Coliftings and Gorenstein injective modules (Q1279742)

An \(R\)-module \(M\) is called Gorenstein injective if \(\text{Ext}^i_R(Q, M)\) and \(\text{Ext}^R_i(Q, M)\) vanish for all \(i\geq 0\), where \(Q\) is an \(R\)-module of finite injective or projective dimension and \(R\) is a commutative noetherian ring. This goal of this article is to develop Maranda type results for Gorenstein injective modules. When \(x\) is an \(R\)-regular element that is not regular on the \(R\)-module \(M\), let \(r_x(M)\) be the smallest integer \(r\) such that \(x^r\cdot \text{Ext}^1(\;,M)=0\). The authors show that if \(r_x(M)\) is finite, \(r\geq r_x(M)\geq 0\), \(r\geq r_x(N)\geq 0\), and \(M\) and \(N\) are strongly indecomposable Gorenstein injective \(R\)-modules such that \(\text{Hom}_R({R/x^rR}, M)\cong \text{Hom}_R({R/x^rR}, N)\), then \(M\cong N\) or \(N\cong S(M)\cong S^2(N)\), where \(S^i(M)\) is the \(i\)-th cosyzygy of \(M\). In addition, if \(r_x(M)\) is finite, \(r\geq r_x(M)\geq 0\) and \(M\) is strongly indecomposable then either \(\text{Hom}_R({R/x^rR}, M)\) is indecomposable or \(\text{Hom}_R({R/x^rR}, M)\cong L\oplus S_{R/x^rR}(L)\) for some indecomposable \(L\). Consequently, if \(x\) is \(R\)-regular and \(E, E'\) are indecomposable injective \(R\)-modules with \(\text{Hom}_R({R/x^rR}, E)\cong \text{Hom}_R({R/x^rR}, E')\not= 0\), then \(E\cong E'\). Let \(R\rightarrow S\) be a ring homomorphism and \(L\) an \(S\)-module. An \(R\)-module \(M\) is called a ``colifting'' of \(L\) to \(R\) if \(L\cong \text{Hom}_R(S,M)\) and \(\text{Ext}^i_R(S,M)=0\) for all \(i\geq 1\). The module \(M\) is a ``weak colifting'' of \(L\) if \(L\) is a direct summand of \(\text{Hom}_R(S,M)\) and \(\text{Ext}^i_R(S,M)=0\) for all \(i\geq 1\). An example is provided of a module \(M\) that is a weak colifting of \(L\), but is not a colifting of \(L\). For an \(R\)-regular element \(x\), write \(\overline R ={R/xR}\). The authors show that if \(M\) is a strongly indecomposable Gorenstein injective module over an \(n\)-Gorenstein ring \(R\) and \(M\) is a colifting of a nonzero \(\overline R\)-module \(L\) such that \(x\cdot \text{Ext}^1(\;,M)=0\), then \(S_R(\text{Hom}_R(\overline R, S^{-1}(M)))\) is a reduced Gorenstein injective \(R\)-module and \(G_R(L)\cong M\oplus S^{-1}(M)\) where \(G_R(L)\) and \(S^{-1}(M)\) denote the Gorenstein injective envelope of the \(R\)-module \(L\) and the kernel of the injective cover of \(M\), respectively. An \(R\)-module \(M\) is an ``essential colifting'' of an \(\overline R\)-module \(L\) if \(M\) is a colifting of \(L\) and the \(R\)-imbedding \(L\subseteq M\) is an essential extension. The authors show that when \(R\) is \(n\)-Gorenstein and \(M\) is an essential colifting of an \(\overline R\)-module to \(R\), then \(\text{Hom}_R(\overline R, G_R(M))\cong G_{\overline R}(\text{Hom}(\overline R, M))\). This leads to the result that if \(R\) is an essential colifting of \(L\) then R is a colifting of \(G_{\overline R}(L)\) and of \(G_R(L)/ L\). The paper concludes with characterizations for essentially coliftable \(\overline R\)-modules over 2-Gorenstein rings and for weakly coliftable \(\overline R\)-modules.