Noetherian families of modules over \(k[[x]]\) and application. (Familles noethériennes de modules sur \(k[[x]]\) et applications.) (French) Zbl 0858.13009

This paper develops two previous notes published by the authors in C. R. Acad. Sci., Paris, Sér. I 301, 423-426 and 475-478 (1985; Zbl 0607.13017). Their aim is to study the so-called Noetherian families of modules over \(K[[x]]\), \(x=(x_1, x_2,\dots, x_n)\). More precisely, let \((A,\Gamma)\) be a pair where \(A\) is a (commutative and unitary) \(K\)-algebra; \(K\) being a commutative field of characteristic 0 and \(\Gamma\) a subset of the maximal spectrum of \(A\). A family \({\mathcal N}\) of submodules of \(K[[x]]^p\) is called Noetherian (and parameterized by \((A,\Gamma)\)) if there is a submodule \(N\) of \(A[[x]]^p\) such that \({\mathcal N}=(N_\gamma )_{\gamma\in\Gamma}\) where \(N_\gamma\) is the induced submodule in \(\gamma\) by \(N\).
In section 1, quite a few examples of Noetherian families are given. The most important results of the paper are given in section 6 where it is said that every family of modules canonically deduced from a Noetherian one is a Noetherian family. For example Noetherian families will be those gotten by intersection, sum or division of modules; ideal roots; or by prime ideals of modules appearing in a primary decomposition. Sections 3, 4, and 5 are devoted to the proofs which use as a basic tool a certain local regular ring defined in section 2.
Finally, the implicit function theorem of Artin for Noetherian families of equations and the existence of a uniform bound for the Łojasiewicz exponent of a Noetherian family of ideals are shown in sections 7 and 8.


13F25 Formal power series rings
13E99 Chain conditions, finiteness conditions in commutative ring theory


Zbl 0607.13017