22 problems

Let $R$ be a complete Gorenstein local ring that is not an isolated singularity. Say that $R$ has finite CM-plus type if it has only finitely many indecomposable maximal Cohen–Maca…

Let $f\in R=k[z_0,\dots,z_n]$ be a homogeneous polynomial. Let $\operatorname{MCM\mathchar`-rank}(f)$ be the minimum of the integers $r$ for which there is a reduced, homogeneous m…

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…

Dao–Kurano finiteness conjecture. Under some mild assumptions, $R$ has only finitely many maximal Cohen–Macaulay $R$-modules of rank one up to isomorphism.

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

Let $A_\varepsilon$ be the skew quadric hypersurface and let $S_\varepsilon$ be the associated graded algebra. Let $\mathcal{E}_\varepsilon$ be the point scheme of $S_\varepsilon$,…

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.…

Let $R$ be a complete Noetherian local domain, and fix an integer $N$. A maximal Cohen–Macaulay module is rigid if $\operatorname{Ext}^1_R(M,M)=0$. The rank of an $R$-module is its…

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 $R$.

Frobenius-transform realization conjecture. A new condition involving local cohomology for the existence of a small MCM should be realizable inside $\mathbf{F}_{*}M$ for some, poss…

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

A ring $R$ is weakly F-split if it satisfies the weak F-splitting condition defined in the paper. Weak F-splitting conjecture. Every complete ring is weakly F-split. This would ext…

Ext-module conjecture. For every integer $i$ with $0\leq i\leq c$, the module

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…