory commutes with filtered colimits of rings, we have
$\phi^{*}K_{i}=K_{i}(R)$ where $R=\operatorname{colim}R_{λ}$ for
$(\operatorname{Spec}(R_{λ})\to X)_{\lambda\in\Lambda}$ the proobject
corresponding to $\phi^{*}$. Let $\mathfrak{N}\subset R$ be the ideal of
nilpotent elements. Then, for $i<0$, we get
$K_{i}(R)=K_{i}(R/\mathfrak{N})=0$, 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 $R/\mathfrak{N}$ is a valuation ring. This completes the
proof. ∎

######  Remark 8.4.

The proof of Theorem 8.3 shows the following. Note that we use Corollary 5.3
to reduce to the smaller class of those procdh local rings $R$ with
$\operatorname{length}Q(R)$ and $\dim R$ finite.

######  Proposition 8.5.

Take $\mathbb{F}=\mathbb{F}_{p}$ or $\mathbb{Q}$, set
$\mathcal{C}=\mathrm{Spt}$ or $D(\mathbb{Z})$, and let
$\mathcal{E}\in\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}},\mathcal{C})$
be a presheaf. Consider the following conditions.

  1. (Desc)

For each Noetherian $\mathbb{F}$-scheme $X$ the restriction
$\mathcal{E}|_{\mathrm{Sch}_{X}}$ is a procdh sheaf.

  2. (Fin)

$\mathcal{E}$ is finitary, in the sense that it preserves filtered colimits of
$\mathbb{F}$-algebras.

  3. (BB)≥N

For each procdh local ring $R$ with $\operatorname{length}Q(R)$ and $\dim R$
finite, the procdh stalk $\mathcal{E}(R)$ is homologically bounded below $N$.
That is, $π_{i}\mathcal{E}(R)=0$ for $i<N$.

If $\mathcal{E}$ satisfies (Desc), (Fin), and (BB)N then for every Noetherian
$\mathbb{F}$-scheme $X$ there is a natural equivalence

$a_{\mathrm{procdh}}(F_{\geq N})(X)\simeq F(X).$ (29)

We will produce a number of presheaves that satisfy the conditions of
Proposition 8.5 using the following definition.

######  Definition 8.6.

Let $\mathbb{F}$ and $\mathcal{C}$ be as in Proposition 8.5. For
$\mathcal{E}\in\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}},\mathcal{C})$,
define

${\operatorname{Nil}\mathcal{E}:=\operatorname{fib}(\mathcal{E}\to
a_{\mathrm{cdh}}\mathcal{E})\in\mathrm{PSh}(\mathrm{Sch}_{\mathbb{F}}^{\mathrm{qcqs}},\mathcal{C}),}$
(30)

where
$a_{\mathrm{cdh}}:\mathrm{PSh}(\mathrm{Sch}_{\mathbb{F}}^{\mathrm{qcqs}},\mathcal{C})\to\mathrm{Shv}_{\mathrm{cdh}}(\mathrm{Sch}_{\mathbb{F}}^{\mathrm{qcqs}},\mathcal{C})$
is the cdh sheafification functor.

###  8.2 Negative cyclic homology

Recall that for $k\to R$ a morphism of commutative rings, one defines
$\operatorname{HC}^{-}(R/k)=HH(R/k)^{hS^{1}}$ 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 $\mathbb{Z}$-indexed filtration,212121It is not necessarily
exhaustive in general, but is exhaustive if $X/k$ is $L_{X/k}$ has Tor-
amplitude contained in $[0,1]$.

$\biggl{\\{}Fil_{\operatorname{HKR}}^{n}\operatorname{HC}^{-}(X/k)\biggr{\\}}_{n\in\mathbb{Z}}\;\text{on
$\operatorname{HC}^{-}(X/k)$}$ (31)

and natural equivalences

$\operatorname{gr}_{Fil_{\operatorname{HKR}}}^{n}\operatorname{HC}^{-}(X/k)\simeq\widehat{L\Omega}^{\geq
n}_{X/k}[2n].$ (32)

