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Let $A$ be a Hopf algebra over a field $k$, and let $B$ be a normal subHopf algebra of $A$. Suppose we have an $A$-free coresolution of $k$ over the form $F_n=K_n \otimes_k A$. Kochman claims that $F \Box_{A//B} A$ is then an $A$-free coresolution of $k \Box_{A//B} A$. I would appreciate any help with the following questions. Why does cotensoring with $A$ over $A//B$ preserve exactness here? Is $- \Box_{A//B} A$ always an exact functor?

(I am attempting to follow Kochman's book in a computation of the homotopy of some Thom spectra, which is where the above comes up)

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A short way of stating this is:is $A$ coflat over $A/\!/B$? – Mariano Suárez-Alvarez Dec 17 '11 at 4:20
(You want this in general, or are your Hopf algebras non-negatively graded and cocommutative, or do you have some other hypothesis?) – Mariano Suárez-Alvarez Dec 17 '11 at 4:23
Kochman states this is true in general, but the application I am interested in is where $A$ is the dual of the Steenrod algebra (commutative but not cocommutative, non-negatively graded). – Vitaly Lorman Dec 17 '11 at 17:14
There is some stuff in Appendix 1 of 'Complex Cobordism and the Stable Homotopy Groups of Spheres' by Ravenel that develops this kind of stuff, although a quick glance couldn't find your exact problem. Baring that, I suggest your recent questions have been Mathoverflow level - you will certainly get useful responses there (and I would be interested myself!) – Juan S Dec 19 '11 at 23:08
Thanks for the advice! I posted it on MO and got an answer: – Vitaly Lorman Dec 29 '11 at 2:43

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