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Cambridge Studies in Advanced Mathematics : Local Cohomology: An Algebraic Introduction with Geometric Applications Series Number 136 - M. P. Brodmann

Cambridge Studies in Advanced Mathematics

Local Cohomology: An Algebraic Introduction with Geometric Applications Series Number 136

Hardcover Published: 18th April 2019
ISBN: 9780521513630
Number Of Pages: 505

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This second edition of a successful graduate text provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, including in multi-graded situations, and provides many illustrations of the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Serre's Affineness Criterion, the Lichtenbaum-Hartshorne Vanishing Theorem, Grothendieck's Finiteness Theorem and Faltings' Annihilator Theorem, local duality and canonical modules, the Fulton-Hansen Connectedness Theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. The book is designed for graduate students who have some experience of basic commutative algebra and homological algebra and also experts in commutative algebra and algebraic geometry. Over 300 exercises are interspersed among the text; these range in difficulty from routine to challenging, and hints are provided for some of the more difficult ones.

Industry Reviews

Review of the first edition: '... Brodmann and Sharp have produced an excellent book: it is clearly, carefully and enthusiastically written; it covers all important aspects and main uses of the subject; and it gives a thorough and well-rounded appreciation of the topic's geometric and algebraic interrelationships ... I am sure that this will be a standard text and reference book for years to come.' Liam O'Carroll, Bulletin of the London Mathematical Society
Review of the first edition: 'The book is well organised, very nicely written, and reads very well ... a very good overview of local cohomology theory.' Newsletter of the European Mathematical Society
Review of the first edition: '... a careful and detailed algebraic introduction to Grothendieck's local cohomology theory.' L'Enseignement Mathematique
'... the book opens the view towards the beauty of local cohomology, not as an isolated subject but as a tool helpful in commutative algebra and algebraic geometry.' Zentralblatt MATH
'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
"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

Preface to the First Edition
Preface to the Second Edition
Notation and conventions
The local cohomology functors
Torsion modules and ideal transforms
The Mayer-Vietoris sequence
Change of rings
Other approaches
Fundamental vanishing theorems
Artinian local cohomology modules
The Lichtenbaum-Hartshorne Theorem
The Annihilator and Finiteness Theorems
Matlis duality
Local duality
Canonical modules
Foundations in the graded case
Graded versions of basic theorems
Links with projective varieties
Castelnuovo regularity
Hilbert polynomials
Applications to reductions of ideals
Connectivity in algebraic varieties
Links with sheaf cohomology
Bibliography
Index
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780521513630
ISBN-10: 0521513634
Series: Cambridge Studies in Advanced Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 505
Published: 18th April 2019
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 23.11 x 15.24  x 3.05
Weight (kg): 0.84
Edition Number: 2
Edition Type: Revised

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