Sequences that Preserve the Homological Degree

Abstract

The homological degree is a cohomological degree generalizing the classical degree function of a module. Given a finitely generated graded module M, over a standard graded ring A, this note is concerned with the question of when, for an element x in the augmentation ideal of A that is not in any associated prime of M other than the augmentation ideal itself, the homological degree of M is equal to the homological degree of M/xM. This question is answered when the dimension of M is one or two.

Key Words:

• Arithmetic degree
• Cohomological degrees
• Homological degree

2000 Mathematics Subject Classification:

• Primary 13D99
• Secondary 13A02

ACKNOWLEDGMENTS

The first named author would like to acknowledge the hospitality of the Nanoscience Technology Center of the University of Central Florida for their hospitality during the writing of this article. Both authors wish to thank Jürgen Herzog for conversations regarding this work.

Notes

Communicated by R. Villarreal.