ON PRO-CDH DESCENT ON DERIVED SCHEMESSHANE KELLY, SHUJI SAITO, AND GEORG TAMME
Abstract. We prove a ‘pro-cdh descent’ result for suitably connective localizing invariants
and the cotangent complex on arbitrary qcqs derived schemes. As an application, we deducea generalised Weibel vanishing for negative K-groups of non-Noetherian schemes.

Contents
1. Introduction
2. Preliminaries on derived algebraic geometry
3. Modifications of derived schemes
4. Pro-cdh descent for connective localizing invariants
5. Pro-cdh descent for the cotangent complex and motivic cohomology
6. Generalised Weibel vanishing
References

1
5
11
13
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1. Introduction
The algebraic K-groups of a blowup BlZ (X) of a quasi-compact, quasi-separated (qcqs, forshort) scheme X in a regularly immersed center Z have been computed by Thomason [Tho93](see also [CHSW08, Prop. 1.5]). In particular, the blowup squareE

BlZ (X)

Z

X

gives rise to a cartesian square of algebraic K-theory spectra and hence to a long exact
sequence of algebraic K-groups. If the closed immersion Z ,→ X is not regular, this will
fail in general. However, if X is Noetherian, and one takes infinitesimal information into
account, one still gets a cartesian square. More precisely, if Z(n) and E(n) denote the n-th
infinitesimal thickenings of Z in X and E in BlZ (X), respectively, the square of pro-spectra
K(X){K(Z(n))}n

K(BlZ (X))

{K(E(n))}n

Date: July 5, 2024.
The first author is supported by JSPS KAKENHI Grant (19K14498) and Bilateral Joint Research Projects(120213206). The second author is supported by JSPS Grant-in-aid (B) #20H01791 representative Shuji Saito.
The third author is partially supported by the DFG through TRR 326 (Project-ID 444845124).
12

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

is weakly cartesian and hence induces a long exact sequence of pro-abelian groups. Notehere that the pro-systems only depend on the underlying topological space of Z (and E)
as a closed subset of X (and BlZ (X), respectively) but not on the particular subscheme
structure. In fact, more generally any abstract blowup square of Noetherian schemes gives
rise to a weakly cartesian square of K-theory pro-spectra. In this form, this result, oftencoined “pro-cdh descent”,1 is first proven in [KST18b], but there are important precursors[KS02, Cor06, Kri10, GH06, GH11, Mor16a, Mor18]. A variant for Noetherian ANS stacks
has been considered in [BKRS22]. These pro-cdh descent results play a central role in the
resolution of Weibel’s K-dimension conjecture [KST18b] (and [BKRS22] for stacks) and thedevelopment of a continuous K-theory of rigid spaces [Mor16b, KST18a, KST23].
Given the rising interest in non-Noetherian schemes, which appear naturally, for example,
when working over a perfectoid base ring such as OCp , it is an obvious question, whether, orin which form, pro-cdh descent also holds in this general setting. The following example, avariant of which was constructed by Dahlhausen and the third author [DT22] precisely forthat purpose and which was independently studied by the first two authors [Kel24, Footnote2] with respect to the procdh topology, shows that pro-cdh descent as formulated above does
not hold for general abstract blowup squares of non-Noetherian schemes.
Example 1.1. Let R be a valuation ring of dimension at least 2. Let 0 ( p ( m be prime
ideals in R, and choose x ∈ m \ p and y ∈ p \ {0}. Then xn divides y for all n ≥ 1 and foreach n the squareSpec(R/(xn ))Spec(R/(y))
idSpec(R/(xn ))Spec(R/(xy))

is an abstract blowup square. If the induced square of K-theory pro-spectra would be weakly
cartesian, the map K(R/(xy)) → K(R/(y)) would be an equivalence. However, as y 2 = 0
in R/(xy), 1 + y is a unit and defines a non-trivial element in the kernel of K1 (R/(xy)) →K1 (R/(y)).
A posteriori, this is not surprising: A key point in [KST18b] was to prove Thomason’s
blowup formula for derived blowups in quasi-smooth centers together with the fact that forevery closed subscheme of an affine Noetherian scheme there exists a quasi-smooth derived
closed subscheme with the same underlying classical scheme. The condition that all (derived) rings and schemes appearing be Noetherian then allowed to pass from derived schemes
back to classical schemes. For instance, for a Noetherian commutative ring A and elements
f1 , . . . , fr in A generating the ideal I ⊆ A, the pro-system of (Koszul type) derived quotients{A//f1n , . . . , frn }n is equivalent to the pro-system of discrete rings {A/I n }n . This suggested
that some variant of the above pro-cdh descent statement, where one takes “derived” infinitesimal thickenings everywhere, could still be true. This is precisely what we achieve in thispaper. Our main K-theoretic result is the following.
Theorem A. Let f : Y → X be a proper, locally almost finitely presented morphism of
qcqs derived schemes which is an isomorphism outside a closed subset Z ⊆ |X| whose open
1We remark that it would be more appropriate to call this “pro-cdh excision” since it is not precisely adescent statement for some topology. However, together with Nisnevich excision, it does imply actual (Čech)
descent for the procdh topology of [KS24], see Theorem 6.1 there. For this reason, and to be in line with theexisting literature we stick to the term “pro-cdh descent.”

ON PRO-CDH DESCENT ON DERIVED SCHEMES

3complement is quasi-compact. Denote by XZ∧ and YZ∧ the formal completion of X (respectively
Y ) along Z (respectively f −1 (Z)) viewed as ind-derived schemes. Then the square of prospectra
K(X)

K(XZ∧ )K(Y )

K(YZ∧ )

is weakly cartesian.
Let us point out that this result is not specific to K-theory. It holds more generally forevery localising invariant which is k-connective in the sense of [LT19] for some integer k; see
Theorem 4.5 for the general statement. For instance, it also applies to THH, TC, and rational
negative cyclic homology.
Remark 1.2. One may ask whether the same statement also holds for spectral schemes. As
our proof makes essential use of the theory of derived blowups of derived schemes developed
by Khan and Rydh [KR18], we cannot treat this more general case with our methods.
We also prove the analogous result for the cotangent complex, see Theorem 5.4. By thearguments of [EM23] one then obtains pro-cdh descent in the above form for motivic cohomology (Corollary 5.5). We thank Matthew Morrow for suggesting this result for the cotangent
complex and indicating the application to motivic cohomology.
In [KST18b], pro-cdh descent of K-theory was used to derive Weibel’s conjecture on thevanishing of negative K-groups of a Noetherian scheme X below − dim(X). It is an obvious
question what we can say about negative K-groups if we merely assume X to be qcqs; seee.g. Morrow’s Oberwolfach talk [Mor23]. In this generality, one can at least prove a vanishing
result for Weibel’s homotopy K-theory KH(X). Let vdim(X) denote the valuative dimensionof a scheme X introduced by Jaffard [Jaf60] (see [EHIK20] or [KS24, §7.1] for accounts). It
coincides with the Krull dimension if X is Noetherian. Furthermore, write Lcdh for the cdh
sheafification functor on presheaves of spectra and K≥0 for the presheaf of connective algebraic
K-theory.
Theorem 1.3. Let X be a qcqs scheme of finite valuative dimension.
(1) KH−i (X) = 0 for all i > vdim(X).
(2) The natural maps Lcdh K≥0 → Lcdh K → KH are equivalences.
Though we are mainly interested in (1), we include (2) for the sake of completeness. For
schemes essentially of finite type over a field of characteristic 0, this was first proven in [Hae04].
For general Noetherian X, (1) was first proven in [KS17] and (2) in [KST18b]. The fact that
KH is a cdh sheaf is [Cis13]. In the general case, the theorem follows easily from recent
results on the cdh-topology [EHIK20] and the K-theory of valuation rings [KM21, KST21].
In fact, the proof given under Noetherian assumptions in [KM21] still works, and for thereader’s convenience we reproduce this proof at the end of the paper. Alternatively, the proofof [KS17] respectively [KST18b] works mutatis mutandis, using the fact that a blowup does
not increase the valuative dimension (this is not necessarily true for the Krull dimension).
The previous theorem suggests that for any qcqs scheme X, the negative K-groups vanish
below − vdim(X). In this direction, we prove the following.

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SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

Theorem B (Theorem 6.1). Let X be a qcqs (derived) scheme. Assume that the underlying
topological space of the spectrum of every local ring of every irreducible component of X is
Noetherian. Then the following hold.
(1) K−i (X) = 0 for all i > vdim(X).
(2) For all i ≥ vdim(X) and any integer r ≥ 0, the pullback map K−i (X) → K−i (ArX ) is
an isomorphism. In other words, X is K− vdim(X) -regular.
Remarks 1.4.
(1) The assumption is for instance satisfied for any qcqs derived scheme of
Krull dimension ≤ 1. It is also stable under passing to schemes essentially of finite
type over X. In particular, the theorem holds for all schemes essentially of finite typeover a valuation ring.
(2) For general qcqs Q-schemes, (1) was announced by Elmanto and Morrow [Mor23,
Thm. 4]. Moreover, they prove that the canonical map K−i (X) → KH−i (X) is
an isomorphism for i ≥ dim(X). Our result implies this only for i ≥ vdim(X).
Conversely, their isomorphism does not imply our assertion (2).
(3) We do not know if the valuative dimension is really needed in the vanishing bound. I.e.,
we don’t know an example of a scheme X with vdim(X) > dim(X) and K− vdim(X) (X) 6=
0.
Structure of the arguments and the paper. The proof of Theorems A and B follows theoutline of [KST18b], but requires substantially more input from derived geometry. The proofof Theorem A is reduced to the two special cases of derived blowups and finite morphisms,
respectively. This reduction is achieved by a structural result about modifications of derived
schemes, Theorem 3.2, which we view as our main contribution here. In its proof, we needseveral preliminary results from derived algebraic geometry which we discuss in Section 2.
The case of Theorem A for derived blowups could essentially be proved as in [KST18b]. We
here present a stronger result with a simplified proof due to Antieau ([Ant18]; we reproduce
his proof in Proposition 4.1). For the case of finite morphisms in Theorem A, compared to[KST18b] we give a simple proof of a more general result (Proposition 4.2). This follows quite
directly from [LT19].
The proof of our generalised Weibel vanishing statement is contained in Section 6. As weare working with non-Noetherian schemes, this needs some additional input.
Related work. Another approach to pro-cdh descent statements has been proposed byClausen and Scholze. It is based on condensed mathematics [Sch] and Efimov’s theory of
localizing invariants of large categories [Efi24].
In forthcoming work, the first two authors introduce a “procdh topology” on qcqs derived
schemes. It will follow from Theorem A (or 4.5) that every localizing invariant which is
k-connective for some integer k satisfies descent for that topology.
Acknowledgement. We thank Christian Dahlhausen for discussions about pro-cdh descentin the non-Noetherian setting which led to the variant of Example 1.1 and which was oneof the starting points of this project. We are grateful to Ben Antieau for allowing us toinclude his pro-excision result for derived blowups, which is stronger than the one obtained in[KST18b]. We thank Matthew Morrow for bringing our attention to the cotangent complex
and further comments and Mauro Porta for related discussions.

