Lectures on Cyclic Homology by D. Husemoller

By D. Husemoller

The ebook is a simple advent to the topic, divided into 3 elements. the 1st, Chapters 1 and a couple of, is heritage fabric on specified and Hochschild homology for the start reader. In Chapters three, four, five 3 definitions of cyclic homology are thought of, its invariance less than Morita equivalence, its relation to Lie algebra homology, and the Connes' B operator. The 3rd half, Chapters 6 and seven, relates cyclichomology to differential varieties and exhibits how the Chern personality takes values in cyclic homology. integrated is the classical Hochschild-Kostant-Rosenberg theorem referring to differential kinds to Hochschild homology.

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Let [A, M] denote the graded k-submodule generated by all elements of the form [a, x] = ax − ǫ(θ, θ′ )xa where a ∈ Aθ,x ∈ Mθ′ . As a graded k-module we denote by M αβ = M/[A, M]. If f : M → M ′ is a morphism of A-bimodules, the f ([A, M]) ⊂ 30 [A, M ′ ] and f induces on the quotient f αβ : M αβ → M ′ αβ , and this defines a functor A ModA → Θ(k) which is the largest quotient of an Abimodule M such that the left and right actions are equal. It is a kind of abelianization, in the sense that for the A-bimodule A the result A/[A, A] is just the abelianization of the Lie algebra Lie(A).

49 Proof. 4). This proves the proposition. 3. 2) being exact leads to the following isomorphism involving (C∗ (A)1−T , b) namely (C∗ (A)1−T , b) → im(N) ⊂ (C∗ (A), b′ ). Further, we have a morphism of the assembled double complex into 49 the complex of covariants CC∗ (A) → C∗ (A)1−T , b which also maps the double complex filtration arising from the vertical grading into the degree filtration. In other words for Ci (A) → F pCn (A)1−T F pCCn (A) = i≤p,i+ j=n where F p Cn (A)1−T    Cn (A)1−T =  0 for p ≤ n for p > n.

The first is based on Appendix 2 of the rational homotopy theory paper of Quillen [1969]. 1. On the category of cocommutative differential Hopf algebras A over k, the natural morphism H(P(A)) → P(H(A)) is an isomorphism where x ∈ P(A) means ∆(x) = x ⊗ 1 + 1 ⊗ x. Proof. Quillen proves rather directly that for a differential Lie algebra L with universal enveloping U(L) differential Hopf algebra that U(H(L)) → H(U(L)) is an isomorphism. Now U and P are inverse functors between differential Lie algebras and cocommutative differential Hopf algebras by a basic structure theorem of Milnor and moore 1965.

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