The finitistic dimension of a Nakayama algebra

Abstract Let A be an artin algebra, Gelinas 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-pro A = del A 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 .