12 problems

Normality and Cohen–Macaulayness conjecture. Every isotonian algebra $K[P,Q]$ is a normal Cohen–Macaulay domain.

Let $\Gamma=\operatorname{Ind}(\Delta)$, where $\Delta$ is a unit-interval simplicial complex with $\dim(\Delta)>1$. Let $R_{\Gamma}$ be the associated toric ring, let $t$ be its t…

Authors–Shibata conjecture. The following conditions are equivalent:

Let $P$ be a poset, let $\widehat{P}$ be its augmented poset, let $\mathcal{Q}(\widehat{P})$ denote the set of maximal chains of $\widehat{P}$, and let $\mathfrak{C}_P$ and…

Let $C_{2s+1}$ be the cycle graph on $2s+1$ vertices, and let $K[C_{2s+1}]$ denote its toric ring. Write its $h$-vector as a sequence with entries $h_1,h_2,\ldots,h_{s-1}$ appearin…

Let $\mathcal{P}$ be a $(0,1)$-polytope, and let $k[\mathcal{P}]$ be the toric ring generated by $\mathcal{P}$. A ring $R$ is trivial if it can be constructed from polynomial rings…

Let $\mathcal{P}$ be an integral convex polytope, and let … be its toric ring. A graded algebra is strongly Koszul in the sense of Herzog, Hibi, and Restuccia. Herzog–Hibi–Restucci…

Normality–$K_5$-minor conjecture. The following conditions are equivalent:

A toric edge ring is the monoid algebra associated to the edge monoid of a finite graph, and its depth is its commutative-algebraic depth. Depth-seven conjecture. Every toric edge…

Let $P$ be a finite poset that is a disjoint union of chains. Write ${\mathcal J}(P)$ for the poset of order ideals of $P$, and let $K[\mathcal O(P)]$, $K[\mathcal C(P^)]$, and…

Let $Q$ be the affine semigroup associated with the cyclic polytope construction in the paper, and let $K[Q]$ be its semigroup algebra over the field $K$. Write $n$ and $d$ for the…

Simplicity conjecture. If ${\mathcal P}$ is a simple collection of cells, then