Algebraic Geometry for Associative Algebras

by Van Oystaeyen; Freddy

• ISBN13: 9780824704247
• ISBN10: 082470424X
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-06-06
• Publisher: CRC Press

Summary

This work focuses on the association of methods from topology, category and sheaf theory, algebraic geometry, noncommutative and homological algebras, quantum groups and spaces, rings of differential operation, Cech and sheaf cohomology theories, and dimension theories to create a blend of noncommutative algebraic geometry. It offers a scheme theory that sustains the duality between algebraic geometry and commutative algebra to the noncommutative level.

Author Biography

Freddy Van Oystaeyen is a Professor of Mathematics at the University of Antwerp, UIA, Belgium.

Table of Contents

Preface	p. iii
Introduction	p. 1
The Noncommutative Site
Affine Schematic Algebras	p. 5
Proj and Schematic Algebras	p. 16
Schemes on the Noncommutative Site	p. 37
Structure Sheaves and their Sections
Serre's Global Section Theorem on the Noncommutative Site	p. 53
The Quantum Site	p. 63
Quantum Sections. Examples	p. 70
Almost Commutative Geometry	p. 77
Sheaves of Localizations	p. 78
Sheaves of Microlocalizations and Quantum Sections	p. 85
Quantized Formal Schemes	p. 91
Regular Algebras
Some Facts about Dimensions	p. 103
Regularity Conditions	p. 117
Short Survey of Results on Low Dimensional Regular Algebras	p. 127
Valuations and Divisors
Valuations of Skewfields	p. 139
Filtrations for Totally Ordered Groups	p. 145
Valuation Filtrations and Extensions	p. 151
Reductions of Algebras	p. 158
Introduction of a Divisor Theory on Weyl Fields	p. 173
Cohomology Theories
Cohomology of Schematic Algebras	p. 189
Cech Cohomology	p. 200
A Functorial Approach
Group Objects	p. 207
Quantum Groups	p. 213
Multiplicative Matrices	p. 222
Braided Categories	p. 225
Some Invariants of Braided Categories and Quantum Groups	p. 235
Formalizing the Topology
The Lattice-Type Approach	p. 243
Skew Grothendieck-Topologies	p. 255
Elements of Sheaf Theory over Skew Topologies	p. 262
References	p. 271
Index	p. 281
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