Coordinate ring of an affine toric variety ($\mathbb{C}[{\sigma }^{\vee }\cap M]$)

The coordinate ring of $U_{\sigma }$ is generated by the characters ${\chi }^m$ for lattice points $m\in {\sigma }^{\vee }\cap M$, with multiplication ${\chi }^m{\chi }^{m^{\prime }}={\chi }^{m+m^{\prime }}$.