2 forgot to include exterior algebras

This question is inspired by Hartshorne's exercise II.5.7 (c-d): the problem reads: Let $0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow0$ be a short exact sequence of locally free sheaves. Then for any $r$ there is a finite filtration $F$ of $S^r(\mathcal{F})$ (the sheaf of symmetric algebras over $\mathcal{O}_X$) such that $F^pS^r(\mathcal{F})/F^{p+1}S^r(\mathcal{F})$ is isomorphic to $S^p(\mathcal{F}')\otimes S^{r-p}(\mathcal{F}'')$ (and similarly for exterior algebras).

So basically, I had this feeling that a spectral sequence might be lurking around somewhere; my thinking is this: if we can write down a spectral sequence of a graded complex that converges to $S^r(\mathcal{F})$, then we would naturally be guaranteed a grading of this object which agrees with the $E_\infty$-page. So is there some sort of way to write down a complex so that we heuristically have something like $E_2^{p,q}\Rightarrow S^{p+q}(\mathcal{F})$ where if we go through and read off the sum of the diagonals of the $E_\infty$-page, we get something like $S^{p}(\mathcal{F}')\otimes S^{r-p}(\mathcal{F}'')$?

1

# Spectral sequence of symmetric or exterior algebras?

This question is inspired by Hartshorne's exercise II.5.7 (c-d): the problem reads: Let $0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow0$ be a short exact sequence of locally free sheaves. Then for any $r$ there is a finite filtration $F$ of $S^r(\mathcal{F})$ (the sheaf of symmetric algebras over $\mathcal{O}_X$) such that $F^pS^r(\mathcal{F})/F^{p+1}S^r(\mathcal{F})$ is isomorphic to $S^p(\mathcal{F}')\otimes S^{r-p}(\mathcal{F}'')$.

So basically, I had this feeling that a spectral sequence might be lurking around somewhere; my thinking is this: if we can write down a spectral sequence of a graded complex that converges to $S^r(\mathcal{F})$, then we would naturally be guaranteed a grading of this object which agrees with the $E_\infty$-page. So is there some sort of way to write down a complex so that we heuristically have something like $E_2^{p,q}\Rightarrow S^{p+q}(\mathcal{F})$ where if we go through and read off the sum of the diagonals of the $E_\infty$-page, we get something like $S^{p}(\mathcal{F}')\otimes S^{r-p}(\mathcal{F}'')$?