38 problems

Let $A$ be an algebra, and let $\tau$-tilting theory be considered over a field $\mathbb{K}$. An algebra is of finite representation type when it has only finitely many isomorphism…

Etingof–Ostrik's conjecture. An algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras.

Telescope Conjecture for module categories. Every such cotorsion pair is of finite type.

Let $A$ be a finite-dimensional algebra, and let $A\text{-mod}$ denote its module category. An abelian category is D-standard if every pseudo-identity on its bounded derived catego…

Existence conjecture. There exists a non-almost-semi-inner automorphism $\varphi\in\operatorname{Aut}(_{L}\mathcal{M}^{0})$.

Suppose that $V$ is a simple vertex operator algebra and that a finite group $G$ acts on $V$. Let $V^G$ denote the subalgebra of $G$-invariants, called the $G$-orbifold of $V$. For…

Let $A$ be a ring and let ${\mathcal C}$ be an $A$-coring, meaning an $A$-bimodule equipped with counit and comultiplication maps satisfying the coring axioms. The coring…

The conjecture. The category $\mathcal M(p)$ is equivalent to the category of modules over $\mathbf A^{(p)}\times\mathbb Z_{<p}^r$ for an appropriate $r$ depending on $p$.

Correspondence conjecture. The table below, obtained using the preceding information together with heuristics, gives the correct correspondence between module categories and modula…

Let ${\mathcal C}$ be a rigid monoidal category with finitely many irreducible objects. A module category over ${\mathcal C}$ is called indecomposable if it is not equivalent to a…

Magnitude conjecture. 1. We have

Let $\mathcal{C}$ be the braided tensor category of finite-dimensional modules over $u_q(\mathfrak{g})$, where $q$ is a root of unity of odd order and the category is equipped with…

Let $u_q(\mathfrak{g})$ be a general small quantum group, and let $\mathbf{vect}$ denote its canonical module category of finite-dimensional vector spaces. Non-full exactness conje…

Let ${\bf G}$ be the group under consideration, let $\Bbbk$ be the coefficient field, and let $\mathscr{X}({\bf G})$ be the abelian category of $\Bbbk {\bf G}$-modules defined by t…

Let $R\longrightarrow A$ be a homomorphism of associative rings. Proposition establishes five equivalent conditions: Becker-contraacyclicity for complexes of $R$-cotorsion left $A$…

Let $H$) be a finite-dimensional quasitriangular Hopf algebra, let $\tilde H$ be its formal ribbon extension, and let $N$ be a finite-dimensional indecomposable $\tilde H$-module.…

Let $A$ be a domain and let $\mathrm{Mod}_A^{\mathrm{m} gen}$ be the Serre quotient of the category of equivariant $A$-modules by its torsion subcategory. Let $G$ be the automorphi…

Let $V$ be a meromorphic open-string vertex algebra and let $W$ be a $V$-bimodule. Write $\widehat{H}^1(V,W)$ for the first Hochschild-like cohomology and $Z^1(V,W)$ for the space…

Countable Telescope Conjecture for module categories. The cotorsion pair $(\mathcal{A},\mathcal{B})$ is of countable type.

Let $V$ be a (meromorphic open-string) vertex algebra and let $W$ be a $V$-bimodule. Write $H^1(V,W)$ for the first cohomology and $Z^1(V,W)$ for the space of zero-mode derivations…

Let $V$ be a vertex operator algebra such that all irreducible ordinary $V$-modules are $C_1$-cofinite, and let $W$ be a generalized $V$-module. C1-cofiniteness and finite-length c…

Let $\Gamma$ be the fixed data defining the unitary $*$-2-category $\operatorname{TLJ}(\Gamma)$. A MW-type bipartite balanced $\Gamma$-fair graph is a bipartite balanced $\Gamma$-f…

A $VIC(\mathbb{Z})$-module is a functor from the category $VIC(\mathbb{Z})$; it is finitely generated if it is generated by finitely many elements under the morphisms of this categ…

Classification conjecture. Theorem classification is in fact a classification of module categories for $\mathcal{C}^m$; equivalently, every module category for $\mathcal{C}^m$ is e…

Trivalent cell-system conjecture. The trivalent cell system in any canonical module category with nimrep graph $\sigma_{2m}$ is equivalent to one of those presented in the classifi…