Sergey Gorchinskiy

Parameterized Picard–Vessiot extensions and Atiyah extensions

Abstract Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is to the Galois theory of linear differential equations with parameters. Namely, we show the existence of a parameterized Picard–Vessiot extension and, therefore, the Galois correspondence for many differential fields with, possibly, non-differentially closed fields of constants, that is, fields of functions of parameters. Other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic, which will appear in a separate paper. This application is based on differential categories developed in the present paper, and not just differential algebraic groups and their representations.

Geometric phantom categories

In this paper we give a construction of phantom categories, i.e. admissible triangulated subcategories in bounded derived categories of coherent sheaves on smooth projective varieties that have trivial Hochschild homology and trivial Grothendieck group. We also prove that these phantom categories are phantoms in a stronger sense, namely, they has trivial K-motives and, hence, all their higher K-groups are trivial too.

Motives and representability of algebraic cycles on threefolds over a field

We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibered by surfaces with algebraic H^2. This gives another new examples of three-dimensional varieties whose motives are finite-dimensional.

Isomonodromic differential equations and differential categories

Abstract We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between the Gauss–Manin connection and parameterized differential Galois groups.

A higher-dimensional Contou-Carrère symbol: local theory

We construct a higher-dimensional Contou-Carr\`ere symbol and we study its various fundamental properties. The higher-dimensional Contou-Carr\`ere symbol is defined by means of the boundary map for

Positive Model Structures for Abstract Symmetric Spectra

K

Higher-dimensional Contou-Carrère symbol and continuous automorphisms

-groups. We prove its universal property. We provide an explicit formula for the higher-dimensional Contou-Carr\`ere symbol over

Tangent space to Milnor K-groups of rings

\mathbb Q

Transcendence degree of zero-cycles and the structure of Chow motives

and we prove integrality of this formula. A relation with the higher-dimensional Witt pairing is also studied.

Continuous homomorphisms between algebras of iterated Laurent series over a ring

We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorn’s sense, of a certain positive projective model structure on spectra, where positivity basically means the truncation of the zero level. The localization is by the set of stabilizing morphisms or their truncated version.