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D. Kaledin

Publications and source records attributed to D. Kaledin.

At least 19 recordsLinked to original sources
arXiv2025

On 2-categories of extensions

This is essentially an illustration for the general technology of homotopical enhancements developed recently in arxiv:2409.17489. We take the derived category of an abelian category, and we look at the full subcategory spanned by complexes of length 2. This has a natural refinement to a 2-category that we call "the 2-category of extensions". However, just using the triangulated structure on the derived category is not enough to obtain this refinement. In this short note, we first construct the 2-category of extensions by hand -- that is, using abelian category techniques -- and then show how it can be recovered very easily and naturally in the enhanced formalism of arxiv:2409.17489.

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arXiv2024

Taming large categories

No new results. This is a short overview of the standard machinery of filtered colimits and accessible categories, written in parallel to a homotopically enhanced version available as Section 7.6 in arXiv:2409.17489.

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arXiv2024

Enhancement for categories and homotopical algebra

We develop foundations for abstract homotopy theory based on Grothendieck's idea of a "derivator". The theory is model-independent, and does not depend on model categories, nor on simplicial sets. It is designed to accomodate all the usual potential applications, such as e.g. enhancements for derived categories of coherent sheaves, in a way that is as close as possible to usual category theory.

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arXiv2022

Mackey profunctors

We develop the theory of Mackey profunctors, a version of Mackey functors for profinite groups.

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arXiv2022

What do abelian categories form?

This is mostly an overview. Given finitely presentable abelian categories $A$ and $B$, we sketch the construction of an abelian category of continuous functors from $A$ to $B$ that has nice $2$-categorical behaviour and gives an explicit model for the stable category of stable functors between the derived categories of $A$ and $B$. The construction is absolute, thus allows one to recover not only Hochschild but also Mac Lane Cohomology.

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arXiv2021

Bokstedt periodicity generator via K-theory

For a prime field $k$ of characteristic $p > 2$, we construct the B\"okstedt periodicity generator $v \in THH_2(k)$ as an explicit class in the stabilization of $K$-theory with coefficients $K(k,-)$, and we show directly that $v$ is not nilpotent in $THH(k)$. This gives an alternative proof of the "multiplicative" part of B\"okstedt periodicity.

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arXiv2021

Trace theories, Bokstedt periodicity and Bott periodicity

We flesh out the theory of "trace theories" and "trace functors" sketched in arXiv:1308.3743, extend it to a homotopical setting, and prove a reconstruction theorem claiming that a trace theory is completely determined by the associated trace functor. As an application, we consider Topological Hoshschild Homology $THH(A,M)$ of a algebra $A$ over a perfect field of positive characteristic, with coefficients in a bimodule $M$, and prove two comparison results. Firstly, we give a very simple algebraic model for THH in terms of Hochschild-Witt Homology WHH of arXiv:1604.01588 (and we also identify $TP(A)$ with the periodic version $WHP(A)$ of WHH). Secondly, we prove that $THH(A)$ is identified with the zero term of the conjugate filtration on the co-periodic cyclic homology $\overline{HP}(A)$ of arXiv:1509.08784, and the isomorphism sends the Bokstedt periodicity generator to the Bott periodicity generator. We also give an independent proof of Bokstedt periodicity that is somewhat shorter than the usual ones.

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arXiv2020

Adjunction in 2-categories

We give an overview of the parts of arXiv:2004.04279 that deal with 2-categories, up to and including adjunction, and explain how the Segal-type approach to 2-categories adopted there is related to the more standard approaches. As an application, we construct the derived Morita and the Fourier-Mukai 2-categories over a Noetherian ring, and show how to embed one into the other.

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arXiv2019

Spectral algebras and non-commutative Hodge-to-de Rham degeneration

We revisit the non-commutative Hodge-to-de Rham Degeneration Theorem of the first author, and present its proof in a somewhat streamlined and improved form that explicitly uses spectral algebraic geometry. We also try to explain why topology is essential to the proof.

