Integral and graded quasi-hereditary algebras. I
type: Item
instance of: scholarly article
title: Integral and graded quasi-hereditary algebras. I
zbMATH Open document ID: 0699.16015
DOI: 10.1016/0021-8693(90)90169-O
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: The paper extends the notion of a quasi-hereditary algebra from finite dimensional algebras to finitely generated algebras over a commutative Noetherian ring. Moreover, the authors begin the study of graded quasi- hereditary algebras. Let k be a commutative Noetherian ring and let A be a k-algebra, which is a finitely generated projective k-module. An ideal J in A is called a heredity ideal if \(J^ 2=J\), J is a projective right R-module and the k-algebra \(End_ A(J_ A)\) is semisimple. The algebra A is called quasi-hereditary if there exists a sequence \(0=J_ 0\subseteq J_ 1\subseteq...\subseteq J_ t=A\) of ideals in A such that \(J_ i/J_{i-1}\) is a heredity ideal in \(A/J_{i-1}\). Several results for quasi-hereditary Noetherian algebras are established in analogy with the finite dimensional case. In particular it is shown that gl.dim \(A\leq 2t-2+gl.\dim k\). Moreover, if \(J_ i=Ae_ iA\) where \(e_ i\) is an idempotent in A such that \(gl.\dim (e_ iAe_ i)\) is finite then there is a full recollement diagram \[ D^ b(mod A/J_ i)\quad D^ b(mod A)\quad D^ b(mod e_ iAe_ i). \] It is shown in the paper that the q-Schur algebras \(S_ q(n,r,k)\) are quasi-hereditary if \(q^{1/2}\in k\). A positively Z-graded k-algebra A is called a graded quasi-hereditary algebra if A is a quasi-hereditary k-algebra in the sense above. In this case each of the idempotent ideals \(J_ i\) in the defining sequence is in fact homogeneous. Elementary properties of graded quasi-hereditary algebras are established and several examples are presented. In connection with recent work of Beilingson and Ginsberg the concept of a formal quasi hereditary algebra is briefly discussed.
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 4146190
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: finitely generated algebras
zbMATH Keywords: graded quasi- hereditary algebras
zbMATH Keywords: heredity ideal
zbMATH Keywords: projective right R-module
zbMATH Keywords: quasi- hereditary Noetherian algebras
zbMATH Keywords: full recollement diagram
zbMATH Keywords: q-Schur algebras
zbMATH Keywords: idempotent ideals
author: Edward Cline
author: Brian J. Parshall
author: Leonard L. Scott
reviewed by: Daniel Simson
MaRDI profile type: Publication
cites work: Filtrations of cohomology modules for Chevalley groups
cites work: Q4056148
cites work: Maximal Orders
cites work: The Brauer Group of a Commutative Ring
cites work: Representation Theory of Artin Algebras I
cites work: On Maximally Central Algebras
cites work: Q5562621
cites work: Q3661504
cites work: Q3933105
cites work: Kazhdan-Lusztig conjecture and holonomic systems
cites work: Q3808315
cites work: Algebraic stratification in representation categories
cites work: Q3804882
cites work: Q3922829
cites work: The q ‐Schur Algebra
cites work: q-Tensor Space and q-Weyl Modules
cites work: Quasi-hereditary algebras
cites work: Every Semiprimary Ring is the Endomorphism Ring of a Projective Module Over a Quasi-Hereditary Ring
cites work: The Modular Representation Theory of q-Schur Algebras
cites work: The Centers of Semi-Simple Algebras Over a Commutative Ring, II
cites work: Q5682108
cites work: Q3955560
cites work: Polynomial representations of \(GL_n\)
cites work: Q3994341
cites work: Semisimple algebras over a commutative ring
cites work: Note on Relative Homological Dimension
cites work: A \(q\)-analogue of \(U(\mathfrak{gl}(N+1))\), Hecke algebra, and the Yang-Baxter equation
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Q3035669
cites work: Q3900232
cites work: Q3816243
cites work: Singularities of closures of K-orbits on flag manifolds
cites work: Q5611879
cites work: Graded and filtered rings and modules
cites work: Q4731369
cites work: Q3972287
cites work: Q3495525
cites work: Koszul Resolutions
cites work: Q4061092
cites work: Modular Permutation Representations
cites work: Q3808313
description: scientific article; zbMATH DE number 4146190