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An excursion into \(p\)-adic Hodge theory: from foundations to recent trends. (English, French) Zbl 1430.14001
Panoramas et Synthèses 54. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-913-5/pbk). xii, 268 p. (2019).

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Publisher’s description: This volume offers a progressive and comprehensive introduction to \(p\)-adic Hodge theory. It starts with Tate’s works on \(p\)-adic divisible groups and the cohomology of \(p\)-adic varieties, which constitutes the main concrete motivations for the development of \(p\)-adic Hodge theory. It then moves smoothly to the construction of Fontaine’s p-adic period rings and their apparition in several comparison theorems between various \(p\)-adic cohomologies. Applications and generalizations of these theorems are subsequently discussed. Finally, Scholze’s modern vision on p-adic Hodge theory, based on the theory of perfectoids, is presented.
The articles of this volume will be reviewed individually.
Indexed articles:
Freixas i Montplet, Gerard, An introduction to Hodge-Tate decompositions, 1-17 [Zbl 1443.14023]
Caruso, Xavier, An introduction to \(p\)-adic period rings, 19-92 [Zbl 1443.13018]
Brinon, Olivier, Filtered \((\varphi,N)\)-modules and semi-stable representations, 93-129 [Zbl 1443.14021]
Yamashita, Go, An introduction to \(p\)-adic Hodge theory for open varieties via syntomic cohomology, 131-157 [Zbl 1443.14024]
Hattori, Shin, Integral \(p\)-adic Hodge theory and ramification of crystalline representations, 159-203 [Zbl 07219334]
Brinon, Olivier; Andreatta, Fabrizio; Brasca, Riccardo; Chiarellotto, Bruno; Mazzari, Nicola; Panozzo, Simone; Seveso, Marco, An introduction to perfectoid spaces, 207-265 [Zbl 1443.14022]
Caruso, Xavier, Introduction, vii-xii [Zbl 1442.14005]
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14G45 Perfectoid spaces and mixed characteristic