17 problems

Hochster's direct summand conjecture. The sequence splits whenever $R$ is regular. Equivalently, the inclusion $R\to S$ is pure.

A local Shimura variety is associated with a local Shimura datum $(G,\mu,b)$, and its infinite-level space is obtained by passing to the inverse limit over compact open level subgr…

Geometric Sen perfectoidness conjecture. The object $X_\infty$ of $X_{\mathrm{pro\acute{e}t}}$ is perfectoid if and only if $\kappa_{X_\infty}^*$ is surjective, equivalently if and…

Let $X$ be a perfectoid boundary carrier equipped with a condensed sheaf/module $\mathcal{F}$ of sources and operators, and let $X^{\flat}$ be its tilt. Assume the holographic gene…

Global perfectoidness conjecture. This inverse limit is representable by a perfectoid space. Perfectoidness at infinite $p$-level is a central representability question for Shimura…

Perfectoidness conjecture. The space $\mathcal{M}_{(G,\mu,b),\infty}$ is representable by a perfectoid space. This is known for local Shimura varieties of Abelian type; the paper g…

Let $X/C$ be a smooth proper connected variety of pure dimension $d$ with globally generated $1$-forms, and fix a point $x\in X(C)$. Let … be a closed subgroup. The canonical map s…

Let $K$ be a complete discrete valuation field extension of $\mathbb{Q}_p$ with perfect residue field, let $\overline{K}$ be an algebraic closure of $K$, and set…

Let $O_E$ be the reflex field's valuation ring, and consider the perfectoid local models introduced by Scholze and Weinstein as v-sheaves. Scholze–Weinstein's representability conj…

Let $n$ be a positive integer and, for each integer $d$ with $0<d<n$, let $W_{n,d}$ be the space defined in the paper from the corresponding local geometric construction. Perfectoi…

Let $I=\{1,2\}$, let $M$ be a $\Phi$-equivariant $F$-isocrystal $(M,\varphi,\varphi_1,\varphi_2)$ in the category…

Kummer pro-étale extension conjecture. The construction developed in the section should extend to suitable Kummer pro-étale sites with logarithmic perfectoid subdomains forming a b…

Let $S\in\operatorname{Perf}_{\mathbb{F}_q}$ and let $\mathscr{E}$ be a $G$-bundle on $X_S$. Let $B$ be a Borel subgroup of $G$, and consider the moduli space of reductions of…

Let $E$ be the local field in the paper, let $\operatorname{Gr}$ be the corresponding affine Grassmannian, and let … be the Beauville–Laszlo morphism. Beauville–Laszlo uniformizati…

Let $K$ be a perfectoid field, let $(A,A^+)$ be a complete affinoid $K$-algebra for which $A^+$ has the $p$-adic topology, and set $X=\operatorname{Spa}(A,A^+)$. Assume that $X$ ha…

Let $K$ be a perfectoid field, let $(A,A^+)$ be a complete affinoid $K$-algebra, and set $X=\operatorname{Spa}(A,A^+)$. Assume that $X$ has a cover by rational subsets…

Let $K$ be a perfectoid field with tilt $K^\flat$, let $X\subset \mathbb{P}^n_K$ be a smooth closed subscheme, and let $\tilde{X}\subset (\mathbb{P}^n_K)^\mathrm{ad}$ be a small op…