## Calculations of Homotopy Coinvariants

Does anyone know of a reference in which homotopy coinvariants of some coaction in the category of spectra are explicitly calculated? Particularly in the case of Hopf-Galois extensions of ring spectra.

Thanks!

-----(EDIT)------ I should mention that I later realized the above concept is not in John Rognes' monograph. It is in Kathryn Hess' paper on homotopical Hopf-Galois extensions and Fridolin Roth's recent dissertation. My apologies to anyone who went looking in Rognes' paper.

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What do you mean by the homotopy coinvariants of a coaction? If you have actually a $G$-Galois extension of ring spectra $R\to S$, then $S_{hG} \simeq S^{hG} \simeq R$. Thus, for example, $KU_{hC_2} \simeq KO$, as shown (or rather sketched) in Rognes's monograph on Galois theory. – Lennart Meier Feb 1 at 20:17
Are you sure that his $S^{hcoH}$ isn't equivalent to what Lennart is suggesting? – Sean Tilson Feb 1 at 21:15
Sorry, deleted some comments I had made that were mistaken. And thanks Lennart, yeah, I am specifically referring to the sort of homotopy coinvariants discussed in Kathryn Hess' paper arxiv.org/pdf/0902.3393.pdf – Jon Beardsley Feb 1 at 23:31
After looking at Kathryn Hess' paper I definitely understand why you want some examples! As a general hint: Perhaps you could try in the future to give the definitions of the non-standard terms in your question. There may be someone who is not familiar with the terms and can give nevertheless an answer; moreover it benefits the more casual reader. – Lennart Meier Feb 2 at 3:04
Yeah that's a good point. I didn't really think about it, and I'm actually not really familiar enough with the standard literature to even realize someone might be confused! But thanks, yeah, I guess basically we can just write it down as the best homotopical-ization of the cotensor of comodule algebras over a (homotopical) Hopf-Algebra. I suppose that's more or less exactly what it is, and Hess shows that in nice cases, this can indeed be made homotopical. – Jon Beardsley Feb 2 at 19:09