Séries de Poincaré et modules inertes. (Poincaré series and inert modules)
Let f: (S,\({\mathfrak n})\to (R,{\mathfrak m})\) be a homomorphism of noetherian local rings, which induces the identity map on their common residue field k, and let M be an R-module. The paper investigates the relation of the Poincaré series of M over R, \(P^ M_ R(t)=\sum_{i\geq 0}\dim_ kTor^ R_ i(M,k)t^ i \), and a similar formal power series over S. Of course, in order for \(P^ M_ R(t)\) to be defined, one needs the dimensions of the corresponding Tor's to be finite, and the author proves that this condition is equivalent to the finiteness of \(\dim_ kExt^ i_ R(k,M)\) for \(i\in {\mathbb{Z}}\), and to the finiteness of \(\dim_ kH(K\otimes_ RM)\), when K is the Koszul complex over a minimal system of generators of \({\mathfrak m}\). He also shows that if \(P^ M_ S(t)\) is defined, then so is \(P^ M_ R(t)\), and that the converse holds when S/\({\mathfrak m}S\) is artinian. These results partly depend on a rather general change of rings spectral sequence constructed in the paper, which in particular yields a coefficientwise inequality: \(P^ k_ R(t)\cdot P^ M_ S(t)\gg P^ k_ S(t)\cdot P^ M_ R(t).\) The author says an R-module M is inert by f if the inequality above becomes an equality. The study of inert modules, which is the main object of the paper, is linked to earlier work on the homology of local rings. Thus, it is shown that the homomorphism f is large [that is, the induced map \(f_*: Tor^ S(k,k)\to Tor^ R(k,k)\) is surjective, cf. \textit{G. Levin}, Math. Scand. 46, 209-215 (1980; Zbl 0472.13010)] if and only if every finite type R-module M is inert by f. On the other hand, if f is small [that is, \(f_*\) is injective, cf. the reviewer, J. Algebra 50, 400-453 (1978; Zbl 0395.13005)] then M is inert by f if and only if the induced map \(Tor^ S(M,k)\to Tor^ R(M,k)\) is injective. A particularly nice application of the ideas and techniques of the paper is given in the proof of the following result: If the ring R is Golod, and the module M is finitely generated, then the sequence \(\{\dim_ kTor^ R_ i(M,k)| i>\dim_ k{\mathfrak m}| {\mathfrak m}^ 2\}\) is constant if R is a hypersurface ring, and is strictly increasing otherwise.
scientific article; zbMATH DE number 4179508
scientific article