Approximations of complete modules by complete big Cohen-Macaulay modules over a Cohen-Macaulay local ring
Let \(\mathcal T\) be a class of objects in an additive category \(\mathcal A\). A morphism \(f: M \rightarrow T\) with \(M \in {\mathcal A}\) and \(T \in {\mathcal T}\) is called a \(\mathcal T\)-\textit{preenvelope} if \(\text{Hom}_{\mathcal A}(T,T^{\prime}) \rightarrow \text{Hom}_{\mathcal A}(M,T^{\prime})\) is surjective, \(\forall T^{\prime} \in {\mathcal T}\). A \(\mathcal T\)-preenvelope is called a \(\mathcal T\)-\textit{envelope} when any endomorphism \(h\) of \(T\) with \(h \circ f = f\) is an automorphism. Dually, one has the notion of \(\mathcal T\)-\textit{cover}. Now, let \((A,\mathfrak{m},k)\) be a Cohen-Macaulay noetherian local ring, \(\mathcal X\) the class of complete big Cohen-Macaulay \(A\)-modules plus the null module and \(\mathcal Y\) the class of complete \(A\)-modules of finite injective dimension. In the paper under review, the author shows that any complete \(A\)-module, not necessarily finitely generated, admits a surjective \(\mathcal X\)-cover with kernel in \(\mathcal Y\) and an injective \(\mathcal Y\)-envelope with cokernel in \(\mathcal X\). This result extends to complete modules the theory developed by \textit{M. Auslander} and \textit{R. O. Buchweitz} [Mém. Soc. Math. Fr. 38, 5--37 (1989; Zbl 0697.13005)] for finitely generated modules. The above result allows the author to extend to complete modules some other known facts concerning finitely generated ones. For example, she shows that every element of \({\mathcal X} \cap {\mathcal Y}\) is of the form \(W\otimes_{\widehat A}F\), where \(W\) is the dualizing module of \(\widehat A\) and \(F\) is a complete flat \(A\)-module, thus extending a result of \textit{R. Y. Sharp} [Proc. Lond. Math. Soc., III Ser. 25, 303--328 (1972; Zbl 0244.13015)]. The author also shows that if \(M\) is a complete \(A\)-module and \((0) \neq Y \in {\mathcal Y}\) then: \[ \text{dim}\, A = \text{inf}\, \{ i\, | \, \text{Ext}^i_A(k,M) \neq 0\} + \text{sup}\, \{ i\, | \, \text{Ext}^i_A(M,Y) \neq 0 \}. \] This relation extends a result of \textit{C. Peskine} and \textit{L. Szpiro} [Publ. Math. Inst. Hautes Ét. Sci. 42, 77--119 (1972; Zbl 0268.13008)].
scientific article; zbMATH DE number 5600720
scientific article