-   We have a triangle equivalence ${\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)\simeq{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}{\operatorname{End}\nolimits}_\Lambda(T))$.

-   $\Lambda$ has global dimension $d$.

The construction of $T$ is as follows: First we define a $\Lambda$-module $P$ by $$P_i:=\left[\begin{array}{c}
{{\mathcal O}}\\ {{\mathcal O}}\\ \vdots\\ {{\mathcal O}}\\ {{\mathcal O}}\end{array}\right]\in{\operatorname{\mathsf{mod}}}\Lambda_i\ \mbox{ and }\ 
P:=P_1\otimes_{{{\mathcal O}}}\cdots\otimes_{{{\mathcal O}}}P_n\in{\operatorname{\mathsf{mod}}}\Lambda.$$ This is a direct summand of the $\Lambda$-module $\Lambda$ and can be described as $$P=\Lambda e,\ \mbox{ where }\ e:=e_1\otimes\cdots\otimes e_n\in H^0({\mathbb{P}}^d, \Lambda)
\ \mbox{ for }\ 
e_i:=\left[\begin{array}{ccccc}
1&0&\cdots&0&0\\
0&0&\cdots&0&0\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&\cdots&0&0\\
0&0&\cdots&0&0
\end{array}\right]\in H^0({\mathbb{P}}^d, \Lambda_i).$$ Next for each $i=1,\ldots,n$, we define an invertible $\Lambda_i$-bimodule $J_i$ by $$J_i:=\left[\begin{array}{ccccccc}
{{\mathcal O}}&{{\mathcal O}}&{{\mathcal O}}(-L_i)&\cdots&{{\mathcal O}}(-L_i)&{{\mathcal O}}(-L_i)&{{\mathcal O}}(-L_i)\\
{{\mathcal O}}&{{\mathcal O}}&{{\mathcal O}}&\cdots&{{\mathcal O}}(-L_i)&{{\mathcal O}}(-L_i)&{{\mathcal O}}(-L_i)\\
{{\mathcal O}}&{{\mathcal O}}&{{\mathcal O}}&\cdots&{{\mathcal O}}(-L_i)&{{\mathcal O}}(-L_i)&{{\mathcal O}}(-L_i)\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\
{{\mathcal O}}&{{\mathcal O}}&{{\mathcal O}}&\cdots&{{\mathcal O}}&{{\mathcal O}}&{{\mathcal O}}(-L_i)\\
{{\mathcal O}}&{{\mathcal O}}&{{\mathcal O}}&\cdots&{{\mathcal O}}&{{\mathcal O}}&{{\mathcal O}}\\
{{\mathcal O}}(L_i)&{{\mathcal O}}&{{\mathcal O}}&\cdots&{{\mathcal O}}&{{\mathcal O}}&{{\mathcal O}}\end{array}\right]$$

The following can be easily checked:

\[obs:what J does\] $J_i^\ell\otimes_\Lambda P_i$ is the $(1-\ell)$-th (modulo $p_i$) column of $\Lambda_i\otimes_{{\mathcal O}}{{\mathcal O}}(\lceil
\ell/p_i \rceil L_i)$, where $\lceil x\rceil$ is the smallest integer $a$ satisfying $a\geq x$.

We define an autofunctor $(-)({\vec x}_i)$, which we abbreviate simply by $({\vec x}_i)$, of ${\operatorname{\mathsf{mod}}}\Lambda_i$ by $({\vec x}_i):=J_i\otimes_{{\mathcal O}}-\colon{\operatorname{\mathsf{mod}}}\Lambda_i\to{\operatorname{\mathsf{mod}}}\Lambda_i$. Then we extend this action to ${\operatorname{\mathsf{mod}}}\Lambda$ by first introducing an invertible 