Simplicial degrees of functors

Abstract

The main result of this paper is to show that if G∗ is a simplicial group of finite length, then H_nG∗ also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and H_nG∗ is the simplicial abelian group given by [k][map ]H_nG_k. 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 π_iG∗=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.