Local Cohomology: An Algebraic Introduction with Geometric Applications

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This book provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, and provides many illustrations of applications of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.
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Editorial Reviews

From the Publisher
"From the point of view of the reviewer (who learned all his basic knowledge about local cohomology reading the first edition of this book and doing some of its exercises), the changes previously described (the new Chapter 12 concerning canonical modules, the treatment of multigraded local cohomology, and the final new section of Chapter 20 about locally free sheaves) definitely make this second edition an even better graduate textbook than the first. Indeed, it is well written and, overall, almost self-contained, which is very important in a book addressed to graduate students."
Alberto F. Boix, Mathematical Reviews
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Product Details

Meet the Author

M. P. Brodmann is Emeritus Professor in the Institute of Mathematics at the University of Zurich.

R. Y. Sharp is Emeritus Professor of Pure Mathematics at the University of Sheffield.

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Table of Contents

Notation and conventions
1 The local cohomology functors 1
2 Torsion modules and ideal transforms 17
3 The Mayer-Vietoris Sequence 47
4 Change of rings 66
5 Other approaches 82
6 Fundamental vanishing theorems 102
7 Artinian local cohomology modules 123
8 The Lichtenbaum-Hartshorne Theorem 136
9 The Annihilator and Finiteness Theorems 152
10 Matlis duality 179
11 Local duality 197
12 Foundations in the graded case 216
13 Graded versions of basic theorems 237
14 Links with projective varieties 265
15 Castelnuovo regularity 277
16 Bounds of diagonal type 294
17 Hilbert polynomials 312
18 Applications to reductions of ideals 325
19 Connectivity in algebraic varieties 342
20 Links with sheaf cohomology 374
Bibliography 407
Index 410
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