UID:
almafu_9959238884602883
Umfang:
1 online resource (xxii, 491 pages) :
,
digital, PDF file(s).
Ausgabe:
Second edition.
ISBN:
1-316-08882-0
,
1-107-47180-X
,
1-139-04405-2
,
1-139-78179-0
,
1-139-77576-6
,
1-139-79318-7
,
1-139-77880-3
,
1-139-77728-9
Serie:
Cambridge studies in advanced mathematics ; 136
Inhalt:
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.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Series Page; Title; Copyright; Dedication; Contents; Preface to the First Edition; Preface to the Second Edition; Notation and conventions; 1 The local cohomology functors; 1.1 Torsion functors; 1.2 Local cohomology modules; 1.3 Connected sequences of functors; 2 Torsion modules and ideal transforms; 2.1 Torsion modules; 2.2 Ideal transforms and generalized ideal transforms; 2.3 Geometrical significance; 3 The Mayer-Vietoris sequence; 3.1 Comparison of systems of ideals; 3.2 Construction of the sequence; 3.3 Arithmetic rank; 3.4 Direct limits; 4 Change of rings
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4.1 Some acyclic modules4.2 The Independence Theorem; 4.3 The Flat Base Change Theorem; 5 Other approaches; 5.1 Use of Čech complexes; 5.2 Use of Koszul complexes; 5.3 Local cohomology in prime characteristic; 6 Fundamental vanishing theorems; 6.1 Grothendieck's Vanishing Theorem; 6.2 Connections with grade; 6.3 Exactness of ideal transforms; 6.4 An Affineness Criterion due to Serre; 6.5 Applications to local algebra in prime characteristic; 7 Artinian local cohomology modules; 7.1 Artinian modules; 7.2 Secondary representation; 7.3 The Non-vanishing Theorem again
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8 The Lichtenbaum-Hartshorne Theorem8.1 Preparatory lemmas; 8.2 The main theorem; 9 The Annihilator and Finiteness Theorems; 9.1 Finiteness dimensions; 9.2 Adjusted depths; 9.3 The first inequality; 9.4 The second inequality; 9.5 The main theorems; 9.6 Extensions; 10 Matlis duality; 10.1 Indecomposable injective modules; 10.2 Matlis duality; 11 Local duality; 11.1 Minimal injective resolutions; 11.2 Local Duality Theorems; 12 Canonical modules; 12.1 Definition and basic properties; 12.2 The endomorphism ring; 12.3 S2-ifications; 13 Foundations in the graded case
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13.1 Basic multi-graded commutative algebra13.2 *Injective modules; 13.3 The *restriction property; 13.4 The reconciliation; 13.5 Some examples and applications; 14 Graded versions of basic theorems; 14.1 Fundamental theorems; 14.2 *Indecomposable *injective modules; 14.3 A graded version of the Annihilator Theorem; 14.4 Graded local duality; 14.5 *Canonical modules; 15 Links with projective varieties; 15.1 Affine algebraic cones; 15.2 Projective varieties; 16 Castelnuovo regularity; 16.1 Finitely generated components; 16.2 The basics of Castelnuovo regularity; 16.3 Degrees of generators
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17 Hilbert polynomials17.1 The characteristic function; 17.2 The significance of reg2; 17.3 Bounds on reg2 in terms of Hilbert coefficients; 17.4 Bounds on reg1 and reg0; 18 Applications to reductions of ideals; 18.1 Reductions and integral closures; 18.2 The analytic spread; 18.3 Links with Castelnuovo regularity; 19 Connectivity in algebraic varieties; 19.1 The connectedness dimension; 19.2 Complete local rings and connectivity; 19.3 Some local dimensions; 19.4 Connectivity of affine algebraic cones; 19.5 Connectivity of projective varieties; 19.6 Connectivity of intersections
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19.7 The projective spectrum and connectedness
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English
Weitere Ausg.:
ISBN 0-521-51363-4
Weitere Ausg.:
ISBN 1-299-84235-6
Sprache:
Englisch
URL:
https://doi.org/10.1017/CBO9781139044059
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