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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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