arXiv subjects

Alexander I. Efimov

Publications and source records attributed to Alexander I. Efimov.

At least 19 recordsLinked to original sources

arXiv2026

Theorem of the heart for Weibel's homotopy $K$-theory

In this paper we prove the theorem of the heart for Weibel's homotopy $K$-theory $KH.$ Namely, if $\mathcal{C}$ is a small stable $\infty$-category with a bounded $t$-structure, then the realization functor $D^b(\mathcal{C}^{\heartsuit})\to \mathcal{C}$ induces an equivalence of spectra $KH(\mathcal{C}^{\heartsuit})\xrightarrow{\sim}KH(\mathcal{C}).$ In a certain sense this result is dual to the Dundas-Goodwillie-McCarthy theorem. We deduce the dévissage theorem for $KH$ of abelian categories, also on the level of spectra (in all degrees). More generally, we prove these results for dualizable categories with nice $t$-structures and for the so-called coherently assembled abelian categories. The proof is heavily based on another new result, which is a much stronger version of Barwick's theorem of the heart. Its special case states the following: if $\mathcal{C}$ is a small stable category with a bounded $t$-structure, such that for some $n\geq 1$ the realization functor induces isomorphisms on $\operatorname{Ext}^{\leq n}$ between the objects of $\mathcal{C}^{\heartsuit},$ then the map $K_j(\mathcal{C}^{\heartsuit})\to K_j(\mathcal{C})$ is an isomorphism for $j\geq -n-1,$ and a monomorphism for $j = -n-2.$ Moreover, we prove that these estimates are sharp, even for dg categories over a field. In particular the naive $K$-theoretic theorem of the heart fails for $K_{-3}.$

math.KT↗

arXiv2025

Rigidity of the category of localizing motives

In this paper we study the category of localizing motives $\operatorname{Mot}^{\operatorname{loc}}$ -- the target of the universal finitary localizing invariant of idempotent-complete stable categories as defined by Blumberg-Gepner-Tabuada. We prove that this (presentable stable) category is rigid symmetric monoidal in the sense of Gaitsgory and Rozenblyum. In particular, it is dualizable. More precisely, we prove a more general version of this result for the category $\operatorname{Mot}^{\operatorname{loc}}_{\mathcal{E}}$ -- the target of the universal finitary localizing invariant of dualizable modules over a rigid symmetric monoidal category $\mathcal{E}.$ We obtain general results on morphisms and internal $\operatorname{Hom}$ in the categories $\operatorname{Mot}^{\operatorname{loc}}_{\mathcal{E}}$ of localizing motives. As an application we compute the morphisms in multiple non-trivial examples. In particular, we prove the corepresentability statements for $\operatorname{TR}$ (topological restriction) and $\operatorname{TC}$ (topological cyclic homology) when restricted to connective $\mathbb{E}_1$-rings. As a corollary, for a connective $\mathbb{E}_{\infty}$-ring $R$ we obtain a $\operatorname{TR}(R)$-module structure on the nil $K$-theory spectrum $NK(R).$ We also apply the rigidity theorem to define refined versions of negative cyclic homology and periodic cyclic homology. This was announced previously in \cite{E24b}, and certain very interesting examples were computed by Meyer and Wagner in \cite{MW24}. Here we do several computations in characteristic $0,$ in particular showing that in seemingly innocuous situations the answer can be given by an interesting algebra of overconvergent functions.

math.KT↗

arXiv2025

Some remarks on Quillen's Dévissage theorem

In this paper we give a different proof of Quillen's Dévissage theorem using Barwick's theorem of the heart. The key ingredient is a certain short exact sequence of dg categories, which is closely related with the Auslander-type construction for nilpotent extensions which was used in the papers of Kuznetsov-Lunts \cite{KL15}, Land-Tamme \cite{LT19} and the author \cite{E20}.

