Auslander-Buchweitz Approximations of Equivariant Modules

by Mitsuyasu Hashimoto

• ISBN13: 9780521796965
• ISBN10: 0521796962
• Format: Paperback
• Copyright: 2000-11-20
• Publisher: Cambridge University Press

Summary

This book focuses on homological aspects of equivariant modules. It presents a new homological approximation theory in the category of equivariant modules, unifying the Cohen-Macaulay approximations in commutative ring theory and Ringel's theory of delta-good approximations for quasi-hereditary algebras and reductive groups. The book provides a detailed introduction to homological algebra, commutative ring theory and homological theory of comodules of co-algebras over an arbitrary base. It aims to overcome the difficulty of generalising known homological results in representation theory. This book will be of interest to researchers and graduate students in algebra, specialising in commutative ring theory and representation theory.

Table of Contents

Introduction	xi
Conventions and terminology	xv
Background Materials	1	(130)
From homological algebra	1	(36)
Yoneda's lemma	1	(2)
Adjoint functors and limits	3	(2)
Exact categories	5	(4)
Derived categories and derived functors	9	(3)
Extensions and Ext groups	12	(3)
The cobar resolution	15	(3)
Grothendieck categories	18	(2)
Grothendieck topology and sheaf theory	20	(4)
Noetherian categories and locally noetherian categories	24	(1)
Semisimple objects in a Grothendieck category	25	(3)
Full subcategories of an abelian category	28	(1)
X-approximations and the Auslander-Buchweitz theory	29	(8)
From commutative ring theory	37	(34)
Flat modules and pure maps	37	(4)
Mittag-Leffler modules	41	(3)
Faithfully flat morphisms and descent theory	44	(2)
The I-depth	46	(2)
Cohen-Macaulay, Gorenstein, and regular rings	48	(2)
Local cohomology	50	(1)
Ring-theoretic properties of morphisms	51	(5)
Betti numbers, Bass numbers and complete intersections	56	(1)
Resolutions of perfect modules	57	(2)
Dualizing complexes and canonical modules	59	(2)
The duality of proper morphisms and rational singularities	61	(4)
Summary of open loci results	65	(2)
Normal flatness	67	(4)
Hopf algebras over an arbitrary base	71	(30)
Coalgebras and bialgebras	72	(1)
Hopf algebras	73	(2)
Comodules	75	(1)
Sweedler's notation	76	(1)
Bicomodules, Hom and &ctimes;	76	(5)
The restriction and the induction	81	(5)
Locally noetherian property	86	(1)
The dual algebra of a coalgebra	87	(3)
The dual coalgebra of an algebra	90	(1)
Rational modules	91	(2)
FPCP coalgebras and IFP coalgebras	93	(2)
&ctimes; and Hom of modules and comodules over a Hopf algebra	95	(3)
The dual Hopf algebra	98	(1)
Module algebras and comodule algebras	99	(1)
Coalgebras and comodules over a scheme	100	(1)
From representation theory	101	(21)
Group schemes as faisceaux	101	(1)
Rational respresentations of an algebraic group	102	(3)
Algebraic tori	105	(1)
Maximal tori, Borel subgroups, and reductive groups	106	(1)
Split reductive groups	107	(3)
General linear groups	110	(1)
Representations of reductive groups over an algebraically closed field	111	(2)
Universal module functors	113	(2)
Tilting modules	115	(1)
Cotilting modules	116	(6)
Basics on equivariant modules	122	(9)
Cocommutative Hopf algebra actions	122	(2)
TorA and ExtA as A#U-modules	124	(3)
(G,A)-modules	127	(4)
Equivariant Modules	131	(26)
Homological aspects of (G, A)-modules	131	(8)
Construction of ExtA	131	(5)
Equivariant modules of a split torus	136	(1)
FPCP groups and IFP groups	137	(2)
Matijevic--Roberts type theorem	139	(18)
Stability of various loci	139	(5)
Universal density of hyperalgebras	144	(2)
A generalization to equivariant sheaves	146	(7)
Matijevic--Roberts type theorem	153	(4)
Highest Weight Theory	157	(72)
Highest weight theory over a field	157	(21)
Weak split highest weight coalgebras	157	(5)
Weak highest weight theory	162	(7)
Highest weight coalgebras and good comodules	169	(5)
Weak highest weight coalgebras and good filtrations	174	(4)
Donkin systems	178	(20)
U-acyclicity of flat complexes	178	(6)
The definition and the existence of a Donkin system	184	(7)
Basic properties of the Donkin system	191	(7)
Ringel's theory over a field	198	(7)
Ringel's approximation over a field	198	(4)
Tilting modules over a field	202	(3)
Ringel's theory over a commutative ring	205	(24)
Tilting modules over a commutative ring	205	(8)
Minimal Ringel's approximations over local rings	213	(2)
Cohen---Macaulay analogue of u-good module	215	(4)
Cohen---Macaulay Ringel's approximation	219	(3)
Applications to split reductive groups	222	(2)
Good modules of a general linear group	224	(5)
Approximations of Equivariant Modules	229	(32)
Approximations of (G, A)-modules	229	(21)
Graded G-algebras	229	(6)
Reductive group actions on graded algebras	235	(4)
Relative Ringel's approximation	239	(6)
Relative Cohen---Macaulay Ringel's approximation	245	(5)
An application to determinantal rings	250	(11)
Resolutions of determinantal rings	250	(4)
Buchsbaum---Rim type resolutions	254	(2)
Kempf's construction	256	(5)
Glossary	261	(6)
Bibliography	267	(10)
Index	277