Numerical aspects of complexes of finite homological dimensions (Q6597271)

Let \((R,\mathfrak{m},k)\) be a commutative noetherian local ring, and \(X\) an \(R\)-complex. Then the depth of \(X\) is given by \N\[\N\operatorname{depth}_{R}(\mathfrak{m},X)= -\sup\left(\operatorname{RHom}_{R}(k,X)\right),\N\]\Nand for any \(i\in \mathbb{Z}\), the \(i\)th Bass number of \(X\) is given by \N\[\N\mu_{R}^{i}(\mathfrak{m},X)= \operatorname{rank}_{k}\left(\operatorname{Ext}_{R}^{i}(k,X)\right).\N\]\NIf \(X\) is homologically bounded, then the type of \(X\) is given by \N\[\Nr_{R}(X)= \mu_{R}^{\operatorname{depth}_{R}(\mathfrak{m},X)}(\mathfrak{m},X).\N\]\NThe authors show that if \(X\) is homologically bounded and finite with \(G_{C}\text{-dim}_{R}(X)< \infty\), and \(C\) is a semidualizing \(R\)-complex, then \N\[\Nr_{R}(X)= \nu_{R}\left(\operatorname{Ext}_{R}^{G_{C}\text{-dim}_{R}(X)-\inf(C)}(X,C)\right)r_{R}(C)\N\]\Nwhere \(\nu_{R}(-)\) denotes the minimal number of generators. Additionally, they prove that if \(M\) and \(N\) are finitely generated \(R\)-modules such that \(\operatorname{Ext}_{R}^{i}(M,N)=0\) for \(i\gg 0\), and \(\operatorname{id}_{R}\left(\operatorname{Ext}_{R}^{i}(M,N)\right)< \infty\) for every \(i\geq 0\), then \(\operatorname{pd}_{R}(M)< \infty\) and \(\operatorname{id}_{R}(N)< \infty\). These findings extend the recent results of Ghosh and Puthenpurakal.