A new non-A1-invariant motivic cohomology for qcqs schemes over Z is built from a global filtration on topological cyclic homology and is shown to relate to algebraic K-theory, étale cohomology, and syntomic cohomology.
Periodic Cyclic Homology over Q
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abstract
Let $X$ be a derived scheme over an animated commutative ring of characteristic 0. We give a complete description of the periodic cyclic homology of $X$ in terms of the Hodge completed derived de Rham complex of $X$. In particular this extends earlier computations of Loday-Quillen to non-smooth algebras. Moreover, we get an explicit condition on the Hodge completed derived de Rham complex, that makes the HKR-filtration on periodic cyclic homology constructed by Antieau and Bhatt-Lurie exhaustive.
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Motivic cohomology of mixed characteristic schemes
A new non-A1-invariant motivic cohomology for qcqs schemes over Z is built from a global filtration on topological cyclic homology and is shown to relate to algebraic K-theory, étale cohomology, and syntomic cohomology.