$X\in{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)$, there exists only finitely many $i\in{\mathbb{Z}}$ such that ${\operatorname{Hom}\nolimits}_{{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)}(T,X[i])\neq0$ since $\Lambda$ has finite global dimension by Proposition \[global dimension\]. Thus ${\mathscr{C}}:=\operatorname{\mathsf{add}}\{ T[i]\mid i\in{\mathbb{Z}}\}$ is a contravariantly finite subcategory of ${\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)$. In particular, ${\mathscr{C}}*\cdots*{\mathscr{C}}$ ($n$ times) is also contravariantly finite for all $n\ge0$.

Let $E:={\operatorname{End}\nolimits}_\Lambda(T)$. Then $\operatorname{\mathsf{thick}}T$ is triangle equivalent to ${\mathsf{K}}^{{\operatorname{b}\nolimits}}(\operatorname{\mathsf{proj}}E)$ [@Ke]. Since $E$ has finite global dimension $n$, we have $$\operatorname{\mathsf{thick}}T={\mathscr{C}}*\cdots*{\mathscr{C}}\ \ \ (n+1\ \mbox{times})$$ (see e.g. [@KK Proposition 2.6]). In particular, $\operatorname{\mathsf{thick}}T$ is contravariantly finite. Applying [@IY Proposition 2.3(1)], for any $X\in{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)$, there exists a triangle $Y\to X\to Z\to Y[1]$ with $Y\in\operatorname{\mathsf{thick}}T$ and ${\operatorname{Hom}\nolimits}_{{\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)}(U,Z)=0$ for any $U\in\operatorname{\mathsf{thick}}T$. By Proposition \[generate\], we have $Z=0$ and $X\simeq Y\in\operatorname{\mathsf{thick}}T$. Thus the assertion follows.

\(b) We already observed ${\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}\Lambda)=\operatorname{\mathsf{thick}}T\simeq{\mathsf{K}}^{{\operatorname{b}\nolimits}}(\operatorname{\mathsf{proj}}E)$. This is ${\mathsf{D}}^{{\operatorname{b}\nolimits}}({\operatorname{\mathsf{mod}}}E)$ since $E$ has finite global dimension.

Explicit correspondence between GL orders on ${\mathbb{P}}^d$ and GL weighted ${\mathbb{P}}^d$ {#correspondence}
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A graded Morita equivalence
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In this section we show, given a ring graded by a commutative group, how to modify the ring, so that it is graded by a subgroup.

Let $G$ be an abelian group and $A$ be a $G$-graded ring. We denote by $\operatorname{\mathsf{Mod}}^GA$ the category of $G$-graded $A$-modules, and by ${\operatorname{\mathsf{mod}}}^GA$ the category of finitely generated 