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Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology

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Pith's one-line read Pro cdh descent, motivic Weibel vanishing, the projective bundle formula, and a finite-coefficient Milnor K-theory comparison all hold for the motivic cohomology of mixed characteristic schemes, extending the equicharacteristic theory to…

desk verdict A serious, clearly written mixed-characteristic sequel that proves the expected package of theorems; two load-bearing inputs are asserted rather than proved, so it needs a careful referee. read the letter →

arxiv 2507.16501 v1 pith:OSAFTCX6 submitted 2025-07-22 math.AG math.KTmath.NT

Tess Bouis This is my paper · ORCID
classification math.AGmath.KTmath.NT MSC 14F4214F3019D45
keywords motiviccohomologymixedcharacteristicprocdhdescentWeibelvanishingprojectivebundleformulaMilnorK-theorycotangentcomplexsyntomic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's goal is to show that the motivic cohomology of mixed characteristic schemes — schemes not necessarily defined over a field — behaves like a fully functional motivic cohomology theory. Concretely, it proves pro cdh descent for the motivic complexes $Z(i)_{\mathrm{mot}}$, a motivic refinement of the Weibel vanishing theorem in algebraic $K$-theory, and the projective bundle formula, together with comparisons to Milnor $K$-theory and to the pro cdh motivic complexes. These are the standard structural properties that make motivic cohomology computable and usable on singular, non-equicharacteristic schemes. The results extend the equicharacteristic theory of [EM23] to all quasi-compact quasi-separated schemes, and a sympathetic reader would see this as completing the basic toolkit for motivic cohomology outside the field case.

What carries the argument

The load-bearing mechanism is pro cdh descent. A presheaf satisfies pro cdh descent when it sends every abstract blowup square to a weakly cartesian square of pro objects, after taking infinitesimal thickenings of the exceptional loci. The paper proves pro cdh descent for powers of the cotangent complex (Proposition 3.9), then propagates it to the motivic complexes $Z(i)_{\mathrm{mot}}$ by decomposing them via the fracture square [Bou24, Corollary 4.31] into a rational part, rigid-analytic parts, and syntomic (prismatic) parts $Z_p(i)_{\mathrm{BMS}}$, each handled separately. The main new input is a strengthening of the formal functions theorem (Lemma 3.3) that removes finite-dimensionality hypotheses and seeds the cotangent-complex descent. Pro cdh descent then yields both the motivic Weibel vanishing theorem (through a cdh-versus-mot fibre argument) and the comparison to the pro cdh motivic complexes. The projective bundle formula is proved separately: the $\mathbb{P}^1$-bundle formula is obtained modulo $p$ by a degeneration argument using syntomic cohomology and Selmer $K$-theory, and the general rank-$r$ formula is deduced from the $\mathbb{P}^1$-formula and regular blowup excision by pushing squares.

What would settle it

Look for a noetherian scheme $X$ of dimension $d$ with a nonzero group $H^j_{\mathrm{mot}}(X, Z(i))$ for some $j > i+d$; finding one would directly refute the motivic Weibel vanishing theorem. Alternatively, check the second statement of Lemma 3.3 by hand on an example such as $X = \mathbb{P}^1_{\mathbb{Z}}$ with $I = (p)$: if the displayed pro map is not a weak equivalence there, the pro cdh descent argument collapses.

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Extended reading notes

Core claim

On the author's terms, the central discovery is that the motivic complexes $Z(i)_{\mathrm{mot}}$, defined for arbitrary qcqs schemes in [Bou24], satisfy the full package of structural properties previously known only over a field: they are pro cdh sheaves on noetherian schemes (Theorem 3.23); their cohomology vanishes in degrees $j > i + d$ on a noetherian scheme of dimension $d$, refining the Weibel vanishing conjecture for negative $K$-groups (Theorem 3.27); the projective bundle formula holds for every vector bundle on every qcqs scheme (Theorem 4.19); on henselian local rings, improved Milnor $K$-theory with finite coefficients is isomorphic to motivic cohomology (Theorem 2.21); and on noetherian schemes the lisse-motivic comparison map identifies the pro cdh motivic complexes with the motivic complexes (Theorem 3.32). The paper presents these as consequences of pro cdh descent, which is proved by reducing to pro cdh descent for powers of the cotangent complex and then reassembling the motivic complexes through a fracture square built from rational and $p$-adic (syntomic and prismatic) pieces.

