Completions of commutative rings and modules
The completions of \(R\)-modules over a commutative ring relative to a torsion theory \(\sigma\) are studied. The main result is the description of completion as a direct product of classical completions on local rings obtained by localizing at prime ideals. Let \(\sigma\leq \tau\) be torsion theories in \(R\text{-mod}\), \(R\) be \(\sigma\)-noetherian and \(\sigma\) be stable. The \((\sigma, \tau)\)- completion of a module \({}_ R M\) is defined by \(M^{(\sigma, \tau)}= \varprojlim \{ Q_ \sigma (M/N)\mid M/N\in {\mathcal I}_ \tau \}\). The following sets of prime ideals determine \(\sigma\): \({\mathcal Z} (\sigma)= \{{\mathfrak p}\in \text{Spec } R\mid R/{\mathfrak p}\in {\mathcal I}_ \sigma\}\), \({\mathcal K} (\sigma)= \{{\mathfrak p}\in \text{Spec } R\mid R/{\mathfrak p}\in {\mathcal F}_ \sigma\}\) and \({\mathcal C} (\sigma)\), the set of maximal elements in \({\mathcal K} (\sigma)\). For some \({\mathfrak p}\in {\mathcal C} (\sigma)\) let \(\pi_{\mathfrak p}\) be the torsion theory determined by \({\mathcal Z} (\pi_{\mathfrak p})= {\mathcal Z} (\sigma)\cup \{{\mathfrak p\}}\). Then \((R_{\mathfrak p})^{(\sigma, \pi_{\mathfrak p})}= \widehat {R}_{\mathfrak p}\), the ordinary \({\mathfrak p}R_{\mathfrak p}\)-completion of the local ring \((R_{\mathfrak p}, {\mathfrak p} R_{\mathfrak p})\). Similarly \(M^{(\sigma; \pi_{\mathfrak p})}= \widehat {M}_{\mathfrak p}\) for every \(\sigma\)-finitely generated \(R\) module \(M\). For arbitrary \(\sigma\) the torsion theory determined by the set \({\mathcal Z} (\sigma^ 1)= {\mathcal Z} (\sigma)\cup {\mathcal C} (\sigma)\) is denoted by \(\sigma^ 1\) and the \((\sigma, \sigma^ 1)\)-completion is studied. It is shown that if \(M\) is a \(\sigma\)-finitely generated module, then \(M^{(\sigma, \sigma^ 1)} \cong \prod \{\widehat {M}_{\mathfrak p}\mid {\mathfrak p}\in {\mathcal C} (\sigma)\}\) and this permits to prove that the functor of \((\sigma, \sigma^ 1)\)-completion is exact on \(\sigma\)- finitely generated modules. Moreover, is \(\sigma\) perfect then \(R^{(\sigma, \sigma^ 1)} \otimes_ R M\cong M^{(\sigma, \sigma^ 1)}\) for every \(\sigma\)-finitely generated module \(M\).
scientific article; zbMATH DE number 755541
scientific article