27 problems

Takahashi's conjecture. Then $R$ is Cohen–Macaulay.

Let $R$ be an analytic Cohen–Macaulay local ${\mathcal C}$-algebra of countable Cohen–Macaulay type. The Schreyer conjecture. $R$ has at most a one-dimensional singular locus, name…

Let $R$ be the polynomial ring in the variables ${\bf x}$, and let $D_1,\ldots,D_d$ be the inversion factors associated to the basic representatives of a general linear…

Bolognini–Macchia–Strazzanti's conjecture. $J_G$ is Cohen–Macaulay if and only if $G$ is accessible.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive dimension, and suppose that $R$ is unmixed. Let $Q$ be a parameter ideal of $R$. Vasconcelos' Vanishing Conjecture. Th…

Let $L_n^2$ be the graph on $[1,n]$ with edges $\{i,i+1\}$ for $i=1,\ldots,n-1$ and $\{i,i+2\}$ for $i=1,\ldots,n-2$. Let $\Delta_k^t(L_n^2)$ be the associated complex, and let its…

Let $L_n$ be the path graph on vertices $\{x_1,\ldots,x_n\}$, and let $\Delta_k^t(L_n)$ denote the associated complex. The preceding result establishes the claim when $n=2k-1$. Lin…

Vasconcelos' SCM conjecture. If $I$ has deviation at least $3$, is generically a complete intersection, is not Gorenstein, and the resolution of $S/I$ is pure, then $I$ is not SCM.

Let $R={\mathbb C}[{\bf z}^{(v)}]$, let $I_{v,w}$ be the Kazhdan–Lusztig ideal defining the local chart of $X_w$ at $E^{(v)}_\bullet$, and let $\overline I_{v,w}$ be its ideal of l…

Let $\mathcal{D}=(V(\mathcal{D}),E(\mathcal{D}))$ be a vertex-weighted oriented graph, let $G$ be its underlying graph, and let $S=\mathbb{K}[x_1,\ldots,x_n]$ be the polynomial rin…

Let $K$ be a field, let $R=K[x_1,\ldots,x_n]$, and let $\lambda=(\lambda_1,\ldots,\lambda_l)$ be a partition of $n$ satisfying condition (2) or (3) of the stated classification pro…

Hyperplane-section conjecture.

Comparison conjecture. In general,

The quadratic Cohen–Macaulay Hirsch conjecture. If $S/I$ is Cohen–Macaulay, then $I$ is Hirsch.

Let $s,d,n$ be positive integers, let $\nu_d\mathbb{P}^n$ be the degree-$d$ Veronese re-embedding of projective space, and write $\sigma_s(\nu_d\mathbb{P}^n)$ for its $s$-secant va…

Let $\boldsymbol{\tau}=(\tau_1,\tau_2,\tau_3,\ldots)$ be a partition, and let $X_\lambda$ denote the variety associated with the partition $\lambda$ in the algebra of generalized p…

Let $(R,\mathfrak{m},k)$ be a Gorenstein local ring of dimension $d$. The Ding conjecture. … This conjecture concerns the relationship between Auslander's index and generalized Löe…

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring of dimension $d$, let $I$ be an ideal satisfying the sliding depth condition, and let $\mathfrak{a}\subseteq I$ define an alge…

Degree-two Cohen–Macaulay Hirsch conjecture. If $S/I$ is Cohen–Macaulay, then $I$ is Hirsch.

Faithful big Cohen–Macaulay conjecture.

Negative Hilbert coefficient conjecture.

A Gorenstein ring is a commutative Noetherian local ring whose canonical module is isomorphic to the ring itself, and its Cohen–Macaulay representation type describes the cardinali…

Let $({\bf R},{\mathfrak m})$ be a Noetherian local ring with $\dim {\bf R}\geq 2$, and let $U\subseteq {\mathfrak m}{\bf R}^r$ be a parameter module of ${\bf R}^r$, where $r>0$. H…

Glaz's existence problem. Can one define non-Noetherian Cohen–Macaulay rings so that: (i) the definition agrees with the usual definition for Noetherian rings; (ii) coherent regula…

The Cohen–Macaulay conjecture. If ${\mathcal C}$ is an admissible clutter and ${\mathcal C}$ is unmixed, then ${\mathcal C}$ is Cohen–Macaulay.