Here, $\widehat{L\Omega}_{X/\mathbb{Q}}$ is the Hodge-completed derived de
Rham complex equipped with the Hodge filtration
$\bigl{\\{}\widehat{L\Omega}^{\geq
n}_{X/k}\bigl{\\}}_{n\in\mathbb{N}}$.222222That is,
$\widehat{L\Omega}_{X/k}=\varprojlim_{n}L\Omega_{X/k}^{<n}$ where for affines
$X$, the complex $L\Omega_{X/k}^{<n}$ is the totalisation of the simplicial
chain complex $[r]\mapsto\sigma^{<n}\Omega^{*}_{P_{r}/A}$ for $P_{\bullet}\to
B$ a polynomial $A$-algebra resolution of $B$ and where $σ^{<n}$ refers to the
stupid truncation in cohomological degrees $<n$. The graded pieces of this
filtration are computed by

$gr^{n}\widehat{L\Omega}_{X/k}\simeq\wedge^{n}L_{X/k}[-n]$ (33)

where $L_{X/k}\in\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}}_{k},D(k)))$ is the
cotangent complex, [Bha12, Construction 4.1].

By [EM23, Lem.4.5], the cofibre
${\operatorname{cofib}}(Fil_{\operatorname{HKR}}^{0}\operatorname{HC}^{-}(-/\mathbb{Q})\to\operatorname{HC}^{-}(-/\mathbb{Q}))$
is a cdh sheaf on $\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{Q}}$, so the
canonical morphism
$\operatorname{Nil}Fil_{\operatorname{HKR}}^{0}\operatorname{HC}^{-}(-/\mathbb{Q})\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{Nil}\operatorname{HC}^{-}(-/\mathbb{Q})$
is an equivalence of presheaves. Therefore, applying $\operatorname{Nil}$ to
(31) produces a complete and exhaustive $\mathbb{N}$-indexed filtration

${\biggl{\\{}Fil_{\operatorname{HKR}}^{n}\operatorname{Nil}\operatorname{HC}^{-}(-/\mathbb{Q})\biggl{\\}}_{n\in\mathbb{N}}\;\text{
on $\operatorname{Nil}\operatorname{HC}^{-}(-/\mathbb{Q})$}}$ (34)

with identifications

${\operatorname{gr}^{n}_{Fil_{\operatorname{HKR}}}\operatorname{Nil}\operatorname{HC}^{-}(-/\mathbb{Q})\simeq\operatorname{Nil}\widehat{L\Omega}^{\geq
n}_{-/\mathbb{Q}}[2n].}$ (35)

######  Lemma 8.7.

The presheaves $\operatorname{Nil}\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq
n}\in\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{Q}},D(\mathbb{Q}))$
satisfy (Desc), (Fin), and (BB)≥-n.

###### Proof.

  1. (Desc)

By [Mor16b, Thm.2.10], the
$\wedge^{n}L_{-/\mathbb{Q}}\in\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{Q}},D(\mathbb{Q}))$
are procdh sheaves on $\mathrm{Sch}_{\mathbb{Q}}^{\mathrm{noe}}$. Hence, the
$L\Omega^{<n}_{-/\mathbb{Q}}$, and their limit
$\widehat{L\Omega}_{-/\mathbb{Q}}=\lim_{n}L\Omega^{<n}_{-/\mathbb{Q}}$ are
also procdh sheaves. From this we deduce that the fibres
$\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq
n}=\operatorname{fib}(\widehat{L\Omega}_{-/\mathbb{Q}}\to
L\Omega_{-/\mathbb{Q}}^{<n})$ are procdh sheaves. Since all cdh sheaves are
procdh sheaves, the fibres $\operatorname{Nil}\widehat{L\Omega}^{\geq
n}_{-/\mathbb{Q}}=\operatorname{fib}(\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq
n}\to a_{\mathrm{cdh}}\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq n})$ are procdh
sheaves.

