Weaken the hypotheses in the symmetry–bounds equivalence result (Proposition 5.8)

Determine weaker assumptions than R being Gorenstein under which the following holds: for a local ring R and a finitely generated R-module M, if for all finitely generated R-modules N one has PR(M,N) < ∞ if and only if PR(N,M) < ∞, then b_M < ∞ if and only if cob_M < ∞, where b_M = sup{PR(M,N) | PR(M,N) < ∞} and cob_M = sup{PR(N,M) | PR(N,M) < ∞}.

Background

Proposition 5.8 shows that over a Gorenstein local ring, the symmetry condition PR(M,N) < ∞ ⇔ PR(N,M) < ∞ for all finitely generated N is equivalent to the finiteness of both the Auslander bound b_M and the co-Auslander bound cob_M. Immediately after, the authors suggest that the Gorenstein hypothesis may be stronger than necessary.

The conjecture asks for identifying strictly weaker ring-theoretic conditions under which the same equivalence between the symmetry property and the two bounds’ finiteness remains valid.