cdh-Motivic Cohomology

• cdh-motivic cohomology is a framework that organizes motivic cohomology theories via the cdh topology to capture descent for abstract blow-up squares and singular schemes.
• It constructs both A¹-invariant and non-A¹-invariant theories by cdh-sheafifying motivic complexes, preserving blow-up descent, projective bundle formulas, and spectral sequences linking to K-theory.
• Recent developments with formal schemes and refined pro-cdh techniques yield optimal dimension bounds, sharper vanishing theorems, and universal comparisons with classical arithmetic invariants.

Searching arXiv for the cited papers and recent work on cdh/pro-cdh motivic cohomology. cdh-motivic cohomology is the collection of motivic cohomology theories, descent formalisms, and comparison results organized by the cdh topology, the Grothendieck topology generated by Nisnevich coverings and abstract blow-up squares. In its classical $\mathbb A^1$-invariant form, it is obtained by cdh-sheafifying motivic complexes coming from smooth schemes; in more recent non-$\mathbb A^1$-invariant and pro-cdh forms, it is designed to retain sensitivity to singularities and nilpotents while preserving blow-up descent, projective bundle formulas, and Atiyah–Hirzebruch-type spectral sequences to $K$-theory (Bachmann et al., 13 Aug 2025, Elmanto et al., 2023, Bouis, 2024, Kelly et al., 2024).

1. The cdh topology and the cdh-local motivic setting

The cdh topology refines the Nisnevich topology by adjoining abstract blow-up squares. Concretely, an abstract blow-up square is a cartesian square

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

with $Y \to X$ a finitely presented closed immersion, $X' \to X$ finitely presented and proper, and $X' \setminus Y' \simeq X \setminus Y$. In this sense, the cdh topology is tailored to capture excision for blow-ups and descent for singularities, and a Nisnevich sheaf is a cdh sheaf precisely when it sends abstract blow-up squares to cartesian squares (Bachmann et al., 13 Aug 2025).

At the stable motivic level, the cdh topology does not produce a different motivic homotopy category. For every qcqs base scheme $S$, the cdh-local motivic homotopy category $SH_{\mathrm{cdh}}(S)$ is canonically equivalent to $SH(S)$, so any motivic spectrum can be viewed as cdh-local and therefore satisfies cdh descent (Khan, 2023). This equivalence explains why cdh descent is pervasive in motivic constructions without implying that all concrete motivic complexes on singular schemes are $\mathbb A^1$0-invariant.

The topos-theoretic background is also significant. For a qcqs scheme of finite valuative dimension, its cdh $\mathbb A^1$1-topos is hypercomplete, extending Voevodsky’s theorem beyond the noetherian case (Elmanto et al., 2020). This hypercompleteness is a technical input in valuation-theoretic descent, point detection, and excision arguments.

2. $\mathbb A^1$2-invariant cdh-motivic cohomology

A modern $\mathbb A^1$3-invariant cdh theory is constructed by left Kan extending motivic cohomology from smooth schemes and then cdh-sheafifying. In the notation of Bachmann, Elmanto, Morrow, and collaborators,

$\mathbb A^1$4

with $\mathbb A^1$5 for $\mathbb A^1$6. These complexes are finitary cdh sheaves on qcqs schemes and form the graded pieces of a cdh-motivic filtration on homotopy $\mathbb A^1$7-theory: $\mathbb A^1$8 The same work proves the comparison

$\mathbb A^1$9

and, after base-change stability of slices, identifies $K$0 with the $K$1-invariant motivic cohomology represented by the zeroth slice of $K$2 (Bachmann et al., 13 Aug 2025).

This framework realizes a version of Voevodsky’s slice-filtration program over arbitrary qcqs schemes. It also clarifies the role of cdh descent: cdh-motivic cohomology is not merely a substitute for Nisnevich descent on singular schemes, but the descent theory that makes the filtration on $K$3 computable through blow-ups, valuation rings, and syntomic/prismatic comparison maps (Bachmann et al., 13 Aug 2025).