math.KT↗

arXiv2025

Localizing invariants of inverse limits

In this paper we study the category of nuclear modules on an affine formal scheme as defined by Clausen and Scholze \cite{CS20}. We also study related constructions in the framework of dualizable and rigid monoidal categories. We prove that the $K$-theory (in the sense of \cite{E24}) of the category of nuclear modules on $\operatorname{Spf}(R^{\wedge}_I)$ is isomorphic to the classical continuous $K$-theory, which in the noetherian case is given by the limit $\varprojlim\limits_{n} K(R/I^n).$ This isomorphism was conjectured previously by Clausen and Scholze. More precisely, we study two versions of the category of nuclear modules: the original one defined in \cite{CS20} and a different version, which contains the original one as a full subcategory. For our category $\operatorname{Nuc}(R^{\wedge}_I)$ we give three equivalent definitions. The first definition is by taking the internal $\operatorname{Hom}$ in the category $\operatorname{Cat}_R^{\operatorname{dual}}$ of $R$-linear dualizable categories. The second definition is by taking the rigidification of the usual $I$-complete derived category of $R.$ The third definition is by taking an inverse limit in $\operatorname{Cat}_R^{\operatorname{dual}}.$ For each of the three approaches we prove that the corresponding construction is well-behaved in a certain sense. Moreover, we prove that the two versions of the category of nuclear modules have the same $K$-theory, and in fact the same finitary localizing invariants.

math.KT↗

arXiv2025

K-theory and localizing invariants of large categories

In this paper we introduce and study the so-called continuous $K$-theory for a certain class of "large" stable $\infty$-categories, more precisely, for dualizable presentable categories. For compactly generated categories, the continuous $K$-theory is simply the usual (non-connective) $K$-theory of the full subcategory of compact objects. More generally, we show that any localizing invariant of small stable $\infty$-categories can be uniquely extended to a localizing invariant of dualizable categories. We compute the continuous $K$-theory for categories of sheaves on locally compact Hausdorff spaces. Using the special case for sheaves on the real line, we give an alternative proof of the theorem of Kasprowski and Winges \cite{KW19} on the commutation of $K$-theory with infinite products for small stable $\infty$-categories. We also study the general theory of dualizable categories. In particular, we give an "explicit" proof of Ramzi's theorem \cite{Ram24a} on the $ω_1$-presentability of the category of dualizable categories. Among other things, we prove that dualizability is equivalent to "flatness" in the category of presentable stable categories.

math.KT↗

arXiv2024

Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry.

math.SG↗

arXiv2024

On the Hahn-Witt series and their generalizations

In this paper we study the field of Hahn-Witt series $HW(\overline{\mathbb{F}}_p)$ with residue field $\overline{\mathbb{F}}_p$ (also known as a $p$-adic Malcev-Neumann field \cite{La86, P93}), and its generalizations. Informally, the Hahn-Witt series are possibly infinite linear combinations of rational powers of $p,$ in which the coefficients are Teichmüller representatives, and the set of exponents is well-ordered. They form an algebraically closed extension of $\mathbb{Q}_p,$ with a canonical automorphism $φ,$ coming from the absolute Frobenius of $\overline{\mathbb{F}}_p.$ We prove that the action of $φ$ on the $p$-power roots of unity is given by $φ(ζ)=ζ^{-1},$ answering a question of Kontsevich. More generally, we consider the $π$-typical Hahn-Witt series $HW_{(K,π)}(\overline{\mathbb{F}}_q)$, where $π$ is a uniformizer in a local field $K$ with residue field $\mathbb{F}_q.$ Again, this field is an algebraically closed extension of $K,$ and it has a canonical automorphism $φ_π,$ coming from the relative Frobenius of $\overline{\mathbb{F}}_q$ over $\mathbb{F}_q.$ We prove that the action of $φ_π$ on the maximal abelian extension $K^{ab}$ corresponds via local class field theory to the uniformizer $-π\in K^*.$

