Let $E=(\mathbb{Z}/p\mathbb{Z})^n$, an elementary Abelian p-group. Let $k$ be an algebraically closed field of characteristic 0. There is a good description of $H^*(E,F^{\times})$ where $F$ is a field of characteristic $p$ Is there a description of $H^*(E,k^{\times})$ where $k$ is a field of characteristic 0? The goal is to describe $H^3(E,k^{\times})$ as an $Aut(E)\cong GL_n(\mathbb{F}_p)$-module.
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2$\begingroup$ How does $E$ acts on $k^\times$? $\endgroup$– Denis NardinApr 24, 2014 at 17:42
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$\begingroup$ $E$ acts trivially on $k^{\times}$. $\endgroup$– brippleApr 25, 2014 at 16:09
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1 Answer
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Since your $E$, or its classifying space $\mathbf{B} E$, is $p$-local, the features of $k^\times$ that you want to study are $p$-torsion and $p$-divisibility; you can go about that several ways, but because the group structure on $k^\times$ is multiplication in the field, these translate to studying $p$th roots of unity and $p$th roots of non-roots-of-unity. There should be a coefficients long exact sequence, $$ H^l(E,k^\times) \overset{p}\to H^l(E,k^\times) \to H^l(E,k^\times / k^{\times p}) \overset\beta\to H^{l+1}(E,k^\times) $$
Also, you will have to be much more specific about what field $k$ you're actually looking at; for instance, $\mathbb{Q}$ has almost no $p$th roots, and $\overline{\mathbb{Q}}$ has all of them, while ${\mathbb{Q}}^\mathrm{Ab}$ has $p$-divisible $p$-torsion.
