Martin's Blog

Algebraic tori

Posted by martin on Friday, 08 January 2010 at 16:16

Algebraic tori are the simplest examples of algebraic groups. In this post I will define algebraic tori and give some examples. Later I will write about their character groups and representations, and after that I will be able to talk about Hodge structures.

I have been trying to write a post about algebraic tori for several days, mainly because I was trying to sort out the proof that tori over separably closed fields are split. This is complicated and not very important as in practice I only care about perfect fields, so I have left it out.

Note that the algebraic tori considered here have nothing to do with the complex tori in my last post; indeed the complex points of an algebraic torus are not compact in the usual topology! They are called tori because they play the same role in the theory of algebraic groups as real tori play in the theory of Lie groups.

Algebraic groups

I don’t want to say too much about algebraic groups in general, but it is necessary to have a definition. Finding the right way to define algebraic groups turns out to be quite tricky; the most convenient approach is to use the functor of points.

An algebraic group over a field $k$ is a functor $k\textbf{-Alg} \to \textbf{Grp}$ such that when you compose with the forgetful functor $\textbf{Grp} \to \textbf{Set}$ you get the functor of points of a $k$-variety. (For the purposes of this article, a $k$-variety means a geometrically reduced separated scheme of finite type over $k$.)

If $G$ is an algebraic group over $k$ and $K$ a field extending $k$, then we write $G_K$ for the algebraic group over $K$ obtained by composing the functor $G$ with the forgetful functor $K\textbf{-Alg} \to k\textbf{-Alg}$.

A homomorphism of algebraic groups is a natural transformation between functors $k\textbf{-Alg} \to \textbf{Grp}$; in other words it is a morphism of $k$-varieties which induces a group homomorphism on $A$-points for each $k$-algebra $A$.

An algebraic subgroup of $G$ is an algebraic group $H$ (over $k$) such that for each $k$-algebra $A$, $H(A)$ is a subgroup of $G(A)$.

A linear algebraic group is an algebraic group isomorphic to an algebraic subgroup of $\mathrm{GL}_n$ for some $n$. It is easy to see that the underlying variety of a linear algebraic group is affine. In fact any affine algebraic group is linear, but we won’t need this fact today.

Split tori

The multiplicative group $\mathbb{G}_m$ is the algebraic group (over any field $k$) given by $\mathbb{G}_m(A) = A^\times$ (the invertible elements of $A$). This is a linear algebraic group because it is isomorphic to $\mathrm{GL}_1$.

A split torus over $k$ is an algebraic group isomorphic to $\mathbb{G}_m^r$ for some nonnegative integer $r$. This is a linear algebraic group because it can be realized as the subgroup of diagonal matrices inside $\mathrm{GL}_r$.

General tori

If $k$ is a non-algebraically closed field, then matrices over $k$ may be diagonalisable over $\bar{k}$ but not over $k$ itself. Such matrices still have many of the good properties of matrices diagonalisable over $k$, so they are often useful to work with.

In the same way, when we consider linear algebraic groups, it is often useful to work with groups over $k$ which are only diagonalisable over $\bar{k}$.

Accordingy, we define a torus over $k$ to be a linear algebraic group $T$ over $k$ such that $T_{\bar{k}}$ is a split torus.

By definition, any torus over an algebraically closed field is split. In fact any torus over a separably closed field is split, but the proof is quite complicated.

Quasi-split tori

Let $A$ be a finite-dimensional separable $k$-algebra. Then there is a linear algebraic group $G$ given by $G(B) = \mathbb{G}_m(A \otimes_k B) = (A \otimes_k B)^\times$, denoted $\mathop{\mathrm{Res}_{A/k}} \mathbb{G}_m$.

To check that $G$ is an algebraic variety, choose a basis $\{ v_1, \ldots, v_r \}$ for $A$ over $k$. This induces a ring homomorphism $\phi : A \to M_n(k)$. Now $f = \det \phi(x_1 v_1 + \ldots + x_r v_r)$ is a polynomial in $x_1, \ldots, x_r$.

$A^\times$ is the set of points in $\mathbb{A}^r(k)$ where $f(x_1, \ldots, x_r)$ is invertible, and in general $G(B)$ is the set of points in $\mathbb{A}^r(B)$ where $f(x_1, \ldots, x_r)$ is invertible in $B$. So the points of $G$ are the points of an affine variety over $k$.

Because $A$ is separable, $A \otimes_k \bar{k} \cong \bar{k}^r$, so $G_{\bar{k}} \cong \mathbb{G}_m^r$ and $G$ is a torus.

$G$ is usually not split over $k$: for example if $k$ contains $n$ roots of unity, then any torsion element of $T(k)$, for $T$ a split torus over $k$, can have order at most $n$. But we can choose $A$ to be an extension of $k$ containing a root of unity of order greater than $n$.

A torus of the form $\mathop{\mathrm{Res}_{A/k}} \mathbb{G}_m$ for some finite-dimensional separable $k$-algebra $A$ is called a quasi-split torus.

Tags , , ,

Trackbacks

Use the following link to trackback from your own site:
http://www.martinorr.name/blog/trackbacks?article_id=117

  1. Character groups of algebraic tori From Martin's Blog
    In this post I will return to the subject of algebraic tori. Just as Pontryagin duality classifies locally compact abelian groups through their characters, so algebraic tori are also classified by their characters. In order to account for the arit...

Comments

No comments.

Comments are disabled

Archives

Syndicate