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

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

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

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

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.

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

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

## Comments (0)

There are also: