Extending torsors on the punctured Spec(A inf)

We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine’s ring A inf {A_{\mathrm{inf}}} , extend to the whole spectrum. Using descent we can extend a similar result for the ring 𝔖 {\mathfrak{S}} of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the B dR + {B^{+}_{\mathrm{dR}}} -affine Grassmannian to the Witt vector affine flag variety.

Tate motives on Witt vector affine flag varieties

Relying on recent advances in the theory of motives we develop a general formalism for derived categories of motives with $${\mathbf{Q}}$$ Q -coefficients on perfect $$\infty $$ ∞ -prestacks. We construct Grothendieck’s six functors for motives over perfect (ind-)schemes perfectly of finite presentation. Following ideas of Soergel–Wendt, this is used to study basic properties of stratified Tate motives on Witt vector partial affine flag varieties. As an application we give a motivic refinement of Zhu’s geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.

Affine flag varieties

This chapter reviews affine flag varieties. It generalizes some of the previous results to the case where G over Zp is a parahoric group scheme. In fact, slightly more generally, it allows the case that the special fiber is not connected, with connected component of the identity G? being a parahoric group scheme. This case comes up naturally in the classical definition of Rapoport-Zink spaces. The chapter first discusses the Witt vector affine flag variety over Fp. This is an increasing union of perfections of quasiprojective varieties along closed immersions. In the case that G° is parahoric, one gets ind-properness.

Dual Complex of Log Fano Pairs and Its Application to Witt Vector Cohomology

We prove the contractibility of the dual complexes of weak log Fano pairs. As applications, we obtain a vanishing theorem of Witt vector cohomology of Ambro–Fujino type and a rational point formula in Dimension 3.

A Tannakian Framework for G-displays and Rapoport–Zink Spaces

We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport–Zink functors associated to triples $(G,\{\mu \},[b])$, where $G$ is a flat affine group scheme over ${\mathbb{Z}}_p$ and $\mu$ is a cocharacter of $G$ defined over a finite unramified extension of ${\mathbb{Z}}_p$. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport–Zink spaces given by Bültel and Pappas. As an application, we prove a special case of a conjecture of Bültel and Pappas by showing their definition coincides with that of Rapoport and Zink in the case of unramified EL-type local Shimura data.

-DISPLAYS AND RAPOPORT–ZINK SPACES

Let $(G,\unicode[STIX]{x1D707})$ be a pair of a reductive group $G$ over the $p$-adic integers and a minuscule cocharacter $\unicode[STIX]{x1D707}$ of $G$ defined over an unramified extension. We introduce and study ‘$(G,\unicode[STIX]{x1D707})$-displays’ which generalize Zink’s Witt vector displays. We use these to define certain Rapoport–Zink formal schemes purely group theoretically, i.e. without $p$-divisible groups.