Does quasi-isomorphic complexes of sheaves give the same cohomology groups?

Question

Let $F^{\bullet}$ and $ I^{\bullet }$ be two bounded below complexes of sheaves of $O_X$-modules (on a scheme X) and let $F^{\bullet}\rightarrow I^{\bullet}$ be a quasi-isomorphism between complexes of sheaves. Take the global section functor $\Gamma(X,)$ and take cohomology, is it true that $H^*(\Gamma(X,F^{\bullet}))\cong H^*(\Gamma(X,I^{\bullet}))$?

Answer 1

To flesh out Zhen Lin's initial comment, if $F$ is a sheaf of modules, and $I^\bullet$ is an injective resolution of $F$, then there is a quasi-isomorphism from the complex $F[0]=\cdots \to 0 \to F \to 0 \to \cdots$, and the injective resolution $I^\bullet$ (cf. Stacks Project 013G). By definition, the $i$'th cohomology of the complex $\Gamma(X, I^\bullet)$ is $H^i(X, F)$. On the other hand, the $i$'th cohomology of $\Gamma(X, F[0])$ is trivial for $i > 0$.

As such, if what you said was true, all higher sheaf cohomology would be trivial.