16 problems

Characterization conjecture. An edge ideal $I(G)$ has homological linear quotients if and only if $G$ is co-chordal and $\operatorname{H}_n^c$-free for any $n\geq 6$.

Let $G$ be a graph with edge ideal $I(G)$. Suppose that, for some integer $q\in\mathbb{N}$, $I(G)^q$ has linear quotients with an ordering $\mathcal N^{(q)}:u_1>\dots>u_r$ of its g…

Let $G$ be a finite simple graph, let $S=K[x_1,\ldots,x_n]$, and let $I(G)\subseteq S$ be its edge ideal. A monomial ideal has linear quotients if its minimal generators admit an o…

Linear quotients conjecture for symbolic powers. If $I(G)$ has a linear resolution, then $I(G)^{(k)}$ has linear quotients for every $k\geq 1$.

Soleyman Jahan–Zheng conjecture. If $I$ has componentwise linear quotients, then $I$ has linear quotients.

Let $I\subset S$ be a componentwise polymatroidal ideal in a polynomial ring $S=K[x_1,\dots,x_n]$. Power conjecture. Every power $I^k$ of $I$ has linear quotients. The source gives…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. For each $j\geq 0$, let $I_{\langle j\rangle}$ be the monomial ideal gener…

Simon’s conjecture for monomial ideals. If $G(I)\subset G(\mathfrak{m}^d)$, then $I$ can be extended to $\mathfrak{m}^d$ by linear quotients.

Simon’s conjecture. If $G(I)\subset G(I_{n,d})$, then $I$ can be extended to $I_{n,d}$ by linear quotients.

Lexicographic homological quotients conjecture. For all $k\geq 0$ and all $\ell\geq 1$, the ideal $\operatorname{HS}_k((I(G)^\vee)^\ell)$ has linear quotients with respect to the l…

Powers conjecture. Every positive power of the cover ideal has homological linear quotients:

Let $I$ be a $t$-spread Veronese ideal, and let $\HS_j(I)$ denote its $j$th homological shift ideal. The linear-quotients conjecture. All the homological shift ideals of $t$-spread…

Let $K$ be a field, let $S=K[x_1,x_2,\ldots,x_n]$, let $I\subset S$ be a proper monomial ideal with linear quotients, and write $[n]=\\{1,\ldots,n\\}$. The stronger variable reduct…

Let $K$ be a field, let $S=K[x_1,x_2,\ldots,x_n]$, and let $I\subset S$ be a monomial ideal with linear quotients, meaning that its minimal generators admit an ordering…

Reverse-lexicographical characterization conjecture. The ideal $I$ is polymatroidal if and only if it has linear quotients with respect to the reverse lexicographical ordering of i…

Strong gcd conjecture. Then $I$ satisfies the strong gcd condition.