Epsilon multiplicity for graded algebras

Abstract

The notion of ε-multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that the relative epsilon multiplicity of reduced standard graded algebras over an excellent local ring exists as a limit. We also obtain some important special cases of Cutkosky's results concerning ε-multiplicity, as corollaries of our main theorem.

Epsilon multiplicity for Noetherian graded algebras

“…In our paper, we extend the aforementioned result (Theorem 5 of [8]) by allowing A and B to be reduced Noetherian graded algebras over an excellent local ring R. We now state the main theorem from our paper:…”

“…The question of whether εmultiplicity actually exists as a limit has already been considered by many mathematicians. Some related papers in this direction are [5], [6], [7], [3], [4], [12], [9] and [8]. In [8], the following result was established by the author: Theorem (Theorem 5 [8]).…”

“…Some related papers in this direction are [5], [6], [7], [3], [4], [12], [9] and [8]. In [8], the following result was established by the author: Theorem (Theorem 5 [8]). Suppose that (R, m R ) is an excellent local ring and…”