The Stacks project

Lemma 17.15.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$, $\mathcal{G}$ be $\mathcal{O}_ X$-modules.

  1. If $\mathcal{F}$, $\mathcal{G}$ are locally generated by sections, so is $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$.

  2. If $\mathcal{F}$, $\mathcal{G}$ are of finite type, so is $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$.

  3. If $\mathcal{F}$, $\mathcal{G}$ are quasi-coherent, so is $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$.

  4. If $\mathcal{F}$, $\mathcal{G}$ are of finite presentation, so is $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$.

  5. If $\mathcal{F}$ is of finite presentation and $\mathcal{G}$ is coherent, then $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$ is coherent.

  6. If $\mathcal{F}$, $\mathcal{G}$ are coherent, so is $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$.

  7. If $\mathcal{F}$, $\mathcal{G}$ are locally free, so is $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G}$.

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 17.15.3 the complex $\mathcal{F}_2 \otimes \mathcal{G} \to \mathcal{F}_1 \otimes \mathcal{G} \to \mathcal{F} \otimes \mathcal{G} \to 0$ 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 $\mathcal{F}_2 \otimes \mathcal{G}$ are isomorphic to finite direct sums of $\mathcal{G}$. 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. $\square$


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