Unstable elements in cohomology and a question of Lescot

TL;DR

This paper investigates the finiteness of a numerical invariant related to cohomology over local rings, expanding known classes of rings where this invariant is finite by linking it to stable cohomology and multiplicative structures.

Contribution

It identifies new classes of rings with finite sigma invariant by connecting it to stable cohomology and multiplicative structures, extending previous results.

Findings

Finite sigma(R) for new classes of rings.

Relation between sigma(R) and stable cohomology.

Use of multiplicative structures to analyze finiteness.

Abstract

In his work on the Bass series of syzygy modules of modules over a commutative noetherian local ring $R$, Lescot introduces a numerical invariant, denoted $\sigma(R)$, and asks whether it is finite for any $R$. He proves that this is so when $R$ is Gorenstein or Golod. In the present work many new classes of rings $R$ for which $\sigma(R)$ is finite are identified. The new insight is that $\sigma(R)$ is related to the natural map from the usual cohomology of the module to its stable cohomology, which permits the use of multiplicative structures to study the question of finiteness of $\sigma(R)$.