Proof.
We first prove that the tensor product of locally free \mathcal{O}_ X-modules is locally free. This follows if we show that (\bigoplus _{i \in I} \mathcal{O}_ X) \otimes _{\mathcal{O}_ X} (\bigoplus _{j \in J} \mathcal{O}_ X) \cong \bigoplus _{(i, j) \in I \times J} \mathcal{O}_ X. The sheaf \bigoplus _{i \in I} \mathcal{O}_ X is the sheaf associated to the presheaf U \mapsto \bigoplus _{i \in I} \mathcal{O}_ X(U). Hence the tensor product is the sheaf associated to the presheaf
U \longmapsto (\bigoplus \nolimits _{i \in I} \mathcal{O}_ X(U)) \otimes _{\mathcal{O}_ X(U)} (\bigoplus \nolimits _{j \in J} \mathcal{O}_ X(U)).
We deduce what we want since for any ring R we have (\bigoplus _{i \in I} R) \otimes _ R (\bigoplus _{j \in J} R) = \bigoplus _{(i, j) \in I \times J} R.
If \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0 is exact, then by Lemma 17.16.3 the complex \mathcal{F}_2 \otimes _{\mathcal{O}_ X} \mathcal{G} \to \mathcal{F}_1 \otimes _{\mathcal{O}_ X} \mathcal{G} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G} \to 0 is exact. Using this we can prove (5). Namely, in this case there exists locally such an exact sequence with \mathcal{F}_ i, i = 1, 2 finite free. Hence the two terms \mathcal{F}_2 \otimes _{\mathcal{O}_ X} \mathcal{G} and \mathcal{F}_1 \otimes _{\mathcal{O}_ X} \mathcal{G} are isomorphic to finite direct sums of \mathcal{G} (for example by Lemma 17.16.5). Since finite direct sums are coherent sheaves, these are coherent and so is the cokernel of the map, see Lemma 17.12.4.
And if also \mathcal{G}_2 \to \mathcal{G}_1 \to \mathcal{G} \to 0 is exact, then we see that
\mathcal{F}_2 \otimes _{\mathcal{O}_ X} \mathcal{G}_1 \oplus \mathcal{F}_1 \otimes _{\mathcal{O}_ X} \mathcal{G}_2 \to \mathcal{F}_1 \otimes _{\mathcal{O}_ X} \mathcal{G}_1 \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G} \to 0
is exact. Using this we can for example prove (3). Namely, the assumption means that we can locally find presentations as above with \mathcal{F}_ i and \mathcal{G}_ i free \mathcal{O}_ X-modules. Hence the displayed presentation is a presentation of the tensor product by free sheaves as well.
The proof of the other statements is omitted.
\square
Comments (2)
Comment #6215 by Yuto Masamura on
Comment #6352 by Johan on
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