Cyclic homology and the Lie algebra homology of matrices

579

homology of the double complex â8(A)red defined by an exact sequence

0 -^ ^ (к)„огш -^ ^ (А)погш ^ ^ (A)red -> 0

where â8(A)norm is the normalized version of Connes' double complex described in 1.10. This reduced Connes' complex is the same as ^(A)^^^ except that the diagonal of A's is replaced by Ä's.

PROPOSITION 4.1. One has long exact sequences

- > НСЛк) -^ НСЛА) -^ ЙС^{А) -^ HQ_i(k) ^ -> ЙЛА) -> ЙС^{А) -^ ЙС^_2(А) -> Й^-г(А) -^.

The first follows from the exact sequence defining ^(A)red and the fact that the homology of ^(A)norm is HC^{A). The second exact sequence can be derived as Theorem 1.6. but using the double complex ^{A)rcd-

The reduced theory is a natural thing to consider when dealing with mented algebras. We recall that an augmented algebra A is of the form А = кф1 where I is the augmentation ideal, and that A is isomorphic to the algebra with identity obtained by adjoining an identity to the non-unital ring I. In fact the categories of augmented algebras and non-unital algebras are equivalent in this way.

For an augmented algebra the first exact sequence in the above proposition splits yielding the isomorphism

HC^iA ) = НС^{к)ФНС^{А)

At this point one might define the cyclic homology of non-unital algebra to be the reduced cyclic homology of the corresponding augmented algebra. On the other hand inspection of the arrows in the double complex ^(A) of the first section shows that it makes sense for non-unital rings, hence we can make the definition HC:ic(I) = H5jc(Tot^(I)). The following shows that these two definitions agree.

PROPOSITION 4.2. If A = k®I is an augmented ring, then the complexes ^(I) and ^(A)red cire isomorphic, hence НС^{1) = ЙС:,^(А).

Proof . We define an isomorphism from ^(I) to ^(A)j.ed by where the isomorphism in the middle sends (xq, ..., x„) in I""^^ and (xi,..., x„)