94 problems

Let $R$ be a commutative Noetherian ring such that $\operatorname{Spec}(R)$ is connected, and let $M$ be a non-zero, finitely generated $R$-module of finite projective dimension. F…

Let $A=S/I$ be a Cohen–Macaulay quotient ring of codimension $c$. In each homological term of its minimal free resolution, let $a_i$ be the maximal twist and $b_i$ the minimal twis…

Dynkin-diagram conjecture. The ideal is in the linkage class of a complete intersection if and only if its associated graph is a Dynkin diagram.

Let an extremal ray of the cone of Betti tables be given, and consider its integral points. Eisenbud–Fløystad–Weyman conjecture. Every sufficiently large integral point on an extre…

Let $S$ be the polynomial ring in the paper, and let $d=(d_0,\dots,d_{n+1})\in\mathbb{Z}^{n+2}$ be a strictly increasing degree sequence. A pure resolution of type $d$ is a minimal…

Buchsbaum–Eisenbud conjecture. Every finite free resolution is a free resolution of a three-generated ideal.

Cellular replacement conjecture. The labelled regular cell complex $Z$ can be constructed from $Y$ by replacing each vertex $\mathbf{b}_j$ with a copy $\mathbf{b}_jX$; moreover, th…

Let $\Sigma$ be a finite set and let $M$ be a free partially commutative monoid whose generating set is $\Sigma$. The homological dimension of $M$ is the largest size of a subset o…

Let $S$ be a polynomial ring and $I\subset S$ an ideal. Let $e(S/I)$ denote the multiplicity, and let the lower and upper bounds be the quantities in the Herzog–Huneke–Srinivasan m…

Let $R/I$ be a Cohen–Macaulay standard graded algebra of codimension $r$ with minimal graded free resolution, and define $m_i$ and $M_i$ as the least and greatest shifts in homolog…

Let $K$ be a field, and let $R$ be a homogeneous $K$-algebra of codimension $s$. For a graded minimal free resolution of $R$ over a polynomial ring, let $M_i$ denote the largest de…

Let $S=k[x_1,\ldots,x_n]$, let $\mathfrak a=\langle m_1,\ldots,m_l\rangle$ be a monomial ideal, and set $A=S/\mathfrak a$. Let $R$ be the graded commutative algebra constructed fro…

Let $R=k[x_1,\ldots,x_n]$ and let ${\mathfrak c}\subset R$ be a complete intersection generated by forms all of the same degree. Let $A$ be a relatively compressed level Artinian q…

Let $R=k[x_1,\ldots,x_n]$ and let ${\mathfrak c}\subset R$ be a general complete intersection with fixed generator degrees. Let $A=R/I$ be a relatively compressed level quotient of…

Let $X\subset\mathbb P^n$ be a projective variety of dimension $d\geq 1$, with $\operatorname{reg}(X)=m$ and Hilbert polynomial $P_X$. Let $\delta$ satisfy…

Postnikov–Shapiro minimal-resolution conjecture. For $n=3$ and $\phi$ linear, the displayed complex is the minimal free resolution of $J_\phi$. The conjecture predicts that the min…

Let $I \subset R=k[x_1,\dots,x_n]$ be generated by $r$ general forms, and let $c$ be the maximal socle degree of $R/I$, meaning the last degree in which $R/I$ is nonzero. A non-Kos…

Thin Resolution Conjecture. The minimal free resolution is the minimum resolution consistent with the expected Hilbert function; equivalently, it agrees with the Koszul resolution…

Iarrobino's conjecture. The ghost terms arising as a result of Koszul syzygies should be the only kind of exception to the expected minimal free resolution.

MLS LCS formula. If ${\mathcal A}$ is an MLS arrangement with group $G$, then, for all $k\ge 2$,

Let $I$ be a homogeneous licci ideal of height $g$ in a polynomial ring $S$ over a field. Let $D$ be the maximal last shift in a minimal homogeneous free resolution of $I$, let…

Fix $V$, $L$, and $W$. Let $\mathbf{IGr}(m,L)$ be the isotropic Grassmannian, let $\mathscr{O}_m$ be the structure sheaf of $\operatorname{\mathbf{IKal}}(m)$, and let…

Let $L_r^n$ be a curve singularity consisting of $r$ lines through the origin in $\mathbb{A}^n$, and consider the minimal free resolution of its coordinate ring. Minimal Betti-numb…

For an integer $q\geq 1$, let $\mathcal{S}_q=\mathsf{k}[x_A\colon \emptyset\neq A\subseteq [q]]$, and let $\mathcal{E}_q$ be the ideal generated by … For $r\geq 1$, write…

Let $T^n$ be the $n$-torus, let $R_n$ be the ring appearing in the paper, and let $M$ be an $R_n$-module. Suppose that a constructible sheaf…