Gorenstein symmetry for Artin algebras

Determine whether the projective dimensions of injective modules and the injective dimensions of projective modules are equal for every Artin algebra, equivalently whether spli R = silp R holds in that setting.

Background

The paper introduces the invariants spli R and silp R, measuring respectively the projective lengths of injective modules and the injective lengths of projective modules. It notes that their relationship is unclear for general rings and that, for Artin algebras, equality of these invariants is equivalent to the Gorenstein Symmetry Conjecture. The paper proves equality under stronger coherence hypotheses, but does not resolve the Artin-algebra case in general.

References

The relation between $spli R$ and $silp R$ is unclear for a general ring $R$ and ask whether these two invariants are always equal. In the special case where $R$ is an Artin algebra, the equality $spli R = silp R$ is equivalent to the so-called Gorenstein Symmetry Conjecture.

— Mittag-Leffler Conditions, Gorenstein Modules and Homological Invariants  (2608.13898 - Dai et al., 14 Aug 2026) in Section 4, opening paragraph