22 problems

Let $k$ be the residue field and let $G$ and $(b,)$ be the data defining the integral moduli $v$-sheaf $\mathcal{M}^{\mathop{\mathrm{int}}\nolimits}_{G,b,\mu}$. A formal scheme is…

Motivic filtration conjecture. The spectrum $\mathrm{TP}(R/X(p))$ admits a motivic filtration satisfying

Bhatt–Lurie’s cohomological dimension conjecture. The functor

Let $R$ be a topologically finitely generated and $p$-complete ${\mathcal{O}}_{\mathbf{C}}$-algebra, and let … be an ${\mathcal{O}}_{\mathbf C}$-homomorphism. A rational Hodge--Tat…

Let $k$ be a perfect $\mathbf{F}_p$-algebra and let $A$ be a smooth $k$-algebra. Let … be the fully faithful embedding of Theorem. Taking…

Let $X$ be a smooth scheme over a field of characteristic $p$. Write $X_{\mathrm{fpqc}}$ for its fpqc site, let $T_{X/k}$ be its tangent bundle, and let $T_{X/k}^{\sharp}$ be the f…

Let $\mathrm{Stack}_{\mathrm{Bun},G,X,n}$ be the stack of $G$-bundles used in the paper, and let … and … be the indicated endomorphism-line and local subcategories. Breuil–Schneide…

Let $\widehat{G}^\mathrm{Dr}$ be Drinfeld's formal group over the stack $\bbZ_p^{{\mathlarger{\mathbbl{\Delta}}}}$. Its pullbacks to the $q$-de Rham prism and along the Frobenius m…

Unconditional local Shimura conjecture. The natural transformation

Bhatt–Lurie’s conjecture. The class $\alpha_{{\mathrm F}}({\mathfrak X}_0)$ is the obstruction class for lifting ${\mathrm F}_{{\mathfrak X}_0}$ over $\phi_A:A_1\to A_1$.

Let $k$ be the residue field, let $\mu$ be 1-bounded, and let $\mathscr{Q}\in G\mathchar`-F\mathchar`-\mathrm{Gauge}_\mu(k)$. Let $R_{G,\mu}$ represent the deformation functor on c…

Let $R$ be a complete regular local ring over $W(k)$ with residue field $k$, let $(\mathfrak{S},(\mathcal{E}))=(W(k)[[t_1,\dotsc,t_n]],(\mathcal{E}))$ be a prism of Breuil--Kisin t…

Equivalence conjecture. The functors $D_X$ and $D_X^{\operatorname{perf}}$ each define an equivalence of categories. This conjecture predicts that the comparison between the overco…

Log-prismatic extension conjecture. There exists such a morphism

Formal identity conjecture. One has

Reverse stratification conjecture. The expression

Let ${\mathbb M}$ be a rational Hodge–Tate crystal, with associated enhanced log Higgs bundle $({\mathcal M},\theta_{{\mathcal M}},\phi_{{\mathcal M}})$ and generalised representat…

Quasisyntomic–étale comparison conjecture. The quasisyntomic descent and étale comparison results of Bhatt–Scholze should imply an equivalence in some accurate sense beyond the vag…

Given a smooth $\mathbf{Z}$-algebra $R$ equipped with a framing $\square$, let $q\Omega^\bullet_{(R,\square)}$ be the associated complex of $\mathbf{Z}\llbracket q-1 \rrbracket$-mo…

Let ${\mathscr S}_{\sf K}$ be the integral model of a Shimura variety and let ${\mathscr P}_{\sf K}$ be its universal ${\mathcal G}$-shtuka. Prismatic refinement conjecture. The in…

Let $(A,d)$ be a perfect prism such that $A/d$ is $p$-torsion free, and let $S$ be a smooth $A/d$-algebra. Here $W_n\Omega_{S/(A/d)}^i$ denotes the degree-$i$ de Rham–Witt group, a…

Let $(A,I)$ be a bounded prism, and let $R$ be a derived $p$-complete simplicial $A/I$-algebra such that $\overline{{\mathlarger{\mathbbl{\Delta}}}}_{R/A}$ is concentrated in degre…