Multi-ideal-adic completions of Noetherian rings
Let \(R\) be a commutative ring with \(1\) and let \(\{I_{n}\}_{n=0}^{\infty }\) be a filtration of ideals of \(R\), i.e., \(I_{0}\supseteq I_{1}\supseteq I_{2}\supseteq \dots\). We can make \(R\) a topological space w.r.t. the filtration \(\{I_{n}\}\) by taking the cosets \(r+I_{n},\) \(r\in R\) and \(n\in Z_{+},\) as basic open sets. \(R\) is Hausdorff under this filtration topology if and only if \(\bigcap I_{n}=(0).\) Call \(R\) complete with respect to a filtration \(\{I_{n}\}\) if (a) \(\bigcap I_{n}=(0)\) and (b) every Cauchy sequence in \(R\) converges to an element of \(R.\) In ring theory, the completion of a ring \(R\) w.r.t. a filtration \(\{I_{n}\}\) is a complete ring \(R^{\ast }= \underleftarrow{\lim_{n}}\{R/I_{n}\}\) with a well-defined filtration, and a canonical homomorphism \(\psi :R\rightarrow R^{\ast },\) see chapter 9 of \textit{D. G. Northcott} [Lessons on rings, modules and multiplicities. Reprint of the 1968 hardback ed. Cambridge: Cambridge University Press (2008; Zbl 1185.13001)]. The canonical homomorphism \(\psi \) is injective, rendering ring theoretic completion closest to topological completion, precisely when \( \bigcap I_{n}=(0)\). A filtration \(\{I_{n}\}_{n=0}^{\infty }\) is called multiplicative if \( I_{0}=R \) and \(I_{m}I_{n}\subseteq I_{m+n}.\) A well-known multiplicative filtration on \(R\) is the \(I\)-adic filtration \(\{I^{n}\}_{n=0}^{\infty }\), where \(I\) is a fixed (proper) ideal of \(R.\) Now if \(R\) is noetherian with nonzero Jacobcon radical \(J\), as in the paper under review, then \(\bigcap _{n=0}^{\infty }J^{n}=(0).\) The authors introduce a filtration \( \{I_{n}\}_{n=0}^{\infty }\) requiring that \(I_{n}\subseteq J^{n}\) for each \( n>0\) and that \(I_{nk}\subseteq I_{n}^{k}\) for all \(n,k\geq 0.\) They call the completion resulting from this, somewhat restricted, filtration a multi-adic completion. One of the motivations for this study is that several useful examples can be considered as subrings of \(R^{\ast }/J\) where \(R^{\ast }\) is a multi-adic completion, of a suitable Noetherian ring \(R,\) and \(J\) is an ideal of \( R^{\ast }.\) Some of the relevant papers are: \textit{C. Rotthaus} [Nagoya Math. J. 74, 123--135 (1979; Zbl 0388.13013); Nagoya Math. J. 76, 173--180 (1979; Zbl 0388.13014)], \textit{R. C. Heitman} [Rocky Mt. J. Math. 12, 145--148 (1982; Zbl 0484.13010)], \textit{T. Ogoma} [Jap. J. Math., New Ser. 6, 147--163 (1980; Zbl 0456.13005)] and some old preprints of Jun-Ichi Nishimura. The authors study multi-adic completions. They show that if \(R\) is Noetherian with a filtration satisfying the required restrictions the multi-adic completion \(R^{\ast }\) is Noetherian. They also show that in case \(R\) is local, \(R\) being excellent, Henselian, or universally catenary implies that the multi-adic completion \(R^{\ast }\) has the same property. Reviewer's remarks. The still unpublished papers, ``A few examples of local domains, I, II, III'' by Nishimura, have been mentioned by various authors. I asked around about the status of these papers. I heard that Nishimura has finally decided to publish his work. I even saw something like a set of galley proofs from J. Math. Kyoto Univ. of the first preprint mentioned above, dated 2004. Recently, however, \textit{S. Loepp, Ch. Rotthaus} and \textit{S. Sword} [J. Commutative Algebra 1(4), 647--678 (2009; Zbl 1188.13008)] have written an exposition of Nishimura's construction.
scientific article; zbMATH DE number 5604539
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