process

168 C. VoigtProof. a) follows from the explicit formula for H . b) We compute n H .x ˝ a 0da 1 :::da n / D x .2/ ˝ S 1 .x .1/ / .da 1 :::da n /a 0D x .2/ ˝ S 1 .x .1/ / .a 0 da 1 :::da n /C . 1/ n b H .x .2/ ˝ S 1 .x .1/ / .da 1 :::da n /da 0 /D x .2/ ˝ S 1 .x .1/ / .a 0 da 1 :::da n / C b H H n d.x ˝ a 0da 1 :::da n /which yields the claim. c) follows by applying b H to both sides of b). d) Apply H tob) and use a). e) is a consequence of b) and d). f) We computeXn 1B H b H C b H B H D j H db H Cj D0where we use d) and b).nXn 1b H j H d D X j H .db H C b H d/C H n b H dj D0j D0D id H n .1 b H d/ D id H n . H C db H /D id T C db H T Tdb H D id TFrom the definition of B H and the fact that d 2 D 0 we obtain BH2summarize this discussion as follows.D 0. Let usProposition 6.2. Let A be an H -algebra. The space H .A/ of equivariant differentialforms is a paramixed complex in the category of AYD-modules.We remark that the definition of H .A/ for H D D.G/ differs slightly from thedefinition of G .A/ in [25] if the locally compact group G is not unimodular. However,this does not affect the definition of the equivariant homology groups.In the sequel we will drop the subscripts when working with the operators on H .A/introduced above. For instance, we shall simply write b instead of b H and B insteadof B H .The n-th level of the Hodge tower associated to H .A/ is defined byMn 1 n H .A/ D j H .A/ ˚ n H .A/=b.nC1 H.A//:j D0Using the grading into even and odd forms we see that n H .A/ together with theboundary operator B C b becomes a paracomplex. By definition, the Hodge tower H .A/ of A is the projective system . n H .A// n2N obtained in this way.From a conceptual point of view it is convenient to work with pro-categories in thesequel. The pro-category pro.C/ over a category C consists of projective systems inC. A pro-H -algebra is simply an algebra in the category pro.H -Mod/. For instance,every H -algebra becomes a pro-H -algebra by viewing it as a constant projective system.More information on the use of pro-categories in connection with cyclic homology canbe found in [10], [25].