Algebraic Torus

An algebraic torus over a field is an algebraic group which becomes a direct product of copies of the multiplicative group after extension to an algebraic closure. A split torus over is already isomorphic over to . Over , its complex points are . Its characters are algebraic homomorphisms to , and its cocharacters are algebraic homomorphisms from . For a split torus, both lattices are isomorphic to .

The coordinate ring of the split torus is the ring of Laurent polynomials (Milne 2017, p. 40).

On an affine scheme with an algebraic torus group action, a cocharacter induces a graded algebra structure on its coordinate ring, indexed by integers. The fixed subscheme for this cocharacter has coordinate ring given by the quotient ring by the ideal generated by every nonzero-degree piece in this grading. The quotient ring may retain nilpotent functions (Milne 2017).

For example, let with grading and . The ideal generated by the nonzero-degree pieces is , so the fixed subscheme has coordinate ring . This is the ring of dual numbers, which is not a reduced ring.