6 problems

For a monomial ideal $I$ in a polynomial ring $S$, write $\HS_i(I)$ for its $i$-th homological shift ideal, defined by … An ideal is polymatroidal when its minimal monomial generat…

Let $G$ be a connected graph, and let $I=I_c(G)$ be its complementary edge ideal. For each $i\geq 0$, let $\HS_i(\mathcal{R}(I))$ denote the $i$-th homological shift ideal of the R…

Let $I=I_{\mathbf a,d}$ be the Veronese-type polymatroidal ideal, with the notation $\mathbf a$ and $d$ used in the paper. For $i=1,\ldots,n-1$, let $\operatorname{HS}_i(I)$ denote…

Let $S$ be the polynomial ring in the paper, let $I\subset S$ be a polymatroidal ideal, and let $\operatorname{HS}_i(I^k)$ denote the $i$th homological shift ideal of $I^k$. Associ…

Let $I$ be a polymatroidal ideal, and let $\mathcal{R}(I)$ denote its Rees algebra. For each $i>0$, let $\operatorname{HS}_i(\mathcal{R}(I))$ be the $i$th homological shift algebra…

Crupi–Herzog conjecture. For every $i$ and every positive integer $k$, the ideal $\operatorname{HS}_i\bigl(J(G)^k\bigr)$ has a linear resolution.