8 problems

Takahashi's conjecture. Then $R$ is Cohen–Macaulay.

Let $R$ be a non-Gorenstein local ring, and suppose that there is a nonfree totally reflexive $R$-module. The Takahashi conjecture. There are infinitely many isomorphism classes of…

Let $R$ be a local ring, and let $M$ be a nonzero cyclic $R$-module of finite G-injective dimension. Cyclic module conjecture. If such an $M$ exists, then $R$ is Gorenstein. This c…

Let $R$ be a commutative Noetherian local ring and let $M$ be a finitely generated $R$-module. Assume that $M$ has finite length and that $\operatorname{Gdim}(M)$, the Gorenstein d…

Let $R$ be a commutative Noetherian local ring and let $M$ be a finitely generated $R$-module. Write $\operatorname{Gdim}(M)$ for the Gorenstein dimension,…

Let $R$ be a ring, and let the weak global Gorenstein dimension of $R$ be the supremum of the Gorenstein flat dimensions of its modules. Bennis's symmetry conjecture. The weak glob…

Let $G$ be a group. A finite-dimensional classifying space $\underline{E}G$ for proper actions is a finite-dimensional model for the classifying space of the family of finite subgr…

Let $G$ be a discrete group, and let $\operatorname{Gcd} G$ denote its Gorenstein cohomological dimension and $\operatorname{\underline{cd}} G$ its Bredon cohomological dimension f…