The finitistic dimension of a Nakayama algebra

TL;DR

This paper proves that for Nakayama algebras, the finitistic dimension equals the delooping level, providing a new proof of its symmetry and characterizing it via simple modules' projective and injective dimensions.

Contribution

It establishes the equality of finitistic dimension and delooping level for Nakayama algebras and introduces a permutation of simple modules linking their projective and injective dimensions.

Findings

Finitistic dimension equals delooping level for Nakayama algebras

Symmetry of finitistic dimension between an algebra and its opposite

Characterization of finitistic dimension via simple modules' dimensions

Abstract

If A is an artin algebra, G\'elinas has introduced an interesting upper bound for the finitistic dimension of A, namely the delooping level del A. We assert that for any Nakayama algebra, its finitistic dimension is equal to del A. This yields also a new proof that the finitistic dimension of A and of 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 any natural number z, that the number of simple modules S with e(S) = z is equal to the number of simple modules S' with e(S') = z.