### A lower bound for coherences on the Brown-Peterson spectrum.

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We consider Taylor approximation for functors from the small category of finite pointed sets $\Gamma $ to modules and give an explicit description for the homology of the layers of the Taylor tower. These layers are shown to be fibrant objects in a suitable closed model category structure. Explicit calculations are presented in characteristic zero including an application to higher order Hochschild homology. A spectral sequence for the homology of the homotopy fibres of this approximation is provided.

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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