The Poincaré series of the module of derivations of affine monomial curves

Abstract

Let A be a graded k-algebra and M be a finitely generated A-module. The Poincaré series P A M (z) is the formal power series i dim k Tor A i (k, M)z i . We study the Poincaré series of Der k k[S], the module of derivations of a numerical semigroup ring k[S], and we relate it to the Poincaré series of k over k [S] and to the type of S. We then use this in order to determine the Poincaré series of Der k k [S] or, at least, its rationality, for some classes of examples. We finally give an example of a non-rational P k [S] Der k k[S] (z).