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Maxime Ramzi: Postdoc day, Seminar 4: Galois descent in topological Hochschild homology

Time: Fri 2022-02-18 15.50 - 16.20

Location: Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

Video link: Meeting ID: 921 756 1880

Participating: Maxime Ramzi (University of Copenhagen)

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Abstract: Topological Hochschild homology (THH) is an interesting invariant of rings, related among other things to algebraic K-theory. One of its key properties is étale descent, which implies that one can work étale-locally to understand it. On the other hand, Rognes introduced in 2005 a theory of Galois extensions for ring spectra. Despite the name, Galois extensions of ring spectra are typically not étale, and in particular one can wonder independently of étale descent, whether THH satisfies Galois descent. This has been in part studied by Akhil Mathew, who gave a counterexample to the general statement, but also gave some examples where it is satisfied. In this talk, I will briefly introduce these objects and questions, and I will present a general criterion under which Galois descent holds in THH. This concrete criterion, stated in terms of separability, recovers some of the chromatic examples that Akhil obtained in joint work with Clausen, Naumann and Noel using more sophisticated techniques of “chromatic descent”.