The finitistic dimension of a Nakayama algebra

If A is an artin algebra, G\'elinas has introduced the delooping level of A as an interesting upper bound for the finitistic dimension of A. We assert that for any Nakayama algebra A the finitistic dimension of A is equal to the delooping level. This yields also a new proof that the finitistic dimension of A and its opposite algebra are equal, as shown quite recently by Sen. For a cyclic Nakayama algebra with even finitistic dimension d we show that Omega^d yields a bijection between the indecomposable injective modules I with projective dimension d such that the socle of I has even or infinite projective dimension and the indecomposable projective modules P with injective dimension d such that the top of P has even or infinite injective dimension.