Topological Hochschild homology and integral p -adic Hodge theory

In mixed characteristic and in equal characteristic p we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic K -theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex A Ω constructed in our previous work, and in equal characteristic p to crystalline cohomology. Our construction of the filtration on THH is via flat descent to semiperfectoid rings. As one application, we refine the construction of the A Ω -complex by giving a cohomological construction of Breuil–Kisin modules for proper smooth formal schemes over 𝒪 K , where K is a discretely valued extension of 𝐐 p with perfect residue field. As another application, we define syntomic sheaves 𝐙 p ( n ) for all n ≥ 0 on a large class of 𝐙 p -algebras, and identify them in terms of p -adic nearby cycles in mixed characteristic, and in terms of logarithmic de Rham-Witt sheaves in equal characteristic p .