  2. (Fin)

Above we saw that $\widehat{L\Omega}_{-/\mathbb{Q}}$ is a procdh sheaf. In
fact, it is a cdh sheaf, [EM23, Lem.4.5], so
$\operatorname{Nil}\widehat{L\Omega}_{-/\mathbb{Q}}=0$ leading to an
equivalence

$\operatorname{Nil}\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq
n}\cong\operatorname{Nil}L\Omega_{-/\mathbb{Q}}^{<n}[-1].$ (36)

So it suffices to show finitarity of
$\operatorname{Nil}L\Omega_{-/\mathbb{Q}}^{<n}$. The presheaves
$\wedge^{i}L_{-/\mathbb{Q}}$ are finitary and $L\Omega_{-/\mathbb{Q}}^{<n}$
admits a finite filtration whose graded quotients are shifts of
$\wedge^{i}L_{-/\mathbb{Q}}$, so it follows that the
$L\Omega_{-/\mathbb{Q}}^{<n}$ are finitary. The cdh sheafification functor
preserves finitary sheaves so the
$a_{\mathrm{cdh}}L\Omega_{-/\mathbb{Q}}^{<n}$, and therefore
$\operatorname{Nil}L\Omega_{-/\mathbb{Q}}^{<n}$ are finitary.

  3. (BB)≥-n

By Eq.(36) it suffices to show that $L\Omega_{R/\mathbb{Q}}^{<n}$ is supported
in cohomological degree $\leq n-1$ for all $R$ as in Proposition 8.5. We have

${(a_{\mathrm{cdh}}L\Omega_{-/\mathbb{Q}}^{<n})(R)=(a_{\mathrm{cdh}}L\Omega_{-/\mathbb{Q}}^{<n})(R/\mathfrak{N})=L\Omega_{(R/\mathfrak{N})/\mathbb{Q}}^{<n},}$
(37)

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
$L\Omega_{A/\mathbb{Q}}^{<n}$ for a local $\mathbb{Q}$-algebra $A$ is
supported in degrees $\leq n-1$, we are reduced to showing the surjectivity of
the map

$H^{n-1}(L\Omega_{R/\mathbb{Q}}^{<n})\to
H^{n-1}(L\Omega_{(R/\mathfrak{N})/\mathbb{Q}}^{<n}).$

This holds since the map is identified with
$\Omega^{n-1}_{R/\mathbb{Q}}\to\Omega^{n-1}_{(R/\mathfrak{N})/\mathbb{Q}}$.

∎

######  Proposition 8.8.

Let $X$ be a Noetherian $\mathbb{Q}$-scheme of finite Krull dimension. Then
$\operatorname{Nil}\operatorname{HC}^{-}(-/\mathbb{Q})\in\mathrm{PSh}(\mathrm{Sch}_{X},D(\mathbb{Q}))$
satisfy (Desc), (Fin), and (BB)≥0. Therefore

$a_{\mathrm{procdh}}(\operatorname{Nil}\operatorname{HC}^{-}(-/\mathbb{Q}))_{\geq
0}(X)\simeq\operatorname{Nil}\operatorname{HC}^{-}(X/\mathbb{Q}).$

###### Proof.

The property (Desc) follows from the corresponding property for the graded
pieces $\operatorname{Nil}\widehat{L\Omega}_{-/\mathbb{Q}}^{\geq n}$ of the
filtration (34). For (Fin) and (BB)≥0 consider the spectral sequence induced
by (34)

$E_{2}^{i,j}=H^{i-j}(\operatorname{Nil}\widehat{L\Omega}^{\geq-j}_{X/\mathbb{Q}})\Rightarrow
H^{i+j}\operatorname{Nil}\operatorname{HC}^{-}(X/\mathbb{Q}).$

If $X\in\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{Q}}$ has finite valuative
dimension $d$ with Noetherian underlying topological spaces, Corollary 7.11
and (BB)≥j for $\operatorname{Nil}\widehat{L\Omega}^{\geq-j}_{-/\mathbb{Q}}$
from Lemma 8.7 imply $E_{2}^{i,j}=0$ for $i-j>2d-j$, which implies that the
spectral sequence is bounded. Hence, (Fin) and (BB)≥0 for
$\operatorname{HC}^{-}(-/\mathbb{Q})$ follows from (Fin) and (BB)≥j for
$\operatorname{Nil}\widehat{L\Omega}^{\geq-j}_{-/\mathbb{Q}}$, Lemma 8.7. ∎

###  8.3 Integral topological cyclic homology

For $X\in\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}_{p}}$ write
$\operatorname{TC}(X)$ for the integral topological cyclic homology of $X$. By
[BMS19], for any qcqs $\mathbb{F}_{p}$-scheme $X$, there exists a functorial
complete decreasing $\mathbb{N}$-indexed filtration232323[BMS19] treats quasi-
syntomic rings and it is extended to all $p$-complete rings in [AMMN22].

