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An algebraic torus of dimension $n$ is an algebraic group isomorphic over the base field to $({\mathbb{G}}_m{)}^n$. Its character and cocharacter lattices are dual free abelian groups $M$ and $N$.

Toric geometry studies algebraic varieties containing an algebraic torus as a dense open subset, with the torus action extended to the whole variety. Rational polyhedral fans translate their geometry into lattice combinatorics.