Two applications of change of rings theorems for Poincaré series

Abstract:

Let (R, m, k) be an artinian Gorenstein ring with ${\dim _k}m/{m^2} \geqslant 2$. If ${P^R}$ denotes the Poincaré-series and ${\phi _R}$ denotes the Bass-series of R, then ${\phi _R} = ({P^R} - 1){x^{ - 1}}{(1 - {P^R}{x^2})^{ - 1}}$ with $\bar R = R/0:m$, see Proposition 1. With similar methods as in the proof of Proposition 1 we calculate the Poincaré-series of a certain class of local rings: Let (R, m, k) be a local ring, $({\alpha _{ij}})$ a $n - 1 \times n$-matrix with entries in m, $n \geqslant 3$. Let ${x_1}, \ldots ,{x_n}$ be elements in m and $\mathfrak {A}$ the ideal generated by the maximal minors of $({\alpha _{ij}})$ and by ${\Sigma _j}{\alpha _{ij}}{x_j}(i = 1, \ldots ,n - 1)$. If depth $\mathfrak {A} = n$, then ${P^{R/\mathfrak {A}}} \cdot ({(1 - X)^{n - 1}} - X){(1 + X)^n} = {P^R}$.