16 problems

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 $(R,\mathfrak{m},k)$ be a commutative noetherian local ring, where $\mathfrak{m}$ is its maximal ideal and $k=R/\mathfrak{m}$ its residue field. A finitely generated $R$-module…

Hochster's direct summand conjecture. The sequence splits whenever $R$ is regular. Equivalently, the inclusion $R\to S$ is pure.

Serre's intersection conjectures. The following inequalities and equalities should hold:

Peskine–Szpiro's conjecture. One should have

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…

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…

Finitistic dimension conjecture. The finitistic dimension of every Artin algebra is finite. This is a central open problem in the representation theory of Artin algebras. Its resol…

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

Torsion-free Huneke–Wiegand conjecture. If $M\otimes_R M^*$ is MCM, then $M$ is torsion-free.

Generalized Huneke–Wiegand conjecture. If $M\otimes_R M^*$ is MCM, then $M$ is free.

The new intersection conjecture. One has

Let $R$ be a noetherian ring and let $R\subset S$ be a module-finite extension such that $S$, regarded as an $R$-module, has finite projective dimension. A retraction is an $R$-lin…

Let $R$ be a regular ring and let $M$ be an $R$-module. For an $R$-module $N$ and $x\in N$, define its order ideal by … Suppose that $x$ is a minimal generator of the $i$th syzygy…

Finitistic dimension conjecture. The finitistic dimension of every artin algebra is finite; equivalently, ${\mathrm{fin.dim}}\,(A)<\infty$ for every artin algebra $A$.

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…