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Let $\mathcal{E}$ be a torsion free coherent sheaf on $\mathbb{P}_2$. Then sub-sheaves $\mathcal{E}_1$ and $\mathcal{E}_2$ of $\mathcal{E}$ are saturated if $\mathcal{E}/\mathcal{E}_i$ are torsion free for $i=1,2$. Then is $\mathcal{E}_1+\mathcal{E}_2$ also saturated ie $\frac{\mathcal{E}}{\mathcal{E}_1+\mathcal{E}_2}$ is torsion free? (we take $\mathcal{E}_i$ to be torsion free themselves for $i=1,2$.)

In further assumption if we are given $\mathcal{E}_1\cap \mathcal{E}_2 = 0$ then does it play a roll in the saturation of $\mathcal{E}_1+\mathcal{E}_2 \simeq \mathcal{E}_1 \oplus \mathcal{E}_2$?

I came across a similar claim in the following ref pg-39.

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    $\begingroup$ Welcome new contributor. That is not true without further hypotheses (that might be implicit in Lemma 2.51 of that article -- I am unfamiliar with "$(D,F)$-framable" sheaves). Let $\mathcal{E}$ be the locally free sheaf $\mathcal{O}_{\mathbb{P}^2}\oplus \mathcal{O}_{\mathbb{P}^2}(1)$ on $\mathbb{P}^2$. Let $\mathcal{E}_1$ and $\mathcal{E}_2$ be the graphs of two morphisms from $\mathcal{O}_{\mathbb{P}^2}$ to $\mathcal{O}_{\mathbb{P}^2}(1)$. These are each saturated. However, the cokernel of the sum is torsion-free if and only if $\mathcal{E}_1$ equals $\mathcal{E}_2$. $\endgroup$ Commented Mar 22 at 16:15
  • $\begingroup$ In that article, it appears that the author assumes that $\mathcal{E}$ is semistable with slope equal to the slopes of both $\mathcal{E}_1$ and $\mathcal{E}_2$. That excludes the example in my previous comment. $\endgroup$ Commented Mar 22 at 16:21
  • $\begingroup$ Thanks for the comment! It does help knowing this is dependent on the semi-stability of $\mathcal{E}$ and the slopes being equal! it probably is implicit hypothesis! $\endgroup$ Commented Mar 22 at 16:24

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