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The following claims come from the proof of Proposition 3.10 (Page 66) of D.Huybrechts' Fourier-Mukai Transforms in Algebraic Geometry. Since I couldn't find these results in Hartshorne's Algebraic Geometry and I have no idea how to prove them, I hope someone can give me the details about these claims.

Let $X$ be a Noetherian scheme.

  • Claim 1. If $\mathcal{O}_X=\mathcal{F_1^\bullet}\oplus\mathcal{F_2^\bullet}$, we assume that $\mathcal{F_1^\bullet}$ and $\mathcal{F_2^\bullet}$ are coherent sheaves, for their cohomology is concentrated in degree zero (Maybe we can just assume $\mathcal{F_1^\bullet}=\mathcal{F_1}$, $\mathcal{F_2^\bullet}=\mathcal{F_2}$). Then they are ideal sheaves of certain closed subschemes, say: $X_j\subset X$, $\mathcal{F_j^\bullet}\simeq \mathcal{I}_{X_{j}}$ ($j=1,2$).
  • Claim 2. $\mathcal{I}_{X_{1}}+\mathcal{I}_{X_{2}}\subset \mathcal{I}_{X_{1}\cap X_{2}} $
  • Claim 3. $\mathcal{I}_{X_{1}\cup X_{2}}\subset \mathcal{I}_{X_{1}}\cap \mathcal{I}_{X_{2}}$ and $\mathcal{I}_{X_{1}}\cap \mathcal{I}_{X_{2}}=0$

For Claim 1, my thoughts as follows: Since $\mathcal{O}_X=\mathcal{F_1^\bullet}\oplus\mathcal{F_2^\bullet}$, we have an exact sequence

$$0\longrightarrow \mathcal{F_1^\bullet} \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}_X/\mathcal{F_1^\bullet}=\mathcal{F_2^\bullet} \longrightarrow 0.$$

So, for any open subset of $U$ of $X$,

$$0\longrightarrow \mathcal{F_1^\bullet}(U) \longrightarrow \mathcal{O}_X(U) \longrightarrow \mathcal{O}_X(U)/\mathcal{F_1^\bullet}(U) \longrightarrow 0.$$

is an exact sequence of $\mathcal{O}_X(U)$-modules. Hence $\mathcal{F_1^\bullet}$ is a coherent sheaf of ideal on $X$. By Proposition II 5.9 on Hartshorne's Algebraic Geometry (Page 116), $\mathcal{F_1^\bullet}$ is the ideal sheaf of a closed subscheme. But I am not sure it is correct.

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  • $\begingroup$ 2 and 3: Really no idea? You know the definition of $I_A$? $\endgroup$ – Martin Brandenburg Jul 19 '15 at 20:17
  • $\begingroup$ @Martin Brandenburg,it's the kernel of the closed immersion, but I don't know how to see it with $X_1\cap X_2$ and $X_1\cup X_2$. By the way, what is the union and intersection of two schemes? $\endgroup$ – representation Jul 19 '15 at 21:06
  • $\begingroup$ The intersection $X_1\cap X_2$ should be interpreted as the fibered product $X_1\times_X X_2$, where the maps employed are the closed immersions $X_i\hookrightarrow X$. For the union, think about (just set-theoretically) the fact that if $f|_{X_1} \equiv 0$ and $g|_{X_2}\equiv 0$, then $fg|_{X_1\cup X_2}\equiv 0$. $\endgroup$ – Tabes Bridges Jul 21 '15 at 14:52
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For claim 1, apply first exercise 2.31 to replace $\mathcal{F}_{i}^{\bullet}$ by a complex with a single coherent sheaf $\mathcal{F}_{i}$ in degree 0. Then we have $\mathcal{O}_X \cong \mathcal{F}_{1} \oplus \mathcal{F}_2$. In particular this means that each canonical map $\mathcal{F}_i \to \mathcal{O}_X$ allows us to identify $\mathcal{F}_i$ with a subsheaf of $\mathcal{O}_X$, thus an ideal sheaf. Now use a well-known correspondence between quasicoherent ideal sheaves and closed subschemes.

Claims 2 and the first equality of claim 3 follow from, say, an affine-local argument (and glueing by sheaf axioms). The last equality follows from the fact that the sum is direct (similar to standard linear algebra for direct sum decompositions).

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