Unformatted text preview: (4.2.0.3) s(z)=multiplication by zP~ q on H p'q. Following Well, we put C = s (i). For each z ~C*, the endomorphism s(z) of H c commutes with complex conjugation, and hence comes from an endomorphism still noted s(z) of Hn= Hz R denoting the field of real numbers. (4.2.0.4) A morphism q9 between pure Hodge structures H and H' is a group homomorphism ~Oz: Hz~H'z whose Clinear extension <Pc: Hc* H~: commutes with the action s of C* (4.2.0.3). (Thus between pure Hodge structure of different weights there is only the zero morphism.) (4.2.0.5) The category of pure Hodge structures has an internal Hom and a tensor product  defined in the expected way (cf. [8]). (4.2.0.6) Tate's Hodge structure Z(n) is the rank one Hodge structure of weight  2 n which is purely of type ( n,  n), and whose integral lattice H z is the subgroup (4.2.0.7) (2n i)" Zc C....
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 Fall '11
 NormanKatz
 Vector Space, Hodge, automorphism, pure Hodge structures

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