By R. Göbel, L. Lady, A. Mader
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Extra info for Abelian Group Theory. Proc. conf. Honolulu, 1983
Notice that on @ Cp*O= ( S ( g * ) the operator a is just evaluation of a polynomial at 0 , in other words, it coincides with the operator p defined in the introduction to this chapter. 2) holds on ( S ( g 8 )8 Ah,,)G with p = X . So we can and hence 1~ o dG. 4) as Then Let F := ( E + a)-' and define The series on the right is locally finite since RF lowers filtration degree. Let Q := KU. 9). 2 We will prove that + dGQ QdG = I - n u . 9) as dGQ QdG = I - i o ( x u ) . 10) Cartan's Formula We now do a more careful analysis which will lead to a rather explicit formula for the operator n u .
9). 2 We will prove that + dGQ QdG = I - n u . 9) as dGQ QdG = I - i o ( x u ) . 10) Cartan's Formula We now do a more careful analysis which will lead to a rather explicit formula for the operator n u . 7) since F preserves the bigradation and equals the identity on CovO. We may write . R=S-T on COvO= Abas Thus the maps i and aU are homotopy inverses of one another, and hence induce isomorphisms on cohomology. 9) implies that where and s= . J 1 : T = _,-,O,,g'paa 2 +~ c t .. 3. - l . j . 3+2 Chapter 5.
Now since G acts trivially on H(A), being connected, and K is a subgroup of G, we conclude that K also acts trivially on H(A) even though it need not be connected. 1 applies. We can do a bit more: n o m the inclusion T K we get a morphism of double complexes CK(A) ~ d - 4 ) ~ - ~ ( k *8)H(A). 2 yields HK(A) = H~(A)W. 2 Let G be a connected compact Lie group, T a maximal torus and W its Weyl p u p . 1 Suppose that the restriction map - s ( ~ * ) ~S(k*)K This result can actually be strengthened a bit: The tensor product is btjectzve.
Abelian Group Theory. Proc. conf. Honolulu, 1983 by R. Göbel, L. Lady, A. Mader