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$\mathbb{A}^1$-invariant motivic cohomology of schemes

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Pith's one-line read A well-behaved $\mathbb{A}^1$-invariant motivic cohomology exists for every quasicompact quasiseparated scheme, built from the slice filtration on homotopy $K$-theory.

desk verdict Serious realization of Voevodsky's slice program for qcqs schemes, with an honest conditional hypothesis (Val) for mixed-characteristic mod-p comparisons. read the letter →

arxiv 2508.09915 v1 pith:SADV5OX2 submitted 2025-08-13 math.KT math.AG

Tom Bachmann , Elden Elmanto , Matthew Morrow This is my paper · ORCID
classification math.KTmath.AG MSC 14F4214F2019D5519E15
keywords A1-invariantmotiviccohomologyslicefiltrationhomotopyK-theorycdhtopologyBeilinson-Lichtenbaumsyntomicqcqsschemesspectra
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 claims that every quasicompact quasiseparated scheme — in particular, every scheme of finite type over a field or Dedekind domain — carries a well-behaved $\mathbb{A}^1$-invariant motivic cohomology. The theory is constructed from the slice filtration on the motivic spectrum representing homotopy $K$-theory, so that the graded pieces of a natural filtration on $KH$ are exactly the new cohomology groups. The paper proves this theory has the properties that make motivic cohomology useful: it agrees with the classical cycle complex on smooth varieties, with étale cohomology away from residue characteristics, and with logarithmic de Rham–Witt cohomology in characteristic $p$, and it satisfies cdh descent. One conditional ingredient, a regularity hypothesis about henselian valuation rings of mixed characteristic, is needed only for the optimal mod-$p$ comparisons and for $\mathbb{A}^1$-invariance of the auxiliary cdh-motivic cohomology. If the construction stands, it settles the slice-filtration conjectures identifying the zeroth slice of the motivic sphere with motivic cohomology, and provides a candidate derived category of $\mathbb{A}^1$-invariant motives over arbitrary qcqs bases.

What carries the argument

The load-bearing object is the slice filtration on the stable motivic homotopy category $SH(X)$: the descending tower of full subcategories generated by motives of smooth $X$-schemes with $j$ Tate twists, whose associated graded pieces $s_j E$ are the slices of a motivic spectrum $E$. For $E = KGL$, the motivic spectrum representing homotopy $K$-theory, the zeroth slice $s_0 KGL$ is taken as the definition of the motivic ring spectrum $HZ^A_X$, and the higher slices give the cohomology theories $Z(j)_{\mathbb{A}}$. Two auxiliary constructions carry the technical weight: cdh-motivic cohomology, obtained by cdh sheafifying the left Kan extension of motivic cohomology from smooth $\mathbb{Z}$-s

What would settle it

Find a henselian valuation ring $V$ of mixed characteristic $(0,p)$ and a weight $j$ for which the cofiber of $F_p(j)_{\mathrm{syn}}(V) \to \tau_{\le j} R\Gamma_{\mathrm{ét}}(V[1/p], \mu_p^{\otimes j})$ has nonzero cohomology in degree below $j-1$; equivalently, exhibit a qcqs scheme of finite valuative dimension where the map $Z(j)_{\mathrm{cdh}}(X) \to Z(j)_{\mathrm{cdh}}(\mathbb{A}^1_X)$ is not an equivalence modulo $p$. Such a discovery would refute the key hypothesis and with it the conditional Theorems 1.10 and 1.13, while leaving the unconditional core of Theorem 1.1 intact.

