arXiv subjects

Keiho Matsumoto

Publications and source records attributed to Keiho Matsumoto.

9 recordsLinked to original sources

arXiv2026

Degeneration Theorems of Connes and Feigin--Tsygan Type in Mixed Characteristic, with q-Analogues

We prove mixed-characteristic analogues of the Connes and Feigin--Tsygan degeneration theorem. Let $W=W(k)$ be the Witt vectors of a perfect field of characteristic $p>0$. For a smooth proper variety $X$ over $W$, the de Rham-to-$\HP$ spectral sequence is split degenerate under the small-dimension hypothesis $dim(X/W)<p-1$. More generally, if $X$ is smooth and proper over the ring of integers $O_K$ of a finite extension of $\mathrm{Frac}(W)$ with ramification index $e$, we prove the corresponding split degeneration under $2e dim(X/O_K)<p-1$. Under the same ramification hypothesis, we also prove split degeneration of the $Ainf$-to-$TP$ spectral sequence. Finally, after inverting an explicit factorial, we obtain a topological $q$-de Rham analogue.

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arXiv2026

Voevodsky motives and motives with modulus in positive characteristic

Let $k$ be a perfect field of characteristic $p>0$. In this paper, without assuming resolution of singularities, we prove that the triangulated category of motives with modulus with rational coefficients is equivalent to Voevodsky's triangulated category of motives with rational coefficients $\MDM^\eff(k,\Q)\simeq \DM^\eff(k,\Q).$ Equivalently, after tensoring with $\Q$, the multiplicities of the modulus become invisible in the category of motives with modulus in positive characteristic.

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arXiv2026

A Criterion for Phantomness of dg-categories

We study the question of whether the vanishing of additive invariants characterizes phantomness for smooth proper dg categories admitting geometric realizations. More precisely, let $X$ be a smooth proper variety over a field $k$, and let $\sT\subset \perfdg(X)$ be a $k$-linear admissible full dg subcategory. We construct a non-compact motive $\sM(\sT)\in \DM(k,\Q)$ and show that its $l$-adic realization recovers the $K(1,l)$-local algebraic $K$-theory of $\sT$. Analogous statements are obtained for Betti and de Rham realizations, which recover topological $K$-theory and periodic cyclic homology, respectively. As a consequence, assuming that the Chow motive of $X$ is Kimura-finite, we prove a criterion for phantomness: the vanishing of $L_{K(1,l)}K(\sT_{\overline{k}})_\Q$, of Hochschild homology in characteristic zero, or of rational topological $K$-theory over $\mathbb{C}$ implies that the rational noncommutative motive of $\sT$ vanishes. In this way, our results provide a partial answer to a question raised by Sosna. We also establish a deformation-invariance result for phantomness in smooth proper families.

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arXiv2025

Crystalline representations and $p$-adic Hodge theory for non-commutative algebraic varieties

Let $\mathcal{T}$ be an $\mathcal{O}_K$-linear idempotent-complete, small smooth proper stable $\infty$-category, where $K$ is a finite extension of $\mathbb{Q}_p$. We give a Breuil-Kisin module structure on the topological negative cyclic homology $π_i{\rm TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)$, and prove a $K$-theory version of Bhatt-Morrow-Scholze's comparison theorems. Moreover, using Gao's Breuil-Kisin $G_K$-module theory and Du-Liu's $(φ,\hat{G})$-module theory, we prove the $\mathbb{Z}_p[G_K]$-module $T_{A_{\rm inf}}(π_i{\rm TC}^-(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_p)^{\vee})$ is a $\mathbb{Z}_p$-lattice of a crystalline representation. As a corollary, if the generic fibre of $\mathcal{T}$ admits a geometric realization in the sense of Orlov, we prove a comparison theorem between $K(1)$-local $K$ theory of the generic fibre and topological cyclic periodic homology theory of the special fibre with $B_{\rm crys}$-coefficients, in particular, we prove the $p$-adic representation of the $K(1)$-local $K$-theory of the generic fibre is a crystalline representation, this can be regarded as a non-commutative analogue of $p$-adic Hodge theory for smooth proper varieties proved by Tsuji and Faltings. This is the full version of arXiv:2305.00292, containing additional details and results.

