Algebraic torus

An algebraic group that is isomorphic over some extension of the ground field to the direct product of a finite number of multiplicative groups $G_m$. The group $\hat T$ of all algebraic homomorphisms of an algebraic torus $T$ in $G_m$ is known as the character group of $T$; it is a free Abelian group of a rank equal to the dimension of $T$. If the algebraic torus $T$ is defined over a field $k$, then $\hat T$ has a $G$-module structure, where $G$ is the Galois group of the separable closure of $k$. The functor $T \to \hat T$ defines a duality between the category of algebraic tori over $k$ and the category of $\Bbb Z$-free $G$-modules of finite rank. An algebraic torus over $k$ that is isomorphic to a product of groups $G_m$ over its ground field $k$ is called split over $k$; any algebraic torus over $k$ splits over a finite separable extension of $k$. The role played by algebraic tori in the theory of algebraic groups greatly resembles the role played by tori in the theory of Lie groups. The study of algebraic tori defined over algebraic number fields and other fields, such as finite fields, occupies an important place in problems of arithmetic and in the classification of algebraic groups. Cf. Linear algebraic group; Tamagawa number.

References

[1]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	T. Ono, "Arithmetic of algebraic tori" Ann. of Math. (2) , 74 : 1 (1961) pp. 101 139 MR0124326 Zbl 0119.27801
[3]	T. Ono, "On the Tamagawa number of algebraic tori" Ann. of Math. (2) , 78 : 1 (1963) pp. 47 73 MR0156851 Zbl 0122.39101