Gröbner deformations, connectedness and cohomological dimension

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.