Algebra, Arithmetic, and Geometry: Volume II: In Honor of by Michael Harris (auth.), Yuri Tschinkel, Yuri Zarhin (eds.)

By Michael Harris (auth.), Yuri Tschinkel, Yuri Zarhin (eds.)

Algebra, mathematics, and Geometry: In Honor of Yu. I. Manin includes invited expository and study articles on new advancements coming up from Manin’s extraordinary contributions to arithmetic.

Contributors within the moment quantity: M. Harris, D. Kaledin, M. Kapranov, N.M. Katz, R.M. Kaufmann, J. Kollár, M. Kontsevich, M. Larsen, M. Markl, L. Merel, S.A. Merkulov, M.V. Movshev, E. Mukhin, J. Nekovár, V.V. Nikulin, O. Ogievetsky, F. Oort, D. Orlov, A. Panchishkin, I. Penkov, A. Polishchuk, P. Sarnak, V. Schechtman, V. Tarasov, A.S. Tikhomirov, J. Tsimerman, ok. Vankov, A. Varchenko, A. Vishik, A.A. Voronov, Yu. Zarhin, Th. Zink.

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Extra resources for Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu. I. Manin

Example text

Kaledin Connes [C2] improved on the definition by introducing the abelian category of so-called cyclic vector spaces. However, the passage from A to its associated cyclic vector space A# is still done by an explicit ad hoc formula. It is as if we were to know the bar-complex that computes the homology of a group, without knowing the definition of the homology of a group. This situation undoubtedly irked many people over the years, but to the best of my knowledge, no satisfactory solution has been proposed, and it may not exist; indeed, many relations to the de Rham homology notwithstanding, it is not clear whether cyclic homology properly forms a part of homological algebra at all (to the point that for instance in [FT], the word “homology” is not used at all for HC q(A), and it is called instead additive K-theory of A).

A. a. a. pseudofunctor in the original terminology of Grothendieck) from Λ to the category of categories. ” To work with weak functors, it is convenient to follow Grothendieck’s approach in [Gr]. Namely, instead of considering a weak functor directly, we define a category C# in the following way: its objects are pairs [n], Mn of an object [n] of Λ and an object Mn ∈ C n , and morphisms from [n ], Mn to [n], Mn are pairs f, ιf of a map f : [n ] → [n] and a bimodule map ιf : f! (Mn ) → Mn . A map f, ιf is called co-Cartesian if ιf is an isomorphism.

Then the pair C, tr is homologically clean, for any trace functor tr. 3) P = P1 P2 · · · Pn ∈ Shv(C n ) for projective P1 , . . , Pn ∈ C ⊂ Shv(C), and these projectives automatically satisfy the condition (i). To check (ii), one decomposes f : [n] → [n ] into a surjection p : [n] → [n ] and an injection i : [n ] → [n ]. Since the tensor product of projective objects is projective, (p! )! 3), so we may as well assume that f is injective. Then one can find a left-inverse map f : [n ] → [n], f ◦ f = id; since P = (f!

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