Let $A$ be a graded $k$-algebra and $M$ be a finitely generated $A$-module. The Poincar\'e series $P_{M}^A(z)$ is the formal power series $\sum_{i}\dim_{k}\tor_{i}^A(k,M)z^i$. We study the Poincar\'e series of $\der_{k} k[S]$, the module of derivations of a numerical semigroup ring $k[S]$, and we relate it to the Poincar\'e series of $k$ over $k[S]$ and to the type of $S$. We then use this in order to determine the Poincar\'e 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 $P_{\der_{k} k[S]}^{k[S]}(z)$.

Poincaré series of the module of derivations of affine monomial curves

MICALE, VINCENZO
2004-01-01

Abstract

Let $A$ be a graded $k$-algebra and $M$ be a finitely generated $A$-module. The Poincar\'e series $P_{M}^A(z)$ is the formal power series $\sum_{i}\dim_{k}\tor_{i}^A(k,M)z^i$. We study the Poincar\'e series of $\der_{k} k[S]$, the module of derivations of a numerical semigroup ring $k[S]$, and we relate it to the Poincar\'e series of $k$ over $k[S]$ and to the type of $S$. We then use this in order to determine the Poincar\'e 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 $P_{\der_{k} k[S]}^{k[S]}(z)$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11769/24260
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