v-stack - Knowlpedia

Definition

The v-topology on perfectoid spaces is the topology whose covers are the families that are jointly surjective and remain sufficiently surjective after quasi-compact base change; equivalently, it is generated by maps detected by extensions of maps from valuation spectra. It is finer than the pro-étale topology.

A v-sheaf is a sheaf of sets or spaces for the v-topology. A v-stack is the corresponding stack in groupoids or higher spaces. A v-stack is called small when it admits a surjective map from a perfectoid space with the standard smallness conditions.

Diamonds

A diamond is a v-sheaf that can be presented as a quotient

X/R

of a perfectoid space $X$ by a pro-étale equivalence relation $R$ . Diamonds are therefore special v-sheaves. General v-stacks allow stabilizers and moduli behavior that need not admit a diamond presentation.

Why the v-topology is used

Many $p$ -adic moduli functors become sheaves only after passing to the v-topology. Quotients, descent, untilts, and infinite-level towers are then handled in a single category. The stack of $G$ -bundles on the Fargues–Fontaine curve, local-shtuka spaces, and stacks of L-parameters all use v-stack or closely related diamond formalisms.

References

Peter Scholze, “Étale cohomology of diamonds,” 2017, §§8–12. arXiv.
Peter Scholze and Jared Weinstein, Berkeley Lectures on $p$ -adic Geometry, Annals of Mathematics Studies 207, Princeton University Press, 2020, Chapters 11–12.