a b s t r a c tLet A(C) be the coordinate ring of a monomial curve C ⊆ A n corresponding to the numerical semigroup S minimally generated by a sequence a 0 , . . . , a n . In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr m (A) with respect to the maximal ideal m of A = A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr m (A) in the case a 0 , . . . , a n is a generalized arithmetic sequence.
We give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of any homogeneous ideal in a polynomial ring over a field of characteristic 0.
Abstract. Let I be a homogeneous ideal in a polynomial ring P over a field. By Macaulay's Theorem there exists a lexicographic ideal L = Lex(I) with the same Hilbert function as I. Peeva has proved that the Betti numbers of P/I can be obtained from the graded Betti numbers of P/L by a suitable sequence of consecutive cancellations. We extend this result to any ideal I in a regular local ring (R, n) by passing through the associated graded ring. To this purpose it will be necessary to enlarge the list of the allowed cancellations. Taking advantage of Eliahou-Kervaire's construction, we present several applications. This connection between the graded perspective and the local one is a new viewpoint, and we hope it will be useful for studying the numerical invariants of classes of local rings.
Numerical invariants of a minimal free resolution of a module M over a regular local ring (R, n) can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable n-stable filtrations M of M and to compare the Betti numbers of M with those of the associated graded module gr M (M). This approach has the advantage that the same module M can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Among others, the Koszul modules have this behavior. As a consequence of the main result, we extend some results by Aramova, Conca, Herzog and Hibi on the rigidity of the resolution of standard graded algebras to the local setting.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.