Differential Graded Categories Research Articles

Remarks on the equivalence between differential graded categories and A-infinity categories

Remarks on the equivalence between differential graded categories and A-infinity categories

Derived parabolic induction

The classical parabolic induction functor is a fundamental tool on the representation theoretic side of the Langlands program. In this article, we study its derived version. It was shown by the second author that the derived category of smooth G $G$ -representations over k $k$ , G $G$ a p $p$ -adic reductive group and k $k$ a field of characteristic p $p$ , is equivalent to the derived category of a certain differential graded k $k$ -algebra H G • $H_G^\bullet$ , whose zeroth cohomology is a classical Hecke algebra. This equivalence predicts the existence of a derived parabolic induction functor on the dg Hecke algebra side, which we construct in this paper. This relies on the theory of six-functor formalisms for differential graded categories developed by O. Schnürer. We also discuss the adjoint functors of derived parabolic induction.

Hochschild (Co)homology and Derived Categories

These are slightly expanded notes of lectures given in April 2019 at the Isfahan school and conference on representations of algebras. We recall the formalism of derived categories and functors and survey invariance results for the Hochschild (co)homology of differential graded algebras and categories.

Derived categories of NDG categories

In this paper we study N-differential graded categories and their derived categories. First, we introduce modules over an N-differential graded category. Then we show that they form a Frobenius category and that its homotopy category is triangulated. Second, we study the properties of its derived category and give triangle equivalences of Morita type between derived categories of N-differential graded categories. Finally, we show that this derived category is triangle equivalent to the derived category of some ordinary differential graded category.

Mixed vs Stable Anti-Yetter-Drinfeld Contramodules

We examine the cyclic homology of the monoidal category of modules over a finite dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter-Drinfeld contramodules and the usual stable anti-Yetter-Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter-Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter-Drinfeld contramodules (recently introduced).

On the differential graded Eilenberg–Moore construction

The Eilenberg–Moore construction for modules over a differential graded monad is used to study a question of Balmer regarding existence of an exact adjoint pair representing an exact monad. A Bousfield-like localization for differential graded categories is realized as a special case of this construction using Drinfeld quotients. As applications, we study some example coming from G-equivariant triangulated categories and twisted derived categories.

Relative Calabi–Yau structures

We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

On the perturbation algebra

We introduce a certain differential graded bialgebra, neither commutative nor cocommutative, that governs perturbations of a differential on complexes supplied with an abstract Hodge decomposition. This leads to a conceptual treatment of the Homological Perturbation Lemma and its multiplicative version. As an application we give an explicit form of the decomposition theorem for A∞ algebras and A∞ modules and, more generally, for twisted objects in differential graded categories.

Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of special type. Different properties of both noncommutative schemes and morphisms between them are reviewed and discussed. In addition, the concept of a geometric realization for a derived noncommutative scheme is introduced and problems of existence and construction of such realizations are discussed. Also studied are the construction of gluings of noncommutative schemes via morphisms, along with certain new phenomena such as phantoms, quasi-phantoms, and Krull–Schmidt partners which arise in the world of noncommutative schemes and which enable us to find new noncommutative schemes. The last sections consider noncommutative schemes connected with basic finite- dimensional algebras. It is proved that such noncommutative schemes have special geometric realizations under which the algebra goes to a vector bundle on a smooth projective scheme. Such realizations are constructed in two steps, the first of which is the well-known construction of Auslander, while the second step is connected with the new concept of a well-formed quasi-hereditary algebra, for which there are very special geometric realizations sending standard modules to line bundles. Bibliography: 50 titles.

Matrix factorization of Morse–Bott functions

For a function $W\in \mathbb{C}[X]$ on a smooth algebraic variety $X$ with Morse-Bott critical locus $Y\subset X$, Kapustin, Rozansky and Saulina suggest that the associated matrix factorisation category $\mathrm{MF}(X;W)$ should be equivalent to the differential graded category of $2$-periodic coherent complexes on $Y$ (with a topological twist from the normal bundle of $Y$). I confirm their conjecture in the special case when the first neighbourhood of $Y$ in $X$ is split, and establish the corrected general statement. The answer involves the full Gerstenhaber structure on Hochschild cochains. This note was inspired by the failure of the conjecture, observed by Pomerleano and Preygel, when $X$ is a general one-parameter deformation of a $K3$ surface $Y$.

Spectral sequences for Hochschild cohomology and graded centers of derived categories

The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the failure of the characteristic homomorphism to be injective or surjective, in general. To illustrate this, we discuss modules over the dual numbers, coherent sheaves over algebraic curves, as well as examples related to free loop spaces and string topology.

