Here is a cohomology theory for a Hopf algebra, which I am sure has appeared elsewhere. I met it in the van Est spectral sequence for Hopf algebras. Apologies for my being stupid here, but it would be really helpful if someone would tell me where it comes from, and where to look for results on it!

Let $H$ be a Hopf algebra and $\lambda: F\to H\otimes F$ be a left $H$-comodule, which we write in Sweedler notation $\lambda(f)=f_{[-1]}\otimes f_{[0]}$.

Define $D^{n}=H^{\otimes n+1}\otimes F$ for $n\ge 0$, with the tensor product left $H$-coaction. The differential $ d:D^{n}\to D^{n+1}$ with $ d^2=0$ is defined by $$ d(h_{0}\otimes\ldots\otimes h_{n}\otimes f)\,=\, \sum_{n+1\ge i\ge 0}(-1)^{i}\, h_{0}\otimes\ldots\otimes h_{i-1}\otimes 1_H \otimes h_{i}\otimes\ldots \otimes h_{n}\otimes f\ . $$ As $ d$ is a left $H$-comodule map, we can restrict the complex to the invariants to give $({}^{coH}D^{n}, d)$, and the cohomology of the invariants is what I am looking for. The invariants are taken over the tensor product coaction on all factors. There is an explicit reformulation of the cohomology without taking invariants, and seemingly without requiring an algebra structure, though showing an isomorphism is awkward, and does require Hopf algebra structure:

Define a cochain complex $(G^{*},\bar d )$ by $G^{n}=H^{\otimes n}\otimes F$ for $n\ge 0$ with $\bar d f=1_{H}\otimes f-\lambda(f)$ for $f\in F$ and \begin{eqnarray*} \bar d (h_{1}\otimes\dots\otimes h_{n}\otimes f) &=& 1_{H}\otimes h_{1}\otimes\dots\otimes h_{n}\otimes f\,-\, \Delta(h_{1})\otimes\dots\otimes h_{n}\otimes f\,+\,\dots \cr &&+\,(-1)^{n}\,h_{1}\otimes\dots\otimes \Delta(h_{n})\otimes f\,-\, (-1)^{n}\,h_{1}\otimes\dots\otimes h_{n}\otimes \lambda(f)\ . \end{eqnarray*} I guess that the comment below on the dependence on a coalgebra structure is then correct...

  • $\begingroup$ How do you prove the equivalence of the two cohomologies in your post? $\endgroup$ – user66288 Apr 20 at 5:00

Looks to me as if you have not used the product of your Hopf algebra, and it looks to me that you have written an example of a cotorsion product, as defined by Eilenberg and Moore in their paper Homology and fibrations I. Coalgebras, cotensor product and its derived functors. Comm. Math. Hel. 40(1965), 199--236, available here: http://retro.seals.ch/digbib/view?rid=comahe-002:1965-1966:40::223

  • $\begingroup$ The product appears implicitly in the coaction on the tensor product. I am checking the reference - thanks! $\endgroup$ – Edwin Beggs May 1 '15 at 13:25
  • $\begingroup$ It seems to me that the natural action, under which you would be looking at a cobar construction, would just come from the diagonal on the left most tensor factor H. $\endgroup$ – Peter May May 2 '15 at 1:30
  • $\begingroup$ The required action for the van Est spectral sequence is the full tensor product coaction (this is needed to get a non-trivial dependence on the coaction). I see now that my question was ambiguous on this point. If I remember correctly, the sequence without taking the invariants is acyclic - will check. $\endgroup$ – Edwin Beggs May 2 '15 at 11:16
  • $\begingroup$ OK, I have added more explanation - apologies for that. There is a way to reformulate the complex with no explicit invariants and no algebra structure. $\endgroup$ – Edwin Beggs May 2 '15 at 11:35

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