By Edwin H. Spanier

ISBN-10: 0070995826

ISBN-13: 9780070995826

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For this purpose one has to explain "differentiation of hypercohomology classes". g. [42], [IS]). It turns out that the hypercohomology sheaves 1i DR (X / S) are not only Os-modules but also Ds-modules, where Ds is the non-commutative ring sheaf of differential operators on S. For example, DT is locally generated by regular functions and the coordinate derivations a/Oti, i = 1, ... ,T. For explaining such derivations one uses "relative etal coverings" {Ui / S} of X / S. These are open coverings Ui of X having a decomposition Ui --+ A~ --+ S, where the first morphism is etal over its image, A~ = Ak X S.

The factor group r /§r( A) = 8 4 x Z6 acts on the vector space [§r ( A), 1] of §r ( A)-forms of weight 1. By some geometric considerations and elementary representation theory one can hope to find a set of four special forms in [§r(A) , 1] such that the elements of 8 4 act as permutations on this set, up to a character. 28: tll t2, t3 are linearly independent. 28 Special Functional Equations (i) (ii) ')'* (ti) = j, . ti for i = 1,2,3,4, ')' E §r( A); ')'* (tdtj) = t 1 (i)/t1 (j) for i = 1,2,3,4, ')' E §r with image 1 in 8 4 = §r / §r(yC3); (iii) 8* (ti) = (det 8)2 .

Looking at the tangent space of V at 0 one gets all quotient singularities as surface germs «((:2/G, 0), G a finite subgroup of Gl 2 (C), 0 the image of 0 = (0,0) E ((:2. If G is abelian, then (U, P) is called an abelian (quotient) singularity. A more algebraic approach can be given by blowing up the 0- point of ((:2 in the above situation. More generally, let D = ]p'l and LID be an (affine) line bundle over D, that means the fibres of the (affine) projection L ----? D are isomorphic to the affine line Al = Al(C).

### Algebraic topology by Edwin H. Spanier

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