The Bloch-Esnault-Kerz Fiber Square in p-adic Hodge Theory

Explore a comprehensive lecture on the Bloch-Esnault-Kerz fiber square and its implications in p-adic Hodge theory. Delve into the theorem of Bloch-Esnault-Kerz from 2014, which addresses the formal part of the Fontaine-Messing p-adic variational Hodge conjecture for schemes smooth and proper over an unramified local number ring. Examine the conditions for lifting classes in the rational p-adic Grothendieck group and the role of crystalline Chern classes. Investigate Beilinson's generalization of the equivalence between relative rational p-adic K-theory and cyclic homology. Discover how the Bhatt-Morrow-Scholze unification of p-adic Hodge theory and topological cyclic homology clarified these concepts. Learn about the Antieau-Mathew-Morrow-Nikolaus result and its connection to the Nikolaus-Scholze Tate-Orbit-Lemma. Gain insights into how the cartesian square emerges from the Nikolaus-Scholze Frobenius of ℤ and explore Clausen's proposal for defining the Hodge-Tate period map without calculational input. This 1-hour 12-minute lecture by Lars Hesselholt from Nagoya University and the University of Copenhagen offers a deep dive into advanced mathematical concepts at the intersection of algebraic geometry, number theory, and homological algebra.