264 problems

Tree discriminant magnitude homology conjecture. For $k=3,4$,

For every dg algebra $A$ over a field $k$ such that $\dim_k H^*(A)<\infty$, the regular module $A$ cogenerates the unbounded derived category $D(A)$ of dg $A$-modules; equivalently…

For every prime $p$, let $A_*$ denote the mod-$p$ dual Steenrod algebra and let $\operatorname{GrComod}_{A_*}$ denote the full category of graded $A_*$-comodules. The conjecture as…

For every finite-dimensional quantum complete intersection $A$ over an arbitrary field and every finite-dimensional left $A$-module $M$, if $\operatorname{Ext}^{1}_{A}(M,M)=0$ and…

Let $A$ be a finite-dimensional algebra over a field. Assume that $A$ satisfies the Auslander condition: for every finitely generated left $A$-module $M$, there exists an integer…

Let $A$ be a finite-dimensional algebra over a field $K$, and let $\mathcal{G}\subseteq \mathrm{HH}^*(A)$ be the Gerstenhaber ideal generated by all homogeneous nilpotent elements…

Let $P$ be a group, let $L(P)$ denote its free multiplicative Lie algebra, let $\Gamma_n(P)$ be the normal subgroup generated by Lie-bracket commutators of weight $n$, and let…

For every finite-dimensional algebra $A$ over a field and every finitely generated $A$-module $M$, if $\operatorname{Ext}_A^i(M,M)=0$ for all $i>0$ and $M$ is a generator, equivale…

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Let $(R,\mathfrak m)$ be a Noetherian local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $\operatorname{pd}_R M<\infty$ and $\ell_R(M\otimes_RN)<\infty$. T…

Let $(V,0)$ be an isolated complex hypersurface singularity defined by $f\in\mathbb{C}\{x_1,\ldots,x_n\}$, with $n\ge 2$ and $\operatorname{mult}(f)\ge 3$. Let $A(V)$ be the moduli…

Let $k$ be a field and let $A$ be a finite-dimensional $k$-algebra of finite global dimension. Choose representatives $P_1,\ldots,P_n$ of the isomorphism classes of indecomposable…

For every finite-dimensional associative algebra $A$ over a field $K$, the quotient of its Hochschild cohomology ring by the ideal of nilpotent elements is a finitely generated $K$…

For every annular link $L$, the annular Khovanov homology of $L$ is nonzero in the $k$-grading equal to the wrapping number of $L$:…

Balmer's Nerves of Steel Conjecture. The map $\phi$ is bijective. This conjecture seeks to reconcile the homological and triangular spectra of a rigid tensor-triangulated category.…

Let $\operatorname{\textit{Quilt}}$ be the operad of quilts, and let $H_k(\operatorname{Quilt})$ denote its homology in degree $k$. Acyclicity conjecture for the Quilt operad.…

Let $M$ be a matroid, or let $\mathcal{A}$ be a hyperplane arrangement, and let $A(M)$, respectively $A(\mathcal{A})$, denote its Orlik–Solomon algebra. A matroid or arrangement is…

Let $\Gamma$ be a symmetric numerical semigroup, let $\Bbbk$ be a field, and let $R=\Bbbk[\Gamma]$ be its semigroup algebra. Let $M$ be a 2-generated ideal of $R$, viewed as an $R$…

Let $R$ be a commutative Noetherian ring such that $\operatorname{Spec}(R)$ is connected, and let $M$ be a non-zero, finitely generated $R$-module of finite projective dimension. F…

Odd-primary transfer conjecture. The transfer homomorphism $Tr_h^{\mathscr A_p}(\mathbb F_p)$ is one-to-one for any odd prime $p$ and $1\leq h\leq 4$. This extends the known low-ra…

Let $p$ be prime and let $M$ be a $C_p$-Mackey functor. A projective resolution of $M$ is eventually $4$-periodic if its differentials repeat with period $4$ from some point onward…

Deligne's conjecture. There exists a natural action of the operad $\mathsf{Chains}(C_2)$ on the Hochschild complex $\mathcal{C}^\bullet(\mathcal{A},\mathcal{A})$ for any associativ…

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 $A$ be a finite-dimensional algebra over a field. The condition that all higher extensions from its $k$-dual back to $A$ vanish is … Here $D(A)$ denotes the $k$-dual of $A$. Ta…

Let $R$ be the local ring under consideration and let $M$ be an $R$-module. Generalized Auslander–Reiten conjecture. If … for all $n\gg 1$, then $M$ has finite projective dimension…