No abstract

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 Ktheory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex AΩ 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 AΩ-complex by giving a cohomological construction of Breuil-Kisin modules for proper smooth formal schemes over OK , where K is a discretely valued extension of Qp with perfect residue field. As another application, we define syntomic sheaves Zp(n) for all n ≥ 0 on a large class of Zp-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.