The Stacks project

Lemma 59.102.5. With notation as above.

  1. For $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{ph})$ and an abelian torsion sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$.

  2. For a proper morphism $f : X \to Y$ in $(\mathit{Sch}/S)_{ph}$ and abelian torsion sheaf $\mathcal{F}$ on $X$ we have $a_ Y^{-1}(R^ if_{small, *}\mathcal{F}) = R^ if_{big, ph, *}(a_ X^{-1}\mathcal{F})$ for all $i$.

  3. For a scheme $X$ and $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves the map $\pi _ X^{-1}K \to R\epsilon _{X, *}(a_ X^{-1}K)$ is an isomorphism.

  4. For a proper morphism $f : X \to Y$ of schemes and $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves we have $a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, ph, *}(a_ X^{-1}K)$.

Proof. By Lemma 59.102.4 the lemmas in Cohomology on Sites, Section 21.30 all apply to our current setting. To translate the results observe that the category $\mathcal{A}_ X$ of Cohomology on Sites, Lemma 21.30.2 is the full subcategory of $\textit{Ab}((\mathit{Sch}/X)_{ph})$ consisting of sheaves of the form $a_ X^{-1}\mathcal{F}$ where $\mathcal{F}$ is an abelian torsion sheaf on $X_{\acute{e}tale}$.

Part (1) is equivalent to $(V_ n)$ for all $n$ which holds by Cohomology on Sites, Lemma 21.30.8.

Part (2) follows by applying $\epsilon _ Y^{-1}$ to the conclusion of Cohomology on Sites, Lemma 21.30.3.

Part (3) follows from Cohomology on Sites, Lemma 21.30.8 part (1) because $\pi _ X^{-1}K$ is in $D^+_{\mathcal{A}'_ X}((\mathit{Sch}/X)_{\acute{e}tale})$ and $a_ X^{-1} = \epsilon _ X^{-1} \circ a_ X^{-1}$.

Part (4) follows from Cohomology on Sites, Lemma 21.30.8 part (2) for the same reason. $\square$

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