math.NT↗

arXiv2018

Homotopy finiteness of some DG categories from algebraic geometry

In this paper, we prove that the bounded derived category $D^b_{coh}(Y)$ of coherent sheaves on a separated scheme $Y$ of finite type over a field $\mathrm{k}$ of characteristic zero is homotopically finitely presented. This confirms a conjecture of Kontsevich. We actually prove a stronger statement: $D^b_{coh}(Y)$ is equivalent to a DG quotient $D^b_{coh}(\tilde{Y})/T,$ where $\tilde{Y}$ is some smooth and proper variety, and the subcategory $T$ is generated by a single object. The proof uses categorical resolution of singularities of Kuznetsov and Lunts \cite{KL}, and a theorem of Orlov \cite{Or} stating that the class of geometric smooth and proper DG categories is stable under gluing. We also prove the analogous result for $\mathbb{Z}/2$-graded DG categories of coherent matrix factorizations on such schemes. In this case instead of $D^b_{coh}(\tilde{Y})$ we have a semi-orthogonal gluing of a finite number of DG categories of matrix factorizations on smooth varieties, proper over $\mathbb{A}_{\mathrm{k}}^1$.

math.AG↗

arXiv2018

Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration

We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved by Kaledin \cite{Ka}). In particular, we show that there exists a minimal $10$-dimensional $A_{\infty}$-algebra over a field of characteristic zero, for which the supertrace of $μ_3$ on the second argument is non-zero. As a byproduct, we obtain an example of a homotopically finitely presented DG category (over a field of characteristic zero) that does not have a smooth categorical compactification, giving a negative answer to a question of Toën. This can be interpreted as a lack of resolution of singularities in the noncommutative setup. We also obtain an example of a proper DG category which does not admit a categorical resolution of singularities in the terminology of Kuznetsov and Lunts \cite{KL} (that is, it cannot be embedded into a smooth and proper DG category).

math.AG↗

arXiv2017

Some remarks on L-equivalence of algebraic varieties

In this short note we study the questions of (non-)L-equivalence of algebraic varieties, in particular, for abelian varieties and K3 surfaces. We disprove the original version of a conjecture of Huybrechts \cite[Conjecture 0.3]{H} stating that isogenous K3 surfaces are L-equivalent. Moreover, we give examples of derived equivalent twisted K3 surfaces, such that the underlying K3 surfaces are not L-equivalent. We also give examples showing that D-equivalent abelian varieties can be non-L-equivalent (the same examples were obtained independently in \cite{IMOU}). This disproves the original version of a conjecture of Kuznetsov and Schinder \cite[Conjecture 1.6]{KS}. We deduce the statements on (non-)L-equivalence from the very general results on the Grothendieck group of an additive category, whose morphisms are finitely generated abelian groups. In particular, we show that in such a category each stable isomorphism class of objects contains only finitely many isomorphism classes. We also show that a stable isomorphism between two objects $X$ and $Y$ with $\mathrm{End}(X)=\mathbb{Z}$ implies that $X$ and $Y$ are isomorphic.

math.AG↗

arXiv2017

Categorical formal punctured neighborhood of infinity, I

In this paper we introduce and study the formal punctured neighborhood of infinity, both in the algebro-geometric and in the DG categorical frameworks. For a smooth algebraic variety $X$ over a field of characteristic zero, one can take its smooth compactification $\bar{X}\supset X,$ and then take the DG category of perfect complexes on the formal punctured neighborhood of the infinity locus $\bar{X}-X.$ The result turns out to be independent of $\bar{X}$ (up to a quasi-equivalence) and we denote this DG category by $\operatorname{Perf}(\hat{X}_{\infty}).$ We show that this construction can be done purely DG categorically (hence of course also $A_{\infty}$-categorically). For any smooth DG category $\mathcal{B},$ we construct the DG category $\operatorname{Perf}_{top}(\hat{\mathcal{B}}_{\infty}),$ which we call the category of perfect complexes on the formal punctured neighborhood of infinity of $\mathcal{B}.$ The construction is closely related to the algebraic version of a Calkin algebra: endomorphisms of an infinite-dimensional vector space modulo endomorphisms of finite rank. We prove that the DG categorical construction is compatible with the algebro-geometric one. We study numerous examples. In particular, for the algebra of rational functions on a smooth complete connected curve $C$ we obtain the algebra of adeles $\mathbb{A}_C,$ and for $\mathcal{B}=D^b_{coh}(Y)$ for a proper singular scheme $Y$ we obtain the category $D_{sg}(Y)^{op}$ -- the opposite category of the Orlov's category of singularities. Among other things, we discuss the relation with the papers of Tate \cite{Ta} and Arbarello, de Concini, and Kac \cite{ACK}.

