The Stacks project

Lemma 17.21.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.

  1. If $\mathcal{F}$ is locally generated by sections, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

  2. If $\mathcal{F}$ is of finite type, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

  3. If $\mathcal{F}$ is of finite presentation, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

  4. If $\mathcal{F}$ is coherent, then for $n > 0$ each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$ is coherent.

  5. If $\mathcal{F}$ is quasi-coherent, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

  6. If $\mathcal{F}$ is locally free, then so is each $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$, and $\text{Sym}^ n(\mathcal{F})$.

Proof. These statements for $\text{T}^ n(\mathcal{F})$ follow from Lemma 17.16.6.

Statements (1) and (2) follow from the fact that $\wedge ^ n(\mathcal{F})$ and $\text{Sym}^ n(\mathcal{F})$ are quotients of $\text{T}^ n(\mathcal{F})$.

Statement (6) follows from Algebra, Lemma 10.13.1.

For (3) and (5) we will use Lemma 17.21.4 above. By locally choosing a presentation $\mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F} \to 0$ with $\mathcal{F}_ i$ free, or finite free and applying the lemma we see that $\text{Sym}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$ has a similar presentation; here we use (6) and Lemma 17.16.6.

To prove (4) we will use Algebra, Lemma 10.13.3. We may localize on $X$ and assume that $\mathcal{F}$ is generated by a finite set $(s_ i)_{i \in I}$ of global sections. The lemma mentioned above combined with Lemma 17.21.1 above implies that for $n \geq 2$ there exists an exact sequence

\[ \bigoplus \nolimits _{j \in J} \text{T}^{n - 2}(\mathcal{F}) \to \text{T}^ n(\mathcal{F}) \to \text{Sym}^ n(\mathcal{F}) \to 0 \]

where the index set $J$ is finite. Now we know that $\text{T}^{n - 2}(\mathcal{F})$ is finitely generated and hence the image of the first arrow is a coherent subsheaf of $\text{T}^ n(\mathcal{F})$, see Lemma 17.12.4. By that same lemma we conclude that $\text{Sym}^ n(\mathcal{F})$ is coherent. $\square$

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