42 problems

Let $(R,\mathfrak m,k)$ be a Noetherian local ring with a canonical module $K_R$. Tachikawa's conjecture. If … for all $i>0$, then $R$ is Gorenstein. This is the commutative versio…

Let $(R,\mathfrak{m},k)$ be a commutative noetherian local ring, where $\mathfrak{m}$ is its maximal ideal and $k=R/\mathfrak{m}$ its residue field. A finitely generated $R$-module…

Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring, let $I$ be a perfect ideal of grade $3$, and let $\operatorname{Tor}^R(R/I,\Bbbk)$ be its Tor algebra. The Tor algebra is said…

Let $\mathcal{P}$ be a collection of cells. The Gorenstein characterization conjecture asserts that the following are equivalent: 1. $K[\mathcal{P}]$ is Gorenstein; 2. the $h$-poly…

Hirose–Watanabe–Yoshida conjecture. One always has

Stanley's unimodality conjecture. The $h$-vector of $A$ is unimodal.

Let $R$ be a commutative Noetherian Henselian local ring with maximal ideal $\mathfrak m$ and residue class field $k$, and let $\mathrm{mod}\,R$ denote the category of finitely gen…

Let $(R,\mathfrak m,k)$ be a commutative noetherian local ring with a dualizing complex $\operatorname{D}(R)$, shifted so that $\operatorname{H}_i(\operatorname{D}(R))=0$ for $i<0$…

Tom and Jerry conjecture. Rings with 9×16 resolutions fall into two standard families, called Tom and Jerry, bearing a family resemblance to the Segre embeddings of…

Let $R$ be a commutative Noetherian local ring and let $M$ be a finitely generated $R$-module. Assume that $M$ has finite length and that $\operatorname{Gdim}(M)$, the Gorenstein d…

Domino-stability conjecture. The ring $K[\mathcal{P}]$ is Gorenstein if and only if $\mathcal{P}$ is domino-stable.

Let $R$ be a Cohen–Macaulay local ring with canonical module $\omega$. Jorgensen–Leuschke conjecture. The inequality … implies that $R$ is Gorenstein. The paper states this as an o…

Let $\mathbb{X}$ be a family of projective point sets and let $I(\mathbb{X})$ denote its vanishing ideal. Vaz Pinto's conjecture. The ideal $I(\mathbb{X})$ is Gorenstein if and onl…

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring that has a canonical ideal $\mathcal{C}$. The canonical degree and bi-canonical degree are denoted by $\operatorname{cdeg}(R)$…

Let $X_w$ be a Schubert variety, and let $({\mathfrak S},\prec_I)$ be the poset of permutation intervals. The order ideal of intervals corresponding to points where Schubert variet…

Let $R$ be a Gorenstein ring, and write $\operatorname{index}(R)$ for its index and $\operatorname{gll}(R)$ for its generalized Loewy length. Ding's conjecture. One always has … Th…

Let $R_{\Gamma_{3k+2}}$ be the edge algebra associated with the graph $\Gamma_{3k+2}$. Gorenstein conjecture for $R_{\Gamma_{3k+2}}$. The algebra $R_{\Gamma_{3k+2}}$ is Gorenstein…

Let $G$ be a graph with homogeneous subring $S$, and assume that $S$ is normal. Say that $G$ is unmixed when all its relevant minimal vertex covers have the same cardinality, and l…

Let $G$ be an unmixed graph, let $S$ be its homogeneous subring, and let $B$ be the configuration defining $S$. Write $(\mathbb{R}_+B)^\circ$ for the relative interior of the cone…

Higher-dimensional Huneke–Wiegand conjecture. If $M\otimes_R M^*$ is reflexive, then $M$ is free.

Let $(R,\mathfrak{m})$ be an artinian Gorenstein ring whose embedding dimension is $3$. Ikeda's conjecture. … The conjecture concerns equality between the Dilworth number and the s…

Let $\gamma$ be any $(p,2)$-clan with $p\geq 2$, and let $Y_{\gamma}$ be its orbit closure. lci–Gorenstein equivalence conjecture. $Y_{\gamma}$ is a local complete intersection if…

Let $\gamma$ be a $(p,2)$-clan with $p\geq 2$, and let $Y_{\gamma}$ be its orbit closure. Gorenstein pattern-avoidance conjecture. $Y_{\gamma}$ is Gorenstein if and only if…

Let $Y_{\gamma}$ be an orbit closure, and let $\operatorname{Maxsing}(Y_{\gamma})$ be the set of Bruhat-maximal orbits along which $Y_{\gamma}$ is singular. Let…

Let $A$ be a standard graded Gorenstein integral domain, and write its Hilbert series as $H(A;z)=h(A;z)/(1-z)^{\dim(A)}$, where $h(A;z)$ is the $h$-polynomial. Brenti's conjecture.…