Projectivity of the Witt vector affine Grassmannian

Abstract

We prove that the Witt vector affine Grassmannian, which parametrizes W(k)-lattices in $$W(k)[\frac{1}{p}]^n$$ for a perfect field k of characteristic p, is representable by an ind-(perfect scheme) over k. This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably h-descent results for vector bundles.