ON PRO-CDH DESCENT ON DERIVED SCHEMES

5

2. Preliminaries on derived algebraic geometry
We freely use the language of derived algebraic geometry as developed by Toën–Vezzosi[TV08, TV05], Lurie [Lur18] and others. The following subsections mainly serve to fix somenotation and recall some notions and facts that will be used later on. Readers familiar with
derived algebraic geometry may safely skip this section and only come back when needed.
2.1. Derived rings and schemes. We write CAlg∆ for the ∞-category of simplicial commutative rings [Lur18, §25.1] and refer to its objects as derived rings. There is a forgetful
functor CAlg∆ → CAlgcn , where CAlgcn denotes the ∞-category of connective E∞ -algebrasin spectra. This functor preserves small limits and colimits and is conservative.
A derived scheme is a pair X = (|X|, OX ) consisting of a topological space |X| and asheaf OX of derived rings on |X| such that cl X := (|X|, π0 OX ) is a classical scheme and thehigher homotopy sheaves πi OX are quasi-coherent π0 OX -modules. We call cl X the underlying
classical scheme or the classical truncation of X. Derived schemes form the objects of an∞-category dSch, and similarly as above there is a forgetful functor from derived schemes tospectral schemes. There is also a functor of points approach to derived schemes. In otherwords, there is a fully faithful functor
dSch → Fun(CAlg∆ , Spc),
which coincides with the Yoneda embedding on affine derived schemes [Lur18, §1.6].
A map f : X → Y of derived schemes is called proper, a closed immersion, affine, or finite,
respectively, if the underlying map of classical schemes cl f has the corresponding property. If
U ⊆ |X| is an open subset, then (U, OX |U ) is itself a derived scheme. Such a derived scheme
is called an open subscheme of X which we simply denote by U .
2.2. Quasi-coherent modules. If A is a derived ring, we write Mod(A) for the symmetricmonoidal ∞-category of A-modules (in spectra). For a derived scheme X, we denote by
QCoh(X) the category of quasi-coherent sheaves on X. These are stable ∞-categories withcanonical t-structures, and we denote their connective part by Mod(A)cn and QCoh(X)cn ,respectively. Moreover, they only depend on the underlying spectrum or spectral scheme,
respectively. If X = Spec(A) is affine, we have an equivalence QCoh(X) = Mod(A).
If Z ⊆ |X| is a closed subset, we denote by QCoh(X on Z) the full subcategory of QCoh(X)spanned by those quasi-coherent sheaves which are supported on Z. If X is quasi-compact andquasi-separated (qcqs, for short) and the open complement of Z is compact, then QCoh(X on Z)
is compactly generated and its compact objects coincide with the perfect ones, [Lur18,
Prop. 9.6.1.1], [CMNN, Prop. A.9]. Here a quasi-coherent sheaf on X is called perfect if
and only its restriction to each affine open subscheme U = Spec(A) ⊆ X belongs to thesmallest thick stable subcategory of QCoh(U ) = Mod(A) containing A. We write Perf(X)and Perf(X on Z) for the corresponding subcategories. The arguments used to prove this also
apply to the case of connective sheaves:
Lemma 2.1. Let X be a qcqs derived scheme, and let Z ⊆ |X| be a closed subset withcompact open complement. Then QCoh(X on Z)cn is compactly generated and the inclusion
QCoh(X on Z)cn ,→ QCoh(X)cn preserves compact objects. An object of QCoh(X on Z)cn iscompact if and only if it is perfect (as an object of QCoh(X)).
Proof. The same argument as in the proof of [CMNN, Prop. A.9] proves the first two claims:
In case X is affine, these follow from [Lur18, Prop. 7.1.1.12(e)] and the fact that QCoh(X)cn

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SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

is compactly generated by [Lur18, Prop. 9.6.1.2]. The reduction of the global case to the local
case is done by [Lur18, Ex. 10.3.0.2 (4), Prop. 10.3.0.3, Th. 10.3.2.1 (b)]. It then follows thatan object of QCoh(X on Z)cn is compact if and only if its image in QCoh(X)cn is compact.
The compact objects in the latter category coincide with the perfect, connective OX -modules
by [Lur18, Prop. 9.6.1.2.] again.

2.3. Free algebras. If A is a derived ring, we denote the category of derived A-algebras byCAlg∆
A . Its objects are derived rings B together with a map of derived rings A → B. There iscn which admits a left adjoint, the free algebra
an obvious forgetful functor CAlg∆
A → Mod(A)functor, LSym∗A : Mod(A)cn → CAlg∆
M is a connective A-module, the underlying AA . IfLnnmodule of LSym∗A (M ) is the direct sumn≥0 LSymA (M ), where LSymA (M ) is the n-th
derived symmetric power of M as studied for instance by Quillen [Lur18, Constr. 25.2.2.6],
whence the notation.
These constructions globalise: For a derived scheme X, write CAlg∆
OX for the ∞-categoryof sheaves of derived rings on |X| equipped with a map from OX such that the underlying
sheaf of OX -modules is quasi-coherent. There is an adjunction
LSym∗OX : QCoh(X)cn  CAlg∆
OX : forget,
the forgetful functor is conservative and preserves sifted colimits. Consequently, LSym∗OXsends compact objects to compact objects.
For A ∈ CAlg∆
OX one can form the relative spectrum Spec(A) which comes with an affinemorphism Spec(A) → X.
2.4. Finiteness conditions. A map of derived rings A → B is called locally of finite presentation if B is a compact object of CAlg∆
A . It is called almost of finite presentation if B is analmost compact object CAlg∆
,
i.e.
each
truncation τ≤n B is a compact object of τ≤n CAlg∆AA;
see [Lur04, §3.1] and [Lur18, §4.1] for the analog notions for E∞ -algebras. It turns out that
A → B is almost of finite presentation if and only if the underlying map of E∞ -algebras isalmost of finite presentation; the analog for being locally of finite presentation is wrong.
These finiteness conditions are stable under base change: If A → B is locally or almost offinite presentation and A → A0 is an arbitrary map, then also A0 → B ⊗A A0 is locally or
almost of finite presentation, respectively [Lur17, Rem. 7.2.4.28]. They are also stable undercomposition [Lur17, Rem. 7.2.4.29, Cor. 7.4.3.19].
If A is Noetherian, i.e. π0 (A) is Noetherian in the classical sense and all higher homotopy
groups are finitely generated π0 (A)-modules, then a derived A-algebra B is almost of finite
presentation if and only if B is Noetherian and π0 (B) is a classically finitely generated π0 (A)algebra [Lur04, Prop. 3.1.5] or [Lur17, Prop. 7.2.4.31].
A map f : Y → X of derived schemes is called locally of finite presentation or locallyalmost of finite presentation (lafp, for short) if for all affine open subschemes U = Spec(A) ⊆X and V = Spec(B) ⊆ Y with f (V ) ⊆ U the induced morphism A → B is locally offinite presentation or almost of finite presentation, respectively. As in the affine case, thesenotions are stable under base change and composition, and there is a characterization in the
Noetherian case.
For example, if F is a perfect, connective OX -module, i.e. a compact object of QCoh(X)cn ,then Spec(LSym∗OX (F)) is locally of finite presentation over X. Using this observation, we
prove the following lemma.

ON PRO-CDH DESCENT ON DERIVED SCHEMES

7

Lemma 2.2. Let X be a qcqs derived scheme, and let U ⊆ X be a qc open subset withcomplement Z. Let i : Y → X be closed immersion which is an isomorphism over U andsuch that the underlying map of classical schemes cl Y → cl X is finitely presented. Then there
exists a factorization Y → Y 0 → X of i such that(1) the morphism Y → Y 0 is an isomorphism on underlying classical schemes,(2) the morphism Y 0 → X is a closed immersion locally of finite presentation and an
isomorphism over U .
Proof. Let I = fib(OX → i∗ OY ). As i is a closed immersion and an isomorphism over U ,we have I ∈ QCoh(X on Z)cn . As the morphism of classical schemes cl Y → cl X is classicallyof finite presentation, it follows that the image J of π0 (I) in π0 (OX ) is of finite type. By
Lemma 2.1, we can write I as a filtered colimit I = colimα Iα where each Iα ∈ Perf(X on Z)cn .
As π0 (−) commutes with filtered colimits, we have colimα π0 (Iα ) = π0 (I). As π0 (I) → J
is surjective and J is of finite type, there exists an index α such that the induced map
π0 (Iα ) → J is surjective. By construction, we have the following commutative diagram in
QCoh(X)cn .
Iα

0

OX

i∗ OY

By adjunction, this induces a commutative diagram
LSym∗OX (Iα )

OX

OX

i∗ OY

∆
in CAlg∆
OX . We define A ∈ CAlgOX to be the tensor product OX ⊗LSym∗O (Iα ) OX . The
X

above diagram classifies a morphism A → i∗ OY in CAlg∆
OX . We claim that it induces anisomorphism on π0 . Indeed, we compute2
π0 (A) ∼= π0 (OX ) ⊗♥π0 (LSym∗OX (Iα ))∼= π0 (OX ) ⊗♥Sym

π0 (OX )π0 (OX ) (π0 (Iα ))π0 (OX )∼= π0 (OX )/ im(π0 (Iα ) → π0 (OX ))= π0 (OX )/J∼= π0 (i∗ OY ).
We set Y 0 = Spec(A). By construction, we get the factorization Y → Y 0 → X of i. The
above computation shows that Y → Y 0 is an isomorphism on underlying classical schemes.
Moreover, as Iα is supported on Z, Y 0 → X is an isomorphism over U . Finally, as Iα isperfect, it follows that Y 0 → X is locally of finite presentation, as desired.

