The Stacks project

Remark 114.18.2. We obtain a second topology $\tau _ Y$ on $\mathcal{C}_{X/Y}$ by taking the topology inherited from $Y_{Zar}$. There is a third topology $\tau _{X \to Y}$ where a family of morphisms $\{ (U_ i \to A_ i) \to (U \to A)\} $ is a covering if and only if $U = \bigcup U_ i$, $V = \bigcup V_ i$ and $A_ i \cong V_ i \times _ V A$. This is the topology inherited from the topology on the site $(X/Y)_{Zar}$ whose underlying category is the category of pairs $(U, V)$ as in Lemma 114.18.1 part (3). The coverings of $(X/Y)_{Zar}$ are families $\{ (U_ i, V_ i) \to (U, V)\} $ such that $U = \bigcup U_ i$ and $V = \bigcup V_ i$. There are morphisms of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ X) & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _{X \to Y}) \ar[l] \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ Y) } \]

(recall that $\tau _ X$ is our “default” topology). The pullback functors for these arrows are sheafification and pushforward is the identity on underlying presheaves. The diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \ar[d]^ f & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}) \ar[l]^\pi & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _{X \to Y}) \ar[l] \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (Y_{Zar}) & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _ Y) \ar[ll] } \]

is not commutative. Namely, the pullback of a nonzero abelian sheaf on $Y$ is a nonzero abelian sheaf on $(\mathcal{C}_{X/Y}, \tau _{X \to Y})$, but we can certainly find examples where such a sheaf pulls back to zero on $X$. Note that any presheaf $\mathcal{F}$ on $Y_{Zar}$ gives a sheaf $\underline{\mathcal{F}}$ on $\mathcal{C}_{Y/X}$ by the rule which assigns to $(U \to A/V)$ the set $\mathcal{F}(V)$. Even if $\mathcal{F}$ happens to be a sheaf it isn't true in general that $\underline{\mathcal{F}} = \pi ^{-1}f^{-1}\mathcal{F}$. This is related to the noncommutativity of the diagram above, as we can describe $\underline{\mathcal{F}}$ as the pushforward of the pullback of $\mathcal{F}$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{X/Y}, \tau _{X \to Y})$ via the lower horizontal and right vertical arrows. An example is the sheaf $\underline{\mathcal{O}}_ Y$. But what is true is that there is a map $\underline{\mathcal{F}} \to \pi ^{-1}f^{-1}\mathcal{F}$ which is transformed (as we shall see later) into an isomorphism after applying $\pi _!$.


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