FURTHER PROBLEMS

Summary

Introduction

Participants at the Symposium were invited to submit open problems in commutative algebra for inclusion in a Problem Section in these proceedings, and what follows represents the response to 243 that invitation. In fact, the response was rather limited: perhaps the fact that much of the Symposium was concerned with various well known open problems in the subject provides an explanation for this. Indeed, many of the foregoing articles contain statements of and discussions about various problems and conjectures, and thus it seems appropriate to entitle this section ‘Further problems’.

The material below is arranged into three sections: Section 1 was submitted by Chr. Lech and is concerned with problems on inequalities for couples of local rings; Section 2 was submitted by L.J. Ratliff, Jr. and is concerned with problems on asymptotic prime divisors; and Section 3 is an amalgamation of the remaining problems suggested by various participants.

1. Problems on inequalities for couples of local rings (submitted by Chr. Lech and received on 27 May, 1982)

Simple geometric considerations suggest that a (Noetherian) local ring Q should not be less complicated (or singular) than any of its localizations Q_p. Various aspects of this vague idea can be expressed in terms of certain formal power series that can be associated with a local ring, viz. the Hilbert series, the deviation series, and the μseries.