2here the ⊗♥ indicates the underived tensor product and Sym denotes the classical symmetric algebra

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SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

2.5. Formal completion. Let X be a qcqs derived scheme and let Z ⊆ |X| be a closed
subset whose open complement |X| \ Z is quasi-compact. The formal completion XZ∧ is anind-object of derived schemes with a map i : XZ∧ → X. It is determined by the following
universal property: For any derived ring R, composition with i induces an equivalence of
MapInd(dSch) (Spec(R), XZ∧ ) with the union of components of MapdSch (Spec(R), X) consisting of those morphisms Spec(R) → X that set-theoretically factor through Z; see [GR14,
Prop. 6.5.5] for the existence of XZ∧ as an ind-derived scheme. More concretely, we can write
XZ∧ as the ind-system
XZ∧ = {Z 0 }Z 0 ,→X(1)of (a small cofinal subsystem of) all closed immersions of derived schemes Z 0 ,→ X with
|Z 0 | = Z.
If X = Spec(A) is affine, there exist finitely many elements f1 , . . . , fr ∈ π0 (A) whose zero
locus is Z. In this case, XZ∧ can also be represented by(2)

XZ∧ = {Spec(A//f1α , . . . , frα )}α≥1where the symbol // indicates the derived quotient, i.e. the (derived) tensor product A⊗Z[t1 ,...,tr ]Z where the maps send ti to fiα and 0, respectively; see [Lur04, Prop. 6.1.1] or [Lur18,
Lemma 8.1.2.2].
It follows immediately from the universal property that the formation of the derived completion commutes with base change: If f : Y → X is a quasi-compact map, then Yf∧−1 (Z) 'XZ∧ ×X Y . We therefore also write YZ∧ instead of Yf∧−1 (Z) .
If X is a Noetherian classical scheme, then the formal completion is itself classical, equal
to the classical formal completion. This follows for example from [Lur18, Lemma 17.3.5.7].
2.6. Ample line bundles. Ample line bundles on Noetherian derived schemes have beenstudied by Annala [Ann22]. We need some of the results in the more general setting of qcqs
derived schemes. These are certainly well-known, the proofs are essentially the same as forclassical schemes.
Let X be a qcqs derived scheme. A line bundle L on X is called ample if for any pointx ∈ X there exists an n ≥ 1 and a global section s ∈ π0 Γ(X, L⊗n ) such that the non-vanishinglocus Xs of s is affine and contains x. If f : X → Y is a morphism of derived schemes, then
L is called f -ample if for every affine open subscheme U ⊆ Y the restriction of L to f −1 (U )is ample.
Let now L be any line bundle on X and s ∈ π0 Γ(X, L) a global section. We view the latteras a map s : OX → L. If F is any quasi-coherent sheaf on X, we get a diagram
(3)

⊗s

⊗s

F −−→ F ⊗ L −−→ F ⊗ L⊗2 −−→ . . .

In the case of classical schemes, the following lemma is standard. For Noetherian derived
schemes, it is [Ann22, Lemma 2.6].
Lemma 2.3. In the above situation, there is a canonical equivalencecolim Γ(X, F ⊗ L⊗n ) ' Γ(Xs , F).
nProof. If we take sections over Xs in (3), then all maps become equivalences. Thus the
restriction maps induce a map
colim Γ(X, (3)) → colim Γ(Xs , (3)) ' Γ(Xs , F).

ON PRO-CDH DESCENT ON DERIVED SCHEMES

9

We claim that this map is an equivalence. As X is qcqs, a standard induction reduces us
to the case where X = Spec(A) is affine and L is trivial. So we may assume X = Spec(A),Γ(X, L) = A, and F corresponds to the A-module M = Γ(X, F). In this case, the colimit inquestion identifies with M [s−1 ] which also identifies with Γ(Xs , F) as F is quasi-coherent. 
Exactly as in [Ann22, Lemma 2.11], this can be used to prove the following lemma.
Lemma 2.4. Let f : X → Y be a morphism of qcqs derived schemes, L a line bundle on X,
and (Ui )i∈I an open covering of Y . Write fi for the restricted morphism f −1 (Ui ) → Ui . Thenthe following are equivalent:(1) L is f -ample;(2) for every i ∈ I, the restriction L|f −1 (Ui ) of L is fi -ample.
The existence of an ample line bundle implies the resolution property in the following form:
Lemma 2.5. Let X be a qcqs derived scheme which carries an ample line bundle, and let F
be a connective, quasi-coherent OX -module such that π0 F is of finite type as a π0 OX -module.
Then there exists a vector bundle, i.e. a locally free OX -module of finite rank, E on X together
with a map E  F which is surjective on π0 .
Proof. Choose an ample line bundle L. Replacing L by an appropriate tensor power, we may
assume that there exist finitely many global sections si ∈ π0 Γ(X, L), whose non-vanishingloci Xsi form an affine covering of X. As each Xsi is affine and F is quasi-coherent, wehave isomorphisms π0 Γ(Xsi , F) = π0 Γ(Xsi , π0 F) and similarly for OX in place of F. The
assumption that π0 F is a π0 OX -module of finite type then implies that each π0 Γ(Xsi , F) is a
finitely generated π0 Γ(Xsi , OX )-module. Choose finitely many generators mij ∈ π0 Γ(Xsi , F).
By Lemma 2.3 there exists an integer N such that all the mij extend to global sections of
L⊗(−N ) → F which is surjective on π by
F ⊗ L⊗N . These give rise to a map E :=0
ij L
construction.

2.7. Quasi-smooth closed immersions and derived blowups. Derived blowups werefirst introduced in [KST18b] for affine schemes in order to prove pro-cdh descent of algebraic
K-theory on Noetherian schemes. They were then systematically studied and developed muchfurther by Khan, Rydh, and Hekking [KR18, Hek21].
Let X be a derived scheme, and let Z ,→ X be a quasi-smooth closed immersion [KR18,
2.3.6], i.e., Zariski locally on X, Z ,→ X is the derived pullback of the map {0} ,→ ArZ
for some r and some morphism X → ArZ . Equivalently, Zariski locally on X, Z ,→ X isgiven by Spec(A//f1 , . . . , fr ) ,→ Spec(A) for suitable elements fi ∈ π0 (A). The number r iscalled the virtual codimension of the closed immersion. Then one can form the derived blowup
p : dBlZ (X) → X of X in Z (or with center Z) which is characterized by a universal property:
it classifies virtual Cartier divisors on derived X-schemes; see [KR18, §4.1] for details. The
construction of the derived blowup commutes with arbitrary base change. Locally, if Z is thederived pullback of {0} ,→ ArZ along a map X → ArZ , then dBlZ (X) is the derived pullback of
the classical blowup Bl{0} (ArZ ) → ArZ . By the discussion of finiteness conditions in 2.4, as ArZ
is Noetherian, this in particular implies that the morphism p : dBlZ (X) → X is lafp. Clearly,
p is an isomorphism outside Z and proper.
It follows from the description of the underlying classical scheme of dBlZ (X) in [KR18,
Thm. 4.1.5(vii)] that there is always a closed immersion Blcl Z (cl X) ,→ cl dBlZ (X) and this is
an isomorphism over cl U where U is the open complement of Z.

10

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

The derived blowup dBlZ (X) carries a canonical line bundle O(1) (the dual of the ideal
sheaf defining the universal virtual Cartier divisor) and this line bundle is p-ample. Indeed,by Lemma 2.4 we may work affine-locally on X and hence assume that p : dBlZ (X) → X is
the pullback of the classical blowup Bl{0} (AnZ ) → AnZ , and the line bundle O(1) is the pullbackof the classical canonical line bundle on Bl{0} (An ) which is ample. As being relatively ample
is stable under base change (the proof in [Ann22, Prop. 2.12] works in general), it follows that
O(1) is p-ample.
In presence of an ample line bundle, every closed subset with quasi-compact complement
is the support of a quasi-smooth closed subscheme. More precisely, we have the following
lemma.
Lemma 2.6. Let X be a qcqs derived scheme which carries an ample line bundle. Let Z0 ,→cl
X be a classically finitely presented closed subscheme. Then there exists a quasi-smoothclosed subscheme Z ,→ X whose classical truncation is Z0 .
Proof. This is very similar to [BKRS22, Constr. A.2.2]. Let J = fib(OX → OZ0 ). As OZ0 is
a discrete, quasi-coherent sheaf and OX → OZ0 is surjective on π0 , the sheaf J is connective,
quasi-coherent, and π0 J is the ideal sheaf defining Z0 in cl X. As Z0 ,→ cl X is classicallyof finite presentation, π0 J is a π0 OX -module of finite type. Hence Lemma 2.5 implies theexistence of a vector bundle E on X and a map E → J which is surjective on π0 . Let
V (E) be the geometric vector bundle V (E) := Spec(LSym∗OX (E)) over X. The composition
E → J → OX defines a section s : X → V (E), and we form the fibre product
Z

X
0

i
s

X

V (E).

Equivalently, we have
Z = Spec(OX ⊗LSym∗O (E) OX ).
X

By construction, i : Z ,→ X is quasi-smooth. The same computation as in the proof of
Lemma 2.2 shows that cl Z = Z0 , as desired.

