Unformatted text preview: ~-~ ~)). According to (220.127.116.11), the image 1(0 is none other than the class of the extension (18.104.22.168.6) (via the isomorphism (22.214.171.124 bis)). (126.96.36.199.7) 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 @x-modules is none other than the cup-product 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) (188.8.131.52) 0 -)f* (O~/T) ---) O~C/T (Iog O) --, O~C/S (log D) -, 0 which gives rise, via the A p construction (184.108.40.206), to a short exact sequence 0 -,f* ((2~/r) | (2w (log D) --~ K~ z (Y2w (log D)) (220.127.116.11) --+ O]/s (log D) -, 0....
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- Fall '11
- Algebraic Topology, Homological algebra, exact sequence, short exact sequence, Hodge cohomology