# D. Kaledin's research while affiliated with National Research University Higher School of Economics and other places

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## Publications (59)

Given two finitely presentable Abelian categories and , we outline a construction of an Abelian category of functors from to , which has nice 2-categorical properties and provides an explicit model for a stable category of stable functors between the derived categories of and . The construction is absolute, so it makes it possible to recover not on...

В этом обзоре мы, предполагая заданными две конечно представимые абелевы категории $A$ и $B$, даем набросок конструкции абелевой категории функторов из $A$ в $B$, которая имеет хорошие $2$-категорные свойства и дает явную модель для стабильной категории стабильных функторов между производными категориями $A$ и $B$. Конструкция абсолютная, т. е. поз...

This is mostly an overview. Given small abelian categories $A$ and $B$, we sketch the construction of an abelian category of functors from $A$ to the ind-completion of $B$ 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...

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.

The aim of the paper is to introduce an approach to the theory of 2-categories which is based on systematic use of the Grothendieck construction and the Segal Machine and to show how adjunction questions can be investigated by means of this approach and what its connections are with more traditional approaches. As an application, the derived Morita...

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...

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...

Цель статьи - дать введение в подход к теории $2$-категорий, основанный на систематическом использовании конструкции Гротендика и машины Сигала, и показать, как в этом подходе можно исследовать вопросы сопряженности и как он связан с более традиционными подходами. В качестве приложения мы строим производную $2$-категорию Мориты и $2$-категорию Фурь...

Мы возвращаемся к теореме о вырождении некоммутативного обобщения спектральной последовательности Ходжа-де Рама, доказанной первым автором, и приводим улучшенную и упрощенную версию ее доказательства, которая в явном виде использует спектральную алгебраическую геометрию. Мы также объясняем, почему для доказательства существенно необходимы методы ал...

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.

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.

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...

We give an overview of existing enhancement techniques for derived and trianguated categories based on the notion of a stable model category, and show how it can be applied to the problem of gluing triangulated categories. The article is mostly expository, but we do prove some new results concerning existence of model structures.

This is an overview of my papers arxiv:1602.04254 and arxiv:1604.01588.

We prove the non-commutative Hodge-to-de Rham Degeneration Conjecture of Kontsevich and Soibelman.

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...

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...

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 Pirashvil...

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-Mor...

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.

We give a concise overview of arxiv:0812.2519 and arxiv:1412.3248. The paper
contains all the main results and constructions but no proofs.

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

For an additive Waldhausen category linear over a ring $k$, the corresponding
$K$-theory spectrum is a module spectrum over the $K$-theory spectrum of $k$.
Thus if $k$ is a finite field of characteristic $p$, then after localization at
$p$, we obtain an Eilenberg-Maclane spectrum -- in other words, a chain
complex. We propose an elementary and dire...

Categorical resolution of singularities has been constructed in
arXiv:1212.6170. It proceeds by alternating two steps of seemingly different
nature. We show how to use the formalism of filtered derived categories to
combine the two steps into one. This results in a certain rather natural
categorical refinement of the usual blowup of an algebraic va...

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.

We give a direct interpretation of the Witt vector product in terms of tame residue in algebraic K-theory.

We review some recent results and conjectures saying that, roughly speaking, periodic cyclic homology of a smooth non-commutative algebraic variety should carry all the additional "motivic" structures possessed by the usual de Rham cohomology of a smooth algebraic variety (specifically, an R-Hodge structure for varieties over R, and a filtered Dieu...

We construct a triangulated category of cyclotomic complexes, a homological counterpart of cyclotomic spectra of Bokstedt and Madsen. We also construct a version of the Topological Cyclic Homology functor TC for cyclotomic complexes, and an equivariant homology functor from cycloctomic spectra to cyclotomic complexes which commutes with TC. Then on...

For a finite group $G$, the so-called $G$-Mackey functors form an abelian category $M(G)$ that has many applications in the study of $G$-equivariant stable homotopy. One would expect that the derived category $D(M(G))$ would be similarly important as the "homological" counterpart of the $G$-equivariant stable homotopy category. It turns out that th...

We discuss a p-adic version of Beilinson's conjecture and its relationship with noncommutative geometry.

These are lecture notes from Clay Summer School in Goettingen, in 2006; the lectures were an attempt at an elementary introduction to math.KT/0611623.

We propose a category which can serve as the category of coefficients for the
cyclic homology HC_*(A) of an associative algebra A over a field k. The
construction is categorical in nature, and essentially uses only the tensor
category A-bimod of A-bimodules; objects of our category are A-bimodules with
an additional structure. We also generalize th...

Gabber's Theorem claims that the singular support of a D-module is involutive. We show how to give a conceptually clear proof of this in the context of Hochschild Homology and Cohomology of abelian categories.

