Let be a chart in *Gr*(*N*) and
denote the corresponding lexicographic basis. We will denote by the dual basis of , i.e. .
Clearly the direct product is spanned by
where

In section 3 we already called the fundamental character of *Gr*(*N*),
i.e. assigns to each element of the Grassmann algebra its
``scalar'' part. We keep the same notation for

>From its definition, we see that, considered as a multilinear form on *Gr*(*N*), vanishes on homogeneous elements unless all elements are -numbers. In this case,
is just equal to the product . From a direct computation, it is clear that if and only if *p* is even. For example

When *p* is even, is therefore a Hochschild cocycle, moreover, is also clearly cyclic. It is the b of something (Hochschild cochain) but this Hochschild cochain is not
cyclic. Therefore, is a non trivial cyclic cocycle. However, if we try to associate a current to , we discover that the corresponding current
is strictly zero. What we just computed is nothing else than the cyclic cohomology of complex
numbers: it is trivial when *p* is odd and one- dimensional when *p* is even. What happens, as illustrated above, is that this hierarchy of cocycles is also part of the cyclic cohomology of
Grassmann algebras. For us, this is the uninteresting part and we shall single it out in the
sequel.

Mon May 20 14:40:14 MET DST 1996