2.8. Pushouts of derived schemes. We also need certain pushouts of derived schemes.
These have been studied in [GR17]:
Lemma 2.7. Let i : Y1 → Y10 be a closed immersion of derived schemes which is an isomorphism on underlying topological spaces, and let f : Y1 → Y2 be an affine map in dSch. Thenthe following hold.
(1) The pushout square
i

Y1

f0

f

Y2

Y10

i0

Y2 tY1 Y10

exists in dSch. Write Y20 := Y2 tY1 Y10 . The map Y2 → Y20 is a closed immersion and
an isomorphism on underlying topological spaces. In particular, if Y2 is qcqs, then so
is Y20 .

ON PRO-CDH DESCENT ON DERIVED SCHEMES

11(2) For an affine open subscheme U2 ⊆ Y2 with U1 := f −1 (U2 ) ⊆ Y1 , and the corresponding open subschemes Ui0 ⊆ Yi0 (i = 1, 2), the map
U2 tU1 U10 → U20
is an isomorphism.
(3) If f is an open immersion, then so is f 0 .
(4) Assume i exhibits Y1 as the underlying classical scheme of Y10 . Then i0 : Y2 → Y20 is
an isomorphism on underlying classical schemes.
Proof. Except for (4), this is [GR17, Ch. 1, Cor. 1.3.5]. By (2), we may assume Yi = Spec(Ai )(i = 1, 2) and Y10 = Spec(A01 ) are affine and A1 = π0 (A01 ). By the construction of the pushout,
Y20 = Spec(A02 ) where A02 = A2 ×A1 A01 is the pullback in derived rings. We thus have an exact
sequence of homotopy groups
π1 (A1 ) → π0 (A02 ) → π0 (A2 ) ⊕ π0 (A01 ) → π0 (A1 ),which implies (4) as π1 (A1 ) = 0 and π0 (A0 ) ∼= π0 (A1 ).
1

3. Modifications of derived schemes
In the following, X always denotes a qcqs derived scheme.
Definition 3.1. Let U ⊆ X be a quasi-compact open subscheme. A U -modification of X isa proper morphism f : Y → X which is an isomorphism over U . A closed U -modification isa U -modification which is a closed immersion.
Note that we do not assume that a U -modification induces a bijection of the set of genericpoints. For example, if Z ,→ X is a quasi-smooth closed immersion with |Z| ∩ |U | = ∅, thenthe derived blowup BlZ (X) → X is a U -modification which is moreover lafp (see 2.7). In
fact, derived blowups and lafp closed U -modifications generate all lafp U -modifications in thefollowing sense:
Theorem 3.2. Assume that X carries an ample line bundle. Let U ⊆ X be a quasi-compact
open subscheme. Let f : Y → X be an lafp U -modification of X. Then there exists a commutative diagram
g
Y0
Y
h

f

p
e
X
X
where g is an lafp U -modification of Y , h is an lafp closed U -modification, and p is a derived
blowup with center set-theoretically contained in X \ U . Moreover, Y 0 carries a (p ◦ h)-ample
line bundle.

Proof. As f : Y → X is proper and lafp, the morphism of classical schemes cl f : cl Y → cl X
is proper and classically locally of finite presentation, hence classically of finite presentation.
It is also an isomorphism over cl U . By [RG71, Cor. 5.7.12] (or [Sta23, Tag 081T]), there
exists a cl U -admissible blowup3 Y0 → cl X such that the morphism Y0 → cl X factors throughcl
Y → cl X. For the constructions to come, we need the open immersion cl U → Y0 to be affine.
As this is not necessarily the case, we make a further blowup: As cl U is quasi-compact, there
3i.e., a blowup in a finitely presented closed subscheme of cl X which is set-theoretically contained in thecomplement of cl U ⊆ cl X

12

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

exists a finitely presented closed subscheme T0 ,→ Y0 whose underlying topological space isY0 \ cl U .4 Let Y1 → Y0 be the blowup of Y0 in T0 . Then the canonical open immersion cl U → Y1
is affine as its complement is a Cartier divisor. As the composite of two cl U -admissible
blowups is a cl U -admissible blowup [Sta23, Tag 080L], the composition Y1 → Y0 → cl X isa cl U -admissible blowup, say Y1 = BlS0 (cl X) for some classically finitely presented closed
subscheme S0 ,→ cl X.
As X carries an ample line bundle, Lemma 2.6 implies the existence of a quasi-smoothclosed immersion of derived schemes S ,→ X whose classical truncation is S0 ,→ cl X. Let
e → X be the derived blowup of X in S. Then there is a canonical closed immersion
p: X
cle which is an isomorphism over cl U . Note that Y1 need not be classically of finite
Y1 ,→ X,
presentation over X. However, by [GD71, Cor. 6.9.15] we can write the ideal sheaf defining Y1
cle as a filtered colimit of sub-ideal sheaves of finite type. Passing to relative spectra, wein X
cle can be written as a cofiltered limit of classicallydeduce that the closed immersion Y1 ,→ X
cle all of which are isomorphisms over cl U . As
finitely presented closed immersions Yα → X
clcl
Y → X is classically of finite presentation, there exists an α such that the cl X-morphism
Y1 → cl Y factors through a morphism Yα → cl Y , see [Gro66, Prop. 8.14.2]. Thus, so far, wehave constructed a commutative diagram of classical schemes finitely presented over cl X

Yα

cl

cl

cl

e
X

Y

X

in which all morphisms are isomorphisms over cl U . The lower right corner is the classicalpfe→truncation of the cospan X− X←−Y.
As the composite of the affine open immersion cl U → Y1 with the closed immersion Y1 ,→Yα , the open immersion cl U → Yα is affine, too. By Lemma 2.7 we may hence form the pushout
Y2 = Yα tcl U U of derived schemes, for which we have cl Y 2 = Yα . As p and f are isomorphisms
e g2 : Y2 → Y , and a commutative diagram
over U , we get induced morphisms h2 : Y2 → X,
Y2

g2

h2

e
X

Y
f

p

X.

Note that h2 is a closed immersion, as this only depends on the underlying map of classicalschemes. Moreover, all morphisms in the above diagram are U -modifications. However, h2
and g2 need not be locally of almost finite presentation. In order to remedy this, we consider
e ×X Y . As cl Y → cl X is separated, and as
the induced morphism k2 = (h2 , g2 ) : Y2 → X
h2 is a closed immersion, k2 is a closed immersion, too. It is also an isomorphism over U .
e ×X Y ) is
By construction, the map of underlying classical schemes cl k2 : Yα = cl Y 2 → cl (Xclassically of finite presentation. We can hence apply Lemma 2.2 to obtain a factorizatione ×X Y such that k 0 is locally of finiteof k2 through a closed derived subscheme k 0 : Y 0 ,→ X
e ×X Y →presentation and an isomorphism over U , and cl Y 0 ∼= cl Y 2 . The composite h : Y 0 ,→ X
4For example, one can use absolute Noetherian approximation [TT90, Thm. C.9, C.2] to see this.

ON PRO-CDH DESCENT ON DERIVED SCHEMES

13

e is then lafp and an isomorphism over U . As its underlying map of classical schemesXcle it is a closed immersion. Similarly, the composite g : Y 0 →identifies with cl h2 : cl Y 2 ,→ X,
e ×X Y → Y is lafp and an isomorphism over U . This finishes the construction of the assertedX
commutative diagram.
It remains to prove the claim about ample line bundles. As discussed in 2.7, the canonicale is p-ample. As h is a closed immersion and thusline bundle O(1) on the derived blowup X∗
in particular affine, the pullback h O(1) is then (p ◦ h)-ample.

4. Pro-cdh descent for connective localizing invariants
In this section, we prove our main results on pro-descent for localizing invariants. The
strategy is the same as in [KST18b]: We first prove the result for the special cases of derived
blowups and finite modifications and then use the geometric input from Theorem 3.2 to handle
the general case.
We begin by fixing some notation. Let k be a fixed commutative base ring (e.g. k = Z). If
E is an additive invariant of small k-linear ∞-categories with values in a stable presentable∞-category C, e.g. the ∞-category of spectra, and X is a qcqs derived k-scheme, we write
E(X) for E(Perf(X)). Let Z ⊆ |X| be a closed subset with quasi-compact open complement.
Recall from 2.5 that the formal completion XZ∧ is an ind-derived scheme. Applying E we thus
obtain a pro-object E(XZ∧ ). We write E(X, XZ∧ ) for the relative term fib(E(X) → E(XZ∧ ))in Pro(C).
A version of the following Proposition was first proven in [KST18b]. There all schemes
were assumed to be classical Noetherian schemes, E was K-theory, and the result only gave a
weakly cartesian square of pro-spectra, see below for this notion. In a letter to Kerz, Antieau[Ant18] described a simplification of the proof which at the same time gives a cartesian square.
We thank Ben Antieau for allowing us to include his argument in our paper.
e → X be a derived blowup in some
Proposition 4.1. Let X, Z, and E be as above. Let X
quasi-smooth closed immersion S ,→ X with S set-theoretically contained in Z. Then thesquareE(X)

E(XZ∧ )

e
E(X)

e ∧)
E(X
Z

is cartesian in Pro(C).
Proof. Let r ≥ 1 be the virtual codimension of the derived blowup, and let D be its exceptional
divisor, i.e., the universal virtual Cartier divisor on the derived blowup, so that there is acommutative diagram
j
e
D
X
q

p

i
S
X.
e has a semi-orthogonal decomposition as follows.
Recall from [Kha20, Thm. C] that Perf(X)∗eThe functor p : Perf(X) → Perf(X) is fully faithful, denote its essential image by B(0).
e isFor 1 ≤ k ≤ r − 1, the composed functor j∗ (q ∗ (−) ⊗O OD (−k)) : Perf(S) → Perf(X)D

14

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

fully faithful, denote its essential image by B(−k). Then the sequence of full subcategories
e In particular,(B(0), B(−1), . . . , B(−r+1)) forms a semi-orthogonal decomposition of Perf(X).
there is a decomposition(4)

e ' E(B(0)) ⊕
E(X)

r−1
M

E(B(−k)) ' E(X) ⊕

k=1

r−1
M

E(S).

