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algebraic torus
In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of ''tori'' in Lie group theory (see maximal torus). The theory of tori is in some sense opposite to that of unipotent groups, because tori have rich arithmetic structure but no deformations.
==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 copies of the multiplicative group scheme G''''m''/''S over ''S''. 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 ''GL''1,''U'' = G''m''/''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 G''m''/''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 G''m''/''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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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