Related papers: On the derived ring of differential operators on a…
We consider commutative DG rings (better known as nonpositive strongly commutative associative unital DG algebras). For such a DG ring $A$ we define the notions of perfect, tilting, dualizing, Cohen-Macaulay and rigid DG $A$-modules.…
A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over…
For a self-orthogonal module $T$, the relation between the quotient triangulated category $D^b(A)/K^b({\rm add} T)$ and the stable category of the Frobenius category of $T$-Cohen-Macaulay modules is investigated. In particular, for a…
We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category D(g,h) of g-modules and the category of finite dimensional Uq(g)-modules, q=exp(\pi ih), for h\in C\Q*. Aiming at operator algebraists the result is…
Semi-infinite cohomology is constructed from scratch as the proper generalization of finite dimensional Lie algebra cohomology. The differential d and other operators are realized as universal inner deri- vations of a completed algebra,…
In this paper, we try to answer the following question: given a modular tensor category $\A$ with an action of a compact group $G$, is it possible to describe in a suitable sense the ``quotient'' category $\A/G$? We give a full answer in…
A general notion of a quasi-finite algebra is introduced as an algebra graded by the set of all integers equipped with topologies on the homogeneous subspaces satisfying certain properties. An analogue of the regular bimodule is introduced…
We show that, given a projective regular function f on a smooth quasi-projective variety over C, the corresponding cohomology groups of the algebraic de Rham complex with twisted differential d-df and of the complex of algebraic forms with…
We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant…
We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra $H$, we give an explicit description of…
Solid abelian groups, as introduced by Dustin Clausen and Peter Scholze, form a subcategory of all condensed abelian groups satisfying some ''completeness'' conditions and having favourable categorical properties. Given a profinite ring…
Two rings A and B are said to be derived Morita equivalent if their derived categories of modules are equivalent. By results of Rickard, if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules…
A Lie algebroid on a variety X/k is an extension \alpha: g_X \to T_X of the tangent sheaf both as O_X-module and Lie algebra over the base field, with the obvious compatibilities; and given a Lie algebroid one has its associated ring of…
Let $X$ be a smooth projective and geometrically irreducible curve over the finite field $\mathbb{F}_q$ with $q$ elements and $K$ be its function field. Let $\infty$ be a fixed closed point on $X$ and $A$ be the ring of functions regular…
We show how DG categories arise naturally in noncommutative differential geometry and use them to derive noncommutative analogues of the Bianchi identities for the curvature of a connection. We also give a derivation of formulae for…
This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of…
In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $\ell$,…
We define abelian extensions of algebras in congruence-modular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of R-modules for a ring R. We also define a cohomology theory, which we call clone…
As an algebraic study of differential equations, differential algebras have been studied for a century and and become an important area of mathematics. In recent years the area has been expended to the noncommutative associative and Lie…
We define the derived category of quasi--coherent modules for certain Artin stacks as the homotopy category of two Quillen monoidal model structures on the corresponding category of unbounded complexes of quasi--coherent modules.