13 problems

Boij–Söderberg conjecture. The Betti diagram of any Cohen–Macaulay $R$-module is a non-negative linear combination of pure diagrams, and every pure diagram is a rational multiple o…

Bhatt–Hochster–Ma conjecture. The ring $R$ has a lim Cohen–Macaulay sequence.

Let $R$ be a Cohen--Macaulay local ring, and let $\mathcal{X}$ be a resolving subcategory of $\operatorname{mod} R$. A module is maximal Cohen--Macaulay if its depth equals…

A small Cohen–Macaulay module over a local ring is a finitely generated module whose depth equals the dimension of the ring. The small Cohen–Macaulay conjecture. Every complete loc…

Tau-rigidity conjecture. If $M$ is an indecomposable syzygy, then $M\oplusc3M$ is $c4$-rigid.

Let $(R,\mathfrak{m})$ be a complete local ring. A small Cohen–Macaulay module over $R$ is a nonzero finitely generated $R$-module $S$ such that $\operatorname{depth}(S)=\dim(R)$.…

Cohen–Macaulay descent conjecture. If $M\otimes_R N$ is a Cohen–Macaulay $R$-module, then $N$ is a maximal Cohen–Macaulay $R$-module. This conjecture asks whether Cohen–Macaulaynes…

Conjecture. Theorem 1 would remain valid without any assumptions on the algebras $A$ and $B$ other than their being isomorphic as graded $S$-modules; equivalently, the conditions i…

Let $R$ be a Cohen–Macaulay local ring. Let $\operatorname{\mathcal{X}}$ be a resolving subcategory of finitely generated $R$-modules with finite radius. Dao–Takahashi's conjecture…

Let $G$ be a reductive group with maximal torus $H$ acting on a representation $W$ with weights $\beta_1,\ldots,\beta_d\in X(H)$. Let … and let $\rho$ be half the sum of the positi…

Let $R$ be a local normal domain, and let $R^{sh}$ denote its strict henselization. The group $\operatorname{Cl}(R^{sh})$ is the divisor class group of the strict henselization. St…

Let $R$ be a local normal domain over an algebraically closed field. The divisor class group $\operatorname{Cl}(R)$ is the group of rank-one reflexive classes. Hochster's finitenes…

Let $R$ be a standard graded Gorenstein ring of countable Cohen–Macaulay type, meaning that it has only countably many indecomposable graded Cohen–Macaulay modules up to shifts. Hy…