Hilbert-Samuel polynomials for the contravariant extension functor

Abstract

Abstract. Let (R, m) be a local ring and M and N finite R-modules. In this paper we give a formula for the degree of the polynomial giving the lengths of the modules Ext i R (M, N/m n N ). A number of corollaries are given and more general filtrations are also considered.

Asymptotic behavior of dimensions of syzygies

“…Remark 10. It should be noted that [5,Remark 5.6] claims that using [5, Proposition 5.5] one can show that if R is equidimensional and Conjecture 4 is true, then dim(Ω n (M )) is constant for n ≫ 0. However, [5,Proposition 5.5] requires the assumption that dim(R) 2.…”

“…This question was also explored in the last section of [5]. In [5,Remark 5.2 (i)] it is noted that if R is unmixed and equidimensional, then (dim(Ω i (M ))) ∞ i=0 is constant for i ≫ 0.…”

“…This question was also explored in the last section of [5]. In [5,Remark 5.2 (i)] it is noted that if R is unmixed and equidimensional, then (dim(Ω i (M ))) ∞ i=0 is constant for i ≫ 0. This is clear since the associated primes of any submodule of R βi(M) are also associated primes of R, and are therefore primes of maximal dimension by assumption.…”

Constructing big indecomposable modules

“…It is known that the lengths of the R-modules Ext 1 R (M, R/m n R) are given by a polynomial ξ M,m (n) for all n large enough (see [12], [16]). In [4], the authors give an exact formula for the degree of this polynomial; in particular when M has constant rank on the punctured spectrum, the degree is dim R − 1.…”

Higher Iterated Hilbert Coefficients of the Graded Components of Bigraded Modules

“…The present paper is motivated by Kodiyalam's work [6], the papers by Theodorescu [11], by Katz and Theodorescu [8], [9] and the paper [3]. In these papers it was shown that for finitely generated R-modules M and N over a Noetherian (local) ring R, and for an ideal I ⊂ R such that the length of Tor In these papers bounds are given for the degree of these polynomials.…”