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arXiv2017

Witt vectors as a polynomial functor

For every commutative ring $A$, one has a functorial commutative ring $W(A)$ of $p$-typical Witt vectors of $A$, an iterated extension of $A$ by itself. If $A$ is not commutative, it has been known since the pioneering work of L. Hesselholt that $W(A)$ is only an abelian group, not a ring, and it is an iterated extension of the Hochschild homology group $HH_0(A)$ by itself. It is natural to expect that this construction generalizes to higher degrees and arbitrary coefficients, so that one can define "Hochschild-Witt homology" $WHH_*(A,M)$ for any bimodule $M$ over an associative algebra $A$ over a field $k$. Moreover, if one want the resulting theory to be a trace theory in the sense of arXiv:1308.3743, then it suffices to define it for $A=k$. This is what we do in this paper, for a perfect field $k$ of positive characteristic $p$. Namely, we construct a sequence of polynomial functors $W_m$, $m \geq 1$ from $k$-vector spaces to abelian groups, related by restriction maps, we prove their basic properties such as the existence of Frobenius and Verschiebung maps, and we show that $W_m$ are trace functors in the sense of arXiv:1308.3743. The construction is very simple, and it only depends on elementary properties of finite cyclic groups.

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arXiv2016

Hochschild-Witt complex

In arxiv:1602.04254, we have defined polynomial Witt vectors functor from vector spaces over a perfect field $k$ of positive characteristic $p$ to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild-Witt complex $WCH_*(A)$ for any associative unital $k$-algebra $A$, with homology groups $WHH_*(A)$. We prove that the group $WHH_0(A)$ coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if $A$ is commutative, finitely generated, and smooth, the groups $WHH_i(A)$ are naturally identified with the terms $WΩ^i_A$ of the de Rham-Witt complex of the spectrum of $A$.

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arXiv2016

Spectral sequences for cyclic homology

We prove that for a homologically smooth and proper DG algebra over a field of characteristic 0, the Hodge-to-de Rham spectral sequence degenerates. This has been conjectured by M. Kontsevich and Y. Soibelman arXiv:math/0606241 and proved in arXiv:math/0611623 under a technical assumption. In this paper, the assumption is removed, and the argument is considerably simplified (in particular, it no longer uses Dold-Kan equivalence and simplicial methods). We also analyse the degeneration of the conjugate spectral sequence in positive caracteristic constructed in arXiv:1509.08784.

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arXiv2015

Bokstein homomorphism as a universal object

We give a simple construction of the correspondence between square-zero extensions $R'$ of a ring $R$ by an $R$-bimodule $M$ and second MacLane cohomology classes of $R$ with coefficients in $M$ (the simplest non-trivial case of the construction is $R=M=Z/p$, $R'=Z/p^2$, thus the Bokstein homomorphism of the title). Following Jibladze and Pirashvili, we treat MacLane cohomology as cohomology of non-additive endofunctors of the category of projective $R$-modules. We explain how to describe liftings of $R$-modules and complexes of $R$-modules to $R'$ in terms of data purely over $R$. We show that if $R$ is commutative, then commutative square-zero extensions $R'$ correspond to multiplicative extensions of endofunctors. We then explore in detail one particular multiplicative non-additive endofunctor constructed from cyclic powers of a module $V$ over a commutative ring $R$ annihilated by a prime $p$. In this case, $R'$ is the second Witt vectors ring $W_2(R)$ considered as a square-zero extension of $R$ by the Frobenius twist $R^{(1)}$.

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arXiv2015

Co-periodic cyclic homology

Following an old suggestion of M. Kontsevich, and inspired by recent work of A. Beilinson and B. Bhatt, we introduce a new version of periodic cyclic homology for DG agebras and DG categories. We call it co-periodic cyclic homology. It is always torsion, so that it vanishes in char 0. However, we show that co-periodic cyclic homology is derived-Morita invariant, and that it coincides with the usual periodic cyclic homology for smooth cohomologically bounded DG algebras over a torsion ring. For DG categories over a field of odd positive characteristic, we also establish a non-commutative generalization of the conjugate spectral sequence converging to our co-periodic cyclic homology groups.

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arXiv2015

Cartier isomorphism for unital associative algebras

Given an associative unital algebra $A$ over a perfect field $k$ of odd positive characteristic, we construct a non-commutative generalization of the Cartier isomorphism for $A$. The role of differential forms is played by Hochschild homology classes, and de Rham diferential is replaced with the Connes-Tsygan differential.

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arXiv2015

Trace theories and localization

We show how one can twist the definition of Hochschild homology of an algebra or a DG algebra by inserting a possibly non-additive trace functor. We then prove that many of the usual properties of Hochschild homology survive such a generalization. In some cases this even includes Keller's Localization Theorem.

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