0
$\begingroup$

Given $f : X \rightarrow Y$ , a continuous map, let $0 \rightarrow \Im \rightarrow \mathcal{F} \rightarrow \mathcal{H} \rightarrow 0$ be an exact sequence of sheaves on $Y$. I need to prove that $$ 0 \rightarrow f^{-1}\Im \rightarrow f^{-1} \mathcal{F} \rightarrow f^{-1} \mathcal{H} \rightarrow 0 $$ is an exact sequence of sheaves on $X$.

Now I think one way to prove this is to show that for every $x \in X$ the corresponding sequence of stalks $$ 0 \rightarrow (f^{-1}\Im)_x \rightarrow (f^{-1} \mathcal{F})_x \rightarrow (f^{-1} \mathcal{H})_x \rightarrow 0 $$ is exact. But I am not able to figure out how to show the latter sequence is exact. Any help with this!

1 Answers 1

1

Hint : try to prove that $f^{-1}\mathcal F_x = \mathcal F_{f(x)}$ for any sheaf $\mathcal F$ on $X$.

  • 1
    One should note this is basically the functoriality together with the fact that the inverse image w.r.t the inclusion of a point is by definition the same as taking the stalk. – 2017-01-27