37 problems

Let $R$ be a Gorenstein singularity, let $M$ be a maximal Cohen–Macaulay $R$-module, and set $A:=\operatorname{End}_R(R\oplus M)$. Let $e\in A$ be the idempotent corresponding to…

Loday's proposal. These data should satisfy

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\mathbf{\Gamma}^{[a,b]}(\underline{w}_0)$ and…

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\widehat{\mathbf{\Gamma}}^{[a,b]}(\underline{w}_0)$ be the c…

Let $(S,n,{\bf a})$ be a parametrized surface and let $R(S,n,{\bf a};k)$ be the dga generated by diagrams of $k$ strands between the corresponding collections of arcs. Let $\mu_m$…

Let $\mathcal{O}^{k,n-k}$ be the parabolic subcategory of a regular block of $\mathcal{O}(\mathfrak{gl}_n)$, and consider the $A_{\infty}$-Ext algebra of the Verma modules in this…

Let $R_k$ be the dga defined from $k$ parallel copies of the arc in the local model $S=D^2$, with no singular points and with two boundary stops. Its cohomology algebra is…

Let $d$ be a positive integer, and let $\Pi$ be an $H^0$-finite connective $(d+1)$-CalabiYau dg algebra over a field $k$. Let $\mathcal{C}(\Pi)$ be its $d$-CalabiYau cluster catego…

Balanced dualizing dg-module conjecture. The graded commutativity hypothesis can be removed from the theorem asserting that every Gorenstein dg-algebra as in the setup admits a bal…

Benson's generation conjecture. The category $\mathcal{D}^b(C^*BG)$ is generated as a thick subcategory of $\mathcal{D}(C^*BG)$ by the objects $\operatorname{Hom}_{C_*G}(k,X)$ for…

Let $A\to B$ be the DG algebra homomorphism in the setting preceding the conjecture, and let $N$ be a DG $B$-module. Write $\operatorname{\texttt{(AR1)}}$ and…

A $3$-Calabi--Yau algebra is an algebra with the corresponding Calabi--Yau property, and a quiver with potential consists of a quiver together with a potential whose associated Gin…

Let $R_{\mathfrak{F}}$ be the DG algebra introduced earlier in the paper, and let $n$ denote the parameter occurring in its construction. The dimension of a triangulated category i…

Let $k$ be the base field, let $M\in M_3(k)$, and let $\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ denote the associated cochain DG skew polynomial algebra. Write…

Infinite free-extension naive lifting conjecture. If

Let $A$ be a connective dga and let $S$ be a one-dimensional $A$-module. Let $E\coloneqq {\mathrm{\normalfont\mathbb{R}}}\mathrm{End}_A(S)$ be the derived endomorphism dga of $S$.…

Let $({}_{dg}^{red}Q_b^{{\underline{N}}},0)$ be the reduced dg-quiver Schur algebra and let $({}_{dg}Q_b^{{\underline{N}}},0)$ be the non-reduced dg-quiver Schur algebra. Dg-Morita…

Let $T_b^{\lambda,\underline{N}}$ be the dg-enhanced KLRW algebra and let $({}_{dg}Q_b^{{\underline{N}}},d_N)$ be the dg-quiver Schur algebra, with the two maps of graded dg-algebr…

Let $({}_{dg}Q_b^{{\underline{N}}},d_N)$ be the dg-quiver Schur algebra defined as the dg-endomorphism algebra of the direct sum of the dg-enhanced modules…

Let $G_{\rho}^{{\underline{N}}}$ and $Y_{\rho}^{{\underline{N}}}$ be the dg-enhanced and ordinary cyclic modules constructed from the tensor-product data, with differential $d_N$ o…

Let $k$ be an algebraically closed field, let $X$ be a projective variety with an action of a linear algebraic group $G$, and suppose that $X$ has finitely many $G$-orbits. Let…

Let $\mathscr R$ be a smooth finite-dimensional DG algebra, and let $\mathscr{P}\!\mathit{erf}\!\operatorname{--}\mathscr R$ denote its DG category of perfect modules. A phantom is…

Let $Z$ be a projective scheme, let ${\Bbbk}$ be the base field, and let $\mathscr R$ be a finitely-dimensional DG algebra. Suppose that the DG category…

Let $R$ be the DG-algebra under consideration, and let ${\mathcal F}$ be the full subcategory of ${\mathsf{D}}(R)$ consisting of the relevant DG-flat objects. Let…

Let $k$ be a base field, and let $A$ and $B$ be DG algebras over $k$. For a field extension $k^{\prime}/k$, write $A_{k^{\prime}}=A\otimes_k k^{\prime}$ and…