35 problems

Let $k$ be a field of characteristic $0$, and let ${\mathcal A}$ be a smooth proper dg category over $k$. For every integer $n\geq 1$, consider the complex … where…

Let $M$ be a symplectic manifold, let ${\cal W}$ be its Weyl algebra bundle, and let $(\Omega(M, {\cal W}), \nabla, R)$ be the associated $Q$-algebra. A finitely generated projecti…

Realization conjecture. There exist an exact category $\mathcal{B}$ of finite homological dimension and a group $H$ acting on $\mathsf{D}^{b}_{\mathrm{dg}}(\mathcal{B})$ such that…

Decategorification isomorphism conjecture. The injective algebra homomorphism above is always an isomorphism.

Let $\mathcal{C}$ be a $\mathbb{C}$-linear dg-category. Write $K_*^{\operatorname{top}}(\mathcal{C})$ for its topological $K$-theory groups and $HP_*(\mathcal{C})$ for its periodic…

Let $\mathcal T$ be a triangulated category with $\operatorname{FM} \mathcal T\operatorname{NE}\emptyset$. The proper Fourier–Mukai locus is…

Let $\mathcal{T}$ be a smooth proper dg category, and let $\operatorname{Sym}^n\mathcal{T}$ denote its $n$th symmetric power in the sense of Ganter–Kapranov. Write…

Let $\mathcal{T}$ be a smooth proper dg-category over $\mathbb{C}$. Write $K_{\operatorname{top}}(\mathcal{T})$ for its topological K-theory spectrum,…

A phantom is a nonzero admissible subcategory whose Grothendieck group vanishes. Let $\mathcal{R}$ be a smooth finite-dimensional dg-algebra, and let…

Standard derived equivalence characterization. The following conditions should be equivalent:

Let $\mathcal{A}$ be a small dg category. Its Hodge classes are the subspace … where $\operatorname{Ch}^{\operatorname{top}}$ is the topological Chern character, $j$ is the map fro…

Let $\operatorname{\mathcal{A}}$ be a saturated dg-category over $\operatorname{\mathbb{C}}$, and let $\operatorname{Hdg}(\operatorname{\mathcal{A}})_{\operatorname{\mathbb{Q}}}$ d…

Let $k$ be a perfect field, let $l$ be a prime invertible in $k$, and let $\operatorname{\mathcal{A}}$ be a saturated dg-category over $k$. Let…

Moduli-space conjecture. There exists an étale algebraic space $\operatorname{\mathsf{SOD}}_{\mathcal{D}/U}$ over $U$ with a functorial bijection

Marcolli–Tabuada's noncommutative Standard Conjecture D. If $\mathcal{C}$ is a smooth and proper differential $\mathbb{Z}$-graded category over a field $k$ of characteristic $0$, h…

Let $A$ be an algebra, let $C_{sg}(A,A)$ denote its singular Hochschild cochain algebra, and let $\operatorname{\mathsf{sg}}_{dg}(A)$ be the dg singularity category of $A$. Write…

Let $A_1$ and $A_2$ be DG-categories, and let $F$ and $G$ be a pair of homotopy adjoint functors between them. Let $T$ denote the homotopy monad induced by the composite of the adj…

Let $\mathcal B$ and $\mathcal C$ be small DG categories over a field $\mathrm k$ of characteristic zero. Let…

Let $A$ be a smooth DG algebra. Let $\operatorname{ch}:K_0(A\otimes A^{op})\to (HH_{\bullet}(A)\otimes HH_{\bullet}(A^{op}))_0$ be the Chern character, and let…

Let $B$ be a proper DG algebra over a field $cmathrm k$. The Hochschild homology pairing is a map $HH_{\bullet}(B)\otimes HH_{\bullet}(B^{op})\to \mathrm k$, and let…

Let $X$ be a proper scheme over a perfect field $\mathrm k$. Write $\operatorname{Perf}_{\mathrm{top}}(\widehat{{\mathfrak D}^b_{coh}(X)}_{\infty})$ and…

Let $NcS^{\cdot-\cdot}$ be the category whose objects are dg-categories and whose morphisms are pairs of dg-functors $F,G$ equipped with a natural equivalence $F\simeq G$. Let…

Let $NcS$ be the category whose objects are dg-categories without additional structures, and let $\widetilde{NcS}(S)$ and $DM^{\mathcal{GW}}_{nc}(k)$ be the categories constructed…

Let $\mathcal{A}$ be a smooth and proper dg category, and let $K_0(\mathcal{A})$ be the Grothendieck group of its category of compact objects. Let…

Let $G$ be a reductive group scheme, let $X$ be a space with the quotient simplicial object $[X/G]_{\bullet}$, and let $Cpx$ denote the functor sending a space to the dg-category o…