k=1

Now let Z 0 ,→ X be any closed derived subscheme with |Z 0 | = Z. Note that all ∞-categories
appearing above are in fact Perf(X)-linear, as are the functors between them. In particular, wee along Perf(X) → Perf(Z 0 ). As
can base change the semi-orthogonal decomposition of Perf(X)there are canonical equivalences (as follows from [Lur18, Cor. 9.4.3.8] by passing to compact
objects; see also [BZFN10, Thm. 4.7] with slightly different hypotheses)
e ⊗Perf(X) Perf(Z 0 ) ' Perf(Xe ×X Z 0 ),Perf(X)
Perf(S) ⊗Perf(X) Perf(Z 0 ) ' Perf(S ×X Z 0 ),e ×X Z 0 ) admits a semi-orthogonal decomposition
we conclude that Perf(X(B(0)Z 0 , B(−1)Z 0 , . . . , B(−r + 1)Z 0 )with B(0)Z 0 ' Perf(Z 0 ) and B(−k)Z 0 ' Perf(S ×X Z 0 ) for 1 ≤ k ≤ r − 1. In particular,(5)e ×X Z 0 ) ' E(Z 0 ) ⊕
E(X

r−1
M

E(S ×X Z 0 ).

k=1

Recall from (1) that E(XZ∧ ) ∈ Pro(C) is given concretely as the pro-object {E(Z 0 )}Z 0 ,→X,|Z 0 |=Z
where Z 0 runs through all closed derived subschemes of X with |Z 0 | = Z. As formal come ∧ ) = {E(Xe ×X
pletion is compatible with base change (see 2.5), we similarly have E(XZ0Z )}Z 0 ,→X,|Z 0 |=Z . Comparing the decompositions (4) and (5) it thus suffices to prove that thefunctor induced by pullback
Perf(S) → {Perf(S ×X Z 0 )}Z 0 ,→X,|Z 0 |=Z
is an equivalence of pro-∞-categories. For this, it suffices to check that the map of ind-derived
schemes {S ×X Z 0 }Z 0 ,→X,|Z 0 |=Z → S is an equivalence. But this is clear: By 2.5 again, thesource represents the Z-completion SZ∧ of the target. As S is set-theoretically contained inZ, we clearly have SZ∧ = S.
Let ` be an integer. Recall from [LT19, Def. 2.5] that a spectra valued localizing invariant
E is called `-connective if, for any n-connective map (n ≥ 1) of connective E1 -ring spectra
A → B, the induced map E(A) → E(B) is (n + `)-connective. For example, K-theory,
topological cyclic homology TC and rational negative cyclic homology HN(− ⊗ Q/Q) are
1-connective, THH is 0-connective; see [LT19, Ex. 2.6].
Recall also that a map of pro-spectra {Cα }α → {Dα }α is called a weak equivalence if eachtruncation {τ≤n Cα }α → {τ≤n Dα }α is an equivalence in Pro(Sp) and there are similar notionsof being weakly cartesian, weakly contractible, and so on; see [LT19, Def. 2.27].
Proposition 4.2. Let E be a localizing invariant of small k-linear ∞-categories that is `connective for some integer `. Let f : Y → X be a finite, lafp morphism of qcqs derived

ON PRO-CDH DESCENT ON DERIVED SCHEMES

15

k-schemes which is an isomorphism outside the closed subset Z with |X| \ Z quasi-compact.
Then the commutative square of pro-spectra
E(X)E(XZ∧ )E(Y )E(YZ∧ ).

(6)

is weakly cartesian.
Proof. We first reduce to the case that X is affine: As X is qcqs, we can write X as thecolimit of a finite diagram of open affine subschemes Vi ,→ X. As any localizing invariant
satisfies Zariski descent , we get E(X) = limi E(Vi ) and E(Y ) = limi E(Y ×X Vi ). If Z 0 ,→ X
is a closed subscheme with |Z 0 | = Z, we also have E(Z 0 ) = limi E(Vi ×X Z 0 ). As finite limits
in Pro(C) are computed level-wise and formal completion is compatible with base change, this∧
implies E(XZ∧ ) = limi E((Vi )∧Z ) and similarly for E(YZ ). Replacing X by Vi and Z by Vi ∩ Z
we thus reduce to the case that X is affine.
So assume now that X is affine, say X = Spec(A). As f is finite, also Y is affine, say
Y = Spec(B). Let φ : A → B be the corresponding morphism of derived rings and write
J = fib(A → B). As A and B are connective, J is (−1)-connective. By [Lur18, Cor. 5.2.2.2]
the A-algebra B is almost perfect as an A-module, hence also J is almost perfect as an
A-module, i.e. τ≤n J is a compact object in τ≤n+1 Mod(A)≥−1 = Mod(A)[−1,n] for every n.5
Choose f1 , . . . , fr ∈ π0 (A) whose zero set is Z. Recall from (2) that XZ∧ is then represented
by the ind-derived scheme {Spec(A//f1α , . . . , frα )}α≥1 . Consider the commutative diagram ofderived rings
A

A//f1α , . . . , frα

B

B//f1α , . . . , frα .

(7)

We claim that as pro-system in α, this square is weakly cartesian. The map of vertical fibres(in A-modules) is the canonical map
J −→ J//f1α , . . . , frα ,so we have to prove that this map is a weak equivalence as a pro-system in α. For any
i = 1, . . . , r, the fibre of the map of pro-systems J → {J//fiα }α is the pro-system
(8)

fi

fi

{ J ←− J ←− . . . }.

We show below that this system is weakly contractible. We then get weak equivalences
'
'
'
J −
→ {J//f1α }α and J −→ {J//f2α }α −→ {J//f1α , f2α }α , and so on, so {(7)}α is indeed weakly
cartesian.
The assumption that f be an isomorphism outside Z implies that J[fi−1 ] = 0 for i =1, . . . , r. Note that
fi

fi

fi

J[fi−1 ] = colim(J −
→J −
→J −
→ . . . ).
5The (n + 1)-truncated objects in Mod(A)≥−1 are precisely the n-truncated, (−1)-connective objects in
Mod(A) with respect to the standard t-structure. Hence the usual truncation τ≤n J coming from the t-structureis the categorical (n + 1)-truncation Mod(A)≥−1 → τ≤n+1 Mod(A)≥−1 .

16

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

As the standard t-structure on Mod(A) is compatible with filtered colimits, we have
fi

fi

fi

0 = τ≤n J[fi−1 ] = colim(τ≤n J −→ τ≤n J −→ τ≤n J −→ ...)As τ≤n J is compact in Mod(A)[−1,n] , we have
0 = π0 (map(τ≤n J, τ≤n J[fi−1 ])) ∼= colim π0 (map(τ≤n J, τ≤n J))
which means that there is an N such that the power fiN acts nullhomotopically on τ≤n J. It
follows that (8) is weakly contractible, and hence the pro-system of squares {(7)}α is indeed
weakly cartesian.
Note that for every α the canonical map B ⊗A (A//f1α , . . . , frα ) → B//f1α , . . . , frα is anequivalence. Thus we may apply the variant of [LT19, Thm. 2.32] for `-connective localizing
invariants to deduce that {(7)}α induces a weakly cartesian square of E-theory pro-spectra.

Remark 4.3. In Proposition 4.2 one can actually relax the finiteness assumption if one adds
other hypotheses: As the proof shows, we only need that the pro-systems (8) are weakly
contractible for each i (using notation of the proof). This is satisfied if, on each truncation
τ≤n J, some power of each fi acts null-homotopically.
If X and Y are n-truncated, then also J is n-truncated. It is then enough to assume that
J is perfect to order n in the sense of [Lur18, Def. 2.7.0.1] in order to conclude that the
pro-systems (8) are weakly contractible.
As a special case, if the map f in Proposition 4.2 is a closed immersion of classical qcqs
schemes which is classically finitely presented, then the conclusion of the proposition holds,
i.e. (6) is weakly cartesian.
Remark 4.4. There is a version of the above Proposition for stacks: Let X be a qcqs ANS
derived algebraic stack [BKRS22, A.1], Y → X a finite, locally almost finitely presented
morphism of derived algebraic stacks, and Z ,→ X a closed immersion with quasi-compact
open complement. Let E be any connected localizing invariant in the sense of [BKRS22,Def. C.1.3] (e.g. a 2-connective or a finitary 1-connective localizing [BKRS22, Rem. C.1.5]).
Then the square (6) of pro-spectra is weakly cartesian.
Indeed, the proof of [BKRS22, Thm. 4.2.1] works with the following changes: As in the
proof of Lemma 2.3.2 in op. cit., the proof of our Proposition 4.2 shows that the formallycompleted square
YZ∧

Y

XZ∧

X

(9)

is weakly cocartesian. As in the proof of Theorem 2.4.1 of op. cit. this implies that the squareof derived (pro-)categories induced by (9) is weak pro-Milnor and satisfies weak pro-basechange in the sense of Definitions C.2.4 and C.2.6 there. Hence by Theorem C.3.1 there thesquare (6) is weakly cartesian.
We now come to our main descent result, which in particular includes Theorem A.
Theorem 4.5. Let E be a localizing invariant of small k-linear ∞-categories which is `connective for some integer `. Let X be a qcqs derived k-scheme, U ⊆ X a quasi-compact

ON PRO-CDH DESCENT ON DERIVED SCHEMES

17

open subscheme, and denote by Z the closed subset X \ U . Let f : Y → X be a locally almost
finitely presented U -modification of X. Then the square of pro-spectra
E(X)E(XZ∧ )E(Y )E(YZ∧ )

