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Trivial formal fibres and formal Laurent series. (English) Zbl 0744.13008
All rings are commutative noetherian. If $$R$$ is a local ring, $$\hat R$$ will denote its completion. If $$P\subset R$$ is a prime ideal of $$R$$, $$k(P)$$ will denote the quotient field of $$R/P$$. If $$R$$ is a ring and $$P\subset R$$ a prime ideal, $$T_ P$$ will denote the completion of a free $$R_ P$$-module with the $$P$$-adic topology. If $0\to R\to PE^ 0(R)\to PE^ 1(R)\to \ldots$ is a minimal pure injective resolution of $$R$$ over itself, then each $$PE^ i(R)$$ is uniquely a product $$\prod T_ P$$ $$(P\in\hbox{Spec}(R))$$. The cardinality of the vector space $$T_ P\otimes k(P)$$ is denoted $$\pi_ 1(P,R)$$.
By the algebra of formal Laurent series over a complete local ring $$R$$ it is meant all the symbols $$\sum a_ nx^ n (-\infty<n<\infty)$$ with $$a_ n\in R$$ subject to the condition that $$\lim_{n\to\infty}a_{- n}=0$$. The composition laws are the obvious ones. This ring will be denoted $$R(((x)))$$. The subring $$R((x))\subset R(((x)))$$ consisting of those formal Laurent series $$\sum a_ nx^ n$$ such that for some $$n_ 0$$, $$a_{-n}=0$$ for $$n\geq n_ 0$$ is called the ring of Laurent series. For any ring $$R$$ and any $$R$$-module $$N$$, $$N[x]$$, $$N[[x]]$$ and $$N((x))$$ will denote the obvious modules over $$R[x]$$, $$R[[x]]$$ and $$R((x))$$ respectively.
The authors prove the results: (1) Let $$R$$ be a local ring and let $$P\subset R$$ be a prime ideal, then the following are equivalent:
(i) The canonical homomorphism $$k(P)\to k(P)\otimes \hat R$$ is an isomorphism,
(ii) $$R/P$$ is a complete local ring,
(iii) The natural map $$\hat R_ P\to(\hat R)_ P$$ is an isomorphism,
(iv) $$k(P)\otimes(\hat R/R)=0,$$
(v) $$\pi_ 1(Q,R)=0$$ for all prime ideals $$Q\supset P$$;
(2) If $$R$$ is a complete local ring with maximal ideal $$M$$, then $$R(((x)))$$ is isomorphic to $$R[[x]]_ P$$, where $$P=M\cdot R[[x]]=M[[x]]$$.
##### MSC:
 13F25 Formal power series rings 13D25 Complexes (MSC2000) 18G10 Resolutions; derived functors (category-theoretic aspects)
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