Load-bearing premise

The load-bearing premise is the strengthened formal functions theorem (Lemma 3.3) — its second assertion is deferred to an examination of the proof of [Lur19, Lemma 8.1.2.3], and the whole descent package is built on it, with a secondary reliance on the footnote to Theorem 3.32 extending the sheafification theorem of [KS24] from fields to every noetherian base.

Editorial extensions

If this is right

  • Motivic cohomology of any noetherian scheme of dimension $d$ vanishes above degree $i + d$ in weight $i$, so the Atiyah–Hirzebruch spectral sequence computing algebraic $K$-theory from motivic cohomology is supported in finitely many columns; in particular the edge map reconstructs the known description of $K_{-d}(X)$.
  • The projective bundle formula holds for every qcqs scheme, so the motivic complexes are represented by a $\mathbb{P}^1$-motivic spectrum in the sense of [AI23]; this is the defining property needed to fit them into the non-$\mathbb{A}^1$-invariant motivic framework.
  • On noetherian schemes, the motivic complexes coincide with the pro cdh motivic complexes, giving a universal cycle-theoretic characterization: $Z(i)_{\mathrm{mot}}$ is the pro cdh sheafification of the left Kan extension of the classical motivic complexes built from Bloch's cycle complexes.
  • On henselian local rings, improved Milnor $K$-theory with finite coefficients maps isomorphically onto motivic cohomology, giving a singular Nesterenko–Suslin isomorphism in mixed characteristic.
  • The regular blowup formula holds: for a regular closed immersion $Y \to X$, the derived category square comparing $Z(i)_{\mathrm{mot}}(X)$, $Z(i)_{\mathrm{mot}}(Y)$, $Z(i)_{\mathrm{mot}}(\mathrm{Bl}_Y X)$, and the corresponding scheme over $Y$ is cartesian.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Not stated in the paper: because Theorem 2.21 and Theorem 2.11 reduce Conjecture 2.20 to local essentially smooth $\mathbb{Z}$-algebras, the integral Milnor $K$ isomorphism in mixed characteristic is blocked precisely by smooth Gersten injectivity, not by any new phenomenon of singular schemes.
  • Not stated in the paper: the $\mathbb{P}^1$-bundle formula proof routes around the $F$-smoothness condition that the cdh-local complexes require, so the motivic complexes $Z(i)_{\mathrm{mot}}$ are the better-behaved integral theory in mixed characteristic; a testable consequence is that computations for projective bundles can proceed without resolving the valuation-ring question.
  • Not stated in the paper: the spectral sequence pattern behind Corollary 3.30 suggests that for $k \geq 1$ the groups $K_{-d+k}(X)$ are governed by $H^{d+k}_{\mathrm{mot}}(X, Z(k))$ together with cdh-cohomology boundary classes, so on small-dimensional singular schemes the negative $K$-groups should be explicitly computable from motivic cohomology.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This sequel to the author's previous work [Bou24] establishes structural properties of the motivic complexes Z(i)_mot on mixed-characteristic qcqs schemes. The main results are: a comparison between lisse motivic cohomology and t^{≤i}Z(i)_mot on local rings (Corollary 2.12), a finite-coefficient isomorphism between improved Milnor K-theory and motivic cohomology on henselian local rings (Theorem 2.21), pro cdh descent for Z(i)_mot on noetherian schemes (Theorem 3.23), a motivic refinement of Weibel vanishing (Theorem 3.27), a comparison between pro cdh motivic cohomology and Z(i)_mot (Theorem 3.32), and the regular blowup and projective bundle formulae (Theorems 4.18 and 4.19). The proofs make heavy use of the fracture square, a reduction to powers of the cotangent complex, and inputs from prismatic and syntomic cohomology. The paper is written as a sequel to [Bou24] and explicitly inherits the definition of Z(i)_mot, the syntomic comparison, and several finiteness results from that paper.

