# Projectivity of the Witt vector affine Grassmannian

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## 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.

## Notes

### Acknowledgements

This work started after the authors listened to a talk of Xinwen Zhu on his work at the MSRI, and the authors would like to thank him for asking the question on the existence of \({\mathcal {L}}\). They would also like to thank Akhil Mathew for enlightening conversations related to Sect. 11.2. Moreover, they wish to thank all the participants of the ARGOS seminar in Bonn in the summer term 2015 for their careful reading of the manuscript, and the many suggestions for improvements and additions. The first version of this preprint contained an error in the proof of Lemma 4.6, tracing back to an error in [18, Corollary 3.3.2]; we thank Christopher Hacon, and Linquan Ma (via Karl Schwede) and the anonymous referee for pointing this out. The authors are also indebted to the referee for providing numerous other comments that improved the readability of this paper. Finally, they would like to thank the Clay Mathematics Institute, the University of California (Berkeley), and the MSRI for their support and hospitality. This work was done while B. Bhatt was partially supported by NSF grant DMS 1340424 and P. Scholze was a Clay Research Fellow.

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