Gröbner deformations, connectedness and cohomological dimension |
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Authors: | Matteo Varbaro |
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Institution: | aDipartimento di Matematica, Universitá di Genova, Via Dodecaneso 35, I-16146 Genova, Italy |
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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 in a noetherian ring R and its relation with the connectivity dimension of . 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. |
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Keywords: | Grö bner deformations Connectedness Cohomological dimension |
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