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Concept
Algebraic torus
Summary
In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by , , or , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups . These groups were named by analogy with the theory of tori in Lie group theory (see Cartan subgroup). For example, over the complex numbers the algebraic torus is isomorphic to the group scheme , which is the scheme theoretic analogue of the Lie group . In fact, any -action on a complex vector space can be pulled back to a -action from the inclusion as real manifolds.
Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them such as symmetric spaces and buildings.
In most places we suppose that the base field is perfect (for example finite or characteristic zero). This hypothesis is required to have a smooth group schemepg 64, since for an algebraic group to be smooth over characteristic , the maps must be geometrically reduced for large enough , meaning the image of the corresponding map on is smooth for large enough .
In general one has to use separable closures instead of algebraic closures.
Multiplicative group
If is a field then the multiplicative group over is the algebraic group such that for any field extension the -points are isomorphic to the group . To define it properly as an algebraic group one can take the affine variety defined by the equation in the affine plane over with coordinates . The multiplication is then given by restricting the regular rational map defined by and the inverse is the restriction of the regular rational map .
Let be a field with algebraic closure . Then a -torus is an algebraic group defined over which is isomorphic over to a finite product of copies of the multiplicative group.
In other words, if is an -group it is a torus if and only if for some . The basic terminology associated to tori is as follows.
Official source
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