Morphism of sites and abelian sheaf cohomology

Question

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

Answer 1

Following the suggestion in the comment, I post the answer I found myself, in case it’s useful to anyone sometime.

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.