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On the motivic cohomology of some singular rings

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Pith's one-line read This paper shows that the non-A1-invariant motivic cohomology of singular rings is integrally computable, giving explicit formulas for truncated polynomial algebras over number fields and four other families.

desk verdict A useful first batch of computations in the new non-A1-invariant motivic cohomology, but the headline integral result in Corollary 3.8 rests on a compatibility step that is cited rather than proved. read the letter →

arxiv 2608.05220 v1 pith:JWCMMNTH submitted 2026-08-05 math.AG math.KTmath.NTmath.OA

Tess Bouis This is my paper · ORCID
classification math.AGmath.KTmath.NTmath.OA MSC 14F4219D5514C1519E08
keywords motiviccohomologysingularringsalgebraicK-theorytopologicalcyclichomologyintegralp-adicHodgetheorytruncatedpolynomialalgebrasvaluationC*-algebras
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

This paper proves that the recently introduced non-$\mathbb{A}^1$-invariant motivic cohomology of singular rings is concretely computable in five families where the classical definition of motivic cohomology does not apply. The centerpiece is an integral formula: for $K$ a number field with ring of integers $\mathcal{O}_K$ and any $e\geq 1$, the relative groups $H^n_{mot}(\mathcal{O}_K[x]/(x^e),(x),\mathbb{Z}(i))$ vanish except in degrees $n=1,2$, where $H^1\cong\mathcal{O}_K^{e-1}$ and $H^2$ is a finite group whose order is $((ei)!(i!)^{e-2})^d\cdot|\mathcal{O}_K/D|^{ei-i}$, with $d=[K:\mathbb{Q}]$ and $D$ the different ideal. The same method computes motivic cohomology of finite chain rings, perfect and semiperfect $\mathbb{F}_p$-algebras, henselian valuation rings, and smooth algebras over $C(X;\mathbb{C})$ for compact Hausdorff $X$. The proofs run through the identification of motivic complexes with graded pieces of the weight filtration on topological cyclic homology, with integral $p$-adic Hodge theory replacing trace methods. If correct, these results turn a previously formal theory into an integral computational tool and refine known $K$-theory calculations.

What carries the argument

The central object is the non-$\mathbb{A}^1$-invariant motivic cohomology $Z(i)_{mot}$, defined through trace methods and, for nilpotent thickenings, identified with the $i$-th graded piece of the motivic filtration on integral topological cyclic homology (Lemma 3.1). The computations are assembled by a cartesian square comparing $Z(i)_{mot}$ with syntomic cohomology on the $p$-adic side and derived de Rham cohomology on the rational side, together with cdh descent for motivic complexes. For truncated polynomials, the $p$-adic input is the prismatic computation of $\mathbb{Z}_p(i)^{BMS}(\mathcal{O}_K[x]/(x^e),(x))$ and the rational input is the de Rham computation $\mathbb{Q}^{e-1}[-1]$; the cartesian square then yields the integral groups of Corollary 3.8. For valuation rings, the lisse-motivic comparison map and the Beilinson–Lichtenbaum comparison to truncated étale cohomology play the same load-bearing role.

What would settle it

For $\mathcal{O}_K=\mathbb{Z}$, $e=2$, $i=1$, Corollary 3.8 predicts $H^2_{mot}(\mathbb{Z}[x]/(x^2),(x),\mathbb{Z}(1))$ is a finite group of order $2$ and $H^1_{mot}\cong\mathbb{Z}$; a direct computation of these groups from the defining fibre sequence with $\mathbb{Z}(1)_{TC}$ that yields any different group would disprove the main formula. Similarly, for $\mathbb{Z}/p^2$, Theorem 2.3 predicts that $H^1_{mot}(\mathbb{Z}/p^2,\mathbb{Z}(i))$ has order $(p^i-1)p^i$ for large $i$, which is checkable by an independent syntomic calculation.

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Extended reading notes

Core claim

At the level of the paper's own claims, the main discovery is that the non-$\mathbb{A}^1$-invariant motivic cohomology $Z(i)_{mot}$ gives new integral invariants of singular rings that can be completely determined. Corollary 3.8 states that for a number field $K$, $e\geq 1$, and any $i,n\geq 1$, one has $H^n_{mot}(\mathcal{O}_K[x]/(x^e),(x),\mathbb{Z}(i))\cong\mathcal{O}_K^{e-1}$ for $n=1$, $\cong A_i$ for $n=2$, and $0$ otherwise, where $A_i$ is finite of order $((ei)!(i!)^{e-2})^d\cdot|\mathcal{O}_K/D|^{ei-i}$. The paper also proves that $Z(i)_{mot}(\mathcal{O}_K/\pi^n)$ is concentrated in degree one of order $(q^i-1)q^{i(n-1)}$, that perfection is a $p$-local equivalence for qcqs $\mathbb{F}_p$-schemes, that henselian valuation rings satisfy a lisse-motivic comparison and a Beilinson–Lichtenbaum-type description with finite coefficients, and that smooth algebras over commutative $C^\ast$-algebras are motivically regular. These are stated as refinements of known $K$-theory results, with the integral refinements obtained from the weight filtration on topological cyclic homology and integral $p$-adic Hodge theory.

