## 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.

### MSC:

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

Zbl 0607.13017