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Extended reading notes

Core claim

The paper defines, for each qcqs scheme $X$, the $\mathbb{A}^1$-motivic cohomology groups as shifts of the slices of the motivic spectrum $KGL$: $Z(j)_{\mathbb{A}}(X) = \mathrm{map}_{SH(X)}(1_X, s_j KGL_X)[-2j]$. Its main theorem is that these assemble into a multiplicative family of presheaves of complexes with an Atiyah–Hirzebruch spectral sequence abutting to $KH$, with $\mathbb{A}^1$-invariance, finitary cdh descent, a projective bundle formula, and the expected comparisons to étale cohomology, syntomic cohomology, and the cycle complex. The paper further proves that the unit map $1_X \to KGL_X$ induces an equivalence $s_0(1_X) \simeq HZ^A_X$ in $SH(X)$, where $HZ^A_X := s_0 KGL_X$; this

Load-bearing premise

The load-bearing premise is that for every henselian valuation ring of mixed characteristic $(0,p)$ mapping to the scheme, mod-$p$ syntomic cohomology is essentially determined by the étale cohomology of the $p$-inverted locus — a regularity condition that would follow from F-smoothness and is expected but not yet proved.

Editorial extensions

If this is right

  • An Atiyah–Hirzebruch spectral sequence with $E_2^{i,j} = H^{i-j}_{\mathbb{A}}(X, \mathbb{Z}(-j))$ abuts to $KH^{-i-j}(X)$ for every qcqs scheme, and the filtration is complete when the scheme has finite valuative dimension.
  • On smooth varieties over fields, the new groups agree with higher Chow groups, so the construction is a common generalization of the classical motivic cohomology theories.
  • Away from residue characteristics, $Z(j)_{\mathbb{A}}/\ell \simeq L_{\mathrm{cdh}} \tau_{\le j} R\Gamma_{\mathrm{ét}}(-,\mu_\ell^{\otimes j})$; in characteristic $p$, $Z(j)_{\mathbb{A}}/p^r \simeq R\Gamma_{\mathrm{cdh}}(X, W_r\Omega^j_{\log})[-j]$. The expected comparison isomorphisms therefore hold in equal characteristic unconditionally.
  • The category of $HZ^A_X$-modules inherits a six-functor formalism from $SH(X)$, giving a candidate for the derived category of $\mathbb{A}^1$-invariant motives over any qcqs base.
  • In mixed characteristic, the mod-$p$ comparisons hold conditionally on the key hypothesis about henselian valuation rings; rationally, all comparisons are unconditional.

Reading between the lines

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

  • If the key hypothesis Val$(X,p,j)$ is ever disproved, the failure would be localized to the comparison between mod-$p$ syntomic cohomology and étale cohomology of valuation rings; the $\mathbb{A}^1$-motivic cohomology itself, defined from slices, would remain a viable theory, though it would no longer be known to agree with the cdh-sheafified version in mixed characteristic.
  • The cdh-descent and Milnor-excision properties suggest an effective computational strategy for $KH$ on singular schemes: evaluate the filtration on henselian valuation rings, where the new cohomology reduces to étale and Milnor $K$-theory data; verifying the key hypothesis in low weights would make the mixed-characteristic computations unconditional.
  • The Hermitian variant in Section 10 indicates a parallel story for Grothendieck–Witt groups and the motivic spectrum $KO$, with an 8-periodic refinement of periodicity; if the same machinery works there, it would produce an Atiyah–Hirzebruch spectral sequence from Milnor–Witt motivic cohomology to Hermitian $K$-theory.
  • The paper leaves open integral versions of low-weight vanishing results it proves rationally; the slice definition gives a concrete candidate theory on which those integral statements could be tested.
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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

2 major / 4 minor

Summary. This paper constructs a theory of A1-invariant motivic cohomology for arbitrary qcqs schemes. The theory Z(j)_A is defined as slices of KGL (Definition 4.15) and is shown to provide an Atiyah-Hirzebruch filtration on KH, to be A1-invariant and a finitary cdh sheaf, to satisfy étale and syntomic comparisons in equal characteristic (Theorem 1.1(4)(5)), low-weight identifications, the projective bundle formula, and agreement with Bloch's cycle complex on smooth varieties. It also proves the zeroth-slice theorem s0(1_X) ≃ HZ^A_X and absolute base change (Theorem 1.6). A second theory Z(j)_cdh is introduced; its A1-invariance and the optimal Beilinson-Lichtenbaum comparison in mixed characteristic are proved only conditionally on the conjectural hypothesis Val(X,p,j) of Section 8.1.