(10)

is weakly cartesian.
If X and Y are Noetherian classical schemes, and f is classically of finite type, then f is
lafp (see 2.4). Moreover, the formal derived completions are equivalent to the classical formal
completions (see 2.5). We thus recover the classical pro-cdh descent statement as formulated(for K-theory) for example in [KST18b, Thm. A].
Proof of Theorem 4.5. We first prove the theorem under the additional assumption that Y
carries an f -ample line bundle. Exactly as in the proof of Proposition 4.2 we reduce to thecase that X is affine. Then Y carries an ample line bundle.
By Theorem 3.2, we find an lafp U -modification g : Y 0 → Y such that f ◦ g factors as ane followed by a derived blowup p : X
e → X with center
lafp, closed U -modification h : Y 0 ,→ X
set-theoretically contained in Z. Both of these are isomorphisms over U . By Proposition 4.1
and Proposition 4.2, the maps of relative E-theory pro-spectrae XeZ∧ ) → E(Y 0 , Y 0 ∧E(X, XZ∧ ) → E(X,Z)
are weak equivalences. Hence also the composite(11)∧

E(X, XZ∧ ) → E(Y, YZ∧ ) → E(Y 0 , Y 0 Z )is a weak equivalence. It follows that E(X, XZ∧ ) → E(Y, YZ∧ ) is the inclusion of a direct
summand (in the weak sense). As Y carries an ample line bundle, we can repeat this argument
for the U -modification Y 0 → Y . So also the second map in (11) is the inclusion of a direct
summand. It now follows that both maps are in fact weak equivalences.
We now prove the theorem for general Y . As before, we may assume that X is affine. By
Theorem 3.2 again, there exists an lafp U -modification g : Y 0 → Y such that Y 0 carries anample line bundle relative to X. As X is affine, this line bundle is in fact ample and it is also
ample relative to Y . Hence, by Step 1, the maps of pro-spectra E(X, XZ∧ ) → E(Y 0 , Y 0 ∧Z ) and∧∧00∧E(Y, YZ ) → E(Y , Y Z ) both are weak equivalences. By 3-for-2, also E(X, XZ ) → E(Y, YZ∧ )is a weak equivalence, as desired.

5. Pro-cdh descent for the cotangent complex and motivic cohomology
We fix some base ring k (e.g., k = Z). For X a qcqs derived k-scheme, we denote by
LX ∈ QCoh(X) its (algebraic) cotangent complex relative to k, and by LiX its i-th derived
exterior power (i ≥ 0). If X = Spec(A) is affine, we also write LiA for the A-module Γ(X, LiX )corresponding to LiX . If Y → X is a morphism of qcqs derived k-schemes, we denote by
LY /X its relative cotangent complex and by LiY /X its derived exterior powers. Similarly for amorphism of derived k-algebras A → B.
If Z ⊆ |X| is a closed subset with quasi-compact open complement, we obtain a procompletion along Z functor(−)∧
Z : QCoh(X) → Pro(QCoh(X))

18

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

by pulling back to the ind-derived scheme XZ∧ and pushing forward. The usual formal completion along Z is given by composing the above functor with lim : Pro(QCoh(X)) → QCoh(X).
Lemma 5.1. Let X = Spec(A) be an affine derived k-scheme, Z ⊆ |X| a closed subset withquasi-compact open complement. Let M ∈ Mod(A on Z)aperf be an almost perfect A-module
supported on Z. Then the canonical map
M → MZ∧is a weak equivalence in Pro(Mod(A)).
Proof. This was proven in the proof of Proposition 4.2 (replace J there by M ).



For a pro-system of derived rings {A(α)}α we denote by Pro(Mod)({A(α)}) the ∞-categoryof pro-systems of modules over the pro-ring {A(α)}α (see [LT19, §2.4] for a precise definition).
Lemma 5.2. In the situation of the previous lemma, choose f1 , . . . , fr ∈ π0 (A) defining Z,
and write A(α) = A//f1α , . . . , frα . Then the pro-system{LA(α)/A }α ∈ Pro(Mod)({A(α)})
vanishes. In fact, all transition maps in this pro-system are null-homotopic.
Proof. This follows by base change from the universal case: Let R = k[T1 , . . . , Tr ] be thepolynomial ring over k, let R → k be the map sending all Ti to 0, and let gα : R → A be themap sending Ti to fiα . Then A(α) = k ⊗R,gα A and consequently
LA(α)/A ' Lk/R ⊗R,gα A ' Lk/R ⊗k A(α).
Consider the pro-system {Lk/R }α ∈ Pro(Mod(k)) whose transition maps are induced by themaps R → R sending the Ti to Tiβ . Then
{LA(α)/A }α ' {Lk/R }α ⊗k {A(α)}α .
Hence it suffices to prove that all transition maps Lk/R → Lk/R are null-homotopic in Mod(k).
As R → k has a section, the transitivity triangle yields an equivalence Lk/R ' ΣLR/k ⊗R k.
As R is a polynomial ring, LR/k is discrete, given by the module of Kähler differentials
LΩ1R/k = i RdTi . The map Ti 7→ Tiβ induces dTi 7→ βTiβ−1 dTi in Ω1R/k , and hence the zeromap in Ω1R/k ⊗R k ' Σ−1 Lk/R for β > 1. This proves our claim.
Lemma 5.3. In the situation of Lemma 5.2, the canonical map{LiA ⊗A A(α)}α → {LiA(α) }αis an equivalence in Pro(Mod)({A(α)}).
Proof. The transitivity triangle for the maps A → A(α) and Lemma 5.2 imply the case i = 1.
Passing to derived exterior powers over {A(α)} implies the general case.

The following theorem was suggested by Matthew Morrow. It generalizes [Mor16a, Thm. 2.4]
(see also [EM23, Lemma 8.5]).

ON PRO-CDH DESCENT ON DERIVED SCHEMES

19

Theorem 5.4. Let X be a qcqs derived k-scheme, U ⊆ X a quasi-compact open subscheme,
and denote by Z the closed subset X \ U . Let f : Y → X be a locally almost finitely presented
U -modification of X. Then for every i ≥ 0, the square of pro-spectra (or pro-complexes)
Γ(X, LiX )Γ(XZ∧ , LiX ∧ )Γ(Y, LiY )Γ(YZ∧ , LiY ∧ )

Z

(12)
Z

is weakly cartesian.
Here, similarly as in the previous section, Γ(XZ∧ , LiX ∧ ) denotes the pro-object {Γ(Z 0 , LiZ 0 )}Z 0 ,→X
Z
where Z 0 ,→ X runs through all closed immersions of derived schemes with |Z 0 | = Z.
Proof. By Zariski descent we may assume that X = Spec(A) is affine. Choose f1 , . . . , fr ∈π0 (A) defining Z, and write A(α) = A//f1α , . . . , frα so that XZ∧ = {Spec(A(α))}α≥1 . Abusing
notation slightly, we also write f∗ LiY ∈ Mod(A) for the module Γ(X, f∗ LiY ) = Γ(Y, LiY )corresponding to f∗ LiY ∈ QCoh(X).
By Lemma 5.3 we have
Γ(XZ∧ , LiX ∧ ) ' LiA ⊗A {A(α)}.
Z

Similarly, using affine coverings of Y and the usual induction we getΓ(YZ∧ , LiY ∧ ) ' (f∗ LiY ) ⊗A {A(α)}.
Z

So we have to prove that
LiA

LiA ⊗A {A(α)}

f∗ LiY(f∗ LiY ) ⊗A {A(α)}(13)

is weakly cartesian. The map LiA → f∗ LiY factors as LiA → f∗ f ∗ LiA → f∗ LiY , so it suffices toshow that the two squares
LiA

LiA ⊗A {A(α)}

f∗ f ∗ LiA(f∗ f ∗ LiA ) ⊗A {A(α)}

f∗ LiY(f∗ LiY ) ⊗A {A(α)}

and

(14)
f∗ f ∗ LiA(f∗ f ∗ LiA ) ⊗A {A(α)}

are weakly cartesian.
We first treat the left one. Using the projection formula, we rewrite that square as thetensor product of LiA with the squareA{A(α)}f∗ OY

f∗ OY ⊗A {A(α)}.

(15)

As f is lafp, f∗ OY is almost perfect [Lur18, Thm. 5.6.0.2]. On the other hand, by assumption A → f∗ OY is an isomorphism outside Z. Hence the left vertical fibre in (15) lies in

20

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

Mod(A on Z)aperf . It now follows from Lemma 5.1 that the map on vertical fibres is a weakequivalence, and hence that (15) is weakly cartesian. As LiA is connective, tensoring with
LiA preserves weak equivalences and weakly cartesian squares (see e.g. [LT19, Lemma 2.29]).
Thus the left-hand square in (14) is weakly cartesian.
We now consider the right square in (14). The transitivity triangle for f gives rise to a
finite filtration on LiY whose graded pieces are given byf ∗ LkX ⊗OY LYi−k/X ,k = 0, . . . , i.

It follows that the left vertical cofibre in our square has a finite filtration whose graded pieces
are given byi−kkf∗ (f ∗ LkX ⊗OY Li−kk = 0, . . . , i − 1.
Y /X ) ' LX ⊗A f∗ LY /X ,As f is lafp, the relative cotangent complex LY /X is almost perfect [Lur04, Prop. 3.2.14]
i−kand so are its wedge powers Li−k
Y /X . Again because f is lafp, the direct images f∗ LY /X arealmost perfect [Lur18, Thm. 5.6.0.2]. As f is an isomorphism outside Z, the sheaf f∗ LYi−k/X
is supported on Z for k < i. So we may again apply Lemma 5.1 to deduce that the map
i−kf∗ Li−k
Y /X → (f∗ LY /X ) ⊗A {A(α)}α is a weak equivalence for k = 0, . . . , i − 1 and hence so is
the mapi−kk
LkX ⊗A f∗ Li−k
Y /X → (LX ⊗A f∗ LY /X ) ⊗A {A(α)}α .
Using the above filtration, it follows that the map on vertical fibres in the right-hand squarein (15) is a weak equivalence, and hence that square is also weakly cartesian. This finishes
the proof of the theorem.
Combining Theorem 5.4 with the arguments of [EM23, §8.1] we obtain pro-cdh descent for
Elmanto–Morrow’s motivic cohomology denoted by Z(j)mot (−).
Corollary 5.5. Let F be a prime field. Let X be a qcqs derived F-scheme, U ⊆ X a quasicompact open subscheme, and denote by Z the closed subset X \ U . Let f : Y → X be a locallyalmost finitely presented U -modification of X. Then the square of pro-complexes
Z(j)mot (X)Z(j)mot (XZ∧ )Z(j)mot (Y )Z(j)mot (YZ∧ )is weakly cartesian.
Proof. The proof of Elmanto and Morrow goes through verbatim once we replace their
Lemma 8.5 by the above Theorem 5.4.

