Unformatted text preview: (x  1)(x  2)], with basis (6.8.1.2) 1, 1/7, l/y 2, . .., 1/y "1 . To any nth root of unity ~/~,, we assosciate the C(2)automorphism of X(n; a, b, c) given by XF~ X (6.8.1.3) y~~ ~y. This defines an action of/~, on X(n; a, b, c). Let X denote the "identical" character of /~,, i.e., Z(~)=~, and for any integer L let X(f) denote the character of/4, given by (6.8.1.4) ;((~) (~)= ~e. This basis (6.8.1.2) of A(n; a, b, c) over B is just its "isotypique" decomposition according to the characters of/J,: (6.8.1.5) P(X(E)) A(n; a, b, c)=yeB = lim ye,, C(2)[x]. F~ 9O The de Rham cohomology group H~R(X(n; a, b, c)/C(2)) is the co kernel of the exterior differention map (6.8.1.6) A(n; a, b, c) d , 12~(,;~,b,c)/c~ ~ because X(n; a, b, c) is affine and smooth of relative dimension one over C(2). The module f2]~,;,,h.~)/C~Z) is free over A(n; a, b, c) with basis dx. 7 Inventiones math., VoL 18...
View
Full Document
 Fall '11
 NormanKatz
 De Rham cohomology, 6.8.1.0, 6.8.1.1, 6.8.1.2, 6.8.1.3

Click to edit the document details