Calabi–Yau structures on categories of matrix factorizations

We write out explicit proper Calabi-Yau structures, i. e. non-degenerate cyclic cocycles on the differential graded categories of matrix factorizations of regular functions with isolated critical points. The formulas involve the Kapustin-Li trace and its "higher corrections".

Smooth and proper noncommutative schemes and gluing of DG categories

In this paper we discuss different properties of noncommutative schemes over a field. We define a noncommutative scheme as a differential graded category of a special type. We study regularity, smoothness and properness for noncommutative schemes. Admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes. In this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded categories via bimodules. As a consequence of the main theorem we obtain that for any finite-dimensional algebra with separable semisimple part the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme. Moreover, if the algebra has finite global dimension, then the full subcategory is admissible. We also provide a construction of a smooth projective scheme that admits a full exceptional collection and contains as a subcollection an exceptional collection given in advance. As another application of the main theorem we obtain, in characteristic 0, an existence of a full embedding for the category of perfect complexes on any proper scheme to the category of perfect complexes on a smooth projective scheme.

Exponentially Twisted Cyclic Homology

By a theorem of Bernhard Keller the de Rham cohomology of a smooth variety is isomorphic to the periodic cyclic homology of the differential graded category of perfect complexes on the variety. Both the de Rham cohomology and the cyclic homology can be twisted by the exponential of a regular function on the variety. We explain that the isomorphism holds true in the twisted setting and draw some conclusions on derived invariance of the algebraic Gauss-Manin systems associated with regular functions.

Differential graded categories and Deligne conjecture

We prove a version of the Deligne conjecture for n-fold monoidal abelian categories A over a field k of characteristic 0, assuming some compatibility and non-degeneracy conditions for A. The output of our construction is a weak Leinster (n,1)-algebra over k, a relaxed version of the concept of Leinster n-algebra in Alg(k). The difference between the Leinster original definition and our relaxed one is apparent when n>1, for n=1 both concepts coincide.We believe that there exists a functor from weak Leinster (n,1)-algebras over k to C•(En+1,k)-algebras, well-defined when k=Q, and preserving weak equivalences. For the case n=1 such a functor is constructed in [31] by elementary simplicial methods, providing (together with this paper) a complete solution for 1-monoidal abelian categories.Our approach to Deligne conjecture is divided into two parts. The first part, completed in the present paper, provides a construction of a weak Leinster (n,1)-algebra over k, out of an n-fold monoidal k-linear abelian category (provided the compatibility and non-degeneracy condition are fulfilled). The second part (still open for n>1) is a passage from weak Leinster (n,1)-algebras to C•(En+1,k)-algebras.As an application, we prove in Theorem 8.1 that the Gerstenhaber–Schack complex of a Hopf algebra over a field k of characteristic 0 admits a structure of a weak Leinster (2,1)-algebra over k extending the Yoneda structure. It relies on our earlier construction [30] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra.

Generalized non-commutative degeneration conjecture

We propose a generalization of the Kontsevich–Soibelman conjecture on the degeneration of the Hochschild-to-cyclic spectral sequence for a smooth compact differential graded category. Our conjecture states identical vanishing of a certain map between bi-additive invariants of arbitrary small differential graded 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.

A Diagrammatic Categorification of a Clifford Algebra

We give a graphical calculus for a categorification of a Clifford algebra and its Fock space representation via differential graded categories. The categorical action is motivated by the gluing action between the contact categories of infinite strips.

Morita cohomology

AbstractWe consider two categorifications of the cohomology of a topological spaceXby taking coefficients in the category of differential graded categories. We consider both derived global sections of a constant presheaf and singular cohomology and find the resulting dg-categories are quasi-equivalent and moreover quasi-equivalent to representations in perfect complexes of chains on the loop space ofX.

Non-commutative Hodge structures: Towards matching categorical and geometric examples

The subject of the present work is the de Rham part of non-commutative Hodge structures on the periodic cyclic homology of differential graded algebras and categories. We discuss explicit formulas for the corresponding connection on the periodic cyclic homology viewed as a bundle over the punctured formal disk. Our main result says that for the category of matrix factorizations of a polynomial the formulas reproduce, up to a certain shift, a well-known connection on the associated twisted de Rham cohomology which plays a central role in the geometric approach to the Hodge theory of isolated singularities.

Milnor descent for cohesive dg-categories

AbstractWe show that the functor from curved differential graded algebras to differential graded categories, defined by the second author, sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg-categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.