The Stacks project

Proposition 35.9.4. Let $f : T \to S$ be a morphism of schemes.

  1. The equivalences of categories of Proposition 35.8.9 are compatible with pullback. More precisely, we have $f^*(\mathcal{G}^ a) = (f^*\mathcal{G})^ a$ for any quasi-coherent sheaf $\mathcal{G}$ on $S$.

  2. The equivalences of categories of Proposition 35.8.9 part (1) are not compatible with pushforward in general.

  3. If $f$ is quasi-compact and quasi-separated, and $\tau \in \{ Zariski, {\acute{e}tale}\} $ then $f_*$ and $f_{small, *}$ preserve quasi-coherent sheaves and the diagram

    \[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_ T) \ar[rr]_{f_*} \ar[d]_{\mathcal{F} \mapsto \mathcal{F}^ a} & & \mathit{QCoh}(\mathcal{O}_ S) \ar[d]^{\mathcal{G} \mapsto \mathcal{G}^ a} \\ \mathit{QCoh}(T_\tau , \mathcal{O}) \ar[rr]^{f_{small, *}} & & \mathit{QCoh}(S_\tau , \mathcal{O}) } \]

    is commutative, i.e., $f_{small, *}(\mathcal{F}^ a) = (f_*\mathcal{F})^ a$.

Proof. Part (1) follows from the discussion in Remark 35.8.6. Part (2) is just a warning, and can be explained in the following way: First the statement cannot be made precise since $f_*$ does not transform quasi-coherent sheaves into quasi-coherent sheaves in general. Even if this is the case for $f$ (and any base change of $f$), then the compatibility over the big sites would mean that formation of $f_*\mathcal{F}$ commutes with any base change, which does not hold in general. An explicit example is the quasi-compact open immersion $j : X = \mathbf{A}^2_ k \setminus \{ 0\} \to \mathbf{A}^2_ k = Y$ where $k$ is a field. We have $j_*\mathcal{O}_ X = \mathcal{O}_ Y$ but after base change to $\mathop{\mathrm{Spec}}(k)$ by the $0$ map we see that the pushforward is zero.

Let us prove (3) in case $\tau = {\acute{e}tale}$. Note that $f$, and any base change of $f$, transforms quasi-coherent sheaves into quasi-coherent sheaves, see Schemes, Lemma 26.24.1. The equality $f_{small, *}(\mathcal{F}^ a) = (f_*\mathcal{F})^ a$ means that for any ├ętale morphism $g : U \to S$ we have $\Gamma (U, g^*f_*\mathcal{F}) = \Gamma (U \times _ S T, (g')^*\mathcal{F})$ where $g' : U \times _ S T \to T$ is the projection. This is true by Cohomology of Schemes, Lemma 30.5.2. $\square$

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