Let $X\_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring
$$S = \mathbb{C}[x\_{\rho}\: | \: \rho\in\Sigma(1)]$$
Define $\deg(x\_{\rho}) = D\_{\rho}$.
Now, take a divisor $D = \sum\_{\rho\in\Sigma(1)}a\_{\rho}D\_{\rho}$. Then $H^0(X,D)$ is generated by monomials $X^{m}$ such that $m\in M$ with $<m,u\_{\rho}>\geq -a\_{\rho}$ for all $\rho\in\Sigma(1)$, where $u\_{\rho}$ is the generator of the ray corresponding to $D\_{\rho}$.


Take an $m\in M$ yielding a section of $D$. Is it then true that the monomial $x\_{1}^{<m,u\_1>+a\_1}\dots x\_{k}^{<m,u\_k>+a\_k}$, where $k = |\Sigma(1)|$, belongs to the degree $D$ part of $S$?



Sections of Cartier divisors on toric varieties
That is true. Just note that


$$deg(x\_1^{<m,u\_1>+a\_1}\cdots x\_k^{<m,u\_k>+a\_k})$$


is the class in $Cl(X\_{\Sigma})$ of the divisor 


$$\sum\_{\rho} (<m,u\_\rho>+a\_\rho)D\_\rho = \sum\_{\rho}<m,u\_\rho>D\_\rho + \sum\_{\rho}a\_{\rho} D\_\rho \sim div(\chi^m)+D\sim D$$


