The Poincaré series of the module of derivations of some monomial rings

Abstract

Let $R$ be a quasi-homogeneous $k$-algebra and $M$ be a finitely generated graded $R$-module. The formal power series $\sum_{i}\dim_{k}(\mathrm{tor}_{i}^R(k,M)z^i$ is called the Poincaré series of $M$ and it is denoted by $P_{M}^R(z)$. We remark that the Poincaré series of the module of derivations of a monomial ring is rational and determine it in some cases.