The Stacks project

Lemma 17.21.6. 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 quasi-coherent, then so is each $\text{T}(\mathcal{F})$, $\wedge (\mathcal{F})$, and $\text{Sym}(\mathcal{F})$.

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

Proof. It is not true that an infinite direct sum $\bigoplus \mathcal{G}_ i$ of locally free modules is locally free, or that an infinite direct sum of quasi-coherent modules is quasi-coherent. The problem is that given a point $x \in X$ the open neighbourhoods $U_ i$ of $x$ on which $\mathcal{G}_ i$ becomes free (resp. has a suitable presentation) may have an intersection which is not an open neighbourhood of $x$. However, in the proof of Lemma 17.21.5 we saw that once a suitable open neighbourhood for $\mathcal{F}$ has been chosen, then this open neighbourhood works for each of the sheaves $\text{T}^ n(\mathcal{F})$, $\wedge ^ n(\mathcal{F})$ and $\text{Sym}^ n(\mathcal{F})$. The lemma follows. $\square$

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