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TitleGröbner deformations, connectedness and cohomological dimension
Author(s) Matteo Varbaro
TypeArticle in Journal
AbstractIn this paper we will compare the connectivity dimension c ( P / I ) of an ideal I in a polynomial ring P with that of any initial ideal of I. Generalizing a theorem of Kalkbrener and Sturmfels [M. Kalkbrener, B. Sturmfels, Initial complex of prime ideals, Adv. Math. 116 (1995) 365–376], we prove that c ( P / LT ≺ ( I ) ) ⩾ min c ( P / I ) , dim ( P / I ) − 1 for each monomial order ≺. As a corollary we have that every initial complex of a Cohen–Macaulay ideal is strongly connected. Our approach is based on the study of the cohomological dimension of an ideal a in a noetherian ring R and its relation with the connectivity dimension of R / a . In particular we prove a generalized version of a theorem of Grothendieck [A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), in: Séminaire de Géométrie Algébrique du Bois Marie, 1962]. As consequence of these results we obtain some necessary conditions for an open subscheme of a projective scheme to be affine.
KeywordsGröbner deformations, Connectedness, Cohomological dimension
ISSN0021-8693
URL http://www.sciencedirect.com/science/article/pii/S0021869309000283
LanguageEnglish
JournalJournal of Algebra
Volume322
Number7
Pages2492 - 2507
Year2009
Edition0
Translation No
Refereed No
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