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Isamu Iwanari

Publications and source records attributed to Isamu Iwanari.

At least 19 recordsLinked to original sources
arXiv2025

Categorified Koszul duality of algebras

Koszul duality is a duality between algebras that provides deep connections between seemingly different algebraic objects and has important applications in various areas of mathematics. The classical form of Koszul duality concerns associative algebras. In this paper, we develop a categorified generalization of Koszul duality that treats duality phenomena among monoidal stable $\infty$-categories. In our framework, monoidal stable infinity-categories play the role of associative algebras in the classical theory. We establish Koszul duality results for module infinity-categories associated with Artin algebras and related algebras over the little 2-discs operad. The resulting duality exhibits several structural features, including connections to right and left complete t-structures and to categories of Ind-coherent modules.

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arXiv2022

Moduli theory associated to Hochschild pairs

We consider an $A$-linear stable infinity-category $\mathcal{C}$ and the pair $(\mathcal{HH}^\bullet(\mathcal{C}/A),\mathcal{HH}_\bullet(\mathcal{C}/A))$ of the Hochschild cohomology spectrum (Hochschild cochain complex) and the Hochschild homology spectrum (Hochschild chain complex). The purpose of this paper is to provide a moduli-theoretic interpretation of the algebraic structure on the Hochschild pair of $\mathcal{C}$. The algebraic structure on the Hochschild pair is encoded by means of a two-colored topological operad called Kontsevich-Soibelman operad. The notions of cyclic deformations and equivariant deformations (of the Hochschild chain complex) associated to deformations of $\mathcal{C}$ play a central role.

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arXiv2022

On D-modules of categories I

In this paper, we will provide constructions of D-module structures on the complex computing the periodic cyclic homology of a stable infinity-category defined over a scheme of characteristic zero. We give two methods. The first one is based on a canonical extension of factorization homology to the mapping stack and relation between sheaves on free loop space and D-modules. The second one uses the algebraic structure on Hochschild pairs, its moduli-theoretic interpretation, Kodaira-Spencer morphisms, and the relation between dg Lie algebras and pointed formal stacks.

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arXiv2022

On D-modules of categories II

In our paper "On D-module of categories I", we provide two different methods of constructing D-module structures on the complex computing periodic cyclic homology associated to a family of stable infinity categories. One is based on a canonical extension of factorization homology. Another method uses the pair of Hochschild cohomology and Hochschild homology, Kodaira-Spencer map for a family of stable infinity categories, Koszul dualities,and the relation between dg Lie algebras and pointed formal stacks. In this paper, we prove that two resulting structures coindice.

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arXiv2020

Differential calculus of Hochschild pairs for infinity-categories

In this paper, we provide a conceptual new construction of the algebraic structure on the pair of the Hochschild cohomology spectrum (cochain complex) and Hochschild homology spectrum, which is analogous to the structure of calculus on a manifold. This algebraic structure is encoded by a two-colored operad introduced by Kontsevich and Soibelman. We prove that for a stable idempotent-complete infinity-category, the pair of its Hochschild cohomology and homology spectra naturally admits the structure of algebra over the operad. Moreover, we prove a generalization to the equivariant context.

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arXiv2018

Tannaka duality and stable infinity-categories

We introduce a notion of fine Tannakian infinity-categories and prove Tannakian characterization results for symmetric monoidal stable infinity-categories over a field of characteristic zero. It connects derived quotient stacks with symmetric monoidal stable infinity-categories which satisfy a certain simple axiom. We also discuss several applications to examples.

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arXiv2016

Mixed motives and quotient stacks: Abelian varieties

We prove that the symmetric monoidal category of mixed motives generated by an abelian variety (more generally, an abelian scheme) can be described as a certain module category. More precisely, we describe it as the category of quasi-coherent complexes over a derived quotient stack constructed from a motivic algebra of the abelian variety. We then study the structure of the motivic Galois groups of their mixed motives. We prove that the motivic Galois group is decomposed into a unipotent part constructed from the motivic algebra, and the reductive quotient which is the Tannaka dual of Grothendieck numerical motives.

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arXiv2016

Period mappings for noncommutative algebras

We construct a period mapping for deformations of a differential graded algebra, that generalizes Griffiths' period mapping. It is constructed as a morphism between differential graded Lie algebras which has a moduli-theoretic interpretation, where the domain of the morphism is the shifted Hochschild cocohain complex. We then use this period mapping to give some applications such as unobstructedness of deformations.

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arXiv2013

Bar construction and tannakization

In this note we continue our development of tannakizations of symmetric monoidal infinity-categories, begun in our previous paper. The issue treated in this paper is the calculation of tannakizations of examples of symmetric monoidal stable infinity-categories with fiber functors. We consider the case of symmetric monoidal infinity-categories of perfect complexes on perfect derived stacks. The first main result especially says that our tannakization includes the bar construction for an augmented commutative ring spectrum and its equivariant version as a special case. We apply it to the study of the tannakization of the stable infinity-category of mixed Tate motives over a perfect field. We prove that its tannakization can be obtained from the torus-equivariant bar construction of a commutative differential graded algebra equipped with torus-action. Moreover, under Beilinson-Soule vanishing conjecture, we prove that the underlying group scheme of the tannakization is the conventional motivic Galois group for mixed Tate motives. The case of Artin motives is also included.

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arXiv2012

Monoidal Infinity Category of Complexes from Tannakian Viewpoint

In this paper we prove that a morphism between schemes or stacks naturally corresponds to a symmetric monoidal functor between stable infinity-categories of quasi-coherent complexes. It can be viewed as a derived analogue of Tannaka duality. As a consequence, we deduce that an algebraic stack satisfying a certain condition can be recovered from the stable infinity-category of quasi-coherent complexes with tensor operation.

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arXiv2012

Tannakization in derived algebraic geometry

We give a universal construction of a derived affine group scheme and its representation category from a symmetric monoidal infinity-category, which we shall call the tannnakization of a symmetric monoidal infinity-category. This can be viewed as infinity-categorical generalization of the work of Joyal-Street and Nori. We then apply it to the stable infinity-category of mixed motives equipped with the realization functor of a mixed Weil cohomology and obtain a derived motivic Galois group whose representation category has a universality, and which represents the automorphism group of the realization functor. Also, we present basic properties of derived affine group schemes in Appendix.

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arXiv2010

Stable points on algebraic stacks

This paper is largely concerned with constructing coarse moduli spaces for Artin stacks. The main purpose of this paper is to introduce the notion of stability on an arbitrary Artin stack and construct a coarse moduli space for the open substack of stable points. Also, we present an application to coherent cohomology of Artin stacks.

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arXiv2010

Note on local structure of Artin stacks

In this note we show that an Artin stack with finite inertia stack is etale locally isomorphic to the quotient of an affine scheme by an action of a general linear group.

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arXiv2009

Logarithmic geometry, minimal free resolutions and toric algebraic stacks

In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings.

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arXiv2009

The category of toric stacks

In this paper, we prove that there exists an equivalence between 2-category of smooth Deligne-Mumford stacks with torus-embeddings and actions, and the 1-category of stacky fans. For this purpose, we obtain two main results. The first is to investigate a combinatorial aspect of the 2-category of toric algebraic stacks defined in \cite{I2}. We establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second is to give a geometric characterization theorem for toric algebraic stacks.

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