32 problems

Antieau–Gepner–Heller's theorem of the heart conjecture. The map

Let $S$ be a noetherian excellent finite dimensional base scheme of characteristic $0$. The Ayoub conjecture. The heart of the perverse homotopy t-structure on $\mathrm{DM}(S)$ is…

Let $k$ be the base field, let ${\sf DM}_{\mathrm{eff}}(k)$ be the category of effective motivic complexes, and let ${\sf DM}_{\leq n}(k)$ be its subcategory of $n$-motivic complex…

Comonotone classification conjecture. If $f$ is comonotone, then $N(f)$ is an aisle.

Let $X$ be the scheme under consideration, and let … be the heart of the $t$-structure on the bounded quasi-coherent derived category. Let … be its coherent subcategory, and let…

Let $X=\operatorname{Spec}(R)$ be the spectrum of a commutative Noetherian ring $R$. A perversity on $X$ is a function on the points of $X$ that is monotone and comonotone, where c…

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

Let $X$ be a noetherian scheme. A coherent subset of $X$ is a subset of the underlying topological space of the kind appearing in Krause's classification for commutative noetherian…

Let $\sC$ be a small stable $\infty$-category equipped with a bounded t-structure. A spectrum is connective when its negative homotopy groups vanish. Antieau–Gepner–Heller's connec…

Let $\sC$ be a small stable $\infty$-category equipped with a bounded t-structure, and let $\sC^\heartsuit$ denote its heart. Write $\sD^b(\sC^\heartsuit)$ for the bounded derived…

Let $D\mathcal{M}(k,A)$ be Voevodsky's category of motives with coefficients in $A$, and let $D(A)$ be the derived category of $A$-modules. Given an embedding…

Let $k$ be the base field, let $\mathcal{DM}_c(k)$ denote the triangulated category of constructible Voevodsky motives over $k$, and let the $\ell$-adic realisation be the relevant…

Tilting-bundle equivalence conjecture. Under this derived equivalence, the restriction of the functor to the heart of the exotic t-structure gives an equivalence of abelian categor…

Existence conjecture. There is a nice bounded $t$-structure on $\mathit{DM}_{h,lc}(X,\mathbf{Q})$.

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

Enhancement uniqueness conjecture. The homotopy category $h\mathcal{C}$ admits a unique $\infty$-categorical enhancement.

Motivic $t$-structure conjecture. There is a motivic $t$-structure on $\textup{DM}_{\textup{gm}}(k;\mathbb{Q})$ whose heart has semisimple part $\textup{Num}(k;\mathbb{Q})$, every…

HRS-tilt characterization conjecture. The aisle $cD^{\leq 0}$ contains no nonzero Ext-projective if and only if it is a shift of the HRS-tilt associated with some torsion pair…

Derived-equivalence conjecture. Given an arbitrary weighted projective line $cX$, Assertion holds for $cD=\mathcal{D}^b(cX)$.

Let $DM_{gm}^{eff}(k;\mathbb{Q})$ be the category of compact effective Voevodsky motives over a field $k$ with rational coefficients, and let…

Heart-equivalence conjecture. This natural map should be an equivalence of nonconnective $K$-theory spectra. The assertion would identify the $K$-theory of a boundedly $t$-structur…

Let $E$ be a small stable $\infty$-category equipped with a bounded $t$-structure. Write $\mathrm{K}_{-n}(E)$ for its negative nonconnective $K$-groups. Bounded t-structure vanishi…

Let $DM$ be the triangulated category of mixed motives constructed by Hanamura, Levine, and Voevodsky. The conjectural abelian category of mixed motives is expected to be the heart…

Functor-category conjecture. The category $\underline{H}_{t_{hom}}(S)$ is equivalent to the category of those contravariant additive functors…

Cycle-module equivalence conjecture. The functor $H_{cm}$ yields an equivalence of $\underline{H}_{t_{hom}}(S)$ with a certain abelian category of cycle modules.