Let $X$ be a noetherian and separated scheme and $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $M^{P}(X) = \lbrace \mathcal{F} \in M(X) \hspace{2mm} : Codim(sup(\mathcal{F}), X) \geq p \rbrace$ be the full subcategory of $M(X)$. I want to show that the quotient category $M^{p}(X)/M^{p+1}(X)$ is equivalent to $\bigoplus _{x \in X^{p}} \mathcal{A}(O_{X,x})$, where $\mathcal{A}(O_{X,x})$ denote the category of $O_{X,x}$-module of finite length. Here $X^{p}$ denote the points of codimenison $p$.

The natural functor which seems to be equivalence is given as $L : M^{P}(X) \rightarrow \mathcal{A}(O_{X,x})$ by $L(\mathcal{F})$ = $\oplus (\mathcal{F}_{x_{i}})$ where $x_{i}$ are codimension $p$ points. Clearly this functor has kernel $M^{p+1}(X)$, so it induces a faithful functor $U : M^{p}(X)/M^{p+1}(X) \rightarrow \mathcal{A}(O_{X,x})$. Now I don't know how prove that $U$ is full and essential surjective. Any help would be great. Thanks in advance.

Let $X$ be a noetherian and separated scheme and $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $M^{P}(X) = \lbrace \mathcal{F} \in M(X) \hspace{2mm} : Codim(sup(\mathcal{F}), X) \geq p \rbrace$ be the full subcategory of $M(X)$. I want to show that the quotient category $M^{p}(X)/M^{p+1}(X)$ is equivalent to $\bigoplus _{x \in X^{p}} \mathcal{A}(O_{X,x})$, where $\mathcal{A}(O_{X,x})$ denote the category of $O_{X,x}$-module of finite length. Here $X^{p}$ denote the points of codimenison $p$.

The natural functor which seems to be equivalence is given as $L : M^{P}(X) \rightarrow \bigoplus _{x \in X^{p}} \mathcal{A}(O_{X,x})$ by $L(\mathcal{F})$ = $\oplus (\mathcal{F}_{x_{i}})$ where $x_{i}$ are codimension $p$ points. Clearly this functor has kernel $M^{p+1}(X)$, so it induces a faithful functor $U : M^{p}(X)/M^{p+1}(X) \rightarrow \bigoplus _{x \in X^{p}} \mathcal{A}(O_{X,x})$. Now I don't know how prove that $U$ is full and essential surjective. Any help would be great. Thanks in advance.