This book provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, and illustrates many applications for the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo-Mumford regularity, the Fulton-Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. 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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Book Description
Editorial Reviews
Review
'... a careful and detailed algebraic introduction to Grothendieck's local cohomology theory.' L'Enseignment Mathématique
'The book is well organised, very nicely written, and reads very well ... a very good overview of local cohomology theory.' European Mathematical Society
'I am sure that this will be a standard text and reference book for years to come.' Liam O'Carroll, Bull. London Mathematical Society
'The book is well organised, very nicely written, and reads very well ... a very good overview of local cohomology theory.' European Mathematical Society
'I am sure that this will be a standard text and reference book for years to come.' Liam O'Carroll, Bull. London Mathematical Society
Book Description
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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1 of 2 people found the following review helpful:
4.0 out of 5 stars
Good book, but MUST know algebraic geometry,
This review is from: Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge Studies in Advanced Mathematics) (Hardcover)
Even though the book seems to suggest that it's aimed at a reader without considerable knowledge of algebraic geometry, reality paints a different picture. If one follows just the algebra, one misses the richness and beauty of the geometry that this algebra was called forward to describe. This is the only flaw, but I feel it's a serious one, so it's better to be forewarned. However, if the knowledge of algebraic geometry IS there, the book both enriches and informs.
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Inside This Book
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First Sentence:
The main objective of this chapter is to introduce the a-torsion functor (throughout the book, a always denotes an ideal in a (non-trivial) commutative Noetherian ring R) and its right derived functors H (i > 0), referred to as the local cohomology functors with respect to a. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
affine algebraic cones, local cohomology modules, local cohomology theory, minimal secondary representation, show that reg, proper graded ideal, graded local cohomology, exceeding reg, formal fibres, minimal injective resolution, local cohomology functors, infinite residue field, arithmetic rank, regularity reg, bounding system, complete local domain, inverse family, graded prime ideal, positively graded ring, commutative graded ring, commutes with direct limits, canonical module, connectedness dimension, distinct irreducible components, vanishing theorem Key Phrases - Capitalized Phrases (CAPs): (learn more)
Independence Theorem, Mayer-Vietoris Sequence, Artinian R-module, Grothendieck's Vanishing Theorem, Local Duality Theorem, Annihilator Theorem, Flat Base Change Theorem, Lichtenbaum-Hartshorne Vanishing Theorem, Serre-Grothendieck Correspondence, Shifted Localization Principle, Deligne Isomorphism Theorem, Proof Let, Serre's Affineness Criterion, Hartshorne's Example, Noetherian R-module, Prime Avoidance Theorem, Matlis Duality Theorem, Grothendieck's Finiteness Theorem, Cohen-Macaulay R-module, Five Lemma, Deligne Correspondence, Graded Finiteness Theorem, Bertini-Grothendieck Connectivity Theorem, Grothendieck's Connectedness Theorem, Lichtenbaum-Hartshorne Theorem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The main objective of this chapter is to introduce the a-torsion functor (throughout the book, a always denotes an ideal in a (non-trivial) commutative Noetherian ring R) and its right derived functors H (i > 0), referred to as the local cohomology functors with respect to a. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
affine algebraic cones, local cohomology modules, local cohomology theory, minimal secondary representation, show that reg, proper graded ideal, graded local cohomology, exceeding reg, formal fibres, minimal injective resolution, local cohomology functors, infinite residue field, arithmetic rank, regularity reg, bounding system, complete local domain, inverse family, graded prime ideal, positively graded ring, commutative graded ring, commutes with direct limits, canonical module, connectedness dimension, distinct irreducible components, vanishing theorem Key Phrases - Capitalized Phrases (CAPs): (learn more)
Independence Theorem, Mayer-Vietoris Sequence, Artinian R-module, Grothendieck's Vanishing Theorem, Local Duality Theorem, Annihilator Theorem, Flat Base Change Theorem, Lichtenbaum-Hartshorne Vanishing Theorem, Serre-Grothendieck Correspondence, Shifted Localization Principle, Deligne Isomorphism Theorem, Proof Let, Serre's Affineness Criterion, Hartshorne's Example, Noetherian R-module, Prime Avoidance Theorem, Matlis Duality Theorem, Grothendieck's Finiteness Theorem, Cohen-Macaulay R-module, Five Lemma, Deligne Correspondence, Graded Finiteness Theorem, Bertini-Grothendieck Connectivity Theorem, Grothendieck's Connectedness Theorem, Lichtenbaum-Hartshorne Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
4
books
cite this book:
- Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects (Lecture Notes in Pure and Applied Mathematics) by Alberto Corso on page 174, and page 208
- Abelian Groups, Rings, Modules, and Homological Algebra (Lecture Notes in Pure and Applied Mathematics) by Pat Goeters on page 57
- Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics) by R.K. Lazarsfeld in Back Matter
- Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and Multiplier Ideals (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge ... of Modern Surveys in Mathematics) (v. 2) by R.K. Lazarsfeld in Back Matter
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