[@BD Theorem 1(4)].

We first prove that the support of the ${{\mathcal O}}_{{\mathbb{P}}^d}$-module ${\mathscr Ext}^q_\Lambda(X,Y)$ has dimension at most $d-q$. We need to show that ${\mathscr Ext}^q_\Lambda(X,Y)_x=0$ holds for any point $x\in{\mathbb{P}}^d$ with $\dim x>d-q$. For a closed point $y$ on $x$, we know ${\operatorname{gl.\!dim}\nolimits}\Lambda_y=d$ by (a). Since $\Lambda_x=\Lambda_y\otimes_{{{\mathcal O}}_y}{{\mathcal O}}_x$, we have $${\operatorname{gl.\!dim}\nolimits}\Lambda_x=\dim{{\mathcal O}}_x=d-\dim x$$ (e.g. apply [@IW Theorem 2.17(2)$\Rightarrow$(1)] with $(R,\Lambda):=({{\mathcal O}}_y,\Lambda_y)$ there). Therefore ${\mathscr Ext}^q_\Lambda(X,Y)_x={\operatorname{Ext}\nolimits}^q_{\Lambda_x}(X_x,Y_x)=0$ holds if $q>d-\dim x$, and the assertion follows.

In particular, if $p+q>d$, then $H^p({\mathbb{P}}^d,{\mathscr Ext}^q_\Lambda(X,Y))=0$ holds. Using the local-global spectral sequence $H^p({\mathbb{P}}^d,{\mathscr Ext}^q_\Lambda(X,Y))\Rightarrow {\operatorname{Ext}\nolimits}^{p+q}_\Lambda(X,Y)$ (e.g. [@BD Theorem 1(2)]), we have ${\operatorname{Ext}\nolimits}^i_\Lambda(X,Y)=0$ for all $i>d$.

Now we are ready to prove that $T$ generates the category.

\[generate 2\]

-   ${\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)=\operatorname{\mathsf{thick}}T$.

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

We prepare some notions. Let ${\mathscr{C}}$ and ${\mathscr{C}}'$ be additive subcategories of ${\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)$.

We call ${\mathscr{C}}$ *contravariantly finite* if for any $X\in{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)$, there exists a morphism $f:C\to X$ with $C\in{\mathscr{C}}$ such that $${\operatorname{Hom}\nolimits}_{{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)}(C',C)\xrightarrow{f}{\operatorname{Hom}\nolimits}_{{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)}(C',X)$$ is surjective for any $C'\in{\mathscr{C}}$.

We define an additive subcategory ${\mathscr{C}}*{\mathscr{C}}'$ of ${\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)$ by $${\mathscr{C}}*{\mathscr{C}}':=\{X\in{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)\mid {\rm there\;exists\;a\;triangle\;} C\to X\to C'\to C[1]\mbox{ with }\ C\in{\mathscr{C}},\ C'\in{\mathscr{C}}'\}.$$ If ${\mathscr{C}}$ and ${\mathscr{C}}'$ are contravariantly finite, then so is ${\mathscr{C}}*{\mathscr{C}}'$ (see [@Ch Theorem 1.3]).

\(a) For any 