4 problems

The determinantal-ring conjecture. If $0\neq[M]\in\operatorname{Cl}(R)$, then $M$ is exactly $(m+n-2t+1)$-semidualizing. Hence

Let $(R,\mathfrak{m})$ be a local ring. A finitely generated $R$-module $C$ is semidualizing if the homothety morphism … is an isomorphism and $\operatorname{Ext}_{R}^{i}(C,C)=0$ f…

Let $(R,\mathfrak{m},k)$ be an Artinian local ring, with all modules finite. Let $[C_n]\vartriangleleft\cdots\vartriangleleft[C_1]\vartriangleleft[C_0]$ be a chain in…

Let $R$ be the ring under consideration, let $C$ be a semidualizing module, and let $X$ be any $R$-module. Write $\mathcal{G}_C\mathop{\rm \text{-}\dim}\nolimits X$ and…