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Let $K$ be the constant sheaf on $\mathbb P^{1}$ and $\mathcal O_{x}$ the structure sheaf.

Why can we say that $K/O_{x}$ is isomorphic to the direct sum of all the skyscraper sheaves $K/\mathcal O_{X,x}$ at $x$?

This is mentioned in passing at on page 127, and I can't figure it out.

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This is certainly not true. Just compare the global sections. But the formulation in the link

The quotient sheaf $\mathcal{K}/\mathcal{O}_X$ can be thought of (stalkwise) as the direct sum over all $x \in X$ of the skyscraper sheaf $\mathcal{K}/\mathcal{O}_{X,x}$, so is flasque.

is correct (but also a little bit misleading, as you see). I would suggest a direct computation which shows that $\mathcal{K} / \mathcal{O}_X$ is flasque.

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