Pro cdh Descent: Refinements in Algebraic K-Theory

• Pro cdh descent is a refined framework that replaces traditional blow-up squares with a pro system of infinitesimal thickenings, enabling descent for K-theory and homological invariants.
• It employs pro Mayer–Vietoris theorems to overcome failures in ordinary cdh-descent by ensuring homotopy cartesian squares for invariants such as Hochschild homology, TC, and cyclic theories.
• The approach underpins applications like defining K-theory with compact support, proving vanishing theorems in negative K-theory, and unifying descent across derived, formal, and motivic contexts.

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 (Morrow, 2012, Kelly et al., 2024, Kelly et al., 2024, Bouis, 22 Jul 2025).

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 $\{X' \to X,\; Y \to X\}$ (Morrow, 2012).

The basic obstruction is that algebraic $K$-theory, and likewise $K$0, $K$1, and $K$2, fail to satisfy descent for a single abstract blow-up. The pro-cdh construction remedies this by replacing the centre $K$3 with all infinitesimal thickenings. If $K$4 is the sheaf of ideals of $K$5, one writes

$K$6

for the $K$7th thickening of $K$8, and one works in the pro-category over $K$9 with inverse systems

$K$0

The corresponding pro-cdh covering data are the pro-objects $K$1 (Morrow, 2012).

The same idea extends to derived geometry. For a qcqs derived scheme $K$2 with closed complement $K$3, 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 $K$4 is the ind-scheme whose objects are all closed immersions $K$5 with $K$6, and derived thickenings $K$7 and $K$8 are the pro-infinitesimal neighborhoods entering the descent square (Kelly et al., 2024).

2. Fundamental pro Mayer–Vietoris theorems

For an abstract blow-up square of Noetherian, finite-dimensional $K$9-schemes, relative theories are written as

$\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$0

The foundational results establish that the obstruction to descent disappears after passage to the pro-system of thickenings (Morrow, 2012).

For Hochschild and cyclic homology, the canonical map of pro abelian groups

$\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$1

is an isomorphism for all $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$2 when $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$3 or $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$4. Equivalently, the square of pro-spectra

$\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$5

is homotopy cartesian (Morrow, 2012).

In characteristic $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$6, under $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$7-finite hypotheses, the same pattern holds for $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$8, $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$9, and $X' \to X$0 with $X' \to X$1-coefficients: $X' \to X$2 In characteristic $X' \to X$3, and in characteristic $X' \to X$4 under resolution hypotheses, algebraic $X' \to X$5-theory satisfies the analogous statement: $X' \to X$6 Thus the square

$X' \to X$7

is homotopy cartesian for all $X' \to X$8 (Morrow, 2012).

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 $X' \to X$9, the canonical map

$Y \to X$0

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 (Morrow, 2016).

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 $Y \to X$1 and ideals $Y \to X$2: $Y \to X$3 established for André–Quillen, Hochschild, and cyclic homology, and analogously for $Y \to X$4 and $Y \to X$5. 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 (Morrow, 2012).

In characteristic $Y \to X$6, Haesemeyer’s cdh-descent comparison reduces $Y \to X$7-theory pro-descent to cyclic homology pro-descent. In characteristic $Y \to X$8, one replaces $Y \to X$9 by $X' \setminus Y' \to X \setminus Y$0 and $X' \setminus Y' \to X \setminus Y$1 by the fibre of the trace map $X' \setminus Y' \to X \setminus Y$2. 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 (Morrow, 2012).

The historical account emphasizes a parallel route through excision situations of rings. For finite maps one reduces to a Milnor-square-type setting $X' \setminus Y' \to X \setminus Y$3, 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 $X' \setminus Y' \to X \setminus Y$4 model structure on pro-spectra (Morrow, 2016).

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 $X' \setminus Y' \to X \setminus Y$5, Khan–Rydh’s semi-orthogonal decomposition of $X' \setminus Y' \to X \setminus Y$6 yields

$X' \setminus Y' \to X \setminus Y$7

for $X' \setminus Y' \to X \setminus Y$8-connective localizing invariants $X' \setminus Y' \to X \setminus Y$9. Together with the Land–Tamme pro-excision theorem and a factorization theorem for proper l.a.f.p. $X$0-modifications, this proves that the pro-square

$X$1

is weakly cartesian in $X$2 (Kelly et al., 2024).

4. Compact support and negative $X$3-theory

One of the earliest structural consequences is the well-definedness of $X$4-theory with compact support. If $X$5 is a separated $X$6-variety with a proper compactification $X$7 and boundary $X$8, one defines

$X$9

Pro-cdh descent shows that this spectrum is independent of the choice of $\{X' \to X,\; Y \to X\}$0. Moreover, for a closed immersion $\{X' \to X,\; Y \to X\}$1 with open complement $\{X' \to X,\; Y \to X\}$2, one obtains the fibre sequence

$\{X' \to X,\; Y \to X\}$3

This is the sense in which pro-cdh descent can be interpreted as the well-definedness of compactly supported $\{X' \to X,\; Y \to X\}$4-theory (Morrow, 2012).