Load-bearing premise

The entire paper assumes that the non-$\mathbb{A}^1$-invariant motivic cohomology defined in the author's earlier work is a well-behaved theory—specifically that it satisfies cdh descent, that it agrees with the weight filtration on topological cyclic homology for nilpotent thickenings, and that the lisse-motivic comparison holds for henselian valuation rings—and none of these foundational inputs are reproved here.

Editorial extensions

If this is right

  • Corollary 3.8 gives an unconditional integral formula for the motivic cohomology of truncated polynomial algebras over rings of integers, a setting where the classical $\mathbb{A}^1$-invariant motivic cohomology is not defined.
  • Via the Atiyah–Hirzebruch spectral sequence, the same corollary recovers the known $K$-theory computation of $K(\mathbb{Z}[x]/(x^e),(x))$, now from motivic input.
  • Theorem 2.3 shows that the motivic complex of a finite chain ring is concentrated in one degree with an explicit order, refining the $p$-adic $K$-theory computation used as input.
  • Theorem 4.1 shows that, after inverting $p$, perfection is a motivic equivalence for qcqs $\mathbb{F}_p$-schemes, and that positive-weight motivic complexes of perfect $\mathbb{F}_p$-schemes are $p$-local.
  • Theorem 6.1 upgrades $K$-regularity of $C^\ast$-algebras to motivic regularity for smooth algebras over $C(X;F)$.

Reading between the lines

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

  • The factorial orders in Corollary 3.8 suggest a broader phenomenon: motivic cohomology of monomial quotient rings over number fields may be concentrated in two degrees with orders built from factorials and discriminants, and this is testable on rings like $\mathbb{Z}[x,y]/(x^a-y^b)$.
  • Because the identification with TC weights is formal, any future prismatic computation of $\mathbb{Z}_p(i)^{BMS}$ for a singular ring could be converted immediately into an integral motivic statement without repeating the trace-method argument.
  • For henselian valuation rings, Theorem 5.6 predicts that the image of the symbol map generates $H^i_{mot}$ modulo $p^k$; one could test this numerically for $\mathcal{O}_{\mathbb{C}_p}$ by comparing Milnor $K$-theory symbols with explicit étale classes.
  • The motivic regularity theorem for $C^\ast$-algebras may extend to non-smooth algebras if the $K$-theoretic regularity holds, since the proof only needs a comparison of $K$-theory and $KH$-theory.
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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 / 6 minor

Summary. The paper develops motivic refinements of known algebraic K-theory computations for five classes of singular rings: finite chain rings, truncated polynomial algebras, perfect and semiperfect F_p-algebras, henselian valuation rings, and smooth algebras over C*-algebras. The main results include Theorem 2.3 on finite chain rings, Corollary 3.8 giving an integral computation of relative motivic cohomology of O_K[x]/(x^e) for number fields, Theorem 3.9 for perfectoids, Theorem 4.1 on perfection, Theorems 5.1, 5.4, and 5.6 on valuation rings, and Theorem 6.1 on motivic regularity of C*-algebras. The arguments are presented as consequences of the author's earlier foundational work [Bou24, Bou25] together with trace-method and prismatic calculations by Riggenbach, Antieau-Krause-Nikolaus, and others.

