28 problems

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 $R$ 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 $\mathcal{P}$ 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…

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…

Hirose–Watanabe–Yoshida conjecture. One always has

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 $(R,\mathfrak{m})$ 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 $R_{\Gamma_{3k+2}}$ 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 $\gamma$ 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.…

Let $R$ be a standard graded strongly $F$-regular ring, with $R_0$ an $F$-finite field of characteristic $p>0$. Let $\mathfrak{m}$ be the unique homogeneous maximal ideal of $R$. H…

Let $\mathfrak{h}$ and $\mathfrak{H}$ be the Eisenstein components of the cuspidal and full Hecke algebras, respectively. The ideals $I$ and $I_{\mathfrak{H}}$ are their Eisenstein…

Let $Y_{\gamma}$ be a $K$-orbit closure, and suppose the maximal components of its singular locus are $\mathcal O_{\alpha_1},\ldots,\mathcal O_{\alpha_k}$. Non-Gorenstein-locus con…

Let $S$ be the polynomial ring and let $I_R$ be the ideal associated with a classifying map $\alpha\in\operatorname{SpH}_k(S)$ as in Lemma. For a spinor set $J\cup J^c$, let…

General partial fraction decomposition conjecture. Under these assumptions, $P(t)$ has a unique partial fraction decomposition

Let $R$ be a local ring in the class $\mathbf{G}(r)$, where $r\ge2$. Gorensteinness conjecture. If $R$ is in $\mathbf{G}(r)$ for some $r\ge2$, then $R$ is Gorenstein. The only know…

Faber–Pandharipande's compactified Gorenstein conjecture. The ring $R^*(\overline{M}_{g,n})$ is Gorenstein with socle in degree

Faber–Pandharipande's rational-tail Gorenstein conjecture. The ring $R^*(M_{g,n}^{\mathrm{rt}})$ is Gorenstein with socle in degree