Colocalization and cotilting for commutative noetherian rings (Q402689)

Let \(R\) is a commutative Noetherian ring with unit. For a module \(M\), the notion \(\text{Add}\,M\) denotes the class of all direct summands of (possibly infinite) direct sums of copies of \(M\). Let \(n<\omega\). Recall that a module \(T\) is \(n\)-tilting provided thatNEWLINENEWLINE1) \(T\) has projective dimension \(\leqslant n\);NEWLINENEWLINE2) \(\text{Ext}_R^i(T,T^{(\kappa)})=0\) for each \(i>0\) and all cardinals \(\kappa\);NEWLINENEWLINE3) There exists an exact sequence \(0\rightarrow R\rightarrow T_0\rightarrow T_1\rightarrow\cdots\rightarrow T_n\rightarrow0\) where \(T_i\in\text{Add}\,T\) for each \(0\leqslant i\leqslant n\).NEWLINENEWLINEThe class \(T^{\bot}=\{M\in \text{Mod-}R\mid\text{Ext}_R^i(T,M)=0\text{ for all }i\geqslant 1\}\) is called the \(n\)-tilting class induced by \(T\), where \(T\) is an \(n\)-tilting module. Also, recall that \(n\)-cotilting module and \(n\)-cotilting class are defined dually. In the paper under review, the authors investigate relations between tilting and cotilting modules in Mod-\(R\) and Mod-\(R_{\mathfrak m}\), where \(\mathfrak m\) runs over the maximal spectrum of \(R\) (i.e.\,\({\mathfrak m}\in{\mathfrak m}\text{Spec}(R)\)). For each \(n<\omega\), they construct a 1-1 correspondence between (equivalence classes of) \(n\)-cotilting \(R\)-modules \(C\) and (equivalence classes of) compatible families \(\mathcal{F}\) of \(n\)-cotilting \(R_{\mathfrak m}\)-modules (\({\mathfrak m}\in{\mathfrak m}\text{Spec}(R)\)). It is induced by the assignment \(C\rightarrow (C^{\mathfrak m}| \,{\mathfrak m}\in{\mathfrak m}\text{Spec}(R))\), where \(C^{\mathfrak m}=\text{Hom}_R(R_{\mathfrak m},C)\) is the colocalization of \(C\) at \({\mathfrak m}\), and its inverse \(\mathcal{F}\longrightarrow\prod_{F\in \mathcal{F}}F\). They also construct a similar correspondence for \(n\)-tilting modules using compatible families of localizations.