Stable equivalence of selfinjective algebras of tilted type (Q1127980)

Let \(B\) be a tilted algebra not of Dynkin type and consider \(\widehat B\) its repetitive algebra. Denote by \(\nu_{\widehat B}\) the Nakayama automorphism of \(\widehat B\) and let \(\varphi\) be a positive automorphism of \(\widehat B\). Then the algebra given by the quotient \(\widehat B/(\varphi\nu_{\widehat B})\) is called a selfinjective algebra of tilted type. In the paper under review, the authors characterize these algebras. One of the results proven can be stated as follows: A selfinjective algebra \(A\) is stably equivalent to a selfinjective algebra \(\Lambda\) of tilted type if and only if \(A\) is socle equivalent to a selfinjective algebra \(\Lambda\) of tilted type. As a consequence they get the following corollary. Let \(A\) and \(\Lambda\) be two stably equivalent selfinjective algebras of tilted type. Then \(A\) and \(\Lambda\) have the same number of pairwise nonisomorphic simple modules.