Excellent Extensions and Homological Conjectures

In this paper, we introduce the notion of excellent extensions of rings. Let Γ be an excellent extension of an Artin algebra Λ, we prove that Λ satisfies the Gorenstein symmetry conjecture (resp., finitistic dimension conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture) if and only if so does Γ. As a special case of excellent extensions, when G is a finite group whose order is invertible in Λ acting on Λ and Λ is G-stable, we prove that if the skew group algebra ΛG satisfies strong Nakayama conjecture (resp., generalized Nakayama conjecture), then so does Λ.

AUSLANDER GENERATORS AND HOMOLOGICAL CONJECTURES

Let A be an artin algebra with representation dimension not more than 3. Assuming that AV is an Auslander generator and M ∈ addAV, we show that both findim(EndAM) and findim(EndAM)op are finite, and consequently the Gorenstein symmetry conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for EndAM.