3. Non-$K$4-invariant theories and pro-cdh descent

Elmanto–Morrow constructed a non-$K$5-invariant motivic cohomology theory for equicharacteristic qcqs schemes by gluing the cdh-local filtration on $K$6 to trace-theoretic filtrations on $K$7. The weight-$K$8 complex $K$9 is the $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$0th graded piece of the resulting motivic filtration on $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$1, and it is a Nisnevich sheaf, finitary, and equipped with an Atiyah–Hirzebruch spectral sequence

$\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$2

In characteristic $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$3 it fits into a pullback square involving $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$4, while in characteristic $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$5 it fits into a pullback square involving syntomic complexes $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$6. It is explicitly not $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$7-invariant on singular schemes; rather,

$\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$8

and the theory detects nonreduced structure, for example through explicit relative calculations on $\begin{tikzcd} Y' \ar[r] \ar[d] & X' \ar[d] \ Y \ar[r] & X \end{tikzcd}$9 (Elmanto et al., 2023).

The mixed-characteristic extension constructs $Y \to X$0 for qcqs schemes by means of a global filtration on $Y \to X$1 whose graded pieces refine derived de Rham and Bhatt–Morrow–Scholze syntomic cohomology. For noetherian schemes, these complexes satisfy pro-cdh descent: for any abstract blow-up square and any $Y \to X$2, the induced pro-square

$Y \to X$3

is weakly cartesian (Bouis, 2024).

A further refinement in mixed characteristic proves that the motivic complexes $Y \to X$4 are characterized on noetherian schemes by finitariness, Nisnevich descent, pro-cdh descent, and normalization on local rings; equivalently,

$Y \to X$5

for every noetherian $Y \to X$6 and $Y \to X$7 (Bouis, 22 Jul 2025). This universal characterization places pro-cdh descent at the center of the theory rather than as an auxiliary excision property.

4. Formal schemes, pro-cdh topology, and optimal dimension bounds

Kelly–Saito introduced a pro-cdh topology on formal schemes in order to remove a defect in the earlier scheme-level pro-cdh topology. On locally noetherian formal schemes, pro-cdh coverings are generated by Nisnevich coverings and formal abstract blow-up coverings

$Y \to X$8

where $Y \to X$9 is the formal completion along a closed subscheme $X' \to X$0 and $X' \to X$1 is a locally algebrizable admissible proper finite type morphism, an isomorphism over $X' \to X$2. The scheme-level pro-cdh $X' \to X$3-topos is recovered from the formal-scheme level by imposing continuity covers $X' \to X$4, corresponding to the condition

$X' \to X$5

This makes the formal theory strictly finer and better adapted to nilpotents and completions (Kelly et al., 2024).

The principal theorem is an optimal homotopy-dimension bound. If $X' \to X$6 is a formal scheme with $X' \to X$7, then

$X' \to X$8

hence for any abelian sheaf $X' \to X$9,

$X' \setminus Y' \simeq X \setminus Y$0

This remedies the earlier $X' \setminus Y' \simeq X \setminus Y$1 bound on the scheme-level pro-cdh $X' \setminus Y' \simeq X \setminus Y$2-topos and is obtained by comparison with a Nisnevich–Riemann–Zariski site whose inverse system of Nisnevich topoi has homotopy dimension $X' \setminus Y' \simeq X \setminus Y$3 (Kelly et al., 2024).

The improved bound has immediate motivic consequences. Kelly–Saito derive a topos-theoretic interpretation of Weibel’s vanishing and of Elmanto–Morrow’s motivic cohomology bounds via the descent spectral sequence

$X' \setminus Y' \simeq X \setminus Y$4

For nonconnective $X' \setminus Y' \simeq X \setminus Y$5-theory this yields $X' \setminus Y' \simeq X \setminus Y$6 for $X' \setminus Y' \simeq X \setminus Y$7, and for Elmanto–Morrow motivic complexes over $X' \setminus Y' \simeq X \setminus Y$8 or $X' \setminus Y' \simeq X \setminus Y$9 it yields

$S$0

under the stated hypotheses (Kelly et al., 2024). In mixed characteristic, the same vanishing range is established in motivic form: $S$1 when $S$2 is noetherian of dimension $S$3 (Bouis, 22 Jul 2025).

5. Structural theorems and comparison isomorphisms

A central structural property is the projective bundle formula. For the mixed-characteristic theory, if $S$4 is a vector bundle of rank $S$5 on a qcqs scheme $S$6, then

$S$7

with the $S$8th summand induced by pullback and multiplication by powers of the motivic first Chern class of $S$9 (Bouis, 22 Jul 2025). Equicharacteristic and mixed-characteristic constructions also prove regular blow-up formulas, giving cartesian squares for regular closed immersions and thereby Mayer–Vietoris and Gysin-type consequences (Elmanto et al., 2023, Bouis, 2024).

