$\Lambda$-bimodule $I_i$ by $$I_i:=\Lambda_1\otimes_{{{\mathcal O}}}\cdots\otimes_{{{\mathcal O}}}J_i\otimes_{{{\mathcal O}}}\cdots\otimes_{{{\mathcal O}}}\Lambda_n$$ and defining an autofunctor $({\vec x}_i)$ of ${\operatorname{\mathsf{mod}}}\Lambda$ for each $i=1,\ldots,n$ by $$({\vec x}_i):=I_i\otimes_\Lambda-\colon{\operatorname{\mathsf{mod}}}\Lambda\to{\operatorname{\mathsf{mod}}}\Lambda.$$ By a simple matrix multiplication, one can easily check the following:

Since $\overbrace{J_i\otimes_{\Lambda_i}\cdots\otimes_{\Lambda_i}J_i}^{p_i}=\Lambda_i\otimes_{{\mathcal O}}{{\mathcal O}}(L_i)$, we have $\overbrace{I_i\otimes_\Lambda\cdots\otimes_\Lambda I_i}^{p_i}=\Lambda\otimes_{{\mathcal O}}{{\mathcal O}}(L_i)$ and

$$\label{pixi}
    (p_i{\vec x}_i)=-\otimes_{{\mathcal O}}{{\mathcal O}}(L_i)\simeq -\otimes_{{\mathcal O}}{{\mathcal O}}(1).$$

Next we introduce the following rank $1$ group: $${\mathbb{L}}={\mathbb{L}}({\bf p}) = {\langle {\vec x}_1,\cdots,{\vec x}_n,{\vec c}\rangle}/({p_i{\vec x}_i-{\vec c}}\;|\;
1\leq i\leq n)$$ By , we get an objectwise action of ${\mathbb{L}}$ on ${\operatorname{\mathsf{mod}}}\Lambda$.

Now we denote by ${\mathbb{L}}_+$ the submonoid of ${\mathbb{L}}$ generated by ${\vec x}_1,\ldots,{\vec x}_n$, and we regard ${\mathbb{L}}$ as a partially ordered set by: ${\vec x}\le {\vec y}$ if and only if ${\vec y}-{\vec x}\in{\mathbb{L}}_+$. If we let $[0,d{\vec c}]:=\{{\vec x}\in{\mathbb{L}}\mid0\le{\vec x}\le d{\vec c}\}$ then the $\Lambda$-module $$\label{define T}
T:=\bigoplus_{{\vec x}\in[0,d{\vec c}]}P({\vec x})$$ gives a tilting bundle in Theorem \[tilting\].

A proof of Theorem \[tilting\] is given in the rest of this section. The basic idea is to reduce our problem for ${\operatorname{\mathsf{mod}}}\Lambda$ to the corresponding one in ${\operatorname{\mathsf{mod}}}{{\mathcal O}}$ and use the following Beilinson’s result:

\[beilinson\][@Be] $\bigoplus_{i=0}^d{{\mathcal O}}(i)$ is a tilting bundle in $\operatorname{\mathsf{coh}}{\mathbb{P}}^d$.

In particular we have

-   $H^i({\mathbb{P}}^d,{{\mathcal O}}(\ell))=0$ for all $i>0$ and 