Eventual vanishing of unstable cohomology for syzygies of a fixed module

Determine whether, for a given finitely generated module M over a commutative noetherian local ring R, there exists an integer n0 such that U(Ω^n M) = 0 for all n ≥ n0, i.e., whether the kernel of Ext_R(k, Ω^n M) → ẐExt_R(k, Ω^n M) eventually vanishes with n.

Background

The subspace U(M) is defined as the kernel of Ext_R(k,M) → ẐExt_R(k,M), measuring instability of cohomology classes. Lescot’s invariant σ(R) asks for a uniform bound n working for all finitely generated modules; the authors also raise the weaker module-specific question of eventual vanishing of U(Ω^n M).

While the authors prove finiteness of σ(R) for many classes of rings, they explicitly note that they do not currently have an answer to the weaker, module-specific question in general.