Two monoidal structures on Satake category in mixed characteristic

Abstract

Fargues and Scholze proved the geometric Satake equivalence over the Fargues-Fontaine curve. This can be transferred to the geometric Satake equivalence concerning a Witt vector affine Grassmannian via nearby cycle. On the other hand, Zhu proved the geometric Satake equivalence concerning a Witt vector affine Grassmannian. In this paper, we explain the coincidence of these two geometric Satake equivalences, including the coincidence of the two symmetric monoidal structures on the Satake category.

27 pages. Revised the statement and proof of Lemma 1.2 in the old version and deleted Theorem 1.3 and Section 6. Added Proposition 1.1. To appear in Journal de Théorie des Nombres de Bordeaux