## Completions of reduced local rings with prescribed minimal prime ideals.(English)Zbl 1421.13003

As the authors clearly explain in the introduction, this paper generalizes some recent results of Arnost et al. on the relationship between local rings and their completion. The streem in which this paper can be located is the characterization of complete local rings $$T$$ that turn out to be the completion of a subring $$A$$ with prescribed properties. Charters and Loepp considered the case of a local integral domain; they found necessary and sufficient conditions for a collection $$G$$ of prime ideals of $$T$$, with finitely many maximal elements, to be the general formal fiber of a subdomain $$A$$ with completion $$T$$. When $$T$$ contains the rational field $$Q$$, their result was extended by Arnost et al. to the case of a reduced local ring $$A$$, with semilocal formal fibre at each of its minimal prime ideals. In the present paper the authors get rid of the condition that $$T$$ contains the rationals and allow the formal fibers of $$A$$ to have a countable number of maximal elements.

### MSC:

 13B35 Completion of commutative rings 13F25 Formal power series rings 13J05 Power series rings 13J10 Complete rings, completion

### Keywords:

completions of local rings; minimal prime ideals
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