${\biggl{\\{}Fil_{\operatorname{BMS}}^{n}\operatorname{TC}(X)\biggl{\\}}_{n\in\mathbb{N}}\text{
on }\operatorname{TC}(X)}$ (38)

with associated graded quotients

$\operatorname{gr}_{Fil_{\operatorname{BMS}}}^{n}\operatorname{TC}(X)\simeq{\mathbb{Z}(n)^{\operatorname{syn}}}(X)[2n]$

for a natural object
${\mathbb{Z}(n)^{\operatorname{syn}}}\in\mathrm{PSh}(\mathrm{Sch}_{\mathbb{F}_{p}},D(\mathbb{Z}_{p}))$
called the syntomic complex.

######  Lemma 8.9.

The presheaf
$\operatorname{Nil}{\mathbb{Z}(n)^{\operatorname{syn}}}\in\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}_{p}},\mathrm{Spt})$
satisfies (Desc), (Fin) and (BB)≥-n.

###### Proof.

  1. (Desc)

This property for $\operatorname{Nil}{\mathbb{Z}(n)^{\operatorname{syn}}}$
follows from the fact that ${\mathbb{Z}(n)^{\operatorname{syn}}}$ is a procdh
sheaf on $\mathrm{Sch}^{\mathrm{noe}}_{\mathbb{F}_{p}}$ by [EM23, Th.8.6] and
Proposition 6.3.

  2. (Fin)

The argument is taken from the proof of [EM23, Th.4.24(4)]. First, we claim
$\operatorname{Nil}{\mathbb{Z}(n)^{\operatorname{syn}}}[\frac{1}{p}]=0$. Using
the fact that $Fil_{\operatorname{BMS}}^{\bullet}\operatorname{TC}$ from (38)
naturally splits after inverting $p$, the claim is reduced to
$\operatorname{Nil}\operatorname{TC}[\frac{1}{p}]=0$. To prove this, we use
the following pullback square in
$\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}},\mathrm{Spt})$

${\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
12.16664pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-27.97221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
19.65271pt\raise 5.15279pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.15279pt\hbox{$\scriptstyle{{\operatorname{tr}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 43.16515pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
43.16515pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{TC}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
53.38737pt\raise-27.97221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{KH\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.3759pt\raise-31.9583pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.84723pt\hbox{$\scriptstyle{{\operatorname{tr}}^{cdh}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
36.16664pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
36.16664pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{a_{\mathrm{cdh}}\operatorname{TC}}$}}}}}}}\ignorespaces}}}}\ignorespaces,}$
(39)

where $KH$ is the homotopy $K$-theory and ${\operatorname{tr}}$ is the
cyclotomic trace ([BHM93], [DGM13]), and ${\operatorname{tr}}^{cdh}$ is
induced by ${\operatorname{tr}}$ via the equivalence $KH\simeq
a_{\mathrm{cdh}}K$, [KST18b, Th.6.3], [Cis13], [KM21].

The pullback square follows from the latter equivalence and the fact that the
fiber of ${\operatorname{tr}}$ is a cdh sheaf by [LT19, Th. A.3]. By (39),
$\operatorname{Nil}\operatorname{TC}[\frac{1}{p}]=0$ follows from
$\operatorname{fib}(K\to KH)[\frac{1}{p}]=0$ by [TT90, Th. 9.6]. Thus, it
suffices to show that ${\mathbb{Z}(n)^{\operatorname{syn}}}(-)/p$ is finitary
noting that the cdh sheafification of a finitary presheaf is finitary. By
[EM23, Lem.4.16], ${\mathbb{Z}(n)^{\operatorname{syn}}}(-)/p$ admits a finite
increasing filtration whose graded pieces are some shifts of
$\wedge^{i}L\Omega_{-/\mathbb{F}_{p}}$ with $i\leq n$, so the desired
assertion follows from the finitarity of
$\wedge^{i}L\Omega_{-/\mathbb{F}_{p}}$.

