Pullback of an invertible sheaf through an isomorphism

Question

Consider an isomorphism of schemes $(f,f^{\#})(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$. Moreover let $\mathcal F$ be an invetible sheaf on $Y$ and let $f^{*}\mathcal{F}$ be its pullback.

Is it true that $\chi(\mathcal{F})=\chi(f^{*}\mathcal{F})$? Clearly $\chi(\cdot)$ is the Euler-Poincaré characteristic of the sheaf.

Answer 1

Yes, because $f^\ast$ is an exact functor if $f$ is an isomorphism.