### On the cohomology of some Hopf algebroids and Hattori-Stong theorems.

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We consider brave new cochain extensions F(BG +,R) → F(EG +,R), where R is either a Lubin-Tate spectrum E n or the related 2-periodic Morava K-theory K n, and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for E n and K n these extensions are always faithful in the K n local category. However, for a cyclic p-group ${C}_{{p}^{r}}$, the cochain extension $F(B{C}_{{p}^{r}+},{E}_{n})\to F(E{C}_{{p}^{r}+},{E}_{n})$ is not a Galois...

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