Margolis homology

General Introduction

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Online References

Voevodsky: Motivic cohomology with Z/2 coeffs. Mentions the motivic analogue of Margolis homology, and some background on the topological version. See pp 14.

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Paper References

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Definition

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Properties

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Standard theorems

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Open Problems

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Connections to Number Theory

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Computations and Examples

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History and Applications

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Some Research Articles

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Other Information

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: H0(M, x) = (ker f)/(im f^{p-1}) and H1(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 H0(M, x) coincides with H1(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.

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General Introduction