---
title: 'Pro cdh Descent: Refinements in Algebraic K-Theory'
url: https://www.emergentmind.com/topics/pro-cdh-descent
type: topic
---

# Pro cdh Descent: Refinements in Algebraic K-Theory

Searching arXiv for recent and foundational papers on pro cdh descent.
Pro cdh descent is the refinement of cdh descent obtained by replacing a single abstract blow-up square with the full inverse system of infinitesimal thickenings of its centre and exceptional locus. In this form, algebraic $K$-theory, Hochschild and cyclic homology, topological Hochschild homology, and topological cyclic homology satisfy a pro Mayer–Vietoris property even though they generally fail descent for a single abstract blow-up square. The resulting framework links formal functions, pro-excision, and geometric blow-up squares, and it underlies constructions such as $K$-theory with compact support, vanishing theorems in negative $K$-theory, and later extensions to qcqs derived schemes, formal schemes, and mixed-characteristic motivic cohomology [1211.1813][2407.04378][2409.14295][2507.16501].

## 1. Cdh descent and its pro refinement

On the category of Noetherian schemes, the cdh-topology is generated by Nisnevich covers and abstract blow-up squares. An abstract blow-up square is a cartesian square
\[
\begin{tikzcd}
Y' \ar[r] \ar[d] & X' \ar[d] \\
Y \ar[r] & X
\end{tikzcd}
\]
in which $X' \to X$ is proper, $Y \to X$ is a closed immersion, and $X' \setminus Y' \to X \setminus Y$ is an isomorphism. In the ordinary cdh-site, such a square is declared to be a cover of $X$ by the family $\{X' \to X,\; Y \to X\}$ [1211.1813].

The basic obstruction is that algebraic $K$-theory, and likewise $HC$, $THH$, and $TC$, fail to satisfy descent for a single abstract blow-up. The pro-cdh construction remedies this by replacing the centre $Y \subset X$ with all infinitesimal thickenings. If $I \subset \mathcal{O}_X$ is the sheaf of ideals of $Y$, one writes
\[
rY := \operatorname{Spec}(\mathcal{O}_X/I^r)
\]
for the $r$th thickening of $Y$, and one works in the pro-category over $X$ with inverse systems
\[
\cdots \to rY \to (r-1)Y \to \cdots \to Y.
\]
The corresponding pro-cdh covering data are the pro-objects $\{X' \to X\} \cup \{rY \to X\}_{r \ge 1}$ [1211.1813].

The same idea extends to derived geometry. For a qcqs derived scheme $X$ with closed complement $Z$, the pro-cdh “topology” is described as a formal device that keeps track not just of a single blow-up square but of its entire system of infinitesimal thickenings. The formal completion $X^\wedge_Z$ is the ind-scheme whose objects are all closed immersions $Z' \to X$ with $|Z'|=Z$, and derived thickenings $\{Z_n\}_n$ and $\{E_n\}_n$ are the pro-infinitesimal neighborhoods entering the descent square [2407.04378].

## 2. Fundamental pro Mayer–Vietoris theorems

For an abstract blow-up square of Noetherian, finite-dimensional $k$-schemes, relative theories are written as
\[
E(X,Y)=\operatorname{hofib}(E(X)\to E(Y)).
\]
The foundational results establish that the obstruction to descent disappears after passage to the pro-system of thickenings [1211.1813].

For Hochschild and cyclic homology, the canonical map of pro abelian groups
\[
\varprojlim_r E_n(X,rY)\xrightarrow{\simeq} E_n(X',Y')
\]
is an isomorphism for all $n \in \mathbb{Z}$ when $E=HH$ or $HC$. Equivalently, the square of pro-spectra
\[
E(X)\to E(X'),\qquad \{E(rY)\}_r \to \{E(rY')\}_r
\]
is homotopy cartesian [1211.1813].

