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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Two applications of change of rings theorems for Poincaré series


Authors: Jürgen Herzog and Manfred Steurich
Journal: Proc. Amer. Math. Soc. 73 (1979), 163-168
MSC: Primary 13H99; Secondary 16A60
MathSciNet review: 516457
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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}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0516457-0
PII: S 0002-9939(1979)0516457-0
Article copyright: © Copyright 1979 American Mathematical Society