Some recent advances in topological Hochschild homology
Bulletin of the London Mathematical Society
We give an account of the construction of the Bhatt-Morrow-Scholze motivic filtration on topological cyclic homology and related invariants, focusing on the case of equal characteristic 𝑝 and the connections to crystalline and de Rham-Witt theory. M S C 2 0 2 0 19D55 (primary), 13D03 (secondary) INTRODUCTION Let 𝑋 be a smooth quasi-projective scheme over a field 𝑘. In this case, one has the algebraic 𝐾theory spectrum 𝐾(𝑋) of 𝑋, defined by Quillen  using the exact category of vector bundles
... on 𝑋 (in general, one should use perfect complexes as in  ). One can think of 𝐾(𝑋) as a type of 'cohomology theory' for the scheme 𝑋, analogous to the topological 𝐾-theory of a compact topological space. With this in mind, the following fundamental result gives an analog of the classical Atiyah-Hirzebruch spectral sequence relating topological 𝐾-theory to singular cohomology. Theorem 1.1 (The motivic filtration on algebraic 𝐾-theory, [51, 90]). There is a functorial, convergent, decreasing multiplicative filtration Fil ⩾ * 𝐾(𝑋) and identifications gr 𝑖 𝐾(𝑋) ≃ ℤ(𝑖) mot (𝑋)[2𝑖] for 𝑖 ⩾ 0. Here the ℤ(𝑖) mot (𝑋), called the motivic cohomology of 𝑋, are explicit cochain complexes introduced by Bloch  (see also  ) in terms of algebraic cycles on 𝑋 × 𝔸 𝑛 𝑘 for 𝑛 ⩾ 0. In particular, we have that The London Mathematical Society acknowledges support from a legacy by Frank Gerrish in the publication of expository articles and surveys with open access.