Significance. If correct, this is a landmark contribution: it provides a well-behaved A1-invariant motivic cohomology for all qcqs schemes, with a spectral sequence to homotopy K-theory, and it resolves several of Voevodsky's slice conjectures in a non-vacuous form. The unconditional core is supported by a detailed 130+ page argument, and the paper is unusually transparent about what is conditional. No ad hoc free parameters enter; the theory is characterized by a universal property. The main caveat is that the mixed-characteristic comparisons advertised in the abstract are not unconditional: Theorem 1.13 depends on Val(X,p,j), which is explicitly conjectural. In addition, Remark 1.7 notes that the paper's definition makes Voevodsky's Conjectures 1 and 7 vacuous; the genuine content is Corollary 9.7. These caveats do not undermine the unconditional theorems, but they narrow the scope from what the abstract suggests.

major comments (2)
  1. [§8.1, Theorems 1.10/1.13] The key hypothesis Val(X,p,j) is unproved and conjectural (Example 8.6). It is load-bearing for Theorem 1.10 (conditional A1-invariance of Z(j)_cdh modulo p) and for Theorem 1.13 (the optimal Beilinson-Lichtenbaum comparison in mixed characteristic). The abstract and Theorem 1.1(5) refer to the mixed-characteristic comparison without stating this assumption. Since Val is not established, the blanket claim of étale/syntomic comparisons 'in the style of the original conjectures' for arbitrary qcqs schemes is only conditional. Please qualify the abstract and the theorem statements, and make the conditional status of Theorem 1.13 visible in the introduction and in the cross-reference from Theorem 1.1(5).
  2. [Remark 1.7 / Corollary 9.7] The paper defines HZ^A_X := s0(KGL_X) and then notes in Remark 1.7 that Voevodsky's Conjectures 1 and 7 become vacuous under this definition. The paper should not be described as establishing those conjectures; the non-vacuous result is the equivalence s0(1_X) ≃ HZ^A_X of Corollary 9.7. This is a legitimate reformulation, but it is currently flagged only in a remark. The introduction and abstract should state clearly that the paper proves a substitute theorem, not a literal verification of the original conjectures.
minor comments (4)
  1. [Abstract and Theorem 1.1(5)] The mixed-characteristic syntomic comparison is stated unconditionally in Theorem 1.1(5) via the cross-reference 'see Theorem 1.13'. Since Theorem 1.13 is conditional on Val, the cross-reference should carry the hypothesis explicitly, and the abstract should say 'conditionally on the key hypothesis Val' for the mixed-characteristic comparisons.
  2. [Definition 4.17] There is a typo: the display 'H^i_{A,cdh}(X,Z(j)) := H^i(Z(j)_A(X))' should read 'H^i(Z(j)_{A,cdh}(X))'.
  3. [Section 10] Section 10 is described as a 'speed-run' and says 'we omit some details' with respect to the Hermitian K-theory analogues. If Theorem 10.22 is advertised as a main result, the omissions make verification difficult. Either include the omitted arguments or clearly label the section as a sketch.
  4. [Remark 1.7] The observation that [156, Conjectures 1 and 7] are vacuous should also appear in the introduction's summary of Voevodsky's slice conjectures, not only in a remark after Theorem 1.6, to avoid overstating the historical claim.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the paper defines A1-motivic cohomology as slices of KGL and openly notes that this makes two Voevodsky conjectures vacuous; all non-formal comparisons rest on external theorems or are explicitly conditional.

  1. self definitional [Definition 4.15; Remark 1.7]
    "For a qcqs scheme X, its A1-motivic cohomology is defined by Z(j)_A(X) := map_SH(X)(1_X, s_j KGL_X)[-2j] ... Taking HZ^A_X as the definition of HZ, then [156, Conjectures 1, 7] are vacuous, while Theorem 1.6 establishes [156, Conjectures 10, 17]."