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arXiv2025

Solid realization of motives with modulus

We construct a covariant realization functor, denoted \textsc{Solidm}, from the category of motives with modulus to the derived category of solid modules in the sense of Clausen--Scholze. For any smooth modulus pair (X, D), the dual of Solidm(X, D) recovers the Hodge realization of Kelly--Miyazaki for (X, D). Using Ren's pro-solid comparison theorem, we give an explicit description of Solidm(X, D) and compute Solidm of the cone of M(U, D restricted to U) $\to$ M(X, D), in the setting where X is a smooth proper variety over a field, D $\subset$ X is a simple normal crossings divisor, and U $\subset$ X is an open immersion. We identify the result via the formal completion of X along the complement X $\setminus$ U.

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arXiv2024

Towards the $p$-adic Hodge theory for non-commutative algebraic varieties

We construct a K-theory version of Bhatt-Morrow-Scholze's Breuil-Kisin cohomology theory for $\sO_K$-linear idempotent-complete, small smooth proper stable infinity-categories, where $K$ is a discretely valued extension of $\Q_p$ with perfect residue field. As a corollary, under the assumption that $K(1)$-local K theory satisfies the Künneth formula for $\sO_K$-linear idempotent-complete, small smooth proper stable $\infty$-categories, we prove a comparison theorem between $K(1)$-local K theory of the generic fiber and topological cyclic periodic homology theory of the special fiber with $\Bcry$-coefficients, and $p$-adic Galois representations of $K(1)$-local K theory for $\sO_K$-linear idempotent-complete, small smooth proper stable $\infty$-categories are semi-stable. We also provide an alternative K-theoretical proof of the semi-stability of p-adic Galois representations of the p-adic étale cohomology group of smooth proper varieties over $K$ with good reduction. is a short, This is a short preliminary version of the work that was later expanded in 2309.13654.

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arXiv2023

Derived invariants and motives, Part I, Integral Grothendieck Riemann-Roch and non-commutative motives

The goal of this series of papers is to give a new non-commutative approach to problems about the density of reductions such as the conjecture of Joshi-Rajan, and the generalization of the conjecture of Serre. In this paper, we prove integral Grothendieck Riemann-Roch which was proved by Papas in the case ch$(k)=0$. As a corollary we prove an integral analogue of Kontsevich's comparison theorem, and we show that if a smooth projective variety $X$ has a full exceptional collection then there is an explicit formula of the motive of $X$ up to bounded torsion.

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arXiv2023

Derived invariants and motives, Part II integral derived invariants and some applications

In this paper we construct new derived invariants with integral coefficients using the theory of motifs, and give several applications. Specifically, we obtain the following results: For complex algebraic surfaces, we prove that certain torsion in the abelianized fundamental group is a derived invariant. We prove that the collection of Hodge-Witt cohomology groups is a derived invariant. In particular, Hodge-Witt reduction and ordinary reduction are preserved by derived equivalence when the characteristic is sufficiently large. Finally, using the techniques of non-commutative algebraic geometry, we prove that Serre's ordinary density conjecture is true for cubic $4$-folds which contain a $\mathbb{P}^2$.

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arXiv2021

Gysin triangles in the category of motifs with modulus

In this paper, we study a Gysin triangle in the category of motives with modulus. We can understand this Gysin triangle as a motivic lift of the Gysin triangle of log-crystalline cohomology due to Nakkajima and Shiho. After that we compare motives with modulus and Voevodsky motives. The corollary implies that an object in the category of motives with modulus decomposes into a $p$-torsion part and a Voevodsky motive part. We can understand the corollary as a motivic analogue of the relationship between rigid cohomology and log-crystalline cohomology.

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