A note on cofinite modules over Noetherian complete local rings (Q6073809)

Let \(R\) be a commutative noetherian ring, and \(\mathfrak{a}\) an ideal of \(R\). An \(R\)-module \(M\) is said to be \(\mathfrak{a}\)-cofinite if \(\mathrm{Supp}_{R}(M)\subseteq \mathrm{Var}(\mathfrak{a})\) and \(\mathrm{Ext}^{i}_{R}\left(R/ \mathfrak{a},M\right)\) is a finitely generated \(R\)-module for every \(i\geq 0\). Now assume that \((R,\mathfrak{m})\) is a commutative noetherian \(\mathfrak{m}\)-complete local ring, and \(\mathfrak{a}\) is a proper ideal of \(R\) such that the category of all \(\mathfrak{a}\)-cofinite \(R\)-modules is an abelian subcategory of the category of all \(R\)-modules. Then for each \(\mathfrak{a}\)-cofinite \(R\)-module \(M\), the author shows that \(\dim\left(R/ \left(\mathfrak{a}+\mathrm{ann}_{R}(M)\right)\right)= \dim_{R}(M)\). As a consequence of this result, it is shown that if \(\mathfrak{b}\) is an ideal of \(R\) such that for each \(\mathfrak{p}\in \mathrm{MinAss}(R)\), we have \(\dim\left(R/ (\mathfrak{b}+\mathfrak{p})\right)\leq 1\) or \(\mathrm{cd}(\mathfrak{b},R/ \mathfrak{p})\leq 1\), then \(\dim\left(R/ \left(\mathfrak{b}+\mathrm{ann}_{R}(M)\right)\right)= \dim_{R}(M)\) for every \(\mathfrak{b}\)-cofinite \(R\)-module \(M\).