15 problems

Let $P$ be a regular local ring with algebraically closed residue field and let $f\in P$. Define … where $\operatorname{Sing}(P/f)$ denotes the singular locus of $P/f$. Buchweitz–G…

Residual-intersection conjecture. If $I$ is unmixed and $J:I$ is a geometric $(\ell-1)$-residual intersection of $I$, then, for all $\rho\gg0$,

Unbounded-rank small maximal Cohen–Macaulay conjecture. For every integer $N$, there exists a complete local domain $R$ that does not admit small maximal Cohen–Macaulay modules of…

Schoutens's conjecture. Every complete local ring admits a very small maximal Cohen–Macaulay module.

Let $R$ be a local complete intersection domain, and let $M,N\in\operatorname{mod} R$. Assume that $R$ is an isolated singularity. Tensor-product maximal Cohen–Macaulay conjecture.…

Let $R$ be a complete local Cohen–Macaulay ring satisfying Serre's condition $(R_2)$, meaning that $R_{\mathfrak p}$ is regular for every prime $\mathfrak p$ of height at most two.…

Power-integral Cohen–Macaulay conjecture. The ring $A^{\operatorname{pow}}$ is Cohen–Macaulay.

Small MCM conjecture for tame bi-partite toric rings. For some sufficiently high power $q$ of $p$, $R$ admits the small MCM

F-integral Cohen–Macaulay conjecture. The ring $\mathbf{F}_{\!}^{\operatorname{int}}(R)$ is Cohen–Macaulay, and hence is a small MCM algebra over $R$.

Very small MCM conjecture. Every complete local ring admits a very small MCM.

Second Brauer-Thrall conjecture for MCM modules. If $R$ has unbounded CM type and $\mathsf{k}$ is infinite, then $R$ has strongly unbounded CM type.

Generalized Gabber conjecture. If $[N]=0$ in $\overline G(R)_{\mathbb Q}$, then $\operatorname{Hom}_R(N,N)$ is a maximal Cohen–Macaulay $R$-module if and only if $N$ is free.

Let $(A,\mathfrak{m})$ be a local Cohen–Macaulay ring of Krull dimension two. The finite–discrete–tame–wild trichotomy conjecture. The representation type of the category of Cohen–…

Let $(A,\mathfrak{m})$ be a normal surface singularity over the algebraically closed field $k=A/\mathfrak{m}$ of characteristic zero. The tame representation-type conjecture. The r…

Let $A$ be a normal analytic algebra of Krull dimension two over an algebraically closed field of characteristic zero. The quasi-homogeneity conjecture. The isomorphism … is equiva…