Abstract. Let R = K[x 1 , x 2 , . . . , xn] and S = R/I be a homogeneous Kalgebra. We establish bounds for the multiplicity of certain homogeneous K-algebras S in terms of the shifts in a free resolution of S over R. Huneke and we conjectured these bounds as they generalize the formula of Huneke and Miller for the algebras with pure resolution, the simplest case. We prove these conjectured bounds for various algebras including algebras with quasipure resolutions. Our proof for this case gives a new and simple proof of the Huneke-Miller formula. We also settle these conjectures for stable and square free strongly stable monomial ideals I. As a consequence, we get a bound for the regularity of S. Further, when S is not Cohen-Macaulay, we show that the conjectured lower bound fails and prove the upper bound for almost Cohen-Macaulay algebras as well as algebras with a p-linear resolution.
Let m = (m 0 , . . . , mn) be an arithmetic sequence, i.e., a sequence of integers m 0 < · · · < mn with no common factor that minimally generate the numerical semigroup n i=0 m i N and such that m i − m i−1 = m i+1 − m i for all i ∈ {1, . . . , n − 1}. The homogeneous coordinate ring Γm of the affine monomial curve parametrically defined by X 0 = t m 0 , . . . , Xn = t mn is a graded R-module where R is the polynomial ring k[X 0 , . . . , Xn] with the grading obtained by setting deg X i := m i . In this paper, we construct an explicit minimal graded free resolution for Γm and show that its Betti numbers depend only on the value of m 0 modulo n. As a consequence, we prove a conjecture of Herzog and Srinivasan on the eventual periodicity of the Betti numbers of semigroup rings under translation for the monomial curves defined by an arithmetic sequence. * (r) 1 g * (s+1) 2].This will be useful in Section 2.
Abstract. Let k be a field of characteristic 0, R = k[x 1 , . . . , x d ] be a polynomial ring, and m its maximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. Let λ(M) denote the length of an Rmodule M. In this paper, we show thatalways exists. This limit has been shown to be e(I)/d! for m-primary ideals I in a local Cohen-Macaulay ring, where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extension modules may not have polynomial growth.
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