Homotopy Theory of Simplicial Abelian Hopf Algebras

We examine the homotopy theory of simplicial graded abelian Hopf algebras over a prime field Fp, p > 0, proving that two very different notions of weak equivalence yield the same homotopy category. We then prove a splitting result for the Postnikov tower of such simplicial Hopf algebras. As an application, we show how to recover the homotopy groups of a simplicial Hopf algebra from its André-Quillen homology, which, in turn, can be easily computed from the homotopy groups of the associated simplicial Dieudonné module. This paper is divided into two parts. The first and larger part, is also the more abstract; in it we undertake a thorough examination of the homotopy theory of simplicial graded abelian Hopf algebras over Fp, p > 0. The second part is a calculational application, relying heavily on the first part and intended partly to demonstrate that the homotopy theory of simplicial Hopf algebras deserves consideration. In this second part we show that the homotopy groups of a simplicial abelian Hopf algebra support a very rich and rigid structure. This has implications for the cohomology spectral sequence of a variety of cosimplicial spaces. (See, for example, the work of the second author [20], or Dwyer’s spectral sequence as explained in [2, §4].) To explain our results in more detail, fix a prime p, and let HA be the category of graded, bicommutative Hopf algebras A over Fp which are connected in the sense that A0 ∼= Fp. The objects inHA are the abelian objects in the category CA of graded connected coalgebras over Fp; hence, we call an object in HA an abelian Hopf algebra. Let sHA be the category of simplicial objects in HA. If f : A → B is a morphism in sHA there are two obvious ways to specify when f is a weak equivalence. On the one hand, if A ∈ sHA, it is, among other things, a graded abelian group and, as such, has graded homotopy groups π∗A ∼= H∗(A, ∂), where ∂ = Σ(−1)di : An → An−1. We could demand that f be a weak equivalence if π∗f is an isomorphism. On the other hand, HA is an abelian category with enough projectives, so sHA acquires a notion of weak The first author was supported by the National Science Foundation