This is an overview of math.AG/0310186, math.AG/0309290, math.AG/0501247, math.AG/0401002 and math.AG/0504584 written for the Proceedings of the AMS Meeting on Algebraic Geometry, Seattle, 2005.

We introduce a version of the Cartier isomorphism for de Rham cohomology valid for associative, not necessarily commutative algebras over a field of positive characteristic. Using this, we imitate the well-known argument of P. Deligne and L. Iluusie and prove, in some cases, a conjecture of M. Kontsevich which claims that the Hodge-to-de Rham, a.k....

We prove some results on formality for families of DG algebras; in particular, we prove that formality is stable under specialization. The results are more-or-less known, but it seems that there are no published proofs.

We study the problem of deformation quantization for (algebraic) symplectic manifolds over a base field of positive characteristic. We prove a reasonably complete classification theorem for one class of such quantizations; in the course of doing it, we also introduce a notion of a restricted Poisson algebra -- the Poisson analog of the standard not...

We study the local structure of the singularity in the moduli space of sheaves on a K3 surface which has been resolved by K O'Grady in his construction of new examples of hyperkaehler manifolds. In particular, we identify the singularity with the closure of a certain nilpotent orbit in the coadjoint representation of the group $Sp(4)$. We also prov...

The Kodaira-Nakano Vanishing Theorem has been generalized to the relative setting by A. Sommese. We prove a version of this theorem for non-compact manifolds. As an apllication, we prove that the cohomology of a fiber of a symplectic contraction is trivial in odd degrees and pure Hodge-Tate in even degrees.

The projective coordinate ring of a projective Poisson scheme $X$ does not usually admit a structure of a Poisson algebra. We show that when $H^1(X,O_X)=H^2(X,O_X)=0$, this can be corrected by embedding $X$ into a canonical one-parameter deformation. The scheme $X$ then becomes the Hamiltonian reduction of the spectrum of the deformed projective co...

Let $V$ be a finite-dimensional symplectic vector space over a field of
characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that
for any crepant resolution $X \to V/G$, the bounded derived category
$D^b(Coh(X))$ of coherent sheaves on $X$ is equivalent to the bounded derived
category $D^b_G(Coh(V))$ of $G$-equivariant coheren...

This is a write-up of my talk at the Conference on algebraic structures in Montreal, July 2003. I try to give a brief informal introduction to the proof of Y. Ruan's conjecture on orbifold cohomology multiplication for symplectic quotient singularities given in V. Ginzburg and D. Kaledin, math.AG/0212279. Version 2: minor changes, added some refere...

We consider symplectic singularities in the sense of A. Beauville as examples of Poisson schemes. Using Poisson methods, we prove that a symplectic singularity admits a finite stratification with smooth symplectic strata. We also prove that in the formal neighborhood of a closed point in some stratum, the singularity is a product of the stratum and...

We prove that the integral closure of a Poisson algebra $A$ over a field of characteristic 0 is again a Poisson algebra.

We consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. We show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The assumptions are satisfied, for example, for affine and for projective varieties. We also give a classification o...

Unfortunately, some proofs in the first version of this paper were incorrect. In this revised version, some minor gaps are fixed, one serious mistake found. The main theorem is now claimed only under a restrictive technical assumption. This invalidates the application to quotient singularities by the Weyl group of type $G_2$. Everything else still...

A canonical hyperkaehler metric on the total space $T^*M$ of a cotangent bundle to a complex manifold $M$ has been constructed recently by the author (see alg-geom/9710026). This paper presents the results of alg-geom/9710026 in a streamlined and simplified form. The only new result is an explicit formula obtained for the case when $M$ is an Hermit...

We study the deformations of a holomorphic symplectic manifold X, not necessarily compact, over a formal ring. We always assume both X and the symplectic form Ω to be algebraic over
\(\mathbb{C}.\) We show (under some additional, but mild, assumptions on X) that the coarse deformation space of the pair
\(\left\langle {X,\Omega } \right\rangle \) ex...

We consider the quotients $X = V/G$ of a symplectic complex vector space $V$ by a finite subgroup $G \subset Sp(V)$ which admit a smooth crepant resolution $Y \to X$. For such quotients, we prove the homological McKay correspondence conjectured by M. Reid. Namely, we construct a natural basis in the homology space $H_\cdot(Y,\Q)$ whose elements are...

Let $V$ be a complex vector space on which a finite group $G$ acts by linear transformations. Let $W = V \oplus V^*$ be the sum of $V$ with its dual $V^*$. We prove that if the quotient $W/G$ admits a smooth crepant resolution, then the subgroup $G \subset Aut V$ is generated by complex reflections. We also obtain some results on the structure of s...

