Details:
Title  Gröbner deformations, connectedness and cohomological dimension  Author(s)  Matteo Varbaro  Type  Article in Journal  Abstract  In 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.  Keywords  Gröbner deformations, Connectedness, Cohomological dimension  ISSN  00218693 
URL 
http://www.sciencedirect.com/science/article/pii/S0021869309000283 
Language  English  Journal  Journal of Algebra  Volume  322  Number  7  Pages  2492  2507  Year  2009  Edition  0  Translation 
No  Refereed 
No 
