Local Cohomology: An Algebraic Introduction with Geometric ApplicationsCambridge University Press, 15.11.2012 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. |
Inhalt
1 | |
Torsion modules and ideal transforms | 16 |
The MayerVietoris sequence | 47 |
Change of rings | 65 |
Other approaches | 81 |
Fundamental vanishing theorems | 106 |
The Annihilator and Finiteness Theorems | 164 |
Matlis duality | 193 |
Foundations in the graded case | 251 |
Graded versions of basic theorems | 285 |
Links with projective varieties | 331 |
Castelnuovo regularity | 346 |
Hilbert polynomials | 364 |
Applications to reductions of ideals | 388 |
Connectivity in algebraic varieties | 405 |
Links with sheaf cohomology | 438 |
Häufige Begriffe und Wortgruppen
a-torsion algebraically closed field Assume that R I canonical module chapter closed subset Cohen—Macaulay coherent sheaf cohomology modules commutative Noetherian ring Corollary covariant functors deduce defined Definition denote dim R/p dimension direct limits dirn Exercise exists find finitely generated graded finitely generated R-module first follows functors from C(R G Spec(R G-graded Gorenstein local ring graded ideal graded R-module Grothendieck’s Hence homogeneous elements homogeneous isomorphism homomorphic image HomR hypotheses injective R-module injective resolution integer Lemma Let p G maximal ideal morphism natural equivalence natural transformation negative strongly connected Noetherian ring Noetherian topological space non-empty non-Zerodivisor notation phism polynomial ring positively graded projective varieties Proposition prove QT-modules R-isomorphism reg2 regular local ring restriction property result ring homomorphism Rn is positively sequences of covariant sheaf Show strongly connected sequence submodule subring Suppose surjective topology unique Vanishing Theorem