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 learn articles on new advancements bobbing up from Manin’s striking 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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Additional resources for Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu. I. Manin

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2. For any associative unital algebra A over k, its Hochschild, cyclic, and periodic cyclic homologies HH q(A), HC q(A), HP q(A) are defined as the corresponding homologies of the cyclic k-vector space A# : def HH q(A) = HH q(A# ), def HC q(A) = HC q(A# ), def HC q(P ) = HP q(A# ). 2 Cyclic bimodules. 1) HH q = TorAopp ⊗A (A, A), q where Tor is taken over the algebra Aopp ⊗ A (here Aopp denotes A with the multiplication taken in the opposite direction). This has a version with coefficients: if M is a left module over Aopp ⊗ A, in other words, an A-bimodule, one defines Hochschild homology of A with coefficients in M by q HH q(A, M ) = TorAopp ⊗A (M, A).

Every functor in Shv(B) is in fact a direct limit of representable functors, so that Shv(B) is an inductive completion of the abelian category B. Now, if we are given two (small) k-linear abelian categories B1 , B2 , then their product B1 × B2 is no longer abelian. However, we still have the abelian category Shv(B1 × B2 ) of bilinear functors B1opp × B2opp → k-V ect, which are left-exact in each variable, and the same goes for polylinear functors. Moreover, for any right-exact functor F : B1 → B2 between small abelian categories, we have the restriction functor F ∗ : Shv(B2 ) → Shv(B1 ), which is left-exact, and its left-adjoint F!

References [AC] J. Arthur, L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies 120 (1989). [C1] L. Clozel, Motifs et formes automorphes: applications du principe de fonctorialit´e, in L. Clozel and J. S. , Automorphic Forms, Shimura Varieties, and L-functions, New York: Academic Press (1990), Vol I, 77– 160. [C2] L. Clozel, Repr´esentations Galoisiennes associ´ees aux repr´esentations automorphes autoduales de GL(n), Publ. Math. , 73, 97–145 (1991).

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