  3. (BB)≥-n

By the finitarity and Theorem 3.17, it suffices to show that
$\operatorname{Nil}{\mathbb{Z}(n)^{\operatorname{syn}}}(R)\in
D(\mathbb{Z}_{p}))^{\leq n}$ for any henselian local ring $R$ such that the
ideal $\mathfrak{N}\subset R$ of nilpotent elements is finitely generated and
$R/\mathfrak{N}$ is a valuation ring. Similarly to (37), we have

$a_{\mathrm{cdh}}{\mathbb{Z}(n)^{\operatorname{syn}}}(R)=a_{\mathrm{cdh}}{\mathbb{Z}(n)^{\operatorname{syn}}}(R/\mathfrak{N})={\mathbb{Z}(n)^{\operatorname{syn}}}(R/\mathfrak{N}).$

Note that ${\mathbb{Z}(n)^{\operatorname{syn}}}$ and
$a_{\mathrm{cdh}}{\mathbb{Z}(n)^{\operatorname{syn}}}$ are not finitary in
general so the first equality requires the assumption that $\mathfrak{N}$ is
finitely generated. Hence, (BB)≥-n follows from [AMMN22, Th.5.2] noting
$(R,\mathfrak{N})$ is a henselian pair. This completes the proof of the claim.

∎

######  Proposition 8.10.

The presheaf
$\operatorname{Nil}\operatorname{TC}\in\mathrm{PSh}(\mathrm{Sch}_{\mathbb{F}_{p}}^{\mathrm{qcqs}},\mathrm{Spt})$
satisfies (Desc), (Fin), and (BB)≥0. Consequently, for any Noetherian
$\mathbb{F}_{p}$-scheme $X$ with $\dim(X)<\infty$ we have

$a_{\mathrm{procdh}}(\operatorname{Nil}\operatorname{TC})_{\geq
0}(X)\simeq\operatorname{Nil}\operatorname{TC}(X).$

###### Proof.

The desired properties follow from Lemma 8.9 by the same argument deducing
Proposition 8.8 from Lemma 8.7: The property (Desc) follows from (Desc) for
$\operatorname{Nil}{\mathbb{Z}(n)^{\operatorname{syn}}}$ by the filtration
(38). The properties (Fin) and (BB)≥0 for
$\operatorname{Nil}\operatorname{TC}$ follow from (Fin) and (BB)≥j for
$\operatorname{Nil}\mathbb{Z}_{p}(-j)^{syn}$ by using the spectral sequence

$E_{2}^{i,j}=H^{i-j}(\operatorname{Nil}\mathbb{Z}_{p}(-j)^{syn}(X))\Rightarrow\pi_{-i-j}\operatorname{Nil}\operatorname{TC}(X),$

arising from the filtration (38). Note that the spectral sequence is bounded
since Corollary 7.11 and (BB)≥j for
$\operatorname{Nil}\mathbb{Z}_{p}(-j)^{syn}$ from Lemma 8.9 imply
$E_{2}^{i,j}=0$ for $i-j>2d-j$. ∎

##  9 Procdh motivic complex

In this section, we address the following conjecture.

######  Conjecture 9.1 (Beilinson, [Beĭ87], cf. also [EM23, Introduction]).

For any reasonable scheme $X$, there is a natural spectral sequence:

$E^{p,q}_{2}=H^{p-q}_{\mathcal{M}}(X,\mathbb{Z}(-q))\Rightarrow K_{-p-q}(X)$
(40)

where $K_{*}(X)$ is the non-connective algebraic $K$-theory of $X$, and
$H^{i}_{\mathcal{M}}(X,\mathbb{Z}(n))$ 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 $\mathrm{Sm}_{k}$ denote the category of smooth schemes separated of
finite type over a field $k$. There exists a complete multiplicative
decreasing $\mathbb{N}$-indexed filtration
$\biggl{\\{}Fil_{\operatorname{mot}}^{n}K(X)\biggl{\\}}_{n\in\mathbb{N}}$ on
$K(X)$ and identifications of spectra

$\operatorname{gr}_{Fil_{\operatorname{mot}}}^{n}K(X)\simeq\mathbb{Z}(n)^{\operatorname{sm}}(X)[2n]$
(41)

functorial in $X\in\mathrm{Sm}_{k}$.

