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 $σ(R)$, 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 $σ(R)$ is finite are identified. The new insight is that $σ(R)$ 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 $σ(R)$.