Unstable elements in cohomology and a question of Lescot

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 $\sigma(R)$.