Above, $\mathbb{Z}(n)^{\operatorname{sm}}(X)[2n]$ is a chain complex of
abelian groups regarded as a spectrum via the Eilenberg-Maclane functor
$D(\mathbb{Z})\to\mathrm{Spt}$. As a complex of abelian groups, it can be
defined as

$\mathbb{Z}(n)^{\operatorname{sm}}(X)=\underline{C}_{*}(\mathbb{Z}_{tr}(\mathbb{G}_{m}^{\wedge
n}))(X)[-n]\quad\text{for }X\in\mathrm{Sm}_{k},$ (42)

where $\underline{C}_{*}(\mathbb{Z}_{tr}(\mathbb{G}_{m}^{\wedge q}))[-q]$ is
Voevodsky’s $\mathbb{A}^{1}$-invariant motivic complex defined in [SV00]. This
is strictly functorial in $\mathrm{Sm}_{k}$ in the sense that it defines a
functor between the 1-category $\mathrm{Sm}_{k}$ and the 1-category of chain
complexes of abelian groups. Scheme-wise this is shown to be quasi-isomorphic
to Bloch’s cycle complex, [Blo86], in [Voe02, Cor.2]. Of course, another
approach is to just take Eq.(41) as the definition for
$\mathbb{Z}(n)^{\operatorname{sm}}(X)[2n]$.

In what follows, we write $\mathbb{F}=\mathbb{Q}$ or $\mathbb{F}_{p}$ and
continue to let $\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}}$ be the category of
qcqs schemes over $\mathbb{F}$ and $\mathrm{Sch}^{\mathrm{noe}}_{\mathbb{F}}$
be its full subcategory of noetherian schemes. Recently, Elmanto-Morrow [EM23]
extended Theorem 9.2 to all $X\in\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}}$ by
using instead of $\mathbb{Z}(n)^{\operatorname{sm}}$ a new motivic complex

$\mathbb{Z}(n)^{\operatorname{EM}}\in\mathrm{PSh}(\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}},D(\mathbb{Z}))).$

They construct $\mathbb{Z}(n)^{\operatorname{EM}}$ by modifying the cdh
sheafification $\mathbb{Z}(n)^{\mathrm{cdh}}$ of the left Kan extension of
$\mathbb{Z}(n)^{\operatorname{sm}}$ along
$\mathrm{Sm}_{\mathbb{F}}\to\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}}$ by
using Hodge-completed derived de Rham complexes in case
$\mathbb{F}=\mathbb{Q}$ and syntomic complexes in case
$\mathbb{F}=\mathbb{F}_{p}$. The construction is motivated by trace methods in
algebraic $K$-theory using the cyclotomic trace map ${\operatorname{tr}}$ from
(39). 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 $n\geq 0$, we define the _procdh-local motivic complex_

$\mathbb{Z}(n)^{\mathrm{procdh}}:=a_{\mathrm{procdh}}L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}}\in\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}},D\mathbb{Z})),$

as the procdh sheafification of the left Kan extension
$L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}}$ of
$\mathbb{Z}(n)^{\operatorname{sm}}$ along
$\mathrm{Sm}_{\mathbb{F}}\to\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}}$.

######  Remark 9.4.

Note that there have been various constructions of motivic cohomology on
smooth 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 of these.

######  Theorem 9.5.

For $X\in\mathrm{Sch}^{\mathrm{noe}}_{\mathbb{F}}$ with $\dim(X)<\infty$,
there exists a complete multiplicative242424For 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 general class of coefficient categories.
decreasing $\mathbb{N}$-indexed filtration
$\biggl{\\{}Fil_{\mathrm{procdh}}^{n}K(X)\biggl{\\}}_{n\in\mathbb{N}}$ on
$K(X)$ and identifications

$\operatorname{gr}_{Fil_{\mathrm{procdh}}}^{n}K(X)\simeq\mathbb{Z}(n)^{\mathrm{procdh}}(X)[2n].$

###### Proof.

By Bhatt-Lurie (see [EHK+20, Ex. A.0.6]), there is a natural equivalence

$K_{\geq 0}\simeq L^{\operatorname{sm}}(K_{|\mathrm{Sm}_{\mathbb{F}}}),$

where the right hand side is the left Kan extension of
$K_{|\mathrm{Sm}_{\mathbb{F}}}$ along
$\mathrm{Sm}_{\mathbb{F}}\to\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}}$. Using
Theorem 8.3, we obtain a filtration on $K$ by procdh sheafifying the left Kan
extension along $\mathrm{Sm}_{\mathbb{F}}\to\mathrm{Sch}_{\mathbb{F}}$ of
$Fil_{\operatorname{mot}}^{\bullet}K_{|\mathrm{Sm}_{\mathbb{F}}}$ from Theorem
9.2. The identification of the graded pieces is clear. Completeness follows
from the boundedness described in Eq.(45) below. ∎