In characteristic $p>0$, under $F$-finite hypotheses, the same pattern holds for $THH$, $TR^m$, and $TC^m$ with $\mathbb{Z}/p^v$-coefficients:
\[
\varprojlim_r E_n(X,rY;\mathbb{Z}/p^v)\simeq E_n(X',Y';\mathbb{Z}/p^v).
\]
In characteristic $0$, and in characteristic $p$ under resolution hypotheses, algebraic $K$-theory satisfies the analogous statement:
\[
\varprojlim_r K_n(X,rY)\xrightarrow{\simeq} K_n(X',Y').
\]
Thus the square
\[
K(X)\to K(X'),\qquad \{K(rY)\}_r \to \{K(rY')\}_r
\]
is homotopy cartesian for all $n \in \mathbb{Z}$ [1211.1813].

A later general theorem removed the earlier characteristic and resolution restrictions for Noetherian schemes. For any abstract blow-up square of Noetherian schemes and every integer $n$, the canonical map
\[
\{K_n(X,Y_s)\}_s \xrightarrow{\cong} \{K_n(X',Y'_s)\}_s
\]
is an isomorphism of pro abelian groups; equivalently, the associated square of pro-spectra is homotopy cartesian. This formulation is presented as the Kerz–Strunk–Tamme theorem in the historical survey [1612.00418].

These statements are precisely pro Mayer–Vietoris theorems: they assert that blow-up descent holds after adjoining all infinitesimal neighborhoods, not before.

## 3. Proof architecture

The proofs combine formal-functions arguments, pro-excision, and geometric reduction steps. A central input is a formal-functions isomorphism for proper schemes $X \to \operatorname{Spec} A$ and ideals $I \subset A$:
\[
\{H_n(X)\otimes_A A/I^r\}_r \;\cong\; \{H_n(X\times_A A/I^r)\}_r,
\]
established for André–Quillen, Hochschild, and cyclic homology, and analogously for $THH$ and $TC$. The ingredients listed are Artin–Rees for pro-modules, Grothendieck’s formal-functions theorem on coherent cohomology, and hypercohomology spectral sequences compatible with inverse limits [1211.1813].

In characteristic $0$, Haesemeyer’s cdh-descent comparison reduces $K$-theory pro-descent to cyclic homology pro-descent. In characteristic $p$, one replaces $HC$ by $TC$ and $K_{\inf}$ by the fibre of the trace map $K \to TC$. Pro-excision then shows that, for finite centres, the obstruction to excision in Hochschild, cyclic, or topological cyclic theories is killed in the pro-limit; blow-up compatibility upgrades this to cartesianity of the pro Mayer–Vietoris square [1211.1813].

The historical account emphasizes a parallel route through excision situations of rings. For finite maps one reduces to a Milnor-square-type setting $(A \to B, I \subset A)$, invokes pro-excision via Tor-vanishing, and uses Artin–Rees to obtain the needed pro-Tor vanishing for Noetherian rings or quasi-regular ideals. General abstract blow-ups are then reduced to the finite case by affine localization and patching via Zariski descent. The equivalence between homotopy cartesianity of pro-spectra and isomorphisms on all pro-homotopy groups is phrased using the Fausk–Isaksen $T^+$ model structure on pro-spectra [1612.00418].

In the derived setting, the proof strategy is reorganized around two basic cases: derived blow-ups in quasi-smooth centres and finite or closed-immersion modifications. For a derived blow-up $p:\widetilde{X}\to X$, Khan–Rydh’s semi-orthogonal decomposition of $\operatorname{Perf}(\widetilde{X})$ yields
\[
E(\widetilde{X}) \simeq E(X)\oplus \bigoplus_{k=1}^{r-1} E(S)
\]
for $k$-connective localizing invariants $E$. Together with the Land–Tamme pro-excision theorem and a factorization theorem for proper l.a.f.p. $U$-modifications, this proves that the pro-square
\[
E(X)\to E(X^\wedge_Z),\qquad E(Y)\to E(Y^\wedge_Z)
\]
is weakly cartesian in $\operatorname{Pro}(\mathrm{Sp})$ [2407.04378].

