Definition of a character of a linear algebraic group

Question

This might sound like a silly question, but I seem to be getting different definitions from different texts.

Let $k$ be some field and $G$ some linear algebraic group over $k$. Now, the standard definition I have seen, for instance in

Milne, J. S., Algebraic groups. The theory of group schemes of finite type over a field, Cambridge Studies in Advanced Mathematics 170. Cambridge: Cambridge University Press (ISBN 978-1-107-16748-3/hbk; 978-1-316-71173-6/ebook). xvi, 644 p. (2017). ZBL1390.14004.

is that a character is a $k$-group homomorphism $\chi:G\rightarrow \mathbb{G}_m$. But, there is this result in

Platonov, Vladimir; Rapinchuk, Andrei; Rapinchuk, Igor, Algebraic groups and number theory. Volume 1 (to appear), ZBL07671765.

which says that if $G$ is torus, then the following two conditions are equivalent:

1. $G$ is split
2. all its characters are defined over $k$.

Yes, Platonov and Rapinchuk define their characters over a much larger algebriacally closed field $\Omega$, but wouldn't this mean that the two definitions of a character and hence the structure of the character group are different? Am I missing something? Are the definitions somehow similar?

Answer 1

Yes, please don't feel bad about this confusion. It is for semi-historical reasons as KReiser rightly pointed out above, and can be quite confusing.

Namely, let us fix a field $K$ and an (affine) algebraic group $G$ over $K$. Then, for any separable$\color{red}{^{(\ast)}}$ algebraic extension $L/K$ one can form the group of $L$-rational characters

$$X^\ast_L(G):=\mathrm{Hom}(G_L,\mathbb{G}_{m,L}),$$

where these are homomorphisms of group $L$-schemes. There is somewhat varying terminology in this regard, but often people call $X_{K^\mathrm{sep}}$ the group of characters of $G$, and denote it just $X^\ast(G)$ (or $X(G)$). They then might call the elements of $X_K^\ast(G)$ the rational characters of $G$. Although, again, this is author dependent, and so you have to sort of feel out the definition in each case.

To understand the relationship between the rational characters over differing fields, it's useful to introduce some extra structure. If $L/K$ is Galois, with Galois group $\Gamma_{L/K}$, then there is a natural action of $\Gamma_{L/K}$ on $X^\ast_L(G)$ given as follows:

$$\sigma\cdot \chi:= \sigma_G\circ \chi\circ\sigma_{\mathbb{G}_{m,L}}^{-1}.$$

Here for a $K$-scheme $X$ and $\sigma$ in $\Gamma_{L/K}$, I am denoting by $\sigma_X$ the natural morphism $X_L\to X_L$ which is the identity on the $X$-factor and the induced map on $\mathrm{Spec}(L)$ given by $\sigma^{-1}\colon L\to L$ (you can remove the inverse here if you are OK with it being a right, opposed to left, action). It is a good exercise to check that this is a well-defined action (e.g. that $\sigma\cdot\chi$ really can still be made sense of as an element of $X^\ast_L(G)$).

From Galois descent for affine schemes (e.g. see [Poonen,§4.4]) there is a natural identification

$$X^\ast_M(G)=X_L^\ast(G)^{\Gamma_{L/M}},$$

for any tower of extensions $L/M/K$ (with $L/K$ Galois). Here I am being somewhat imprecise and identifying $X^\ast_M(G)$ with a subset of $X_L^\ast(G)$ via the (injective) group map

$$X^\ast_M(G)\to X^\ast_L(G),\qquad \chi\mapsto \chi_L,$$

(the base change map).

In particular, one sees that the natural map

$$X_K^\ast(G)\to X^\ast_{K^\mathrm{sep}}(G)=X^\ast(G),$$

is a bijection if and only if $\Gamma_{K^\mathrm{sep}/K}$ acts trivially on $X^\ast(G)$. As you mentioned, if $G$ is a torus, then this is equivalent to $G\cong \mathbb{G}_{m,K}^n$, i.e. that it is split (and in fact, you can use closely related ideas to define the maximal split subtorus -- see [Springer, Proposition 13.2.4]).

I hope this clarifies things!

$\color{red}{(\ast):}$ This is a little too technical to mention above, but you can really have $L/K$ be any algebraic extension. But, if $L/K$ is purely inseparable then the natural map $X^\ast_K(G)\to X_L^\ast(G)$ is a bijection.

References:

[Poonen] Poonen, B., 2017. Rational points on varieties (Vol. 186). American Mathematical Soc..

[Springer] Springer, T.A. and Springer, T.A., 1998. Linear algebraic groups (Vol. 9). Boston: Birkhäuser.