######  Remark 9.6.

By definition, Eq.(42), we have
$\mathcal{H}_{\operatorname{Zar}}^{i}(\mathbb{Z}(n)^{\operatorname{sm}})=0$
for $i>n$, where
$\mathcal{H}_{\operatorname{Zar}}^{i}(\mathbb{Z}(n)^{\operatorname{sm}})$ is
the Zariski cohomology sheaf of $\mathbb{Z}(n)^{\operatorname{sm}}$. It
implies that for any local $k$-algebra $A$, we have

${H^{i}((L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}})(A))=0\quad\text{for
}i>n.}$ (43)

Thus, Corollary 7.11 implies that for
$X\in\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}}$ of finite valuative dimension
$d$ with Noetherian underlying topological space, we have

${H^{i}(\mathbb{Z}(n)^{\mathrm{procdh}}(X))=0\quad\text{for }i>2d+n.}$ (44)

In particular, when $X$ is noetherian, we have for each $n\in\mathbb{N}$

$π_{-i}Fil_{\mathrm{procdh}}^{n}K(X)=0;\qquad i>2d-n$ (45)

so the induced spectral sequence (40) with

$H^{i}_{\mathcal{M}}(X,\mathbb{Z}(n)):=H^{i}(\mathbb{Z}(n)^{\mathrm{procdh}}(X))$

is bounded.

Now, it is a natural question if two constructions
$\mathbb{Z}(n)^{\operatorname{EM}}$ and $\mathbb{Z}(n)^{\mathrm{procdh}}$
coincide.

######  Theorem 9.7.

Assume given
$\mathcal{Z}(n)\in\mathrm{PSh}(\mathrm{Sch}_{\mathbb{F}}^{\mathrm{qcqs}},D(\mathbb{Z})))$
and consider the following conditions.

  1. (a)

There is a natural comparison map

$\psi:\mathbb{Z}(n)^{\operatorname{sm}}\to\mathcal{Z}(n)_{|\mathrm{Sm}_{\mathbb{F}}}$

in $\mathrm{PSh}(\mathrm{Sm}_{\mathbb{F}},D(\mathbb{Z})))$, whose induced map
$\phi:L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}}\to\mathcal{Z}(n)$
has the properties that $\phi(R)$ is an equivalence for all procdh local rings
$R$ with $\operatorname{length}Q(R)$ and $\dim R$ finite.

  2. (b)

$\mathcal{Z}(n)$ is finitary. That is,

$\mathcal{Z}(n)(\lim_{\lambda}P_{λ})=\underset{\lambda}{\operatorname{colim}}\mathcal{Z}(n)(P_{λ})$

for any cofiltered system $(P_{λ})_{λ\in\Lambda}$ in
$\mathrm{Sch}^{\mathrm{qcqs}}_{\mathbb{F}}$ with affine transition morphisms.

  3. (c)

$\mathcal{Z}(n)$ is a procdh sheaf on Noetherian schemes. That is,

$\mathcal{Z}(n)\in\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}^{\mathrm{noe}}_{\mathbb{F}},D(\mathbb{Z}))).$

If (a), (b), and (c) are satisfied then, $\phi$ induces an equivalence
$\mathbb{Z}(n)^{\mathrm{procdh}}(X)\simeq\mathcal{Z}(n)(X)$ for any
$X\in\mathrm{Sch}^{\mathrm{noe}}_{\mathbb{F}}$.

###### Proof.

For procdh sheaves of chain complexes
$\mathcal{E}\in\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}^{\mathrm{noe}}_{\mathbb{F}},D(\mathbb{Z}))$,
we have a descent spectral sequence

$E_{2}^{p,q}=H^{p}_{\mathrm{procdh}}(X,\mathcal{H}_{\mathrm{procdh}}^{q}\mathcal{E})\Rightarrow
H^{p+q}\mathcal{E}(X),$

which converges by finiteness of cohomological dimension, Cor.7.11. Here
$\mathcal{H}_{\mathrm{procdh}}^{q}\mathcal{E}$ is the procdh sheafification of
the presheaf of abelian groups $X\mapsto π_{-q}\mathcal{E}(X)$. So since
$\mathcal{Z}(n)$ and $\mathbb{Z}(n)^{\mathrm{procdh}}$ are procdh sheaves,
Assumption (c), it suffices to show that the morphism

