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 [18], 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 [10]. As consequence of these results we obtain some necessary conditions for open subscheme of a projective scheme to be affine.