Lescot’s conjecture on finiteness of σ(R) via stability of cohomology

Determine whether, for every commutative noetherian local ring R, there exists an integer n ≥ 0 such that for all finitely generated R-modules M the natural map Ext_R(k, Ω^n M) → ẐExt_R(k, Ω^n M) is injective; equivalently, establish that the Lescot invariant σ(R) is finite for all local rings R.

Background

The paper studies the relationship between usual and stable cohomology of modules over a commutative noetherian local ring R, focusing on the map Ext_R(k,M) → ẐExt_R(k,M) and the subspace U(M) of unstable elements (its kernel).

Lescot introduced the invariant σ(R) as the least n such that W(Ω^n M)=0 (equivalently U(Ω^n M)=0) for all finitely generated R-modules M. Recasting this in terms of stable cohomology yields the conjectural assertion that there exists n (independent of M) making Ext_R(k, Ω^n M) → ẐExt_R(k, Ω^n M) injective for all M, i.e., σ(R) < ∞.

The authors verify the conjecture for several classes of rings (e.g., Gorenstein, Golod, generalized Golod, absolutely Koszul algebras, small codepth rings, and certain Veronese subrings), but the general case remains unresolved.