Definition
Given a base scheme S, an algebraic torus over S is defined to be a group scheme over S that is fpqc locally isomorphic to a finite product of multiplicative groups. In other words, there exists a faithfully flat map X → S such that any point in X has a quasi-compact open neighborhood U whose image is an open affine subscheme of S, such that base change to U yields a finite product of copies of GL1,U = Gm/U. One particularly important case is when S is the spectrum of a field K, making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of Gm/L. In general, the multiplicity of this product (i.e., the dimension of the scheme) is called the rank of the torus, and it is a locally constant function on S.
If a torus is isomorphic to a product of multiplicative groups Gm/S, the torus is said to be split. All tori over separably closed fields are split, and any non-separably closed field admits a non-split torus given by restriction of scalars over a separable extension. Restriction of scalars over an inseparable field extension will yield a commutative group scheme that is not a torus.
Read more about this topic: Algebraic Torus
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