PUB - Publikationen an der Universität Bielefeld

Let Abe an artin algebra, Gelinas has introduced an interesting upper bound for the finitistic dimension fin-pro A of A, namely the delooping level delA. We assert that fin-pro A = del A for any Nakayama algebra A. This yields also a new proof that the finitistic dimension of Aand its opposite algebra are equal, as shown recently by Sen. If Sis a simple module, let e(S) be the minimum of the projective dimension of Sand of its injective envelope (one of these numbers has to be finite); and e*(S) the minimum of the injective dimension of Sand of its projective cover. Then the finitistic dimension of Ais 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 hof the simple modules Ssuch that e*(h(S)) = e(S). In particular, this shows for z is an element of 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. (C) 2021 Elsevier Inc. All rights reserved.