# Morphism of sites and abelian sheaf cohomology

Let $$f : \mathcal{C}\to\mathcal{D}$$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi

$$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$

By assumption, $$f^{-1}$$ is an exact functor.

How do we define the induced map on abelian sheaf cohomology

$$H^p(\mathcal{D}, F)\to H^p(\mathcal{C}, f^{-1}F)\ ?$$

On global sections, we have a map $$\Gamma(\mathcal{D}, F)\to \Gamma(\mathcal{C}, f^{-1}F)$$ because $$f^{-1}$$ is exact and then preserves final objects in the topoi.

If I have an injective resolution $$F \to J^{\bullet}$$ in $$Ab(\mathcal{D})$$, then by exactness of $$f^{-1}$$ $$f^{-1}F\to f^{-1}J^{\bullet}$$ is still a resolution.

I’m tempted to consider the induced maps on global sections giving a map of complexes of abelian groups

$$\Gamma(\mathcal{D},J^{\bullet})\to \Gamma(\mathcal{C},f^{-1}J^{\bullet})$$

Cohomology of the left complex is $$H^*(\mathcal{D},F)$$ because each $$J^p$$ is $$\Gamma(\mathcal{D},\cdot)$$-acyclic, but it’s not clear to me that the same is true for $$f^{-1}J^p$$.

• Is $$f^{-1}J^p$$ a $$\Gamma(\mathcal{C},\cdot)$$-acyclic abelian sheaf for every $$p$$?

• If not, then how else do we define the map $$H^p(\mathcal{D}, F)\to H^p(\mathcal{C}, f^{-1}F)\ ?$$

• You should add your answer as an official answer in the box, and then after the wait period, accept it. – David Roberts Feb 13 '19 at 12:14

Since abelian sheaf cohomology on a site is a universal $$\delta$$-functor, having a map $$\Gamma(\mathcal{D},F)\to \Gamma(\mathcal{C}, f^{-1}F)$$ yields a unique map in all cohomological degrees by the universal property of universal $$\delta$$-functors, and that’s it.