The theory also gives concrete blow-up computations. For the blow-up of $\{X' \to X,\; Y \to X\}$5 at the origin, with $\{X' \to X,\; Y \to X\}$6 and $\{X' \to X,\; Y \to X\}$7, pro-descent yields

$\{X' \to X,\; Y \to X\}$8

so the $\{X' \to X,\; Y \to X\}$9-theory of the resolution is recovered from the pro-system of infinitesimal neighborhoods of the origin (Morrow, 2012).

A further consequence is the vanishing half of Weibel’s $K$0-dimension conjecture for the cases treated there: using an abstract blow-up square resolving singularities, one deduces by induction that $K$1 for $K$2 and that $K$3 is finitely generated (Morrow, 2012).

The derived theorem generalizes this pattern from Noetherian classical schemes to qcqs derived schemes. If $K$4 has finite valuative dimension $K$5, then $K$6 when $K$7, and one also obtains that $K$8 is an isomorphism, i.e. $K$9 is $K$00-regular (Kelly et al., 2024).

In the formal-scheme setting, the pro-cdh $K$01-topos on a formal $K$02-scheme $K$03 of Krull dimension $K$04 has homotopy dimension $K$05. For abelian sheaves $K$06 this implies

$K$07

Combined with axioms $K$08, $K$09, $K$10, and $K$11, this yields a topos-theoretic vanishing criterion

$K$12

and, for nonconnective $K$13-theory, recovers Weibel vanishing for Noetherian schemes (Kelly et al., 2024).

5. Derived, formal, and motivic extensions

For qcqs derived $K$14-schemes, pro-cdh descent is formulated for any $K$15-connective localizing invariant in the sense of Land–Tamme, with examples including algebraic $K$16-theory, $K$17, $K$18, and rational negative cyclic homology. If $K$19 is a qcqs derived $K$20-scheme, $K$21 is a quasi-compact open with closed complement $K$22, and $K$23 is a proper l.a.f.p. morphism that is an isomorphism over $K$24, then the square

$K$25

is weakly cartesian in $K$26 for every such invariant $K$27. This includes ordinary blow-ups, finite modifications, and derived blow-ups in quasi-smooth centres (Kelly et al., 2024).

The same paper records broader consequences: pro-cdh descent holds not only for $K$28 but also for $K$29, $K$30, $K$31, $K$32, rational negative cyclic homology, and in particular for the cotangent complex $K$33. It also states that one obtains pro-descent for motivic cohomology $K$34 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 (Kelly et al., 2024).

A distinct extension introduces a pro-cdh topology on locally Noetherian formal schemes. It is generated by Nisnevich coverings and formal abstract-blowup coverings

$K$35

where $K$36 is a formal abstract blowup of $K$37. The resulting $K$38-topos of pro-cdh sheaves of spaces has the optimal homotopy-dimension bound $K$39, thereby remedying the “$K$40”-bound that appeared for the ordinary-scheme pro-cdh topology introduced in the cited earlier work (Kelly et al., 2024).

In mixed characteristic, motivic complexes $K$41 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 $K$42, and for noetherian $K$43 there is a natural equivalence

$K$44

Equivalently, $K$45 is the initial finitary Nisnevich sheaf that agrees with lisse motivic cohomology on local rings and satisfies pro-cdh descent on noetherian schemes (Bouis, 22 Jul 2025).

6. Historical development, adjacent notions, and applications

The historical development begins with pro-excision in low degrees. Bass proved isomorphisms for $K$46, Milnor and Swan analyzed $K$47, and Geller–Weibel identified birelative $K$48. 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 $K$49 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 (Morrow, 2016).

One important application lies in rigid analytic geometry. Raynaud’s theorem identifies qcqs rigid analytic $K$50-varieties with qc formal $K$51-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$52-theory is then used to define continuous $K$53-theory spectra for affinoid algebras and to prove Mayer–Vietoris descent on qcqs rigid spaces (Morrow, 2016).

Another application concerns singular varieties and desingularisations. For a desingularisation $K$54 with exceptional fibre $K$55 and image $K$56, pro-cdh descent yields pro-isomorphisms

$K$57

which are then applied to the codimension filtration $K$58 and to zero-cycles. This is the bridge used to compare the zero-cycle piece of $K$59 of the singular variety with the corresponding groups on the resolution and its modulus thickenings (Morrow, 2014).

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 $K$60, one sets

$K$61

for an open immersion $K$62 with $K$63 proper, and proper cdh-descent shows independence of the compactification (Kuijper, 2022).

The main conceptual clarification is therefore negative rather than positive: pro cdh descent is not the claim that algebraic $K$64-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 (Morrow, 2012).