Nakayama's conjecture and the double dual functors (Q1080928)

The dominant dimension of an algebra R is the biggest n such that the first n terms of a minimal injective resolution of \({}_ RR\) (equivalently \(R_ R)\) are projective: dom.dim R\(=n\). A conjecture of \textit{T. Nakayama} [Abh. Math. Semin. Univ. Hamb. 22, 300-307 (1958; Zbl 0082.030)] states, that a finite dimensional algebra R is QF if dom.dim R\(=\infty\). In this context the author proves for an artinian algebra R the following: In case of dom.dim \(R\geq 2\) resp. \(\geq 3\) R is QF if and only if every simple ideal of R is reflexive resp. if \(Ext(M,R)=0\) for any finite reflexive left R-module M. In case of dom.dim R\(=\infty\) R is QF if and only if there is a positive integer n for each simple left ideal M of R such that \(Ext^{n+1}(Hom_ R(\Omega^ n(M),R)=0\). Here \(\Omega^ n(M)\) denotes the n-th syzygy module of M.