arXiv subjects

Hisato Matsukawa

Publications and source records attributed to Hisato Matsukawa.

4 recordsLinked to original sources

arXiv2026

Reconstruction of Formal Schemes from Categories of Nuclear Modules

We provide a partially functorial and constructive reconstruction procedure for formal schemes from symmetric monoidal categories of nuclear modules. More precisely, for a formal scheme $\mathfrak{X}$, we show that the torsion subcategory $D_{\mathrm{tors}}(\mathfrak{X})$ can be recovered as the maximal strongly compactly generated localizing tensor ideal of Efimov's category $\mathrm{Nuc}^{\mathrm{Ef}}(\mathfrak{X})$, and similarly for the Clausen--Scholze category $\mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})$. Combining this with the Balmer spectrum, we reconstruct $\mathfrak{X}$ from the corresponding symmetric monoidal category of nuclear modules. Moreover, for formal schemes topologically of finite type over a field or over $\mathbb{Z}$, the contravariant functor $\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{Ef}}(\mathfrak{X})$ is fully faithful; in the affine case, the analogous statement holds for $\mathfrak{X} \mapsto \mathrm{Nuc}^{\mathrm{CS}}(\mathfrak{X})$.

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arXiv2025

The Spectrum of Stable Infinity Categories with Actions

We introduce the relative Matsui spectrum, a new invariant associated with a stable $\infty$-category equipped with an action. This construction generalizes both Balmer's tensor triangular spectra and Matsui's triangular spectra, and provides a unified framework for classifying thick submodules. We establish its fundamental properties, including universality, comparison with existing spectra, and descent, and construct a natural morphism to the Balmer spectrum of the base. Applications show that the relative Matsui spectrum recovers the underlying classical geometric spaces from categorical data in various settings: categories of perfect complexes of schemes, twisted derived categories, categories of singularities, and derived matrix factorization categories. Thus the relative Matsui spectrum extends the reach of tensor triangular geometry beyond globally tensorial settings, while preserving geometric intuition.

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arXiv2025

Proper Fourier-Mukai partners of abelian varieties and points outside the Fourier-Mukai loci in Matsui spectra

We prove that any proper Fourier-Mukai partner of an abelian variety is again an abelian variety, by analyzing the Matsui spectrum of the derived category. This result was previously obtained by Huybrechts and Nieper-Wisskirchen in the case of smooth projective varieties. Our proof, however, extends the result to proper schemes using entirely different techniques. More generally, we show that any scheme of finite type that is derived equivalent to an open subscheme of an abelian variety is itself an open subscheme of an abelian variety. We also study the structure of the Matsui spectrum outside the Fourier-Mukai locus. For certain proper schemes, we show that the set of points lying outside the Fourier-Mukai locus in the Matsui spectrum has cardinality at least equal to that of the base field. This suggests the existence of additional geometric structures, such as moduli spaces, beyond the derived-equivalent part. As an application, we provide new counterexamples to conjectures of Ito, which predicted that the Serre invariant locus coincides with the Fourier-Mukai locus. While counterexamples involving K3 surfaces of Picard number one were previously given by Hirano-Ouchi, our examples arising from simple abelian varieties of dimension greater than two are the first of their kind.

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arXiv2024

Cofinality Theorems of Infinity Categories and Algebraic K-Theory

In this paper, we establish a theorem that proves a condition when an inclusion morphism between simplicial sets becomes a weak homotopy equivalence. Additionally, we present two applications of this result. The first application demonstrates that cofinal full inclusion functors of (\infty)-categories are weak homotopy equivalences. For our second application, we provide an alternative proof of Barwick's cofinality theorem of algebraic (K)-theory.

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