Asymptotic growth of algebras associated to powers of ideals

Abstract

Abstract. We study generalized symbolic powers and form ideals of powers and compare their growth with the growth of ordinary powers, and we discuss the question when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.

Asymptotic multiplicities

“…It was first proven in the case when E = I is a homogeneous ideal and F = R is a standard graded normal k-algebra in our paper [6] with Hà, Srinivasan and Theodorescu. It is proven when R is regular, essentially of finite type over a field of characteristic zero, E = I is an ideal in F = R, and the singular locus of Spec(R/I) is the maximal ideal in our paper [7] with Herzog and Srinivasan.…”

“…The theorem is proven for E of low analytic deviation in [7], for the case of ideals, and by Ulrich and Validashti [31] for the case of modules; in the case of low analytic deviation, the limit is always zero. A generalization of this problem to the case of saturations with respect to non m-primary ideals is investigated by Herzog, Puthenpurakal and Verma in [13]; they show that an appropriate limit exists for monomial ideals.…”

On a transform of an acyclic complex of length 3

“…n 3 ·ℓ R ( I (n) /I n ) (cf. [1], [6]), coincides with 1/2 . Throughout this paper, R is a 3-dimensional Cohen-Macaulay local ring with the maximal ideal m .…”

“…which is the invariant called ǫ-multiplicity of I (cf. [1], [6]). Let us maintain the same notations as in Section 4.…”

Multiplicities and Rees valuations

“…In [4], [3], and [7] this limit is shown to exist in many special cases. In particular, in [4] it is shown to exist for graded ideals in a finitely generated standard graded algebra of depth greater than one over a field of characteristic zero.…”