Auslander-Reiten theory for modules of finite complexity over self-injective algebras

Abstract

cxM := inf{d ∈ N | there exists c ∈ R such that βi(M) ≤ c · id−1 for all i 0}, where the infimum of the empty set equals ∞. Observe that cxM = 0 if and only if M is projective. Next, cxM = 1 if and only if the Betti numbers of M are bounded. Moreover, if M is either Ω or τ -periodic, then cxM = 1. If R is the group algebra of a finite group, then cxM <∞ for each R-module M . It is known that cx(ΩM) = cxM and cx(τM) = cxM for each Rmodule M . If 0→ A1 → A2 → A3 → 0 is an exact sequence, then cxAi ≤ max{cxAj | j ∈ {1, 2, 3} {i}} for each i ∈ {1, 2, 3}. The above properties imply, that if C is a connected component of the Auslander–Reiten quiver of R, then there exists d ∈ N such that cxM = d for allM in C which are nonprojective. The following example is due to Rainer Schulz. Let A := k〈x, y〉/(x, y, xy + qyx), where q 6= 0 and q is not a root of 1. If M := A/(x + qy), then βi(M) = 1 for all i ∈ N and M is not Ω-periodic (but M is τ -periodic).