I am trying to understand the proof of Proposition 4 in
S. Ullom, Integral normal bases in Galois extensions of local fields, Nagoya Math. J. Volume 39 (1970), 141-148. The PDF is available here:
http://projecteuclid.org/euclid.nmj/1118798052
It appears that the following result is used, but I'm afraid that I don't quite see the proof.
Let $k$ be a field of characteristic $p$ and let $G$ be a finite $p$-group. Let $W$ be a left $k[G]$-module such that $\dim_k W = |G|$. Suppose that $\dim_k W^{G} = 1$. Then $\dim_k W_G = 1$.
Here $W^G$ denotes invariants and $W_G$ denotes coinvariants.
I have to admit that I know relatively little about representation theory in characteristic $p$. One idea would be to consider the dual representation $\hat{W}$, but I only got so far:
$\dim_k W^{G} = 1 \implies W$ is indecomposable $\implies \hat{W}$ is indecomposable. But maybe this is not the right approach. If I could show that the Tate cohomology groups of $W$ vanish, then of course the desired result drops out, but I think this is rather strong medicine.
Is anyone able to give a proof of the above claim? I suspect the solution is fairly easy, but I just don't see it at the moment.
Best Answer
The point is that you can inject $W$ into $k[G]$ and because of the equality of dimension this is an iso. This lets you to conclude that the coinvariants are $1$-dimensional.
The injection goes as follows:
$k[G]$ is a projective $k[G]$-module for trivial reasons, so its $k$-linear dual is injective, but $k[G]^*\cong k[G]$. So $k[G]$ is an injective $k[G]$-module.
Now $G$-invariants of $k[G]$ is a one dimensional vector space, which we may identify with $W^G$. Since $k[G]$ is injective, there exists a map $\phi: W\rightarrow k[G]$ which induces an injection on $W^{G}$. This means that $Ker \phi \cap W^G=(Ker \phi)^G=0$, but since $G$ is a $p$-group and we are in char $p$, we get that $Ker \phi=0$.