    The Atiyah-Hirzebruch filtration property (Theorem 1.1(1)) is built into the definition: Z(j)_A is defined as the j-th slice of KGL, and the filtration on KH is the slice filtration on KGL, so the graded pieces are Z(j)_A by construction. The paper explicitly acknowledges this in Remark 1.7, saying that Voevodsky's Conjectures 1 and 7 become vacuous under this definition. This is an admitted self-definitional convention rather than a hidden reduction. The substantive non-vacuous claims, such as s0(1_X) ≃ HZ^A_X and the étale/syntomic comparisons, are not obtained from this definition alone; they are proved using external results over fields and descent/base-change arguments.

full rationale

The central construction is admittedly definitional: A1-motivic cohomology is stipulated to be the slices of the KGL spectrum, so the associated graded pieces of the slice filtration on KH are equal to the new theory by definition. The paper does not disguise this; Remark 1.7 explicitly states that two of Voevodsky's conjectures become vacuous under this convention. Thus the only circular-looking step is an acknowledged convention, not a hidden derivation. The principal independent content, especially Theorem 1.6 identifying the zeroth slice of the motivic sphere with HZ^A_X, is proved by reducing to Levine's external theorem over fields and then extending by base change, not by assuming the conclusion. The comparisons to étale and syntomic cohomology rely on external developments (Bhatt-Mathew, Bhatt-Lurie, Geisser-Levine, Rost-Voevodsky) and are not fitted from the target data. The key hypothesis Val(X,p,j) of Section 8.1 is explicitly conjectural and is used to make Theorems 1.10 and 1.13 conditional; this is a transparent assumption, not a circular reduction. Self-citations occur, but they are not load-bearing for the main unconditional theorems, which are anchored in external machine-checkable or independently established results. Accordingly, the circularity score is low: 2, reflecting only the acknowledged self-definitional framing of the slice-filtration properties.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central claim rests on the standard motivic homotopy theory framework, on two big external pillars (Levine's identification of the zeroth slice over fields, and the Bhatt-Lurie-Bhatt-Mathew theory of syntomic cohomology), on Riou's Adams decomposition for the rational structure, and on the open key hypothesis Val(X,p,j) for the sharpest mixed-characteristic statements. No numbers are fitted to data; the paper is purely structural. The invented entities are new mathematical objects that carry independent falsifiable content through comparison theorems with established invariants, so they do not behave like ad hoc fittings.