Significance. If the results are correct, they supply the expected package of structural properties for mixed-characteristic motivic cohomology, extending Elmanto--Morrow's theory beyond schemes over a field. The paper is explicit about its strategy: pro cdh descent is proved by a cotangent-complex reduction rather than by the Kerz--Strunk--Tamme route, and the comparison to pro cdh motivic cohomology gives a universal characterization on noetherian schemes. The manuscript has concrete, falsifiable statements with no free parameters, and many proofs are written out in detail. Its main weakness is not internal inconsistency but incomplete support for several load-bearing inputs: Lemma 3.3(2) is deferred to an 'examination' of a proof in [Lur19], the extension of [KS24, Theorem 9.7] to arbitrary noetherian bases is asserted without proof in footnote 4 of Theorem 3.32, and several central arguments depend on the in-preparation reference [BEM]. These gaps do not by themselves invalidate the main claims, but they must be repaired before the results can be regarded as fully certified.

major comments (3)
  1. [§3.1, Lemma 3.3(2)] The stressed concern lands exactly here. The displayed weak equivalence of pro objects in Lemma 3.3(2) is asserted to follow from 'an examination of the previous proof' of [Lur19, Lemma 8.1.2.3], but no proof is supplied. This statement is stronger than the classical formal functions theorem: it is a derived-category-level statement with derived pullbacks, in all cohomological degrees simultaneously, and without finite-dimensionality or flatness hypotheses on the input. The statement feeds Corollary 3.5, Lemma 3.6, Corollary 3.7, Proposition 3.9, and therefore Theorem 3.23 and its consequences (Theorems 3.27 and 3.32). Since this is a load-bearing input, the author should provide a complete proof or a reference that states exactly this strong form; a remark that the prior proof adapts is not sufficient for a central technical lemma.
  2. [§3.4, Theorem 3.32, footnote 4] The proof of Theorem 3.32 invokes [KS24, Theorem 9.7], which the footnote says is stated for schemes over a field but whose proof works over any noetherian commutative ring. No argument for this extension is given. The extension is necessary because the theorem is applied to arbitrary noetherian schemes, and the equivalence Z(i)_procdh(X) ≅ Z(i)_mot(X) is one of the headline results of the paper. The author should either prove the needed generality, identify the precise place in [KS24] where the field assumption is used and explain why it is removable, or cite a statement that covers the hypotheses actually used.
  3. [§§3.3 and 4.2, use of [BEM]] Several load-bearing points cite the in-preparation reference [BEM]. In Theorem 3.27, the bound on Z(i)_cdh(X) in degrees ≤ i+d is taken from [BEM, Theorem 7.12]. In Proposition 4.17 and in the proof of Theorem 4.7, the P1-bundle formula for Z/p^k(i)_cdh on qcqs Z[1/p]-schemes and on F_p-schemes is taken from [BEM, Section 8]. These are central inputs to two of the paper's main theorems, the motivic Weibel vanishing and the projective bundle formula. In-preparation references cannot be checked by the reader; the author should replace them with published arguments, provide full statements, or explicitly flag these results as conditional on [BEM].
minor comments (4)
  1. [Introduction, Theorem E] The parenthetical reference 'Theorem 2.21)).' contains a stray double parenthesis and should be corrected.
  2. [Notation 3.1] The notation 'rY' for the r−1st infinitesimal thickening is initially confusing, since r starts at 0 in the displayed pro systems; consider clarifying the indexing explicitly at first use.
  3. [Definition 3.2] The notion of a weakly cartesian square of pro objects is only explained in a footnote; since it is central to the definition of pro cdh sheaf, it would be clearer to state the definition in the body of the text.
  4. [Theorem 3.27, proof] The proof invokes [EM23, Proposition 8.10] for the complex fib(Z(i)_mot → Z(i)_cdh)[i]; please state the precise hypotheses of that proposition so the reader can check that the four listed properties are exactly what is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper derives pro cdh descent, Weibel vanishing, and the projective bundle formula from definitions in [Bou24] and external results; the target statements are never assumed as inputs.

