A property of invertible sheaves

Question

Let $f:X \to Y$ be a morphism of schemes, and let $\mathscr{F}$ be an invertible sheaf on $Y$.

It is clear that if $\mathscr{F}^{\otimes n} \cong \mathscr{O}_Y$ then $(f^*\mathscr{F})^{\otimes n} \cong f^*(\mathscr{F}^{\otimes n}) \cong \mathscr{O}_X$.
Is it true that if $\mathscr{F}^{\otimes n}$ is not trivial (i.e. not isomorphic to $\mathscr{O}_Y$) for any $n$, then $f^*(\mathscr{F}^{\otimes n})$ is also not trivial for any $n$?

2 is not true. Take $f$ to be one of the natural inclusions of $\mathbb{A}^1$ into $\mathbb{P}^1$. — Daniel, Oct 23 '23 at 16:34
@Daniel That looks like an answer to me - might I suggest that you record it as such below? — KReiser, Oct 23 '23 at 17:43

score 2 · Accepted Answer · answered Oct 23 '23 at 18:49

There are a number of ways of justifying (1). See here, for example. (2) however is not true.

If $f: \mathbb{A}^1 \to \mathbb{P}^1$ is any map (say, one of the canonical inclusions) then the line bundle $f^* \mathcal{O}(n)$ is trivial for all $n \in \mathbb{Z}$, since $\mathbb{A}^1$ is a regular scheme and is in particular factorial.

A property of invertible sheaves

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