math.AG↗

arXiv2015

Cyclic homology of categories of matrix factorizations

In this paper, we will show that for a smooth quasi-projective variety over $\C,$ and a regular function $W:X\to \C,$ the periodic cyclic homology of the DG category of matrix factorizations $MF(X,W)$ is identified (unde Riemann-Hilbert correspondence) with vanishing cohomology $H^{\bullet}(X^{an},ϕ_W\C_X),$ with monodromy twisted by sign. Also, Hochschild homology is identified respectively with the hypercohomology of $(Ω_X^{\bullet},dW\wedge).$ One can show that the image of the Chern character is contained in the subspace of Hodge classes. One can formulate the Hodge conjecture stating that it is surjective ($\otimes\Q$) onto Hodge classes. For W=0 and $X$ smooth projective this is precisely the classical Hodge conjecture.

math.AG↗

arXiv2015

Derived categories of Grassmannians over integers and modular representation theory

In this paper we study the derived categories of coherent sheaves on Grassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. We prove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of $GL_k.$ This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection \cite{Kap}, which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of $GL_{n-k}.$ The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on $\operatorname{Gr}(k,n).$ We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over $\mathbb{Z},$ and we identify the objects of $D^b(\operatorname{Gr}(k,n)),$ which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in \cite{BLVdB}, by different methods.

math.AG↗

arXiv2015

Coherent analogues of matrix factorizations and relative singularity categories

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different "large" versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. A version of the Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the Serre-Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for CDG-modules over quasi-coherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.

math.CT↗

arXiv2015

Generalized non-commutative degeneration conjecture

In this paper we propose a generalization of the Kontsevich--Soibelman conjecture on the degeneration of Hochschild-to-cyclic spectral sequence for smooth and compact DG category. Our conjecture states identical vanishing of a certain map between bi-additive invariants of arbitrary small DG categories over a field of characteristic zero. We show that this generalized conjecture follows from the Kontsevich--Soibelman conjecture and the so--called conjecture on smooth categorical compactification.

math.AG↗

arXiv2015

Mac Lane (co)homology of the second kind and Wieferich primes

In this paper we investigate the connection between the Mac Lane (co)homology and Wieferich primes in finite localizations of global number rings. Following the ideas of Polishchuk-Positselski \cite{PP}, we define the Mac Lane (co)homology of the second kind of an associative ring with a central element. We compute this invariants for finite localizations of global number rings with an element $w$ and obtain that the result is closely related with the Wieferich primes to the base $w.$ In particular, for a given non-zero integer $w,$ the infiniteness of Wieferich primes to the base $w$ turns out to be equivalent to the following: for any positive integer $n,$ we have $HML^{II,0}(\mathbb{Z}[\frac1{n!}],w)\ne\mathbb{Q}.$ As an application of our technique, we identify the ring structure on the Mac Lane cohomology of a global number ring and compute the Adams operations (introduced in this case by McCarthy \cite{McC}) on its Mac Lane homology.

math.AG↗

arXiv2014

Homological mirror symmetry for punctured spheres

We prove that the wrapped Fukaya category of a punctured sphere ($S^2$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.

math.AG↗

arXiv2011

Quantum cluster variables via vanishing cycles

In this paper, we provide a Hodge-theoretic interpretation of Laurent phenomenon for general skew-symmetric quantum cluster algebras, using Donaldson-Thomas theory for a quiver with potential. It turns out that the positivity conjecture reduces to the certain statement on purity of monodromic mixed Hodge structures on the cohomology with the coefficients in the sheaf of vanishing cycles on the moduli of stable framed representations. As an application, we show that the positivity conjecture (and actually a stronger result on Lefschetz property) holds if either initial or mutated quantum seed is acyclic. For acyclic initial seed the positivity has been already shown by F. Qin \cite{Q} in the quantum case, and also by Nakajima \cite{Nak} in the commutative case.

math.AG↗