# On a question of Buchweitz about ranks of syzygies of modules of finite length

## TL;DR

This paper investigates Buchweitz's question on the ranks of syzygies of finite length modules over local rings, establishing conditions under which the ring must be a hypersurface and exploring the two-dimensional case.

## Contribution

The paper proves that if Buchweitz's inequality holds for Gorenstein rings, then the ring is a hypersurface, and shows the converse in dimension two.

## Key findings

- If the inequality holds, R is a hypersurface.
- In dimension two, the converse also holds.
- Provides new insights into the structure of Gorenstein rings.

## Abstract

Let R be a local ring of dimension d. Buchweitz asks if the rank of the d-th syzygy of a module of finite lengh is greater than or equal to the rank of the d-th syzygy of the residue field, unless the module has finite projective dimension. Assuming that R is Gorenstein, we prove that if the question is affrmative, then R is a hypersurface. If moreover R has dimension two, then we show that the converse also holds true.