16 problems

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…

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 ${\bf G}$ 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 $A$ 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…

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…

Let $\mathfrak{u}$ be a compact semisimple Lie algebra with $*$-compatible Chevalley generators $\{e_r,f_r,h_r\mid r\in I\}$ of $\mathfrak{g}=\mathfrak{u}^{\mathbb{C}}$. Let…

Let $\mathscr{X}_{m,n}$ be the endomorphism algebra of the mixed tensor product considered in the paper, let $K(\lambda)$ be a projective module indexed by a bipartition $\lambda$,…

Let $\mathcal{R}\in\operatorname{CAlg}_{\operatorname{Fin}}$ and let $\mathcal{L}$ be a cyclic commutative algebra over $\mathcal{R}$, meaning a commutative $\operatorname{Fin}$-al…

For $p\geq 3$, let ${\mathcal V}^{(p)}$ be the generalized vertex algebra introduced by … Consider its modules in the category $\mathcal{O}$. Module-category conjecture. For every…

Finite-presentation conjecture. The module $Z$ is finitely presented.

Let $A$ be a ring, and let $\operatorname{Fpinj} A$ be the category of fp-injective $A$-modules. Write $\mathbf K(\operatorname{Inj} A)$ for the homotopy category of complexes of i…

Let $\Lambda$ be an artin algebra, and let $(\mathcal{A},\mathcal{B})$ be a hereditary cotorsion pair in the category of all left $\Lambda$-modules. A cotorsion pair is of finite t…

Let $\Lambda$ be an artin algebra, and let $(\mathcal{A},\mathcal{B})$ be a hereditary cotorsion pair in the category of all left $\Lambda$-modules. A cotorsion pair is hereditary wh…