On flat and Gorenstein flat dimensions of local cohomology modules (Q2810719)

Let \(\mathfrak a\) be an ideal of a Noetherian local ring \(R\) and \(C\) a semidualizing \(R\)-module (i.e. \(\mathrm{Ext}_R^i(C,C)=0\) for all \(i>0\) and the homothety map \(R\rightarrow \mathrm{Hom}_R(C,C)\) is an isomorphism).NEWLINENEWLINELet \(\mathcal{X}\) be a class of \(R\)-modules. For an \(R\)-module \(N\), \(\mathcal{X}\)-\(\dim_RN\) is defined to be the infimum of the non-negative integers \(n\) for which there exists an exact \(R\)-complex NEWLINE\[NEWLINE0\rightarrow X_n\rightarrow X_{n-1}\rightarrow \cdots \rightarrow X_1\rightarrow X_0\rightarrow N\rightarrow 0NEWLINE\]NEWLINE in which each \(X_i\) belongs to \(\mathcal{X}\) and \(X_n\neq 0\). If \(\mathcal{X}\) is the class of flat \(R\)-modules, Gorenstein flat \(R\)-modules and \(C\)-Gorenstein flat \(R\)-modules, then \(\mathcal{X}\)-\(\dim_RN\) is denoted by \(fd_RN\), \(Gfd_RN\) and \(G_C-fd_RN\); respectively.NEWLINENEWLINELet \(T(N)\) denote any of three invariants \(fd_RN\), \(Gfd_RN\) and \(G_C\)-\(fd_RN\). Let \(M\) be an \(R\)-module and \(n\) a non-negative integer. Assume that \(T(M)<\infty\) and \(H_{\mathfrak a}^i(M)=0\) for all \(i\neq n\). The authors show that NEWLINE\[NEWLINET(H_{\mathfrak a}^n(M))\leq T(M)+nNEWLINE\]NEWLINE with the equality when \(M\) is finitely generated. Using this, they establish several characterizations for Cohen-Macaulay modules and Gorenstein rings.