Mysterious property of $\mathbb{Q}$

Question

Hi,

I am currently working through the paper by Bousfield and Gugenheim on rational homotopy theory, and have come to a point where they show why it is important to work over $\mathbb{Q}$, and not just any field (remark 9.7). They assert that if $Hom_k(Hom_{\mathbb{Z}}(G, k), k)=G$ for some non-trivial abelian group $G$, and some field $k$ [edit: of characteristic $0$] (thanks to Fernando Muro for pointing out that we need characteristic $0$), then we must have $k=\mathbb{Q}$ (and also that $G$ is finite dimensional as a $\mathbb{Q}$-vector space, which is clear enough). I can't quite see what property of $\mathbb{Q}$ this reduces to, and it really makes my head spin trying to think through it. The paper is now available online, just google "on pl de rham theory and rational homotopy type".

Any suggestions would be great.

Thanks

Brian

Answer 1

First, $k$ needs to be assumed of characteristic $0$. The mysterious property is that $\mathbb{Q}$ is a prime field:

If $Hom_k(Hom_Z(G, k), k)=G$, $G$ has to be a $k$-vector space, say $G=kT$ for a basis $T$. $Hom_Z (kT;k)$ is a $k$-vector space.

Let me first argue that $T$ has to be finite. Since $Hom_k (kT;k) \to Hom_Z (kT;k)$ is injective, the dimension of $Hom_Z (kT;k)$ is at least the dimension of $(kT)^{\ast}$. Thus it follows that the dimension of $kT$ is at least the dimension of the bidual of $kT$. Thus can only be true if $T$ is finite. We can assume that $T$ has one element; therefore

$$dim_k (Hom_k (Hom_Z (k,k);k))=1.$$

If $k$ is not a prime field, then $dim (Hom_Z(k,k))>1$, which contradicts the above formula.

The same argument works in characteristic $p$ and shows that $k$ has to be $\mathbb{F}_p$.