Closed subschemes and sheaves

Question

Let $X$ be a Noetherian scheme and $Y$ an integral closed subscheme of $X$. Why the structure sheaf $\mathcal{O}_Y$ is indecomposable? (Indecomposable means that it cannot be expressed as a non-trivial product of sheaves)

Answer 1

the closed immersion $i:Y \hookrightarrow X$ corresponds to the structure sheaf exact sequence $$0 \rightarrow \mathcal{I}\rightarrow \mathcal{O}_X \rightarrow i_*\mathcal{O}_Y \rightarrow 0$$ The structure sheaf of $\mathcal{O}_Y$ is hence given by a quotient of $\mathcal{O}_X$ by the ideal sheaf $\mathcal{I}$. $Y$ being integral is equivalent to $Y$ being reduced and irreducible. $Y$ being integral means that vor every affine open $\operatorname{Spec} A \subseteq Y$, $A$ is an integral domain. So "affine locally", this exact sequence above comes from taking a quotient by a prime ideal $\mathcal{I}$. This quotient is integral, so it cannot split into a product.