In this short note we construct two families of examples large stratifying systems in module categories algebras. The first consist on infinite size the category an algebra A. second family show that a finite system dimensional A can be arbitrarily comparison to number isomorphism classes simple A-modules. We both are built using well-established results higher homological algebra.

We study the homotopy category K(InjA) of all injective A-modules InjA and derived category D(ModA) of the category ModA of all A-modules, where A is finite dimensional algebra over an algebraically closed field. We are interested in the algebra with discrete derived category (derived discrete algebra. For a derived discrete algebra A, we get more concrete properties of K(InjA) and D(ModA). The...

We characterize the derivation d : A→ ΩDer(A) by a universal property introducing a new class of bimodules. Abridged English Version In this Note, A denotes an associative algebra over K = R or C with a unit 1l. In the Note [5], a graded differential algebra ΩDer(A) with Ω 0 Der(A) = A was introduced with the notation ΩD(A). It is one of the aims of this Note to show that, by introducing a new ...

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category O associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category O and its parabolic generalizations for classical Lie superalgebras are categories with fu...

Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of “derived” categories have been introduced in the literature. In this note, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for exam...

The recent researchs show that C2-cofiniteness is a natural conditition to consider a vertex operator algebra with finitely many simple modules. Therefore, we extended the tensor product theory of vertex operator algebras developed by Huang and Lepowsky without assuming the compatibility condition nor the semisimplicity of grading operator so that we could apply it to all vertex operator algebr...

1.1. Algebra case. Let A be a filtered quantization of a Z-graded finitely generated Poisson algebra A. By A -mod we denote the category of finitely generated A-modules. A basic tool to study such modules is to reduce them to finitely generated A-modules that can be studied by means of Commutative algebra/ Algebraic geometry. Given an A-module M , one introduces the notion of a good filtration ...

Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of “derived” categories have been introduced in the literature. In this note, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for exam...

After the ring theoretic study of diierential operators in positive characteristic by S.U. Chase and S.P. Smith, B. Haastert started investigation of D-modules on smooth varieties in positive characteristic, and the work of R. BBgvad followed. The purpose of this paper is to complement some basics for further study. We x an algebraically closed eld k of positive characteristic p. All the variet...