Significance. If correct, these results are valuable: they provide the first computations of non-A1-invariant motivic cohomology for singular rings and give new information even about K-theory, such as the nilpotence reformulation in Remark 2.4 and the motivic refinement of Riggenbach's theorem. The paper is careful to compare with existing K-theoretic results, and the rational parts are derived from de Rham cohomology rather than fitted to the expected answer. However, the central claims depend on the author's own preprints [Bou24, Bou25], which are not yet independently verified, and the proof of the headline Corollary 3.8 contains an unverified compatibility step. The paper is likely to be of interest to specialists in motivic cohomology and K-theory if those foundations and the compatibility argument are supplied.

major comments (3)
  1. [§3, Corollary 3.8] The proof of Corollary 3.8 combines the p-adic equivalence from [Rig25] and the rational equivalence from Theorem 3.6 via the cartesian square [Bou24, Cor 4.31]. The assertion that "the previous two equivalences are compatible by construction" and that the only nontrivial compatibility follows from [BL22, Construction 5.5.3] is the load-bearing point of the integral computation. The manuscript does not display the comparison map, does not explain how [BL22, Construction 5.5.3] applies to the relative pair (O_K[x]/x^e,(x)) with the Nygaard filtration on Breuil-Kisin-twisted absolute prismatic cohomology, and does not rule out an automorphism or filtration-shift ambiguity that could change the finite group A_i or its order. This step needs to be proved in detail.
  2. [§5, Theorem 5.1] The proof of Theorem 5.1 is simply a citation to the proof of [Bou25, Lemma 3.25], and the lisse-motivic comparison map is only defined by reference to [Bou25, Definition 2.1]. Since Theorem 5.1 is a headline result of Section 5 and [Bou25] is an unpublished preprint, the paper should either include a full proof, or at least state the relevant lemma and give a self-contained argument, or clearly signal that this result is conditional on the acceptance of [Bou25].
  3. [§3, Lemma 3.1] Lemma 3.1 identifies relative motivic cohomology with the i-th graded piece of the motivic filtration on TC by citing [Bou24, Remark 3.21]. This lemma is foundational for all Section 3 results, but the cited remark is not reproduced and the proof is one sentence. A more explicit statement of the relevant property from [Bou24] would help readers verify the identification and assess the dependence on unpublished foundations.
minor comments (6)
  1. [Contents] The table of contents line "T runcated polynomials" should read "Truncated polynomials".
  2. [§2, Theorem 2.3] The condition "for every integer i≥4 p n,1" is confusing; the comma and the stray "1" appear to be a footnote marker, and the intended inequality should be typeset as i ≥ 4pn.
  3. [§2, Lemma 2.1] In the proof of Lemma 2.1, "F_q" should be typeset as \mathbb{F}_q for consistency with the rest of the paper.
  4. [§3, Corollary 3.8] The symbol "Q′ p∈P" in the cartesian square is not defined; it appears to denote a restricted product, but this should be stated explicitly.
  5. [§3, Theorem 3.6] The hypothesis that the cotangent complex L_{(R⊗_Z Q)/Q} vanishes is invoked to conclude that all positive powers of this cotangent complex vanish; this implication should be spelled out for the reader.
  6. [§6, Theorem 6.1] The abstract uses C(X;\mathbb{C}) while Section 6 uses C(X;F) for a general characteristic zero local field; the notation should be harmonized.

Circularity Check

2 steps flagged · score 4.0 of 10

The numerical results are benchmarked against independent K-theory, but the motivating object is defined in the author's own preprints and Section 5's main theorem is quoted from the same author's earlier work.

  1. self citation load bearing [Section 5, Theorem 5.1 and its proof]
    "Theorem 5.1. Let V be a henselian valuation ring. Then for every integer i ≥ 0, the motivic complex Z(i)_mot(V) ∈ D(Z) is in degrees at most i, and the lisse-motivic comparison map [Bou25, Definition 2.1] Z(i)_lisse(V) → Z(i)_mot(V) is an equivalence in the derived category D(Z). Proof. The second statement already appears in the proof of [Bou25, Lemma 3.25]. As in [Bou25, Lemma 3.25] or [Bou25, Corollary 2.12], the first statement is then a consequence of [Gei04, Corollary 4.4]."

    Theorem 5.1 is the central identification for the valuation-ring section, but it is not proved in this paper: the proof consists of citing the same author's earlier preprint [Bou25]. Example 5.2, Proposition 5.3, and Theorem 5.6 all inherit their key input from this imported equivalence. Because [Bou25] is a same-author, non-machine-checked preprint, the Section 5 derivation is load-bearing self-citation rather than an independently established mathematical fact.

  2. self definitional [Section 3, Lemma 3.1 and its proof]
    "Let R be a commutative ring, and e ≥ 1 be an integer. Then for every integer i ≥ 0, the natural map Z(i)_mot(R[x]/(x^e),(x)) → Z(i)TC(R[x]/(x^e),(x)) is an equivalence ... where Z(i)TC denotes the i-th graded piece of the motivic filtration on integral topological cyclic homology TC, as defined in [Bou24, Section 2.3]. Proof. This is a direct consequence of [Bou24, Remark 3.21], and the fact that cdh sheaves are invariant under nilpotent extensions."

