9 problems

Let $R=\mathsf{k}[X_1,\ldots,X_n]$ be a polynomial ring over a field. A graded $R$-module has a pure resolution when the nonzero graded shifts in its minimal free resolution occur…

Let $R$ be a standard graded algebra over a field $\mathsf{k}$. Let $\mathbb{B}_{\mathbb{Q}}(R)$ be the rational cone generated by Betti tables of finitely generated graded $R$-mod…

Let $R$ be the polynomial ring in $n$ variables, and let $\mathbf{c}=(c_p,\dots,c_0)$ be a strictly decreasing sequence of integers with $p\leq n$. A pure Betti diagram has shift t…

For the $d$-uple Veronese embedding of $\mathbb P^2$, let $\beta(\mathbb P^2,b;d)$ denote its Betti table with twist parameter $b$, and let its Boij–Söderberg coefficients be the c…

Recursive-coefficient conjecture. For $\varepsilon<s<\varepsilon+c$, one has

Let $L$ be an Artinian lex-ideal in the polynomial ring $\mathbf{k}[x,y,z]$. The Boij–Söderberg decomposition of $L$ is the decomposition into pure diagrams described by Theorem 2.…

Let $L=x(x,y,z^t)+J$ be an Artinian lex-segment ideal in $\mathbf{k}[x,y,z]$, where $J$ is as in the surrounding discussion and the case is excluded from the proved result for Theo…

Betti-cone conjecture. The cone $\mathrm{B}_{\mathbb{Q}}(Q)$ is an $(n+1)$-dimensional cone, and its closure is defined by the following $(n+2)$ extremal rays:

Let $A$ be the relevant symmetric algebra with a group action, and let $M$ be a finite-length equivariant $A$-module with equivariant minimal free resolution ${\bf F}_\bullet$. For…