37 problems

Brauer–Thrall conjecture. The following statements hold:

Let $R$ be a non-Gorenstein local ring, and suppose that there is a nonfree totally reflexive $R$-module. The Takahashi conjecture. There are infinitely many isomorphism classes of…

Direct-sum-of-uniforms conjecture. $E$ is a direct sum of uniform modules.

Fix $m\geq 0$. Let $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $A_m^{(n)}$ be the set of $(n+1,m)$-excellent polynomials, meaning polynomials $f$ for which $\delta_{n+1}(f)$ is a polynom…

Uniform generation conjecture. There is a $k\in\mathbb{N}$ such that $\mathcal O_{\mathbf{A},\mathbf{R}}$ is uniformly generated by $k$-ary $(\mathbf{A},\mathbf{R})$-minors.

Let $\Lambda$ be a finite-dimensional algebra. A module $M\in\operatorname{mod}\Lambda$ is a brick if $\operatorname{End}_{\Lambda}(M)$ is a division algebra. The algebra $\Lambda$…

Let $A_n$ be the $n$th Weyl algebra, and let a finitely generated $A_n$-module be torsion when it has rank zero. A module is cyclic if it is generated by one element. Stafford's co…

Let $\widehat{\mathfrak{b}}$ be the algebra whose modules are under consideration, and let $\widehat{V}$ and $\widehat{W}$ be the $\widehat{\mathfrak{b}}$-submodules appearing in t…

Let $\mathcal{X}$ be a class of modules. A class is covering if every module $M$ has an $\mathcal{X}$-cover, namely an $\mathcal{X}$-precover $f\colon X\to M$ such that every…

Let $M$ be a closed oriented $3$-manifold, and let $\mathcal S_{2,\infty}(M)$ denote its Kauffman bracket skein module over $\mathbb Z[A^{\pm 1}]$. Marché's decomposition conjectur…

Converse Wakamatsu tilting conjecture. If $|T|=|\Lambda|$, then $T$ is Wakamatsu tilting.

Weak maximal self-orthogonal conjecture. Every projectively Wakamatsu tilting module is maximal self-orthogonal.

Maximal self-orthogonal conjecture. Every Wakamatsu tilting module is maximal self-orthogonal.

Let $A$ be an infinite-dimensional unital Banach algebra. Dales–Zelazko conjecture. Then $A$ has a maximal left ideal which is not finitely generated. The conjecture concerns the e…

Let $p$ be prime, let $T$ be a non-solvable transitive subgroup of $\mathrm{Sym}(p)$, and let $D$ be the diagonal module used in the monomial-group classification. A module is unis…

Let $p$ be prime, let $T$ be a non-solvable transitive subgroup of $\mathrm{Sym}(p)$, and let $D$ be the diagonal subgroup appearing in the monomial group…

Let $p$ be prime, let $D_{\{p\}'}$ denote the relevant diagonal module, and let $T$ be a non-solvable transitive subgroup of $\mathrm{Sym}(p)$. Write $N_{\mathrm{Sym}(p)}(T)$ for t…

Splitting conjecture. There exist $R$ and $\mathbb{G}$ such that $\operatorname{Ext}(S^G;\Omega^1_{eq}(\hat{P};\hat{\mathfrak{g}}))$ is injective as an $R[\mathbb{G}]$-module, ever…

Contramodule Bass-flat equivalence conjecture. The following conditions are equivalent to each other and to the conditions listed in the theorem: (i$^\flat$) all Bass flat left…

Let $\mathfrak{A}$ be an order in a finite-dimensional semisimple algebra, and let $M$ be a discrete object of $\mathsf{LCA}_{\mathfrak{A}}$. Projective resolutions of $M$ in…

Let $(\mathcal{T},\mathcal{F})$ be an elementary torsion pair in $\operatorname{Mod}\textrm{-}{R}$, where $R$ is a ring. The torsion class $\mathcal{T}$ is said to have enough pure…

Bongartz–Ringel accessibility conjecture. Every indecomposable module of finite length over any algebra is accessible.

Let $V$ be a finite-dimensional irreducible $\widetilde \square_q$-module, where $\widetilde \square_q$ and $\boxtimes_q$ are the algebras appearing in the paper, with generators…

Let $R$ be a complete $d$-dimensional local ring of positive characteristic, and let $M$ be a $d$-dimensional $R$-module. Write $\mathbf{F}_{*}^nM$ for the $n$-fold Frobenius trans…

Let $q\geqslant 2$ be a finite field size and let $n\geqslant 1$. Write $\operatorname{unc}(n,q)$ for the number of uncyclic matrices in $\operatorname{M}(n,q)$, where an uncyclic…