Gorenstein injective, Gorenstein flat modules and the section functor
Let \({\mathfrak{a}}\) be an ideal of a commutative noetherian ring \(R\) with a non-zero identity. For any two \(R\)-modules \(M\) and \(N\), the \(i\)-th generalized local cohomology module of \(M\) and \(N\) with respect to \({\mathfrak{a}}\) is defined by \(H^i_{{\mathfrak{a}}}(M,N):=\underset{n}{\varinjlim}\text{Ext}_R^i(M/{\mathfrak{a}}^n M,N)\). This notion was defined by J. Herzog in 1974. The aim of the paper under review is to investigate generalized local cohomology modules in conjunction with the theory of Gorenstein homological dimensions. The theory of Gorenstein homological dimensions was initiated by \textit{E. E. Enochs} and \textit{O. M. G. Jenda} [Math. Z. 220, No. 4, 611--633 (1995; Zbl 0845.16005)]. An \(R\)-module \(N\) is said to be Gorenstein injective if there exists an exact sequence \[ E_{\bullet}: \dots {\longrightarrow} E_1{\longrightarrow} E_0{\longrightarrow} E^0{\longrightarrow} E^1{\longrightarrow} \dots \] of injective \(R\)-modules such that \(N\cong \ker(E^0{\longrightarrow} E^1)\) and that \(\Hom_R(E,E_{\bullet})\) is exact for all injective \(R\)-modules \(E\). Also, an \(R\)-module \(N\) is said to be Gorenstein flat if there exists an exact sequence \[ F_{\bullet}: \dots {\longrightarrow} F_1{\longrightarrow} F_0{\longrightarrow} F^0{\longrightarrow} F^1{\longrightarrow} \dots \] of flat \(R\)-modules such that \(N\cong \ker(F^0{\longrightarrow} F^1)\) and that \(F_{\bullet}\otimes_RE\) is exact for all injective \(R\)-modules \(E\). Then, the notions Gorenstein injective (respectively, flat) dimension of an \(R\)-module \(N\) is defined as the infimum of the length of Gorenstein injective (respectively, flat) resolutions of \(N\). Assume that \(M\) is finitely generated and Krull dimension of \(R\) and projective dimension of \(M\) are finite. It is proved that any Gorenstein injective \(R\)-module is \(H^0_{{\mathfrak{a}}}(M,\cdot)\)-acyclic. Therefore for any \(R\)-module \(N\), the modules \(H^i_{{\mathfrak{a}}}(M,N)\)'s can be computed by using Gorenstein injective resolutions of \(N\). As an application, it is shown that over a Gorenstein local ring \((R,{\mathfrak{m}})\), the following are equivalent: \noindent i) \(H^i_{{\mathfrak{m}}}(M,N)=0\) for all finitely generated \(R\)-module \(M\), all Gorenstein injective \(R\)-module \(N\) and all \(i>0\). \noindent ii) \(H^i_{{\mathfrak{m}}}(R/{\mathfrak{a}},N)=0\) for all ideals \({\mathfrak{a}}\) of \(R\), all Gorenstein injective \(R\)-module \(N\) and all \(i>0\). \noindent iii) \(R\) is regular. Let \((R,{\mathfrak{m}})\) be a \(d\)-dimensional Cohen-Macaulay local ring possessing a canonical module. In the last section of the paper, the author studied the Auslander and Bass classes of \(R\)-modules. Among other things, it is proved that if \(N\) is a Gorenstein flat \(R\)-module, then \(H^d_{{\mathfrak{m}}}(N)\) belongs to the Auslander class.
scientific article; zbMATH DE number 5186661