Laurent coefficients and Ext of finite graded modules

Abstract

Let $R=\bigoplus_{n\ges0}R_n$ be a graded commutative ring generated over a field $K=R_0$ by homogeneous elements $x_1,\dots ,x_e$ of positive degrees $d_1,\dots ,d_e$. The Hilbert-Serre Theorem shows that for each finite graded $R$--module $M=\bigoplus_{n\in\BZ}M_n$ the {\it Hilbert series\/} $\sum_{n\in\BZ}(\rank_K M_n)t^n$ is the Laurent expansion around $0$ of a rational function $$H_M(t)=\frac{q_M(t)}{{\prod }_{i=1}^e(1-t^{d_i})}$$ with $q_M(t)\in\BZ[t,\ti]$. We demonstrate that Laurent expansions ${\left[M\right]}_z$ of $H_M(t)$ around other points $z$ of the extended complex plane $\overline\BC$ also carry important structural information.