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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 Kahler 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 …