The finitistic dimension of a Nakayama algebra

Let A be an artin algebra, Gélinas has introduced interesting upper bound for the finitistic dimension fin-pro of A, namely delooping level del A. We assert that fin-proA=delA any Nakayama algebra This yields also a new proof and its opposite are equal, as shown recently by Sen. If S is simple module, let e(S) minimum projective injective envelope (one these numbers to finite); e?(S) cover. Then maximum e(S), well e?(S). Using suitable syzygy modules, we construct permutation h modules such e?(h(S))=e(S). In particular, this shows z?N number with e(S)=z equal S? e?(S?)=z.