Winfried Bruns, Udo Vetter (auth.)9780387194684, 0387194681, 3540194681, 9783540194682
Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. Their study has attracted many prominent researchers and has motivated the creation of theories which may now be considered part of general commutative ring theory. The book gives a first coherent treatment of the structure of determinantal rings. The main approach is via the theory of algebras with straightening law. This approach suggest (and is simplified by) the simultaneous treatment of the Schubert subvarieties of Grassmannian. Other methods have not been neglected, however. Principal radical systems are discussed in detail, and one section is devoted to each of invariant and representation theory. While the book is primarily a research monograph, it serves also as a reference source and the reader requires only the basics of commutative algebra together with some supplementary material found in the appendix. The text may be useful for seminars following a course in commutative ring theory since a vast number of notions, results, and techniques can be illustrated significantly by applying them to determinantal rings. |
Table of contents :
Preliminaries….Pages 1-9
Ideals of maximal minors….Pages 10-26
Generically perfect ideals….Pages 27-37
Algebras with straightening law on posets of minors….Pages 38-49
The structure of an ASL….Pages 50-63
Integrity and normality. The singular locus….Pages 64-72
Generic points and invariant theory….Pages 73-92
The divisor class group and the canonical class….Pages 93-104
Powers of ideals of maximal minors….Pages 105-121
Primary decomposition….Pages 122-134
Representation theory….Pages 135-152
Principal radical systems….Pages 153-161
Generic modules….Pages 162-173
The module of Kähler differentials….Pages 174-183
Derivations and rigidity….Pages 184-201
Appendix….Pages 202-218 |
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