YNTOMIC COHOMOLOGY OF DERIVED SCHEMES AND THE MOTIVIC FILTRATION ON TOPOLOGICAL CYCLIC HOMOLOGY

Mathematical Applications of Science Fiction We construct a symmetric monoidal ∞-category of derived syntomic motives Msyn(X) overa derived scheme X. We define the syntomic cohomology RΓsyn(X, Zp(n)) as mapping objects in thiscategory and establish a canonical equivalence between these cohomology groups and the graded piecesof the motivic filtration on the topological cyclic homology TC(X; p). We extend the Bhatt-Morrow-Scholze filtration to the context of derived algebraic geometry using the formalism of quasisyntomicdescent. Finally, we prove that for a quasi-compact quasi-separated derived scheme X, the category ofsyntomic coefficients admits a Tannakian structure, identifying the motivic Galois group in the p-adicsetting.