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Петербургский топологический семинар им. В. А. Рохлина
9 марта 2015 г. 17:30–19:00, г. Санкт-Петербург, ПОМИ, комн. 311 (наб. р. Фонтанки, 27)
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[Analogy between the cyclotomic trace map $K\to TC$ and the Grothendieck trace formula via noncommutative geometry]
I. Amrani
Санкт-Петербургский академический университет — научно-образовательный центр нанотехнологий РАН (Академический университет)
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Аннотация:
There will be a little algebraic geometry, number theory and algebraic K-theory.
We suggest a categorification procedure in order to capture an analogy between Crystalline Grothendieck–Lefschetz trace formula and the cyclotomic trace map $K\to TC$ from the algebraic K-theory to the topological cyclic homology $TC$. First, we categorify the category of schemes to the $(2,\infty)$-category of noncommuatative schemes a la Kontsevich. This gives a categorification of the set of rational points of a scheme. Then, we categorify the Crystalline Grothendieck–Lefschetz trace formula and find an analogue to the Crystalline cohomology in the setting of noncommutative schemes over $\mathbb F_p$. Our analogy suggests the existence of a categorification of the $l$-adic cohomology trace formula in the noncommutative setting for $l$ different from $p$. Finally, we write down the corresponding dictionary.
Язык доклада: английский
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