    The paper's headline integral computation, Corollary 3.8, begins by identifying the non-A1 motivic complex of truncated polynomials with the i-th graded piece of the TC filtration, citing the author's own foundational preprint. Since the non-A1 motivic theory in [Bou24] is built from trace methods, this identification is part of the definitional architecture of the theory rather than an external check. The motivic computation of truncated polynomials is therefore imported from the author's prior definition of motivic cohomology together with Riggenbach's TC computation, rather than derived from an independent motivic first principle.

full rationale

Corollary 3.8's finite group order is not fitted: its p-adic factor comes from Riggenbach's independent prismatic computation [Rig25] and its rational factor from a de Rham computation (Theorem 3.6 and Lemma 3.5), and the result is consistent with the known K-theory of Angeltveit-Gerhardt-Hesselholt. So there is no direct input-equals-output construction of the numerical answer. The circularity burden is that the object being computed, non-A1 motivic cohomology, is only accessible through the author's own preprints: Lemma 3.1 imports the identification with the TC filtration from [Bou24], and Theorem 5.1 is effectively quoted from [Bou25]. These self-citations are load-bearing for the central integral computation and for the valuation-ring section. The asserted compatibility in the proof of Corollary 3.8 ('compatible by construction') is a gap that could affect the integral structure, but it is a correctness risk rather than a circular reduction. On balance the paper has independent content through its external K-theoretic benchmarks, so the score is 4 rather than higher.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper introduces no fitted parameters and no genuinely new entities; the load-bearing content is imported from the author's prior theory and from external K-theory computations. The main axioms are the structural properties of non-A1-invariant motivic cohomology and the p-adic Hodge theory comparisons.

assumptions (9)
  • domain assumption Non-A1-invariant motivic cohomology Z(i)_mot is a well-defined functor satisfying the structural properties proved in [Bou24, Bou25] (cdh descent, cartesian squares, identification with TC filtration).
    Invoked throughout, e.g., Lemma 2.1 (cartesian square [Bou24, Prop3.29]) and Lemma 3.1 ([Bou24, Remark3.21]).
  • domain assumption The cartesian square for Z(i)_mot of finite chain rings relating syntomic cohomology and motivic cohomology of the residue field holds.
    Used in Lemma 2.1 and Theorem 2.3.
  • domain assumption The cartesian square for Z(i)_TC relating derived de Rham cohomology and BMS syntomic cohomology holds, and the compatibility with Nygaard and Hodge filtrations is exactly [BL22, Construction 5.5.3].
    Core of Corollary 3.8 proof.
  • domain assumption Riggenbach's p-adic relative K-theory computations for truncated polynomials over rings of integers and perfectoid rings [Rig25, Thm 6.4, Cor 6.5] are correct.
    Inputs to Corollary 3.8 and Theorem 3.9.
  • domain assumption Frobenius acts by multiplication by p^i on Z(i)_mot of F_p-schemes [EM23, Thm4.34].
    Used in Theorem 4.1 and Corollary 4.3.
  • domain assumption Valuation rings of characteristic p are Cartier smooth over F_p [Bou23, Thm3.4], and the Milnor K-theory comparison [Bou25, Thm2.21] holds for henselian valuation rings.
    Used in Theorems 5.4 and 5.6.
  • domain assumption The lisse-motivic comparison map for henselian valuation rings is an equivalence, as stated in [Bou25, Lemma3.25].
    Theorem 5.1 is essentially this result, with a proof by citation.
  • domain assumption Smooth algebras over C(X;F) are K-regular and K agrees with KH after adding polynomial variables [CT12, Aok24].
    Input to Theorem 6.1.
  • standard math Integral p-adic Hodge theory of BMS19 and BS22 supplies the syntomic cohomology and Nygaard filtration results used in Sections 2, 3, and 5.
    Invoked in the abstract and throughout the proofs as the replacement for trace methods.

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Pith. "Pith review of On the motivic cohomology of some singular rings." pith.science (2026). https://pith.science/paper/JWCMMNTH

@misc{pith2026260805220,
  author       = {Pith},
  title        = {Pith review of: On the motivic cohomology of some singular rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWCMMNTH}},
  note         = {Machine review of arXiv:2608.05220}
}
abstract

Using non-$\mathbb{A}^1$-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic $K$-theory of singular rings, such as rings of the form $\mathbb{Z}/p^n$, $\mathbb{Z}[x]/(x^e)$, and $\mathscr{C}(X;\mathbb{C})$ for $X$ a compact Hausdorff space. These refinements are made possible by the use of integral $p$-adic Hodge theory, as a replacement for the standard use of trace methods in $K$-theory.

