Growth of Bass numbers for direct summands of syzygies of the residue field

Determine whether, for a commutative noetherian local ring R and integer n ≥ 1, the Bass numbers of any nonzero direct summand N of the nth syzygy module Ω^n_R(k) grow exponentially; if so, ascertain the order of this exponential growth.

Background

The authors consider the growth behavior of Bass numbers for syzygies and their direct summands. For R Gorenstein or Golod, both the growth and its order are known (exponential and of the same order as for k).

Despite connections to Avramov’s results on Betti numbers and other structural insights via stable cohomology, the analogous Bass-number question for general local rings R remains unresolved.

References

Let N be a nonzero direct summand of Ωn_R(k) for some n≥ 1. What can one say about growth of the Bass numbers of N? Is it exponential, and if so, what is its order? We know the answers to this question (yes, the growth is exponential and of the same order as that of k) when R is Gorenstein or Golod, but open in general.

— Unstable elements in cohomology and a question of Lescot  (2507.23213 - Iyengar et al., 31 Jul 2025) in Introduction, Question (qu:bass) and following sentence