Algebraic stratification in representation categories
type: Item
instance of: scholarly article
title: Algebraic stratification in representation categories
zbMATH Open document ID: 0659.18011
DOI: 10.1016/0021-8693(88)90123-8
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The paper begins by reviewing the categorical framework set up by \textit{A. A. Beilinson}, \textit{J. Bernstein} and \textit{P. Deligne} [Astérisque 100 (1982; Zbl 0536.14011)] to construct glueings of derived categories: this involves several adjoint functors and compatibility conditions. For the case of an inclusion of module categories defined by a quotient \(B=A/J\), the conditions to set up this framework are reduced to ring-theoretic properties. In particular, if A has finite global dimension the conditions reduce to \(Ext^ n_ A(B,B)=0\) for all \(n>0\). If A is hereditary, it suffices if B is the largest quotient ring all whose composition factors (as A-module) belong to a preassigned set. If A is commutative, the conditions only hold if B is a direct factor. However, if \({\mathcal C}\) is the category of rational modules over a semisimple, simply-connected algebraic group G over an algebraically closed field, and \({\mathcal C}(\Gamma)\) the full subcategory of those all of whose composition factors have highest weight \(\leq\) some \(\lambda\in \Gamma\) (for preassigned finite \(\Gamma)\), then the \({\mathcal C}(\Gamma)\) are isomorphic to categories of modules and their inclusions satisfy the above mentioned conditions.
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 16Exx
zbMATH DE Number: 4077518
zbMATH Keywords: glueings of derived categories
author: Edward Cline
author: Brian J. Parshall
author: Leonard L. Scott
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90123-8
OpenAlex ID: W2031328220
cites work: Characteristic-free representation theory of the general linear group
cites work: Q3933105
cites work: Q3661504
cites work: Q3890859
cites work: Kazhdan-Lusztig conjecture and holonomic systems
cites work: Induced modules and affine quotients
cites work: Induced modules and extensions of representations
cites work: Cohomology, hyperalgebras, and representations
cites work: A Mackey imprimitivity theory for algebraic groups
cites work: Derived categories and Morita theory
cites work: Rational and generic cohomology
cites work: A filtration for rational modules
cites work: On Schur algebras and related algebras. I
cites work: Q5731806
cites work: Complete cohomological functors on groups
cites work: Polynomial representations of \(GL_n\)
cites work: Affine duality and cofiniteness
cites work: Q4143433
cites work: Q3994341
cites work: On the derived category of a finite-dimensional algebra
cites work: Tilted Algebras
cites work: Q3900232
cites work: Q3907748
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description: scientific article; zbMATH DE number 4077518