## 4. Compact support and negative \(K\)-theory

One of the earliest structural consequences is the well-definedness of $K$-theory with compact support. If $X$ is a separated $k$-variety with a proper compactification $\overline{X}$ and boundary $Y=\overline{X}\setminus X$, one defines
\[
K^c(X):=\operatorname{holim}_r K(\overline{X},rY).
\]
Pro-cdh descent shows that this spectrum is independent of the choice of $\overline{X}$. Moreover, for a closed immersion $U \hookrightarrow X$ with open complement $V$, one obtains the fibre sequence
\[
K^c(V)\to K^c(X)\to \operatorname{holim}_r K(r(U\cap Y)).
\]
This is the sense in which pro-cdh descent can be interpreted as the well-definedness of compactly supported $K$-theory [1211.1813].

The theory also gives concrete blow-up computations. For the blow-up of $\mathbb{A}^2=\operatorname{Spec}k[x,y]$ at the origin, with $Y=\{(0,0)\}$ and $X'=\operatorname{Bl}_Y X$, pro-descent yields
\[
\operatorname{holim}_r K_n\!\bigl(\mathbb{A}^2,\operatorname{Spec}k[x,y]/(x,y)^r\bigr)\simeq K_n(X'),
\]
so the $K$-theory of the resolution is recovered from the pro-system of infinitesimal neighborhoods of the origin [1211.1813].

A further consequence is the vanishing half of Weibel’s $K$-dimension conjecture for the cases treated there: using an abstract blow-up square resolving singularities, one deduces by induction that $K_n(X)=0$ for $n<-\dim X$ and that $K_{-\dim X}(X)$ is finitely generated [1211.1813].

The derived theorem generalizes this pattern from Noetherian classical schemes to qcqs derived schemes. If $X$ has finite valuative dimension $d$, then $K_n(X)=0$ when $n<-d$, and one also obtains that $K_{-d}(X)\to K_{-d}(X\times \mathbb{A}^r)$ is an isomorphism, i.e. $X$ is $K_{-d}$-regular [2407.04378].

In the formal-scheme setting, the pro-cdh $\infty$-topos on a formal $S$-scheme $\mathfrak{X}$ of Krull dimension $d$ has homotopy dimension $\le d$. For abelian sheaves $F$ this implies
\[
H^p_{\pro\text{-}cdh}(\mathfrak{X},F)=0 \qquad \text{for all } p>d.
\]
Combined with axioms $(\mathrm{Desc})$, $(\mathrm{Fin})$, $(\mathrm{Rig})_N$, and $(\mathrm{Val})_N$, this yields a topos-theoretic vanishing criterion
\[
\pi_iF(X)=0 \qquad \text{for all } i<-(N+d),
\]
and, for nonconnective $K$-theory, recovers Weibel vanishing for Noetherian schemes [2409.14295].

## 5. Derived, formal, and motivic extensions

For qcqs derived $k$-schemes, pro-cdh descent is formulated for any $k$-connective localizing invariant in the sense of Land–Tamme, with examples including algebraic $K$-theory, $THH$, $TC$, and rational negative cyclic homology. If $X$ is a qcqs derived $k$-scheme, $U\subset X$ is a quasi-compact open with closed complement $Z$, and $f:Y\to X$ is a proper l.a.f.p. morphism that is an isomorphism over $U$, then the square
\[
\begin{tikzcd}
E(X)\ar[r]\ar[d] & E(X^\wedge_Z)\ar[d]\\
E(Y)\ar[r] & E(Y^\wedge_Z)
\end{tikzcd}
\]
is weakly cartesian in $\operatorname{Pro}(\mathrm{Sp})$ for every such invariant $E$. This includes ordinary blow-ups, finite modifications, and derived blow-ups in quasi-smooth centres [2407.04378].

The same paper records broader consequences: pro-cdh descent holds not only for $K$ but also for $THH$, $TC$, $HH$, $HP$, rational negative cyclic homology, and in particular for the cotangent complex $L^i$. It also states that one obtains pro-descent for motivic cohomology $\mathbb{Z}(j)^{\mot}$ via Elmanto–Morrow’s identification with graded pieces of the cotangent complex. A plausible implication is that pro-cdh descent functions as a common descent mechanism for a large class of localizing and deformation-theoretic invariants [2407.04378].

