$\Lambda=\Lambda({\bf L},{\bf p})$ and $L=\Lambda\otimes_{{{\mathcal O}}}{{\mathcal O}}(1)$ which is a $\Lambda$-bimodule. We define a ${\mathbb{Z}}$-graded ring $$B(\Lambda,L):=\bigoplus_{\ell=0}^\infty H^0\left({\mathbb{P}}^d, L^{\otimes_\Lambda\ell}\right).$$

\[B and R\^Zc\] We have an isomorphism of ${\mathbb{Z}}$-graded $k$-algebras: $$B(\Lambda,L)\simeq R^{[{\mathbb{Z}}{\vec c}]}.$$

Clearly we have $$\bigoplus_{\ell=0}^\infty H^0({\mathbb{P}}^d, {{\mathcal O}}(\ell))\simeq 
    S=k[{\bf T}].$$ and so we get a category equivalence $$\Phi:=\bigoplus_{\ell=0}^\infty H^0({\mathbb{P}}^d,-(\ell))
\!:\operatorname{\mathsf{coh}}{\mathbb{P}}^d\simeq {\operatorname{\mathsf{mod}}}^{{\mathbb{Z}}}S/{\operatorname{\mathsf{mod}}}_0^{{\mathbb{Z}}}S=\operatorname{\mathsf{qgr}}S.$$ Since the divisor $L_i$ is the zero set of the polynomial $\ell_i(\bf T)$, the functor $\Phi$ sends the natural inclusion ${{\mathcal O}}(-L_i)\to{{\mathcal O}}$ to the natural inclusion $\ell_i(\bf T) S\to S$. Thus we get $$\begin{aligned}
    B(\Lambda,L)&\simeq
    {\rm T}_{p_1}(S,\ell_1({\bf T}))\otimes_S\cdots\otimes_S {\rm T}_{p_n}(S,\ell_n({\bf T}))\\
    &={\rm T}_{p_1}(S,{X_1^{p_1}})\otimes_S\cdots\otimes_S {\rm T}_{p_n}(S,X_{n}^{p_n})\\
    &=R^{[{\mathbb{Z}}{\vec c}]}.\qedhere
    \end{aligned}$$

To see the role played by $B(\Lambda,L)$ we first need the following more general set up introduced by Artin-Zhang in [@AZ]. Let ${\mathcal C}$ be a $k$-linear abelian category, $P\in{\mathcal C}$ a distinguished object. For any $M\in {\mathcal C}$ we define $H^0(M):={\operatorname{Hom}\nolimits}_{{\mathcal C}}(P,M)$. Assume that:

1.  $P$ is a noetherian object,

2.  $A_0:=H^0(P)={\operatorname{End}\nolimits}_{{\mathcal C}}(P)$ is a right noetherian ring and $H^0(M)$ is a finitely generated $A_0$-module for all $M\in{\mathcal C}$.

Furthermore, let $s$ be a $k$-linear automorphism of ${\mathcal C}$ satisfying the following assumption:

1.  $s$ is *ample* in the following sense:

    1.  for every $M\in{\mathcal C}$, there are 