Voevodsky: Motivic cohomology with Z/2 coeffs. Mentions the motivic analogue of Margolis homology, and some background on the topological version. See pp 14.
List mail from Andrew Salch:
Let k be a finite field of characteristic p and let A be a co-commutative Hopf algebra over k. For each primitive x in A satisfying x^p = 0, one has a pair of Margolis homology functors defined on the category of A-modules: H_0(M, x) = (ker f)/(im f^{p-1}) and H_1(M, x) = (ker f^{p-1})/(im f), where f is the self-map of M given by multiplication by x. If p=2 then H_0(M, x) coincides with H_1(M, x).
Margolis, in his book “Spectra and the Steenrod algebra,” proves that, if A is an exterior algebra or the Steenrod algebra (or a sub-Hopf-algebra of it) at the prime 2, there is a Whitehead theorem for Margolis homology: if a map of bounded-below A-modules induces an isomorphism in all of A’s Margolis homology theories, then the map of A-modules has projective kernel and projective cokernel.
Is it known for what other co-commutative Hopf algebras A this analogue of Whitehead’s theorem holds?
Thanks, Andrew S.
nLab page on Margolis homology