The cyclotomic trace equivalence conjecture for schemes over finite fields

Let XX be a nice scheme over Fp\mathbf{F}_{p}, let K(X)K(X) denote its algebraic KK-theory spectrum, and let TC(X,p)TC(X,p) denote its pp-typical topological cyclic homology. Cyclotomic trace equivalence conjecture. After pp-completion, the cyclotomic trace map

K(X)TC(X,p)K(X)\longrightarrow TC(X,p)

is a weak equivalence on connective covers. This conjecture predicts that, for schemes over Fp\mathbf{F}_{p}, topological cyclic homology provides the noncommutative counterpart of the cohomological side of the crystalline Grothendieck trace formula. Its status is not resolved in the supplied source.

Sources & referencesView supporting material

Primary source

Ilias Amrani, “Analogy between the cyclotomic trace map K TC and the Grothendieck trace formula via noncommutative geometry”, arXiv:1503.00317 (2015).

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