Let $Z$ be a closed subscheme of noetherian scheme $X$ and $U = X-Z$. Let $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $\mathcal{B}$ = $ \lbrace \mathcal{F} \in M(X) : \mathcal{F}_{\lvert U} = 0 \rbrace$ be the full subcategory of $M(X)$. I want to show that the quotient category $M(X)/\mathcal{B}$ is equivalent to $M(U)$.
I defined a functor $L : M(X) \rightarrow M(U)$ by $L(\mathcal{F}) = \mathcal{F}_{\lvert U}$. Clearly this functor has kernel category $\mathcal{B}$ so it induces a faithful functor $U : M(X)/\mathcal{B} \rightarrow M(U)$ and $U$ is also essential surjective since $L$ is. Because every coherent sheaves on $U$ is a restriction of some coherent sheave on $X$. But I am unable to prove that $U$ is full. I did if $X$ is affine scheme but don't know in general. Any help would be great.