Unformatted text preview: ~~ ~)). According to (1.2.2.3), the image 1(0 is none other than the class of the extension (1.2.2.4.6) (via the isomorphism (1.2.1.8 bis)). (1.2.2.4.7) But it is a general fact that the coboundary mapping H~(X, C), H q+l (X, A) associated to any short exact sequence 0~ A, B+ C) 0 of @xmodules is none other than the cupproduct with the element of Ext~x(C, A) (this last operation has a sense, thanks to the isomorphism of functors H q (X, ) ~Ext~, ((gx, )). Q.E.D. 1.3. Application to Hodge Cohomology (1.3.1) Proposition. In the geometric situation of 1.0, consider the short exact sequence (1.1.1) (1.3.1.1) 0 )f* (O~/T) ) O~C/T (Iog O) , O~C/S (log D) , 0 which gives rise, via the A p construction (1.2.1.6), to a short exact sequence 0 ,f* ((2~/r)  (2w (log D) ~ K~ z (Y2w (log D)) (1.3.1.2) + O]/s (log D) , 0....
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 Fall '11
 NormanKatz
 Algebraic Topology, Homological algebra, exact sequence, short exact sequence, Hodge cohomology

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