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Works this paper leans on

16 extracted references · 9 canonical work pages

  1. [7]

    Absolute prismatic cohomology.https://arxiv.org/abs/2201.06120,

    [BL22] Bhargav Bhatt and Jacob Lurie. Absolute prismatic cohomology.https://arxiv.org/abs/2201.06120,

    arXiv

  2. [9]

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

    [Bou25] Tess Bouis. Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology. https://arxiv.org/abs/2507.00000,

    work page arXiv

  3. [11]

    Motivic cohomology of equicharacteristic schemes.https://arxiv

    [EM23] Elden Elmanto and Matthew Morrow. Motivic cohomology of equicharacteristic schemes.https://arxiv. org/abs/2309.08463,

    arXiv

  4. [16]

    RationalK-theory of the dual numbers of a ring of algebraic integers

    [Sou81] Christophe Soulé. RationalK-theory of the dual numbers of a ring of algebraic integers. InAlgebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), volume 854 ofLecture Notes in Math., pages 402–408. Springer, Berlin,

    work page 1980

  5. [1973]

    Values of zeta-functions at nonnegative integers

    [Lic84] Stephen Lichtenbaum. Values of zeta-functions at nonnegative integers. InNumber theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), volume 1068 ofLecture Notes in Math., pages 127–138. Springer, Berlin,

    work page 1983

  6. [1980]

    A procdh topology.https://arxiv.org/abs/2401.02699,

    [KS24] Shane Kelly and Shuji Saito. A procdh topology.https://arxiv.org/abs/2401.02699,

    arXiv

  7. [1986]

    Be˘ ılinson

    10 ON THE MOTIVIC COHOMOLOGY OF SOME SINGULAR RINGS [Be˘ ı87] Alexander A. Be˘ ılinson. Height pairing between algebraic cycles. InCurrent trends in arithmetical algebraic geometry (Arcata, Calif., 1985), volume 67 ofContemp. Math., pages 1–24. Amer. Math. Soc., Providence, RI,

    work page 1985

  8. [1987]

    https://arxiv.org/abs/2508.09915,

    [BEM25] Tom Bachmann, Elden Elmanto, and Matthew Morrow.A 1-invariant motivic cohomology of schemes. https://arxiv.org/abs/2508.09915,

    arXiv

Show all 16 references
  1. [2001]

    Classical

    [Lic73] Stephen Lichtenbaum. Values of zeta-functions, étale cohomology, and algebraicK-theory. InAlgebraicK- theory, II: “Classical” algebraicK-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., Vol. 34...

    1972

  2. [2009]

    On theK-theory ofZ/p n.https://arxiv.org/ abs/2405.04329,

    [AKN24] Benjamin Antieau, Achim Krause, and Thomas Nikolaus. On theK-theory ofZ/p n.https://arxiv.org/ abs/2405.04329,

    arXiv

  3. [2018]

    Higher algebraicK-theory

    [Qui73] Daniel Quillen. Higher algebraicK-theory. I. InAlgebraicK-theory, I: HigherK-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., Vol. 341, pages 85–147. Springer, Berlin-New York,

    1972

  4. [2020]

    Be˘ ılinson

    [Be˘ ı86] Alexander A. Be˘ ılinson. Notes on absolute Hodge cohomology. InApplications of algebraicK-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), volume 55 ofContemp. Math., pages 35–68. Amer. Math. Soc., Providence, RI,

    1983

  5. [2022]

    [Cou23] Kevin Coulembier.K-theory and perfection.https://arxiv.org/abs/2304.01421,

    arXiv

  6. [2023]

    Motivic cohomology of mixed characteristic schemes.https://arxiv.org/abs/2412.06635,

    [Bou24] Tess Bouis. Motivic cohomology of mixed characteristic schemes.https://arxiv.org/abs/2412.06635,

    arXiv

  7. [2024]

    TheK-theory of perfectoid rings.Doc

    [AMM22] Benjamin Antieau, Akhil Mathew, and Matthew Morrow. TheK-theory of perfectoid rings.Doc. Math., 27:1923–1952,

    1923

  8. [2025]

    Beilinson–Lichtenbaum phenomenon for motivic cohomology.https:// arxiv.org/abs/2506.09910,

    [BK25] Tess Bouis and Arnab Kundu. Beilinson–Lichtenbaum phenomenon for motivic cohomology.https:// arxiv.org/abs/2506.09910,

    arXiv

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