Divisorial section ring of a well-formed weighted projective space (${⨁}_{r\ge 0}\mathcal{L}(rD)\cong k[X_0,\dots ,X_n],\mathrm{deg}{X}_i=q_i$)

With $D=\sum a_i{F}_i$ and $\sum q_i{a}_i=1$, the dual toric lattice is $M=\{m\in {\mathbb{Z}}^{n+1}:\sum q_i{m}_i=0\}$. A torus character ${\chi }^m$ is a section of $rD$ exactly when $b_i=m_i+r{a}_i\ge 0$. The vector $b$ has weighted degree $\sum q_i{b}_i=r$, and the correspondence ${\chi }^m\mapsto X^b$ is bijective on bases in every degree and respects multiplication. This gives the displayed graded-ring isomorphism, even when $D$ is a non-Cartier Weil divisor.