The main result of this paper is to show that if
G∗ is a simplicial group of finite
length, then HnG∗
also has finite length. Here the length of a simplicial group means
the length of the corresponding Moore normalization and
HnG∗ is the simplicial abelian
group given by
[k][map ]HnGk.
A similar fact is true if we replace G∗ by a
simplicial ring and we take the algebraic K-functors instead of
group homology.
The origin of such results goes back to the classical paper of Dold
and Puppe (see
Hilfsatz 4·23 of [7]), where the following was
proved: let
formula here
be a functor between abelian categories of degree d,
meaning that the (d+1)st
cross-effect functor in the sense of Eilenberg and MacLane
[9] vanishes. Then
l(TX∗)[les ]dl(X∗) for
any simplicial
object X∗ in A.
Here l(X∗) denotes the length of
X∗. Actually, this property characterizes the
degree of functors in the abelian case
(see Lemma 3·6).
One can modify the notion of the cross-effects in the nonabelian framework
(see
[2]) and define the notion of degree of functors.
One can also use the above inequality
to define the simplicial degree in the nonabelian set-up. However in this
way we get
generally different invariants and the aim of this paper is to establish
the relationship
between the different notions. We show that many classical functors arising
in
homological algebra and K-theory have finite
simplicial degree. Our Conjecture 4·7
claims that this should be always the case.
We remark that if the Moore normalization of a simplicial group
G∗ is zero in
dimensions >k then πiG∗=0
for
i>k as well. Therefore, for a functor T of
simplicial degree d, one has
formula here
Our work can therefore be considered as a new general method proving
vanishing
results. Here is a sample application of the main result.