Thomason filtration via $T(1)$-local $\mathrm{TC}$

Abstract

We construct a natural filtration on $T(1)$-local $\mathrm{TC}$ for any animated commutative rings using prismatic cohomology and descent theory. In the course of the construction, we also study some general properties of prismatic cohomology complexes over perfect prisms after inverting distinguished generators. The construction is intrinsic to $\mathrm{TC}$ and recovers Thomason's spectral sequence for $T(1)$-local algebraic K-theory via the cyclotomic trace map; as a consequence, we also recover the étale comparison for prismatic cohomology.

v3: accepted version, 40 pages; v2: Major updates including new constructions and extensions to all animated commutative rings. Added comparison with the étale Postnikov filtration from algebraic K-theory side and a consequent étale comparison