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

ISBN-10: 0817647465

ISBN-13: 9780817647469

ISBN-10: 0817647473

ISBN-13: 9780817647476

*Algebra, mathematics, and Geometry: In Honor of Yu. I. Manin* comprises invited expository and study articles on new advancements coming 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, okay. Vankov, A. Varchenko, A. Vishik, A.A. Voronov, Yu. Zarhin, Th. Zink.

**Read or Download Algebra, Arithmetic, and Geometry: Volume II: In Honor of Yu. I. Manin PDF**

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

**Example text**

Recall now that C is a k-linear abelian category. To deﬁne Hochschild homology, we have to assume that it is equipped with a right-exact trace functor tr : C → k-Vect; then for any M ∈ C, we set q HH q(M ) = L tr(M ). 2. 1. Proof. For any object [n], Mn ∈ A-bimod# , [n] ∈ Λ, Mn ∈ A⊗n -bimod, let tr( [n], Mn ) = Mn / {av m − mav | v ∈ V ([n]), m ∈ Mn , a ∈ A} , where av = 1⊗1⊗· · ·⊗a⊗· · ·⊗1 ∈ A⊗V ([n]) has a in the multiple corresponding to v ∈ V ([n]), and v ∈ V ([n]) is the next marked point after v counting clockwise.

The functor F! is an extension of the functor F : on Yoneda images Bi ⊂ Shv(Bi ), we have F! = F . And again, the same works for polylinear functors. In particular, given our k-linear abelian tensor category C, we can form the category Shv(C)# of pairs E, [n] , [n] ∈ Λ, E ∈ Shv(C n ), with a map from E , [n ] to E, [n] given by a pair of a map f : [n ] → [n] and either a map E → (f! )∗ E, or map (f! )! E → E; this is equivalent by adjunction. Then Shv(C)# is a biﬁbered category over Λ in the sense of [Gr].

T. Goodwillie’s theorem [Go] claims that if the base ﬁeld k has characteristic 0, the natural map HP q(A) → HP q(A) is an isomorphism, and there is also some information on the behavior of HC q(A). A second type of deformation-theory data includes a commutative kalgebra R with a maximal ideal m ⊂ R. A deformation AR of the algebra A over R is a ﬂat associative unital algebra AR over R equipped with an isomorphism AR /m ∼ = A. In this case, one can form the relative cyclic Rmodule AR# by taking the tensor products over R; thus we have relative homology HH q(AR /R), HC q(AR /R), HP q(AR /R).

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

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