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Topological cyclic homology. (English) Zbl 1473.14038

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 619-656 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: Topological cyclic homology is a manifestation of Waldhausen’s vision that the cyclic theory of Connes and Tsygan should be developed with the initial ring \(S\) of higher algebra as base. Topological cyclic homology receives a map from algebraic \(K\)-theory, called the cyclotomic trace map. The Nikolaus-Scholze approach to topological cyclic homology is also very useful for calculations. The nature of this Frobenius map is much better understood by the work of Nikolaus-Scholze. The chapter explains the definition of the cyclotomic trace map from K-theory to topological cyclic homology. Perfectoid rings are to topological Hochschild homology what separably closed fields are to \(K\)-theory: they annihilate Kähler differentials. The respective homotopy fixed point spectral sequences endow each of the four rings with a descending filtration, which refer to as the Nygaard filtration, and they are all complete and separated in the topology.
For the entire collection see [Zbl 1468.55001].

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
19D50 Computations of higher \(K\)-theory of rings
19D55 \(K\)-theory and homology; cyclic homology and cohomology
55R45 Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity

Citations:

Zbl 1468.55001
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