Generalized complete intersections with linear resolutions

Abstract

We determine the simplicial complexes ∆ whose Stanley-Reisner ideals I∆ have the following property: for all n ≥ 1 the powers I n ∆ have linear resolutions and finite length local cohomologies. Mathematics Subject Classification (2000). Primary 13F55; Secondary 13D02, 13D45.Introduction. Let S = K[X 1 , . . . , X n ] be a standard graded polynomial ring over a field K and m = (X 1 , . . . , X n ). Recall that a graded ideal I ⊂ S is called generalized Cohen-Macaulay or simply FLC (finite local cohomology) when the local cohomology H i m (S/I) has finite length for all i < dim S/I. We find many examples of such ideals in algebraic geometry: the defining ideals of Cohen-Macaulay equidimensional projective schemes over a field are all FLC ideals. However, as far as the authors are concerned, we do not know very much about FLC monomial ideals. The second author gave a combinatorial characterization of FLC monomial ideals [8] as an extension of the well known case of squarefree monomial ideals, i.e., Buchsbaum Stanley-Reisner ideals. On the other hand, the notion of generalized complete intersections (GCI) has been introduced in [4]. A GCI I ⊂ S is a squarefree monomial ideal such that the Stanley-Reisner ring S/I is a complete intersection over the punctured spectrum. All the powers I n (n ≥ 1) of a GCI I ⊂ S are FLC, and a combinatorial characterization of GCI has been given.