We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space $V$ by an action of a finite group $G$ of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if the action of $G$ is generated...

Let $X$ be a hyperkaehler manifold. Trianalytic subvarieties of $X$ are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus $T$, the Hilbert scheme $T^{[n]}$ classifying zero-dimensional subschemes of $T$ admits a hyperkaehler structure. A finite c...

Let $M$ be a Kaehler manifold, and consider the total space $T^*M$ of the cotangent bundle to $M$. We show that in the formal neighborhood of the zero section $M \subset T^*M$ the space $T^*M$ admits a canonical hyperkaehler structure, compatible with the complex and holomorphic symplectic structures on $T^*M$. The associated hyperkaehler metric $h...

A hypercomplex manifold is by definition a smooth manifold equipped with two anticommuting integrable almost complex structures. For example, every hyperkaehler manifold is canonically hypercomplex (the converse is not true). For every hypercomplex manifold M, the two almost complex structures define a smooth action of the algebra of quaternions on...

## Citations

... In terms of stabilization, one observes that by adjunction, the linear span functor V → k[V ] is a comonad, and then the endofunctor of the category Ho(k) given by its stabilization Q q is also a comonad. Since the comonad is non-linear, adding an enhancement to it requires some technology, but whatever technology one uses, enhanced coalgebras over this enhanced comonad are connective spectra, for more-or-less tautological reasons (for example, if one uses "stable model pairs", then this is [Ka9,Theorem 10.6]). However, observe that the whole projective system (5.41) has a structure of a comonad, with the structure maps (5.22), and then so does its stabilization (5.42). ...

... My own most recent encounter with the 2-category formalism was in the course of work on [11] that expands and develops the technology of trace theories and trace functors of [9], with the goal of reaching very concrete applications such as [10], § 3. In [11], the Segal approach is simply adopted by definition, without any explanations or justifications. ...

... My own most recent encounter with the 2-category formalism was in the course of work on [K3] that expands and develops the technology of trace theories and trace functors of [K1], with the goal of reaching very concrete applications such as [K2,Section 3]. In [K3], the Segal approach is simply adopted by definition, without any explanations or justifications. ...

Reference: Adjunction in 2-categories

... Theorem 3.5 [Kal08, Kal17,Mat20]. Let C be a smooth proper S-linear category, where S is a Q-scheme. ...

... To compensate for our failure at the main stated goal, we also prove two comparison theorem. The first one gives a really simple and purely algebraic expression of the Topological Hohchschild Homology T HH q(A, M ) of a k-algebra A with coefficients in an A-bimodule M in terms of the so-called Hochschild-Witt Homology W HH q(A, M ) introduced in [Ka8], [Ka11] (see also an overview in [Ka10]). If M = A is the diagonal bimodule, then we also analyze the circle action on T HH q(A) = T HH q(A, A), and prove that the periodic Topological Cyclic Homology T P q(A) of [H4] coincides with the periodic version W HP q(A) of W HH q(A). ...

... To compensate for our failure at the main stated goal, we also prove two comparison theorem. The first one gives a really simple and purely algebraic expression of the Topological Hohchschild Homology T HH q(A, M ) of a k-algebra A with coefficients in an A-bimodule M in terms of the so-called Hochschild-Witt Homology W HH q(A, M ) introduced in [Ka8], [Ka11] (see also an overview in [Ka10]). If M = A is the diagonal bimodule, then we also analyze the circle action on T HH q(A) = T HH q(A, A), and prove that the periodic Topological Cyclic Homology T P q(A) of [H4] coincides with the periodic version W HP q(A) of W HH q(A). ...

... We recall the following fundamental result of Kaledin [11], see also [22] for a different proof. ...

... where deg u = 2 and deg = 1, 2 = 0. In [15], §6.2, Kaledin generalized the above isomorphism: for every complex V q of flat R-modules and any p he defines a canonical increasing filtration ...

... If M = A is the diagonal bimodule, then we also analyze the circle action on T HH q(A) = T HH q(A, A), and prove that the periodic Topological Cyclic Homology T P q(A) of [H4] coincides with the periodic version W HP q(A) of W HH q(A). The second comparison theorem concerns only the diagonal bimodule case: we prove that T HH q(A) coincides with zero term of the "conjugate filtration" on the co-periodic cyclic homology CP q(A) introduced in [Ka7]. In particular, if one inverts the Bökstedt generator σ, then T HH(A) becomes CP q(A), and the identification sends σ to the Bott periodicity generator u −1 . ...

... q (k/k) by (11.27)). All terms in (11.29) carry the Connes-Tsygan differential (10.19), and the differential for R(k) is easy to compute: it has been shown in [Ka4,Lemma 3.2] that (11.30) ...