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Local rings over which all modules have rational Poincaré series. (English) Zbl 0794.13010
Soient \(R\) un anneau commutatif local d’idéal maximal \(M\) et corps résiduel \(k\), \(E\) un \(R\)-module de type fini et \(P^ R_ E(n)=\sum^ \infty_{n=0} b^ R_ n (E)t^ n\) la série de Poincaré du \(R\)- module \(E\), où \(b^ R_ n(E)=\dim_ k \text{Tor}^ R_ n (E,k)\) sont les nombres de Betti. L’A. démontre les résultats principaux suivants: Si l’algèbre de Lie homotopique \(\Pi^*(R)\) contient une sous-algèbre de Lie libre graduée de codimension finie, il existe un polynôme \(\text{Den}^ R(t) \in \mathbb{Z} [t]\) tel que our tout \(R\)- module de type fini \(E\) la série \(\text{Den}^ R (t) P^ R_ E(t) \in \mathbb{Z} [t]\). Si, de plus, la sous-algèbre de Lie libre contient \(\Pi^ j(R)\) pour \(j>n\), alors \(\text{Den}^ R (t)P^ R_ k(t)\) divise \(\prod_{0 \leq 2i+1 \leq n} (1+t^{2i+1})^{e_{2i+1}}\), où \(e_ i=\dim_ k \pi^ i(R)\) est la deviation d’ordre \(i\) de \(R\). Si la fonction \(b^ R_ E\) est polynômiale pour tout \(R\)-module de type fini \(E\), alors \(R\) est un anneau d’intersection complète de codimension \(\leq q(R)+1\), où \(q(R)=(e(e+1)/2)-\dim_ k (M^ 2/M^ 3)\). Si \(R\) est une intersection complète de multiplicité minimale, alors \(b^ R_ E\) est une fonction polynômiale.

MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13C40 Linkage, complete intersections and determinantal ideals
13H99 Local rings and semilocal rings
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