Filtration with ideal sheaf quotients (Hartshorne, Proof of Theorem III.3.7)

Question

Let $X$ be a Noetherian scheme. Let $f_1,\dots,f_r \in \mathcal{O}_{X}(X)$ be global sections such that $X = X_{f_1} \cup \cdots \cup X_{f_r}$. Define a morphism of sheaves $\mathcal{O}_{X}^r \rightarrow \mathcal{O}_{X}$, given by $(a_1,\dots,a_r) \mapsto a_1 f_1+\cdots+a_r f_r$. By the hypothesis on the $f_i$ this morhpism is surjective (we can see that by looking at the stalks). Let $\mathcal{F}$ be its kernel. Why is there a suitable ordering of the factors of $\mathcal{O}_{X}^r$ and a corresponding filtration of $\mathcal{F}$ of the form \begin{align} \mathcal{F} = \mathcal{F} \cap \mathcal{O}_{X}^r \supset \mathcal{F} \cap \mathcal{O}_{X}^{r-1} \supset \cdots \supset \mathcal{F} \cap \mathcal{O}_{X}, \end{align} where each quotient is a sheaf of ideals on $ \mathcal{O}_{X}$?

Answer 1

Let $K$ be the kernel of $R^r \to R, a \mapsto a_1f_1 + \dotsc a_rf_r$.

Consider some projection $p: R^i \to R$. Note that the kernel of $p$ is some subfactor $R^{i-1}$.

Then $\ker(p_{|K})=K \cap \operatorname{ker}(p)=K \cap R^{i-1}$, i.e. $p_{|K}$ induces an injection $$(K \cap R^i)/(K \cap R^{i-1}) \hookrightarrow R$$ as desired.

In particular the ordering of the factors of $R^r$ does not matter, we can choose any such ordering.

By the way: I have never used that $K$ is the kernel of some map or that the $f_i$ generate $R$. One can do this proof with any submodule $K \subset R^r$.

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More generally, this proof shows: If $A \subset B \subset C$ and $D \subset C$, then $(D \cap B)/(D \cap A)$ injects into $B/A$.