49 problems

Hochster's conjecture. The invariant ring of any linearly reductive group acting on a polynomial ring over a field is Cohen–Macaulay.

Itoh's conjecture. If $A$ is Gorenstein and the third Hilbert coefficient satisfies $e_3^{\mathfrak{a}^*}(A)=0$, then $G_{\mathfrak{a}}(A)^*$ is Cohen–Macaulay.

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…

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

Conca–Herzog–Trung–Valla conjecture. If $R_A(I)$ is a Cohen–Macaulay ring, then there exists a diagonal $\Delta$ such that $R_A(I)_\Delta$ is a Cohen–Macaulay ring.

Let $(R,\mathfrak m)$ be a Cohen–Macaulay local ring of dimension $d$, let $\mu(\mathfrak m)$ denote the minimal number of generators of $\mathfrak m$, and let…

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

Soleyman Jahan's conjecture. Every generic monomial ideal is pretty clean.

Let $G$ be a graph, let $K$ be a field, let $S=K[x_i:i\in V(G)]$, let $I(G)$ denote the ideal associated to $G$, and let $\nu(G)$ denote its matching number. For each positive inte…

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…

Bolognini–Ene–Rinaldo–Saeedi Madani conjecture.

Benedetti–Varbaro conjecture. Every Cohen–Macaulay homogeneous ideal generated in degree two is Hirsch.

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…

Let $S$ be a polynomial ring and let $I\subseteq S$ be a quadratic ideal. Assume that $S/I$ is Cohen–Macaulay. The dual graph of $I$ has vertex set $\operatorname{Min}(I)$, with an…

Hyperplane-section conjecture.

Comparison conjecture. In general,

Ulrich module existence conjecture. Ulrich modules exist over every Cohen–Macaulay local ring $R$.

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

Bolognini–Villarreal conjecture. Every Cohen–Macaulay homogeneous ideal generated in degree two is Hirsch. This paper verifies the conjecture for Cohen–Macaulay binomial edge ideal…

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…