31 problems

Conjecture. For every smooth quasi-projective variety $X$, … The lower bound $\operatorname{Rdim}D^b(X)\geq\dim X$ is known quite generally. The difficult direction is to construct…

Hovey–Palmieri conjectures. The following assertions hold:

Extended Uniqueness Conjecture. If there is an exact equivalence

Freyd's Generating Hypothesis for ${\mathcal{F}}_p$. The functor $\pi_*$ is faithful. Therefore, ${\mathcal{M}}_p$ is a Freyd category, with ${\mathcal{F}}_p$ as its full subcatego…

Generalized telescope conjecture. The assertion is exactly the stable telescope conjecture, or equivalently its ideal-theoretic reformulation, with $\mathbf{t}$ a t-structure in…

Krause's reformulated telescope conjecture. The ideal $I_{\mathbf{t}}$ is generated by identity morphisms of objects in $\mathcal{D}^c$.

Ravenel's telescope conjecture. The decomposition $\mathbf{t}$ is compactly generated.

Faithfulness conjecture. The action of the Artin–Tits group $\mathbb B$ on the triangulated category $\mathcal D$ is faithful.

Ito's conjecture. The Fourier–Mukai spectrum of $\mathcal T$ should coincide with its Serre spectrum:

Let $\mathcal T$ be a triangulated category with $X\in\operatorname{FM}\mathcal T$. A topological space is locally spectral if it has an open cover by spectral spaces, and locally…

Let $X$ be an abelian variety. A smooth projective variety is tt-separated when the relevant Fourier–Mukai copies in its triangular spectrum are pairwise disjoint. Abelian tt-separ…

Let $\mathcal T$ be a triangulated category with $X \in \operatorname{FM} \mathcal T$. The Fourier–Mukai locus $\operatorname{Spec}^{\mathsf{FM}} \mathcal T$ and the Serre-invarian…

Finite-cohomology generation conjecture.

Let $\mathcal{T}$ be a triangulated category admitting a full exceptional sequence of length $n$, and let $\mathfrak{B}_n$ denote the braid group on $n$ strands. The group…

Coefficient comparison conjecture. There are equivalences of triangulated categories

Étale comparison conjecture. There is an equivalence of triangulated categories

Rationality and nonnegativity conjecture. The two Serre dimensions are rational numbers, and the upper Serre dimension is nonnegative:

Enhancement uniqueness conjecture. Then $\mathrm{T}$ admits a unique $\infty$-categorical enhancement.

K-theory non-detection conjecture. There is no natural number $n$ such that, for all such $\mathcal{C}$ and $\mathcal{D}$, an equivalence

Telescope Conjecture. Then

Let $\mathsf{D}$ be a Hom-finite Krull--Schmidt triangulated category and let $\mathsf{H}$ be the heart of a bounded $t$-structure. The category $\mathsf{D}$ is $\mathsf{H}$-discre…

A triangulated category $\mathsf{D}$ is Hom bounded if for every pair of objects $A,B\in\mathsf{D}$, the dimensions of $\operatorname{Hom}(A,B)$ are bounded independently of the ch…

Let $\mathrm C$ be a sesquicategory. A weak recalage–simplicial triangulation equivalence conjecture. The category of weak recalages of $\operatorname{\bf LN}_+(\mathrm C)$ is equi…

Let $\mathscr{T}$ be a compactly generated $\otimes$-triangulated category and let $\mathscr{U}$ be a compactly generated triangulated category, both with all coproducts. Suppose ……

Let $\mathscr{T}$ be a compactly generated triangulated category with all coproducts, and let … be a smashing localization, meaning that its right adjoint preserves coproducts. Wri…