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Let $i:X \hookrightarrow \mathbb{P}^n$ be a smooth projective variety. Let $f:\mathcal{F}_1 \to \mathcal{F}_2$ be an injective morphism of $\mathcal{O}_{\mathbb{P}^n}$-modules that are locally free. Is the induced morphism $i^*:i^*\mathcal{F}_1 \to i^*\mathcal{F}_2$ injective?

Chen
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1 Answers1

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No, this is not true.

Denote by $S=k[X_0,...,X_n]$, $X$ a hypersurface in $\mathbb{P}^n$ defined by an equation $F$ of degree $d$. Then there is an injective morphism $S(-d) \to S$ given by multiplication by $F$. But the induced morphism $(S/(F))(-d) \to S/(F)$ is not injective.

Applying the associated coherent sheaf functor to the modules give a counterexample.

Jana
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    It may be good to know that such a pullback is injective if $\mathcal{F}_2$ is finite locally free and the quotient $\mathcal{F}_1/\mathcal{F}_2$ is locally free (this holds for all schemes, not just $\mathbb{P}^n$, and is treated in Algebraic Geometry I by Goertz & Wedhorn, proposition 8.10). – Herman Rohrbach Jun 10 '14 at 09:30