The finitistic dimension of a Nakayama algebra

Abstract

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