6. Generalised Weibel vanishing
In this section we prove our generalised Weibel vanishing result. As its formulation involves the valuative dimension, we start by recalling the latter. The valuative dimension of acommutative ring was introduced and studied by Jaffard [Jaf60]; we refer to [EHIK20, §2.3]
for an account. For an integral domain A, it is defined asvdim(A) = sup{n | ∃A ⊆ V0 ( V1 ( · · · ( Vn ⊆ Frac(A), Vi valuation ring},

ON PRO-CDH DESCENT ON DERIVED SCHEMES

21

and in general as vdim(A) = sup{vdim(A/p) | p ∈ Spec(A)}. For a classical scheme X, onethen setsvdim(X) = sup{vdim(OX (U )) | U ⊆ X affine open}.
We always have dim(X) ≤ vdim(X) and equality holds if X is Noetherian [Jaf60, Ch. IV,
Thm. 1, Cor. 2 of Thm. 5]. Note that the valuative dimension only depends on the underlying reduced scheme of a scheme. In particular, it makes sense to talk about the valuative
dimension of a closed subset of a scheme, by equipping it with any closed subscheme structure.
Let now X be a derived scheme. The Krull dimension dim(X) of X is defined to be the
Krull dimension of the underlying topological space of X. The valuative dimension vdim(X)is defined to be the valuative dimension of the underlying classical scheme cl X.
The following theorem generalizes [KST18b, Thm. B], which treats the case of Noetherianclassical schemes.
Theorem 6.1. Let X be a qcqs derived scheme. Assume that the underlying topological spaceof the spectrum of every local ring of every irreducible component of cl X is Noetherian. Thenthe following hold.
(1) K−i (X) = 0 for all i > vdim(X).
(2) For all i ≥ vdim(X) and any integer r ≥ 0, the pullback map K−i (X) → K−i (ArX ) is
an isomorphism.
Remarks 6.2.
(1) We point out that Lemma 6.6 on the annihilation of negative K-theory
classes is the only ingredient in our proof of Theorem 6.1 where we need a Noetherian
assumption on the underlying topological space.
(2) Any qcqs scheme of Krull dimension ≤ 1 satisfies the assumptions of the theorem.
(3) The assumptions in the theorem are stable under maps essentially of finite type:
Assume that X satisfies the hypothesis of Theorem 6.1. If f : Y → X is a morphismof qcqs derived schemes such that the morphism cl f of underlying classical schemes isessentially of finite type, then also Y satisfies the hypothesis of the theorem.
Indeed, we may assume that Y and X are classical. Let Y0 be the spectrum of a
local ring of an irreducible component of Y . Then the restriction of f to Y0 factors
through the spectrum X0 of a local ring of an irreducible component of X and theinduced morphism Y0 → X0 is essentially of finite type. Say X0 = Spec(A) and Y0 =Spec(S −1 B) where B is a finitely generated A-algebra and S ⊆ B is a multiplicative
subset. As the topological space |X0 | is Noetherian by assumption, so is | Spec(B)|by [OP68, Cor. 2.6]. As Spec(S −1 B) is homeomorphic to a subspace of Spec(B), it is
Noetherian, too.
For proving Theorem 6.1, we follow the strategy of the proof of [KST18b, Thm. B] though
we need some additional inputs. The following lemma replaces [KS17, Lemma 4] and isessentially due to Scheiderer [Sch92].
Lemma 6.3. Let X be a spectral space, and F a sheaf of abelian groups on X. Let r ≥ 0 be
an integer. Assume that Fy = 0 for all points y ∈ X with dim({y}) > r. Then H n (X, F) = 0
for all integers n > r.
Proof. This is a direct consequence of results of Scheiderer [Sch92], see the proof of Proposition 4.7 there. Let sp• (X) → X be the quasi-augmented simplicial topological space definedin [Sch92, §2]: Its set of n-simplices is given by chains of specialisations
x0  · · ·  xn22SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

of points in X with coincidences between the xi allowed. The topology is induced by theconstructible topology on X n+1 . The quasi-augmentation is given by the canonical map
sp0 (X) → X. Let γn : spn (X) → X be the map sending a chain as above to x0 . By [Sch92,
Rem. 2.5 and Thm. 4.1], the cohomology groups H ∗ (X, F) are computed by the complex
Γ(sp0 (X), γ0∗ F) → Γ(sp1 (X), γ1∗ F) → Γ(sp2 (X), γ2∗ F) → · · ·which arises from the cosimplicial abelian group [n] → Γ(spn (X), γn∗ F). Let
B0 → B1 → B2 → · · ·be the associated normalized subcomplex, which also computes H ∗ (X, F). As in the proof of[Sch92, Prop. 4.7], Bn coincides with the group of those sections a ∈ Γ(spn (X), γn∗ F) whose
supports consist only of non-degenerate simplices x ∈ spn (X). Hence, it suffices to show
(γn∗ F)x = 0 for such x if n > r. So let now x ∈ spn (X) be given by a chain of specialisations
x0  · · ·  xn with pairwise different xi ’s. Then, we must have dim({x0 }) ≥ n > r. Thus theassumption implies (γn∗ F)x = Fx0 = 0 as wanted. This completes the proof of the lemma. 
Lemma 6.4. For a scheme X and a point x ∈ X with closure {x} in X,
vdim(OX,x ) + vdim({x}) ≤ vdim(X).
In particular,vdim(OX,x ) + dim({x}) ≤ vdim(X).
Proof. By definition it suffices to prove the lemma in case X = Spec(A) with A integral. In
this case, the first assertion is [Jaf60, Ch. IV, Prop. 2], the second assertion follows from thefact that dim(Y ) ≤ vdim(Y ) for every scheme Y .
Lemma 6.5. Let X be a qcqs derived scheme of finite valuative dimension. Let F be a sheafof spectra on X for the Zariski topology. Assume that for every x ∈ X, the homotopy groupsof the stalks π−i (Fx ) vanish for i > vdim(OX,x ). Then π−i (F (X)) = 0 for all i > vdim(X).
Note that if F is K-theory, then π−i (Kx ) ∼= K−i (OX,x ) as K-theory commutes with filteredcolimits.
Proof. This is deduced from Lemma 6.3 by the same argument as the proof of [KS17, Prop. 3].
From [CM21, Thm. 3.12] we know that the homotopy dimension of XZar is bounded by the
Krull dimension dim(X), and in particular the ∞-topos of sheaves of spaces on XZar is
hypercomplete. Hence there is a convergent Zariski descent spectral sequence
E p,q = H p (X, Fe−q ) =⇒ π−p−q (F (X)),
2

where Fe−q is the Zariski sheaf of abelian groups on X associated to the presheaf U 7→
π−q (F (U )), and E2p,∗ = 0 unless 0 ≤ p ≤ dim(X). It suffices to show E2p,q = 0 for
p + q > vdim(X). We may assume p ≤ dim(X) so that q > 0. Note that Fe−q,x = π−q (Fx ).
By Lemma 6.3 it now suffices to check that π−q (Fx ) = 0 for all points x with dim({x}) >vdim(X) − q. But for such points x we havevdim(OX,x ) ≤ vdim(X) − dim({x}) < q
by Lemma 6.4, and hence π−q (Fx ) = 0 by assumption.

In the following lemma, given a morphism f : Y → X, we abusively denote the induced
morphism ArY → ArX by the same letter.

ON PRO-CDH DESCENT ON DERIVED SCHEMES

23

Lemma 6.6. Let X = Spec(A) be a classical integral affine scheme whose underlying topological space is Noetherian. Let i > 0, r ≥ 0, and let γ be an element in K−i (ArX ). Then
there exists an irreducible derived scheme Y and a proper lafp morphism f : Y → X which is
an isomorphism over a non-empty open subset U ⊆ X with the property that f ∗ (γ) = 0 in
K−i (ArY ).
Proof. As K-theory commutes with filtered colimits of rings, there exists a subring A0 ⊆ A,
finitely generated as a Z-algebra, and an element γ0 ∈ K−i (ArX0 ) with a∗ (γ0 ) = γ. Here
X0 = Spec(A0 ) and a : X → X0 is the induced morphism.
By [KS17, Prop. 5] there exists a proper birational morphism of schemes f0 : Y0 → X0 suchthat f0∗ (γ0 ) = 0 in K−i (ArY0 ). As X0 is Noetherian, f0 is lafp (see 2.4). Let f1 : Y1 → X be
the derived pullback of f0 along a : X → X0 . Then f1 is proper, lafp, and an isomorphism
over a non-empty open subscheme U ⊆ X. From the commutative diagram
Y1

Y0

f1

f0

X

a

X0

it is clear that f1∗ (γ) = 0 in K−i (ArY1 ).
As cl f 1 is of finite type, the topological space |Y1 | is Noetherian [OP68, Cor. 2.6]. In
particular, the open complement of the closure of |U | in |Y1 | is quasi-compact, and hence
there exists a classically finitely presented closed subscheme Y2 ,→ cl Y 1 whose underlying
topological space |Y2 | is the closure of |U | in |Y1 | (see footnote 4 in the proof of Theorem 3.2 or[KS24, Lemma 4.14(1)] for an alternative argument). Note that Y2 ,→ Y1 is an isomorphism
over U . By Lemma 2.2 there exists a finitely presented closed subscheme Y ,→ Y1 withcl
Y = Y2 . By construction, Y is irreducible, the composite f : Y ,→ Y1 → X is proper, lafp,
and an isomorphism over U , and f ∗ (γ) = 0, as desired.
In the proof, we use the following well known facts about non-positive K-theory on affine(derived) schemes.
Lemma 6.7.
(1) Let A be a derived ring. Then the canonical map K(A) → K(π0 (A)) is
2-connective, i.e. it induces an isomorphism on πi for i ≤ 1 and a surjection on π2 .
(2) Let A be a discrete commutative ring, and let I ⊆ A be a locally nilpotent ideal (i.e.,every element of I is nilpotent). Then the map K(A) → K(A/I) is 1-connective.
(3) Let A be a discrete, commutative ring, X = Spec(A), and let X = X1 ∪ X2 be a closed
covering of X. Write X12 = X1 ∩ X2 . Then there is a long exact sequence
K1 (X) → K1 (X1 ) ⊕ K1 (X2 ) → K1 (X12 ) → K0 (X) → . . .
Proof. (1) For connective K-theory, this is due to Waldhausen [Wal78, Prop. 1.1]. The
general case follows from this together with [BGT13, Thm. 9.53] (or [KST18b, Thm. 2.16]).
Alternatively, see [LT19, Lemma 2.4] for a slightly more general statement.
(2) Writing I as a filtered colimit of nilpotent ideals, we may assume that I itself isnilpotent. Then K0 (I) = 0 by [Wei13, Exc. II.2.5] and hence K1 (A) → K1 (A/I) is surjective by Prop. III.2.3 there. Moreover, K0 (A) → K0 (A/I) is an isomorphism by [Wei13,
Lemma II.2.2]. As the ideal generated by I in any A-algebra is still nilpotent, it then follows
from the definition of negative K-groups [Wei13, Def. III.4.1] that Ki (A) → Ki (A/I) is an
isomorphism for all i ≤ 0.