$\mathcal{H}_{\mathrm{procdh}}^{q}\mathbb{Z}(n)^{\mathrm{procdh}}\to\mathcal{H}_{\mathrm{procdh}}^{q}\mathcal{Z}(n)$

of procdh cohomology sheaves of abelian groups is an isomorphism on Noetherian
schemes. Since the 1-topos $\mathrm{Shv}_{\mathrm{procdh}}(\mathrm{Sch}_{X})$
has enough points, Thm.5.2, and cohomology commutes with filtered colimits,
and both $\mathcal{Z}(n)$ and
$L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}}$ are finitary,
Assumption (b), it suffices to show that for all procdh local rings
$\operatorname{Spec}(R)\to X$ with $R$ as in Assumption (a), the comparison

$H^{q}L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}}(R)\to
H^{q}\mathcal{Z}(n)(R)$

is an isomorphism for all $q\in\mathbb{Z}$. This is precisely Assumption (a).
∎

######  Theorem 9.8 (Elmanto-Morrow).

The presheaf $\mathbb{Z}(n)^{\operatorname{EM}}$ 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
$\mathbb{Z}(n)^{\operatorname{sm}}\to\mathcal{Z}(n)_{|\mathrm{Sm}_{\mathbb{F}}}$
is contained in [EM23, Th.1.1(9)], although we don’t need the full force of
their theorem since we are only asking for existence, not the stronger fact
that the morphism is an equivalence. Now we want to show that the induced map
$L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}}\to\mathcal{Z}(n)$ is
an equivalence on procdh local rings $R$ with $\operatorname{length}Q(R)$ and
$\dim R$ finite.

By [EM23, Th.7.7], it suffices to prove $\mathbb{Z}(n)^{\operatorname{EM}}(R)$
is supported in cohomological degrees $\leq n$ for any procdh local ring $R$.
By [EM23, Th.4.10(2), Th.4.24.(2)], there are fiber sequences (cf. Eq.(30))

$\operatorname{Nil}\widehat{L\Omega}^{\geq
n}_{R/\mathbb{Q}}\to\mathbb{Z}(n)^{\operatorname{EM}}(R)\to\mathbb{Z}(n)^{\mathrm{cdh}}(R)\;\text{
if }\mathbb{F}=\mathbb{Q},$
$\operatorname{Nil}{\mathbb{Z}(n)^{\operatorname{syn}}}(R)\to\mathbb{Z}(n)^{\operatorname{EM}}(R)\to\mathbb{Z}(n)^{\mathrm{cdh}}(R)\;\text{
if }\mathbb{F}=\mathbb{F}_{p},$

where
$\mathbb{Z}(n)^{\mathrm{cdh}}=a_{\mathrm{cdh}}L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}}$.
Note that $\operatorname{Nil}\widehat{L\Omega}^{\geq n}_{R/\mathbb{Q}}$ and
$\operatorname{Nil}{\mathbb{Z}(n)^{\operatorname{syn}}}(R)$ are supported in
degrees $\leq n$ by Lemma 8.7 and Lemma 8.9. Noting that
$\mathbb{Z}(n)^{\operatorname{sm}}$ is finitary, the same argument as Lemma
8.9 gives

$\mathbb{Z}(n)^{\mathrm{cdh}}(R)=\mathbb{Z}(n)^{\mathrm{cdh}}(R/\mathfrak{N})=(L^{\operatorname{sm}}\mathbb{Z}(n)^{\operatorname{sm}})(R/\mathfrak{N}).$

So, $\mathbb{Z}(n)^{\mathrm{cdh}}(R)$ is supported in degrees $\leq n$ by
Eq.(43), which proves the desired assertion. ∎

######  Corollary 9.9.

There is an equivalence
$\mathbb{Z}(n)^{\mathrm{procdh}}\simeq\mathbb{Z}(n)^{\operatorname{EM}}$ in
$\mathrm{PSh}(\mathrm{Sch}^{\mathrm{noe}}_{\mathbb{F}},D(\mathbb{Z})))$.

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