Weak injective and weak flat complexes (Q2816778)

The authors introduce and study a generalization of an injective object in \({\mathcal C}\) the abelian category of chain complexes over a ring \(R\).NEWLINENEWLINEFor complexes \((C, \delta^C)\) and \((D. \delta^D)\) in \({\mathcal C}\), a complex \({\mathcal H}om(C, D)=({\mathcal H}om(C, D)_n, \delta)\) is defined by \({\mathcal H}om(C, D)_n = \prod_i \text{Hom}_R(C_i, D_{n+i})\) and \((\delta_n f)_m = \delta_{n+m}^Df_m - (-1)^n f_{m-1}\delta_m^C\). Let \(\underline{\text{Hom}}(C, D)\) be the complex consisting of cycles of \({\mathcal H}om(C, D)\) with the differential induced by \(\delta^D\). Then the functor \(\underline{\text{Hom}}(C, D)\) gives rise to the right derived functor \(\underline{\text{Ext}}^*(C, D)\) for which \(\underline{\text{Ext}}^i(C, D)\) is a complex for each \(i\in {\mathbb Z}\). With such a functor, the generalization of an injective complex mentioned above is defined as follows. Let \(P\) be a \textit{finitely generated projective} complex; that is, \(P\) is bounded, exact and for each \(n\), the module \(P_n\) is finitely generated and, moreover, the module \(Z_n(P)\) of cycles is projective. By definition, a \textit{super finitely presented complex} \(C\) has an exact sequence \(\cdots \to P^n \to \cdots \to P^0 \to C \to 0\) in \({\mathcal C}\) in which \(P^i\) is finitely generated projective for each \(i\). Then complex \(C\) is called \textit{weakly injective} if \(\underline{\text{Ext}}^1(F, C)=0\) for any super finitely presented complex \(F\).NEWLINENEWLINEThe first theorem of the paper under review asserts that a complex \(C\) is weak injective if and only if \(C\) is exact and each module \(Z_m(C)\) is weak injective in the category of \(R\)-modules in the sense of Gao and Wang [14].NEWLINENEWLINEThe weak injective dimension of a complex \(C\), denoted \(\text{wid} \;C\), is defined to be the infimum of the integers \(n\) such that there exists an exact sequence \(0 \to C \to E^0 \to \cdots \to E^n \to 0\) in which \(E^i\) is weak injective for each \(i\). The dimension relates to the derived functor \(\underline{\text{Ext}}^*( \;, \;)\) as the ordinary injective dimension of a module and the Ext-functor. In fact, for a complex \(C\) in \({\mathcal C}\), the following two conditions are equivalent. (1) \(\text{wid} \;C\leq n\). (2) \(\underline{\text{Ext}}^{n+i}(F, C)=0\) for any super finitely represented complex \(F\) in \({\mathcal C}\) and \(i\geq 1\).NEWLINENEWLINEA dual notion of a weak injective module, a weak flat module, is introduced and considered in the paper. Duality properties between weak injective modules and weak flat modules are also investigated.