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9^ Perfectoid spaces: A survey, to appear in Proceedings of the 2012 conference on Current Developments in Mathematics.
URLﾘﾝｸ(www.math.uni-bonn.de)
PERFECTOID SPACES: A SURVEY 1This work was done while the author was a Clay Research Fellow.
PETER SCHOLZE
Abstract. This paper, written in relation to the Current Developments in Mathematics
2012 Conference, discusses the recent papers on perfectoid spaces. Apart from giving an
introduction to their content, it includes some open questions, as well as complements to
the results of the previous papers.
Contents
1. Introduction 2
2. Perfectoid Spaces 3
2.1. Introduction 3
2.2. Some open questions 9
3. p-adic Hodge theory 13
3.1. Introduction 13
3.2. A comparison result for constructible coefficients 17
3.3. The Hodge-Tate spectral sequence 20
4. A p-adic analogue of Riemann’s classification of complex abelian varieties 25
4.1. Riemann’s theorem 25
4.2. The Hodge-Tate sequence for abelian varieties and p-divisible groups 26
4.3. A p-adic analogue: The Hodge-theoretic perspective 29
4.4. A p-adic analogue: The geometric perspective 30
5. Rapoport-Zink spaces 32
6. Shimura varieties, and completed cohomology 36
References 37