. We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some methods already known about the computation of the Hochschild and cyclic homologies of CDGAs. In the last section of the paper, we analyze the p-local homology of the iterated bar construction of a CDGA (p prime). 1. Introduction. The description of eÆcient algorithms of homological computation might be considered as a very important question in Homological Algebra, in order to use those processes mainly in the resolution of problems on algebraic topology; but this subject also inuence directly on the development of non so closedareas as Cohomological Physics (in this sense, we nd useful references in [12], [24], [25]) and Secondary Calculus ([14], [27], [28]). Working in the context ...

We define a new cohomology for associative algebras which we compute for algebras with units. L.P.T.H.E.-ORSAY 94/10 alg-geom/9404014 1 Laboratoire associ'e au C.N.R.S. 1 Introduction: Definition of the basic cohomology of an associative algebra Let A be an associative algebra over K = R or C and let A Lie be the underlying Lie algebra (with the commutator as Lie bracket). For each integer n 2 N , let C n (A) be the vector space of n-linear forms on A, i.e. C n (A) = (A\Omega n ) . For ! 2 C n (A) and ø 2 C m (A) one defines !:ø 2 C n+m (A) by: !:ø(A 1 ; : : : ; A n+m ) = !(A 1 ; : : : ; A n )ø(A n+1 ; : : : ; A n+m ); 8A i 2 A: Equipped with this product, C(A) = \Phi n C n (A) becomes an associative graded algebra with unit (C 0 (A) = K ). One defines a differential d on C(A) by setting for ! 2 C n (A), A i 2 A d!(A 1 ; : : : ; A n+1 ) = n X k=1 (\Gamma1) k !(A 1 ; : : : ; A k\Gamma1 ; A k A k+1 ; A k+2 ; : : : ; A n+1 ): Indeed, d is the extension as...

Abstract. We show that the entire cyclic cohomology of Banach algebras defined by Connes has the simplicial normalization property. A key tool in the proof is the notion and properties of supertraces on the Cuntz algebra QA. As an example of further applications of this technique we give a proof of the homotopy invariance of entire cyclic cohomology. Key words Simplicial normalization, homotopy invariance, supertraces, Cuntz algebra, entire cyclic cohomology. 1