A distinct extension introduces a pro-cdh topology on locally Noetherian formal schemes. It is generated by Nisnevich coverings and formal abstract-blowup coverings
\[
\{\mathfrak{X}_{/Z}\to \mathfrak{X},\quad f:\mathfrak{Y}\to \mathfrak{X}\},
\]
where $f$ is a formal abstract blowup of $(\mathfrak{X},Z)$. The resulting $\infty$-topos of pro-cdh sheaves of spaces has the optimal homotopy-dimension bound $\le d$, thereby remedying the “$2d$”-bound that appeared for the ordinary-scheme pro-cdh topology introduced in the cited earlier work [2409.14295].

In mixed characteristic, motivic complexes $\mathbb{Z}(i)^{\mot}$ are characterized using pro-cdh descent. The pro-cdh motivic complexes are defined as the pro-cdh-sheafification of the left Kan extension of Bloch’s classical cycle complexes on $\mathrm{Sm}_{\mathbb{Z}}$, and for noetherian $X$ there is a natural equivalence
\[
\mathbb{Z}(i)^{\procdh}(X)\xrightarrow{\simeq}\mathbb{Z}(i)^{\mot}(X).
\]
Equivalently, $\mathbb{Z}(i)^{\mot}$ is the initial finitary Nisnevich sheaf that agrees with lisse motivic cohomology on local rings and satisfies pro-cdh descent on noetherian schemes [2507.16501].

## 6. Historical development, adjacent notions, and applications

The historical development begins with pro-excision in low degrees. Bass proved isomorphisms for $n\le 0$, Milnor and Swan analyzed $K_1$, and Geller–Weibel identified birelative $K_1$. Wodzicki–Suslin supplied an excision criterion in terms of Tor-vanishing. Weibel established a two-dimensional case, Geisser–Hesselholt proved pro versions of the Suslin–Wodzicki criterion, Morrow developed formal-functions theorems and pro-cdh descent in characteristic $0$ and mixed characteristic under resolution hypotheses, and Kerz–Strunk–Tamme established pro-cdh descent for all abstract blow-up squares of Noetherian schemes in all degrees [1612.00418].

One important application lies in rigid analytic geometry. Raynaud’s theorem identifies qcqs rigid analytic $F$-varieties with qc formal $\mathcal{O}_F$-schemes up to admissible blow-up, and any admissible blow-up in the formal category is an abstract blow-up. Pro-cdh descent for $K$-theory is then used to define continuous $K$-theory spectra for affinoid algebras and to prove Mayer–Vietoris descent on qcqs rigid spaces [1612.00418].

Another application concerns singular varieties and desingularisations. For a desingularisation $T:X' \to X$ with exceptional fibre $E$ and image $Y\subset X$, pro-cdh descent yields pro-isomorphisms
\[
\{K(X,Y)\}_r \simeq \{K(X',rE)\}_r,
\]
which are then applied to the codimension filtration $F^dK_0$ and to zero-cycles. This is the bridge used to compare the zero-cycle piece of $K_0$ of the singular variety with the corresponding groups on the resolution and its modulus thickenings [1404.4649].

A nearby but distinct notion is proper cdh-descent for functors that already send abstract blow-up squares to homotopy pullbacks without passing to pro-systems. In that setting one obtains compactly supported extensions of cohomology theories by hypersheaf-theoretic equivalences among smooth proper varieties, proper varieties, and a span category of open immersions followed by proper maps. For such a functor $F$, one sets
\[
F_c(U):=\operatorname{fib}\bigl(F(X)\to F(X\setminus U)\bigr)
\]
for an open immersion $U\hookrightarrow X$ with $X$ proper, and proper cdh-descent shows independence of the compactification [2204.08968].

The main conceptual clarification is therefore negative rather than positive: pro cdh descent is not the claim that algebraic $K$-theory or cyclic theories satisfy ordinary cdh descent for a single abstract blow-up square. The point is precisely that they fail to do so, and that the failure is corrected by retaining all infinitesimal thickenings in a pro-system [1211.1813].

Source: https://www.emergentmind.com/topics/pro-cdh-descent