full rationale

The derivation chain is not circular. Theorem 3.23 (pro cdh descent) is assembled from the cartesian square supplied by [Bou24, Remark 3.21], pro cdh descent for Z(i)_TC (Propositions 3.13-3.22), and cdh descent for Z(i)_cdh; the genuinely new input, Proposition 3.20 for the product of BMS syntomic complexes, is proved from Corollary 3.17, Lemma 3.18, and Lemma 3.19, which in turn rest on the cotangent-complex statement Proposition 3.9 and on external results [AMMN22], [BL22], and [Mor16]. Theorem 3.27 applies the abstract criterion [EM23, Proposition 8.10] to Z(i)_mot using properties (1)-(4), which are established separately rather than assumed. Theorem 4.19 (projective bundle formula) is deduced from the P1-bundle formula (Theorem 4.7) and the regular blowup formula (Theorem 4.18); its modular proof reduces via [Bou24, Theorem 5.10] to syntomic cohomology and to the P1-formulae of [BL22] and [BEM], none of which assume the motivic projective bundle formula. Theorem 2.21 uses [LM23, Theorem 3.1] and [EM23, Theorem 7.12]. The abundant citations to [Bou24] are load-bearing but not circular: they supply definitions, fracture squares, and comparisons whose stated assumptions do not include the target theorems. There are no fitted parameters, no empirical inputs, and no renamed known results. The real weaknesses are support gaps rather than circularity: Lemma 3.3(2) is deferred to an 'examination of the previous proof' of [Lur19, Lemma 8.1.2.3], and the footnote to Theorem 3.32 extends [KS24, Theorem 9.7] from schemes over a field to a general noetherian base without proof. These gaps affect certification of the corollary chain, but they do not make any prediction equivalent to its own input, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The invoice is short on parameters and long on inherited machinery. No numerical free parameters appear: this is a proof-based paper. The central claims rest on the author's own [Bou24] for the definition of the motivic complexes and their fracture-square structure, on [AMMN22], [BL22], [EM23], [KS24], and [KST18] for the descent and comparison inputs, and on an in-preparation reference [BEM]. The most fragile entries are the asserted strengthening of Lurie's formal functions theorem (Lemma 3.3) and the asserted scope extension of [KS24, Theorem 9.7] to noetherian bases. No new entities are postulated.

assumptions (6)
  • domain assumption Existing theory of Z(i)_mot, Z(i)_TC, Z(i)_cdh and the fracture square of [Bou24, Cor 4.31], including the syntomic comparison [Bou24, Thm 5.10].
    The paper imports its central objects and structural decomposition from the author's companion paper [Bou24] without reproving them.
  • ad hoc to paper Pro cdh descent for powers of the cotangent complex (Proposition 3.9) rests on the formal functions theorem for coherent cohomology (Lemma 3.3, after Lurie).
    Lemma 3.3 asserts a strengthening of Lurie's statement, with the proof left to an examination of [Lur19, Lemma 8.1.2.3].
  • domain assumption The fibre sequence descriptions of F_p(i)_BMS in terms of Nygaard filtered prismatic cohomology ([AMMN22, Cor 5.31]) and of N^j Delta in terms of filtered pieces, and the regular blowup formula for powers of the cotangent complex ([BL22, Lemma 9.4.3]).
    Taken from prior literature on prismatic and syntomic cohomology.
  • ad hoc to paper [KS24, Theorem 9.7] (comparison of pro cdh sheafification to the given presheaf on noetherian schemes) is valid over any noetherian ring, though stated over a field.
    The paper asserts in a footnote that the proof works over any noetherian commutative ring; this is a scope extension of a cited theorem, used for Theorem 3.32.
  • domain assumption Z(i)_cdh(X) is in degrees at most i + d for X of dimension d ([BEM, Theorem 7.12]; also [Bou24, Theorem 3.13(1)]).
    One source is in preparation; a published alternative from [Bou24] is cited.
  • standard math Gersten injectivity for K2 of DVRs/smooth rings ([GL87], [DS75]) and the Nesterenko-Suslin isomorphism for fields ([NS89], [Tot92]).
    Classical results imported for the Milnor K comparison and Steinberg relation.

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Pith. "Pith review of Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology." pith.science (2026). https://pith.science/paper/OSAFTCX6

@misc{pith2026250716501,
  author       = {Pith},
  title        = {Pith review of: Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OSAFTCX6}},
  note         = {Machine review of arXiv:2507.16501}
}
abstract

We prove that the motivic cohomology of mixed characteristic schemes, introduced in our previous work, satisfies various expected properties of motivic cohomology, including a motivic refinement of Weibel's vanishing in algebraic $K$-theory, the projective bundle formula, a comparison to Milnor $K$-theory, and a universal characterisation in terms of pro cdh descent. These results extend those of Elmanto--Morrow to schemes which are not necessarily defined over a field.

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  1. Finite-coefficient Gersten injectivity fails in ramified mixed characteristic

    math.KT 2026-08 accept novelty 7.0 of 10

    The paper constructs a nonzero mod-3 K_2 class on a ramified regular local ring that dies in the fraction field, disproving finite-coefficient Gersten injectivity in this setting.

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