7 problems

Let $R$ be a local ring and let $M$ be an $R$-module of finite projective dimension. For every $R$-module $N$ such that $l(M\otimes_RN)<\infty$, finite-projective-dimension decency…

Let $A$ be a finite-dimensional $K$-algebra, and let $S$ be a simple $A$-module. Write $\operatorname{grade} S$ for the grade of $S$. Generalised Nakayama conjecture. For every sim…

Let $A$ be an algebra satisfying the Auslander condition, meaning that it is $n$-Gorenstein for every $n\geq 1$. Auslander–Reiten conjecture. If $A$ satisfies the Auslander conditi…

Huneke–Wiegand quasi-fiber-product conjecture. Gorenstein quasi-fiber product rings satisfy (HW).

Huneke–Wiegand lifting conjecture. If $R/(x)$ satisfies (HW), then $R$ satisfies (HW).

Let $A \rightarrow R$ be a ring homomorphism such that $A$ is a direct summand of $R$, and suppose that $R$ is regular. Cohen–Macaulayness of Direct Summands conjecture. Then $A$ i…

Let $R \rightarrow S$ be an injective ring homomorphism, where $R$ is a regular local ring and $S$ is module-finite over $R$. Direct Summand Conjecture. Then $R$ is a direct summan…