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Vector bundles. --- Fibrés vectoriels. --- K-theorie

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K-theory. --- K-théorie. --- Vector bundles. --- Fibrés vectoriels.

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The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.

Syzygies (Mathematics). --- Syzygies (Mathematics) --- Vector bundles. --- Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Fiber spaces (Mathematics) --- Syzygy theory (Mathematics) --- Categories (Mathematics) --- Rings (Algebra) --- Vector bundles --- Homology theory

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Geometry, Differential --- Géométrie différentielle. --- Generalized spaces --- Espaces généralisés --- Connections (Mathematics) --- Connexions (mathématiques) --- Vector bundles. --- Fibrés vectoriels.

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The aim of this book, which was originally published in 1985, is to cover from first principles the theory of Syzygies, building up from a discussion of the basic commutative algebra to such results as the authors' proof of the Syzygy Theorem. In the last three chapters applications of the theory to commutative algebra and algebraic geometry are given.

Syzygies (Mathematics) --- Rings (Algebra) --- Vector bundles. --- Fiber spaces (Mathematics) --- Algebraic rings --- Ring theory --- Algebraic fields --- Syzygy theory (Mathematics) --- Categories (Mathematics) --- Vector bundles --- 512.55 --- 512.55 Rings and modules --- Rings and modules