24

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME(3) Let I and J be the ideals defining X1 and X2 , respectively. Hence X12 = Spec(A/I +J).
Let A0 = A/I ∩ J. Then A0 sits in a Milnor square
A0

A/I

A/J

A/I + J.

Hence, by [Wei13, Thm. III.4.3], there is a long exact sequence
K1 (A0 ) → K1 (A/I) ⊕ K1 (A/J) → K1 (A/I + J) → K0 (A0 ) → . . .
As X1 ∪ X2 = X, the ideal I ∩ J is contained in the nilradical of A and so is locally nilpotent.
Using (2), we may thus replace A0 by A in the above sequence to get a long exact sequenceof the statement.
Proof of Theorem 6.1. We write N (r) K(X) for the cofiber of the canonical split inclusion
K(X) → K(ArX ) so that we have K(ArX ) ∼= K(X) ⊕ N (r) K(X). Then assertion (2) of thetheorem is equivalent to the statement that N (r) K−i (X) = 0 for all i ≥ vdim(X).
We prove the theorem by induction on d = vdim(X). By Lemma 6.5, applied to K andΣN (r) K respectively, we may assume that X is affine and local. In this case, K−i (ArX ) ∼=K−i (Arcl X red ) for all i ≥ 0 and all r ≥ 0 by Lemma 6.7 (1), (2). So we can assume that X isa classical reduced affine local scheme.
If d = 0, then also the Krull dimension dim(X) is 0. As X is local and reduced, it is the
spectrum of a field. Hence K−i (X) = 0 and for all i > 0 and N (r) K(X) = 0 (by [Wei13,
Thm. II.7.8] and the definition of negative K-groups), as X is regular Noetherian.
Now assume that d is positive and the assertion is proven for all derived schemes satisfying
the assumptions of Theorem 6.1 that are of valuative dimension < d. We first prove theresult for X under the additional assumption that X is irreducible, and hence has Noetherian
topological space.
Let γ be an element in K−i (X) with i > vdim(X) or in N (r) K−i (X) with i ≥ vdim(X).
We have to show that γ = 0. Let f : Y → X be a proper, lafp morphism as provided by
Lemma 6.6. In particular f ∗ (γ) = 0. Let Z be the closed complement of U in X. By Theorem
4.5 we obtain an exact sequence of pro-abelian groups
K−i+1 (YZ∧ ) → K−i (X) → K−i (Y ) ⊕ K−i (XZ∧ )and similarly for N (r) K in place of K. Recall from (1) in 2.5 that K−i (XZ∧ ) is the pro-abeliangroup {K−i (Z 0 )}Z 0 where Z 0 runs through all closed (derived) subschemes Z 0 ,→ X with
|Z 0 | = Z and similarly K−i+1 (YZ∧ ) = {K−i+1 (Y ×X Z 0 )}Z 0 .
As Z does not contain the generic point of X, we have vdim(Z 0 ) < vdim(X) for every Z 0
as above by [EHIK20, Prop. 2.3.2(4)]. Hence K−i (XZ∧ ) vanishes by induction. By [EHIK20,
Prop. 2.3.2(6)] we have vdim(Y ) = vdim(X). As f −1 (Z) does not contain the generic pointof Y , the same argument as before implies that vdim(Y ×X Z 0 ) < vdim(Y ) for all Z 0 as
above. Hence also K−i+1 (YZ∧ ) vanishes by induction. It follows that f ∗ : K−i (X) → K−i (Y )is injective. Similarly, f ∗ : N (r) K−i (X) → N (r) K−i (Y ) is injective. As f ∗ (γ) = 0, we musthave γ = 0, as desired.
We now consider the general case. So X is now a classical, reduced, affine, local scheme ofvaluative dimension d > 0 and every irreducible component of X has Noetherian topologicalspace. For ease of notation, we only consider assertion (1) of the theorem. The proof of (2)ON PRO-CDH DESCENT ON DERIVED SCHEMES

25

is completely parallel. To reduce to the integral case treated above, we apply an argumentfrom [EHIK20, Thm. 2.4.15] as follows: Take an integer i > d = vdim(X) and an elementγ ∈ K−i (X). We wish to show that γ = 0. Let
E = {Z ,→ X reduced, closed | γ|Z 6= 0 ∈ K−i (Z)}.
We need to prove that E = ∅. Note that every Z ∈ E is itself a reduced local affine scheme andthat E is ordered by inclusion. Let (Zλ )λ∈Λ be a descending chain in E and put Z = limλ∈Λ Zλ .
As K-theory commutes with filtered colimits of rings, we have K−i (Z) = colimλ∈Λ K−i (Zλ ).
So if γ|Z = 0, then there exists a λ such that γ|Zλ = 0, which is a contradiction. Hence γ|Z 6= 0
so that Z ∈ E and Z is a lower bound of (Zλ )λ∈Λ . If E 6= ∅, we may apply Zorn’s lemma toconclude that E has a minimal element Z. As K−i (Z) 6= 0, Z must be reducible. Let Z gen be
the set of the generic points of Z equipped with the induced topology from the underlying
topological space of Z. By [HJ65, Cor. 2.4] (see also [EHIK20, Lemma 2.4.14]), there exists
a decomposition Z gen = S1 t S2 with Si closed and non-empty for i = 1, 2. Letting Zi be theclosure of Si in Z with reduced scheme structure, we have Z = Z1 ∪ Z2 and Zi ∩ Z gen = Sifor i = 1, 2. In particular, Z1 ∩ Z2 ∩ Z gen = ∅ so that vdim(Z1 ∩ Z2 ) < d by [EHIK20,
Prop. 2.3.2(4)]. By excision in non-positive K-theory for closed coverings of affine schemes(Lemma 6.7(3)), we have an exact sequence
K−i+1 (Z1 ∩ Z2 ) → K−i (Z) → K−i (Z1 ) ⊕ K−i (Z2 ).
The group on the left-hand side vanishes by induction. As Z was minimal in E, we must haveγ|Zi = 0 for i = 1, 2. Thus we get γ|Z = 0, which is a contradiction. Thus we must have
E = ∅, which completes the proof of Theorem 6.1.

We now give a proof of Theorem 1.3, reproduced from [KM21, Rem. 3.5]. We refer to[EHIK20, §2.1] for a discussion of the cdh topology in this generality but note that the definition of the cdh topology used in [EHIK20] is the one used by Suslin and Voevodsky, [SV00,
Def. 5.7], and the proof that cdh Čech descent is equivalent to cdh excision is Voevodsky’s
proof from [Voe10], rewritten in modern language in [AHW17, Thm. 3.2.5]. Voevodsky’s proof
that cdh Čech descent is equivalent to cdh hyperdescent requires Noetherian hypotheses thatwere lifted in [EHIK20].
Proof. We note that homotopy K-theory is a cdh sheaf. This was first proven by Cisinski[Cis13] for Noetherian schemes of finite dimension, which implies the general statement by absolute Noetherian approximation; alternatively it follows from the fact that KH is truncating,see [LT19, Cor. A.5]. In particular, we get the maps
(16)

Lcdh K≥0 → Lcdh K → KH

which we want to show are equivalences. By [EHIK20, Thm. 2.4.15, Cor. 2.3.3] the ∞-topos of
cdh sheaves of spaces on finitely presented X-schemes is locally of finite homotopy dimension
and of homotopy dimension ≤ vdim(X). This implies that we get a convergent spectral
sequence(17)

E2p,q = H p (Xcdh , π̃−q KH) =⇒ KH−p−q (X).

where π̃q KH denotes the cdh sheafified homotopy groups of KH. A conservative family ofpoints for the cdh topology is given by the spectra of henselian valuation rings [GL01], [GK15,
Thm. 2.3, Thm. 2.6], [EHIK20, Cor. 2.4.19]. As K-theory of valuation rings is connective
and agrees with its homotopy K-theory [KM21, Thm. 3.4], [KST21, Lem. 4.3], and K and26

SHANE KELLY, SHUJI SAITO, AND GEORG TAMME

KH commute with filtered colimits, the maps (16) are equivalences so we get part (2) of
Theorem 1.3. The vanishing
KH−i (X) = 0,

for all i > vdim(X)claimed in part (1) follows from the spectral sequence (17).

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Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku,
Tokyo 153-8914, Japan
Email address: shanekelly@g.ecc.u-tokyo.ac.jp
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku,
Tokyo 153-8914, Japan
Email address: sshuji.goo@gmail.com
Institut für Mathematik, Fachbereich 08, Johannes Gutenberg-Universität Mainz, D-55099
Mainz, Germany
Email address: georg.tamme@uni-mainz.de