assumptions (5)
  • standard math The foundational framework of motivic stable homotopy theory: for each qcqs scheme X, the presentably symmetric monoidal stable infinity-category SH(X) with the universal properties reviewed in Section 3.1 exists.
    Invoked throughout (Definition 4.15, Sections 3-4); the paper reviews rather than proves the framework, citing Morel-Voevodsky [127], Robalo [135], and Hoyois et al. [22].
  • domain assumption Levine's theorem: for fields k, the map s_0(1_k) -> s_0(kgl_k) is an equivalence, i.e., the zeroth slice of the motivic sphere identifies with the zeroth slice of KGL over a field.
    Used repeatedly as the external benchmark to bootstrap the absolute statement s_0(1_X) ≃ HZ^A_X: Proposition 4.11, Corollary 4.52(5), proofs of Theorem 6.3 and Corollary 9.7; quoted from Levine [110, Theorems 6.4.2 and 10.5.1] and Voevodsky [159].
  • domain assumption Key hypothesis Val(X, p, j): for every henselian valuation ring V of mixed characteristic (0, p) with Spec(V) -> X, the mod-p syntomic cohomology F_p(j)_syn(V) is mostly given by the etale cohomology of V[1/p]; expected to follow from F-smoothness of Bhatt-Mathew [31].
    Introduced in Section 8.1; the authors state it remains conjectural. Used for Theorem 1.10 (mod-p A1-invariance of cdh-motivic cohomology) and Theorem 1.13 (optimal Beilinson-Lichtenbaum comparison in mixed characteristic). Not needed for the unconditional core of Theorem 1.1.
  • domain assumption Riou's Adams decomposition: (KGL_X)_Q ≃ direct sum over j of KGL^(j)_X with Adams operators psi^k acting as k^j (Theorem 4.48).
    Quoted from Riou [134]; underpins the rational structure analysis of Section 4.3, including Theorem 4.47 and Corollary 4.52.
  • domain assumption Existence and basic properties of p-adic syntomic cohomology Z_p(j)_syn on qcqs schemes, as summarized in Theorem 5.2 (fpqc descent, BL-type identifications in characteristic p and away from p, projective bundle formula, improved Milnor K-theory in high degrees, Bhatt-Mathew support bound).
    The paper re-expositions parts of the construction via K-theory and TC (Sections 5.1-5.2) but imports the main theorems from [32, 9, 29, 31]; this is the p-adic engine behind the characteristic-p comparisons and the Val hypothesis.
invented entities (3)
  • A1-motivic cohomology Z(j)_A independent evidence
    purpose: Central new cohomology theory for qcqs schemes; defined as the mapping spectrum out of the unit into the j-th slice of KGL (Definition 4.15); the graded pieces of the new multiplicative filtration on KH.
    Matched against external benchmarks: Bloch's cycle complex on smooth schemes (Theorem 1.1(8), Remark 4.14), etale cohomology mod l (Theorem 1.1(4)), syntomic cohomology in characteristic p (Theorem 1.1(5)), and rational Adams eigenspaces (Theorem 4.47(4)).
  • cdh-motivic cohomology Z(j)_cdh independent evidence
    purpose: cdh sheafification of the left Kan extension of Z(j)_A restricted to smooth Z-schemes; auxiliary invariant used to control Z(j)_A and to define the filtration needed for non-A1-invariant motivic cohomology of [68, 41].
    Identified with Spitzweck's motivic cohomology over Z (Remark 4.14, via [16, 145]); satisfies Milnor excision (Section 7.4); agrees with Z(j)_A after A1-localisation (Theorem 1.1(9)).
  • Kato K-theory KKato and mod-p analog F_p(*)Kato independent evidence
    purpose: Introduced in Sections 9.3.5 and 9.3.4 to measure the difference between K-theory and Selmer K-theory (resp. mod-p cdh-motivic cohomology and cdh-sheafified syntomic cohomology); key to the localisation argument in the proof of Theorem 9.1.
    The paper proves a localisation property (Section 9.3.3) that gives them independent content; they are used to separate values on schemes of different characteristics.

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Pith. "Pith review of $\mathbb{A}^1$-invariant motivic cohomology of schemes." pith.science (2026). https://pith.science/paper/SADV5OX2

@misc{pith2026250809915,
  author       = {Pith},
  title        = {Pith review of: $\mathbbA^1$-invariant motivic cohomology of schemes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SADV5OX2}},
  note         = {Machine review of arXiv:2508.09915}
}
abstract

Voevodsky outlined a conjectural programme that his slice filtration in motivic homotopy theory should give rise to a good theory of $\mathbb{A}^1$-invariant motivic cohomology. This paper achieves his vision in the generality of arbitrary quasicompact, quasiseparated schemes, by introducing a theory of $\mathbb{A}^1$-invariant motivic cohomology which is related to Weibel's homotopy $K$-theory via an Atiyah--Hirzebruch spectral sequence, and which we compare to \'etale and syntomic cohomology in the style of the original conjectures of Beilinson and Lichtenbaum. In addition, it is represented by an absolute motivic spectrum and therefore satisfies cdh descent, and modules over it offer a candidate for the derived category of $\mathbb{A}^1$-invariant motives. We establish some of Voevodsky's open conjectures on slices, in particular relating the zeroth slice of the motivic sphere to homotopy $K$-theory. In the final section we prove analogous results for the Hermitian $K$-theory of qcqs schemes on which $2$ is invertible. As an auxiliary tool we introduce cdh-motivic cohomology, defined as the cdh sheafification of the left Kan extension of the motivic cohomology of smooth $\mathbb{Z}$-schemes. We offer a new approach to control the latter, independent of previous work on $\mathbb{A}^1$-invariant motivic cohomology of smooth schemes over mixed characteristic Dedekind domains: our approach is based on recent developments in $p$-adic cohomology, in particular syntomic and prismatic cohomology. The cdh-motivic cohomology is also a necessary ingredient in the last two authors' and Bouis' construction of non-$\mathbb{A}^1$-invariant motivic cohomology of qcqs schemes.

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