The Stacks project

21.27 Comparing two topologies

Let $\mathcal{C}$ be a category. Let $\text{Cov}(\mathcal{C}) \supset \text{Cov}'(\mathcal{C})$ be two ways to endow $\mathcal{C}$ with the structure of a site. Denote $\tau $ the topology corresponding to $\text{Cov}(\mathcal{C})$ and $\tau '$ the topology corresponding to $\text{Cov}'(\mathcal{C})$. Then the identity functor on $\mathcal{C}$ defines a morphism of sites

where $\epsilon _*$ is the identity functor on underlying presheaves and where $\epsilon ^{-1}$ is the $\tau $-sheafification of a $\tau '$-sheaf. See Sites, Examples 7.14.3 and 7.22.3. In the situation above we have the following

1. $\epsilon _* : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_\tau ) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{\tau '})$ is fully faithful and $\epsilon ^{-1} \circ \epsilon _* = \text{id}$,

2. $\epsilon _* : \textit{Ab}(\mathcal{C}_\tau ) \to \textit{Ab}(\mathcal{C}_{\tau '})$ is fully faithful and $\epsilon ^{-1} \circ \epsilon _* = \text{id}$,

3. $R\epsilon _* : D(\mathcal{C}_\tau ) \to D(\mathcal{C}_{\tau '})$ is fully faithful and $\epsilon ^{-1} \circ R\epsilon _* = \text{id}$,

4. if $\mathcal{O}$ is a sheaf of rings for the $\tau $-topology, then $\mathcal{O}$ is also a sheaf for the $\tau '$-topology and $\epsilon $ becomes a flat morphism of ringed sites

   \[ \epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \longrightarrow (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '}) \]

5. $\epsilon _* : \textit{Mod}(\mathcal{O}_\tau ) \to \textit{Mod}(\mathcal{O}_{\tau '})$ is fully faithful and $\epsilon ^* \circ \epsilon _* = \text{id}$

6. $R\epsilon _* : D(\mathcal{O}_\tau ) \to D(\mathcal{O}_{\tau '})$ is fully faithful and $\epsilon ^* \circ R\epsilon _* = \text{id}$.

Here are some explanations.

Ad (1). Let $\mathcal{F}$ be a sheaf of sets in the $\tau $-topology. Then $\epsilon _*\mathcal{F}$ is just $\mathcal{F}$ viewed as a sheaf in the $\tau '$-topology. Applying $\epsilon ^{-1}$ means taking the $\tau $-sheafification of $\mathcal{F}$, which doesn't do anything as $\mathcal{F}$ is already a $\tau $-sheaf. Thus $\epsilon ^{-1}(\epsilon _*\mathcal{F})) = \mathcal{F}$. The fully faithfulness follows by Categories, Lemma 4.24.4.

Ad (2). This is a consequence of (1) since pullback and pushforward of abelian sheaves is the same as doing those operations on the underlying sheaves of sets.

Ad (3). Let $K$ be an object of $D(\mathcal{C}_\tau )$. To compute $R\epsilon _*K$ we choose a K-injective complex $\mathcal{I}^\bullet $ representing $K$ and we set $R\epsilon _*K = \epsilon _*\mathcal{I}^\bullet $. Since $\epsilon ^{-1} : D(\mathcal{C}_{\tau '}) \to D(\mathcal{C}_\tau )$ is computed on an object $L$ by applying the exact functor $\epsilon ^{-1}$ to any complex of abelian sheaves representing $L$, we find that $\epsilon ^{-1}R\epsilon _*K$ is represented by $\epsilon ^{-1}\epsilon _*\mathcal{I}^\bullet $. By Part (1) we have $\mathcal{I}^\bullet = \epsilon ^{-1}\epsilon _*\mathcal{I}^\bullet $. In other words, we have $\epsilon ^{-1} \circ R\epsilon _* = \text{id}$ and we conclude as before.

Ad (4). Observe that $\epsilon ^{-1}\mathcal{O}_{\tau '} = \mathcal{O}_\tau $, see discussion in part (1). Hence $\epsilon $ is a flat morphism of ringed sites, see Modules on Sites, Definition 18.31.1. Not only that, it is moreover clear that $\epsilon ^* = \epsilon ^{-1}$ on $\mathcal{O}_{\tau '}$-modules (the pullback as a module has the same underlying abelian sheaf as the pullback of the underlying abelian sheaf).

Ad (5). This is clear from (2) and what we said in (4).

Ad (6). This is analogous to (3). We omit the details.