The theories admit comparison maps to classical arithmetic invariants in the expected ranges. For henselian local rings $SH_{\mathrm{cdh}}(S)$0, the mixed-characteristic theory proves a finite-coefficient comparison

$SH_{\mathrm{cdh}}(S)$1

which is an isomorphism for all $SH_{\mathrm{cdh}}(S)$2 and $SH_{\mathrm{cdh}}(S)$3 (Bouis, 22 Jul 2025). For $SH_{\mathrm{cdh}}(S)$4 invertible on $SH_{\mathrm{cdh}}(S)$5, the mixed-characteristic motivic complexes satisfy a Beilinson–Lichtenbaum-type comparison

$SH_{\mathrm{cdh}}(S)$6

and there is a corresponding $SH_{\mathrm{cdh}}(S)$7-adic comparison with syntomic complexes in degrees $SH_{\mathrm{cdh}}(S)$8 (Bouis, 2024).

Low weights recover familiar invariants. Weight $SH_{\mathrm{cdh}}(S)$9 is identified with cdh cohomology of the constant sheaf,

$SH(S)$0

and in weight $SH(S)$1 one has the expected identifications

$SH(S)$2

together with a natural first Chern class map $SH(S)$3 (Bouis, 2024). These formulas place cdh-motivic cohomology within the expected formalism of orientations, Chern classes, and projective bundle decompositions while retaining singular sensitivity outside the $SH(S)$4-invariant regime.

6. Related descent theories, weight zero, and interpretive issues

One recurring misconception is that cdh-motivic cohomology is intrinsically $SH(S)$5-invariant. The recent literature makes the opposite point: the non-$SH(S)$6-invariant theories were introduced precisely because algebraic $SH(S)$7-theory fails $SH(S)$8-invariance on singular schemes, and the cdh or pro-cdh formalism supplies the blow-up descent needed to compensate for that failure rather than erase it (Elmanto et al., 2023). A related misconception is that cdh descent by itself always yields optimal vanishing bounds; Kelly–Saito show that the passage to formal schemes and a refined pro-cdh topology is what repairs the earlier $SH(S)$9 homotopy-dimension defect (Kelly et al., 2024).

The cdh topology is also closely related to ldh and valuation-theoretic descent. Under explicit hypotheses $\mathbb A^1$00 and $\mathbb A^1$01, Kelly and Morrow prove that for a $\mathbb A^1$02-linear presheaf with traces,

$\mathbb A^1$03

and they emphasize that motivic cohomology with $\mathbb A^1$04- or $\mathbb A^1$05-coefficients should be regarded as invariant under universal homeomorphisms, not merely nilpotent thickenings (Kelly, 2018). In a complementary direction, Bhatt–Mathew–inspired results show that, for qcqs schemes of finite valuative dimension, cdh hypercompleteness and henselian valuation-ring criteria yield Milnor excision for torsion motivic spectra over fields (Elmanto et al., 2020).

Relative and weight-zero variants further enlarge the scope of the subject. For a closed immersion $\mathbb A^1$06, the spectral sequence for relative homotopy $\mathbb A^1$07-theory has $\mathbb A^1$08-terms given by cdh-hypercohomology of relative motivic complexes, and for smooth affine pairs in top codimension it identifies the relevant relative motivic cohomology with Chow groups with modulus (Krishna et al., 2018). In weight zero, over a trivially valued field of characteristic $\mathbb A^1$09, one has canonical isomorphisms

$\mathbb A^1$10

so singular cohomology of the Berkovich analytification computes weight-zero motivic cohomology (Molokov et al., 2024). This result isolates the cdh topology as the precise bridge between blow-up descent and a topological realization of motivic weight zero.

Taken together, these developments show that cdh-motivic cohomology is best understood not as a single theory, but as a stratified framework. Its $\mathbb A^1$11-invariant branch is represented in the cdh-local motivic homotopy category and computes $\mathbb A^1$12-filtrations; its non-$\mathbb A^1$13-invariant branch encodes singularities, nilpotents, and arithmetic comparison data; and its pro-cdh refinement supplies the formal-geometry input needed for sharp vanishing theorems, Weibel-type bounds, and universal characterizations on noetherian schemes (Bachmann et al., 13 Aug 2025, Bouis, 2024, Bouis, 22 Jul 2025).