35 problems

Let $\Lambda$ be a finite-dimensional $\mathbbm{k}$-algebra, and let $\mathcal{N}$ be the ideal of $HH^{\ast}(\Lambda)$ generated by all homogeneous nilpotent elements. Snashall–So…

Let $\mathcal K$ be a finite tensor category, let $\mathsf{C}$ denote its categorical center, and let … be the comparison map. The comparison-map homeomorphism conjecture. For ever…

Let $k$ be an algebraically closed field of characteristic $p$, let $E$ be an elementary abelian $p$-group, and let $M$ be a finite-dimensional $kE$-module. The rank variety is … w…

Let ${\mathfrak g}$ be a basic simple Lie superalgebra, let $\lambda$ be a weight, and let $L(\lambda)$ be the corresponding finite-dimensional irreducible representation. Write…

Let $\mathfrak{g}$ be a classical Lie superalgebra, let $\mathfrak{e}$ be its detecting subalgebra, and let $L$ be a simple $\mathfrak{g}$-supermodule. Denote by…

Quantum-group divisibility conjecture. Then $\dim Q(\lambda)$ is divisible by $l^a$.

Humphreys' support-variety conjecture. The map ${\mathcal H}$ coincides with Lusztig's bijection.

Let $A$ be a finite-dimensional unipotent Hopf algebra, let $A^{\operatorname{env}}$ be its enveloping Hopf algebra, and let … be the comparison map. The comparison-map conjecture.…

Scheme-theoretic traditional Humphreys conjecture. For every $w\in W_{\mathrm{ext}}^\varnothing$, the coherent sheaf

Relative Humphreys conjecture.

Traditional Humphreys conjecture.

Assume that a local ring $R$ has minimal regular presentation $Q/I$, let $k$ be its residue field, and fix minimal generators $f_1,\ldots,f_n$ of $I$. Let … and let … where each…

Central cohomological support conjecture. The central cohomological support satisfies these conditions, and consequently

Boe's support-variety identification conjecture. These maps should induce isomorphisms of varieties

For a finite-dimensional restricted Lie superalgebra over an algebraically closed field of characteristic $p\geq 3$, let the support variety of a finite-dimensional module be defin…

Standard conjecture. The following assertions hold:

Assume $(A,\mathfrak m)$ is a complete local complete intersection of dimension $d$ and codimension $c$, with algebraically closed residue field $k=A/\mathfrak m$. Let $X$ be an al…

Let $G$ be a quasi-reductive supergroup, and let $M\in\operatorname{Rep}(G)$. Write $\mathfrak{g}_{\mathrm{neat}}$ for the neat locus in the odd part of the Lie superalgebra, and…

Suppose $k$ is algebraically closed. Let $G$ be a height-$1$ infinitesimal unipotent $k$-supergroup scheme, let $M$ be a finite-dimensional rational $G$-supermodule, and let…

Let $k$ be algebraically closed, and let $G$ be an infinitesimal unipotent $k$-supergroup scheme of height $\leq r$. Let $H(G,k)$ denote the cohomology algebra, let…

Let $G$, $G_1$, $W$, ${}^{\mathrm{f}}W^{\mathrm{f}}$, $\mathcal{C}_p$, $\iota_{\Bbbk}$, and the corresponding nilpotent orbit $\mathscr{O}^{\mathbb{C}}_{\mathbf{c}(w)}$ be as in th…

Let $E$con be an elementary abelian $p$-group, let $k$ be an algebraically closed field of characteristic $p$, and write … For a $kE$-module $M$, let $V_G(M)$ be the support variet…

Let $G$ be a semisimple simply connected algebraic group over a field $k$ of characteristic $p>h$, let $G_1$ be its Frobenius kernel, and let $\mathbb{U}_{\zeta}(\mathfrak{g})$ be…

Let $G$ be a semisimple simply connected algebraic group over a field of characteristic $p$, with Frobenius kernel $G_1$, weight lattice $X(T)$, and nilpotent cone…

Carlson's support variety conjecture. The support variety of $M$ should admit a description in terms of cyclic shifted subgroups of the group algebra $kE$, without recourse to coho…