41 problems

Let $S=mathbbm{k}[x_1,\ldots,x_n]$ be a polynomial ring over a field $mathbbm{k}$. A degree sequence is the sequence of generating degrees of a pure resolution, and folding such a r…

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…

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

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…

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 $I$ be a monomial ideal. A minimal generalized Lyubeznik resolution and a minimal generalized Barile–Macchia resolution are resolutions of $I$ with the indicated generalized co…

Exactness conjecture. The chain complex obtained by applying the Chain Complex Construction Procedure results in a free resolution of the cokernel of the initial map. That is, it i…

Let $S$ be a numerical semigroup, let $R$ be its numerical semigroup algebra, and consider the infinite Apéry resolution of the residue field over $R$. Cellular structure conjectur…

Buchsbaum–Eisenbud conjectural resolution. A resolution of $\operatorname{coker}(\varphi^*_{p-1,1})$ should always start out as

Let $C_m$ be the Kunz cone, let $F$ be a face of $C_m$, and let $S$ be a numerical semigroup whose corresponding point lies in $F$. Let $J_S$ be the ideal in the Apéry-resolution c…

Let $C_m$ be the Kunz cone, let $F$ be a face of $C_m$, and let $S$ range over the numerical semigroups whose corresponding points lie in the interior of $F$. For each such $S$, le…

Let $R=k[x,y,z]$ and let $f=(x^2-y^2)z^{d-1} -(x^{d-1}-y^{d-1})x^2 -y^{d+1}$, with $d\geq 2$, and let $J_f$ be its gradient ideal. Syzygy-resolution conjecture. The maximal degree…

Let $R$ be a commutative local or graded ring, and let $M$ be a perfect module with a minimal free resolution of one of the formats … A module is in the BR-linkage class of a Buchs…

Let $I$ be a monomial ideal, let $\mathcal{F}_L$ be a Lyubeznik resolution, and let Algorithm $1$ be the trimming algorithm defined in the source. The resulting complex is compared…

Let $R$ be a polynomial ring over a field $\boldsymbol{k}$, and let $I\subseteq R$ be a monomial ideal. Let $\mathcal{F}_L$ be a Lyubeznik resolution of $R/I$. A Barile–Macchia res…

Let $R$ be a local Gorenstein ring and let $I\subset R$ be a perfect ideal of Dynkin type. A structure map $w^{(i)}_{j,k}(I)$ has strictly positive rank modulo the maximal ideal if…

Let $I$ be a perfect ideal of height $3$ in a Gorenstein local ring $R$ with infinite residue field. For a structure map $w^{(i)}_{j,k}(J)$, write “nonzero modulo the maximal ideal…

Let $R$ be a Gorenstein local ring and let $\mathfrak{f}=(1,m,m+n-1,n)$ be a resolution format realized by some grade $3$ perfect ideal in $R$. A perfect ideal is licci if it belon…

Let $A$ be an Artin Gorenstein ring with codimension $c$ and regularity $r$, and let a Betti table refer to the graded Betti table of a minimal free resolution of $A$. The codimens…

Stronger Buchsbaum–Eisenbud–Horrocks conjecture for syzygy ranks.

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, let $I=(f_1,\dots,f_r)$ be a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal of $S$, and let $\Theta$ be the Jacobian…

Let $K$ be a field, let $R=K[x_1,\ldots,x_{2a+1}]$, and let $I^{\rm Sp}_{(a,a,1)}$ be the Specht ideal associated to the partition $(a,a,1)$. Linear-resolution conjecture. If … the…