The Stacks project

Lemma 82.6.5. In Lemma 82.6.4 if $f$ is proper, then we have

  1. $a_ Y^{-1} \circ f_{small, *} = f_{big, fppf, *} \circ a_ X^{-1}$, and

  2. $a_ Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_ X^{-1}K)$ for $K$ in $D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves.

Proof. Proof of (1). You can prove this by repeating the proof of Lemma 82.5.6 part (1); we will instead deduce the result from this. As $\epsilon _{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. Lemma 82.6.1 shows that $\epsilon _{Y, *} \circ a_ Y^{-1} = \pi _ Y^{-1}$ and similarly for $X$. To show that the canonical map $a_ Y^{-1}f_{small, *}\mathcal{F} \to f_{big, fppf, *}a_ X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that

\begin{align*} \pi _ Y^{-1}f_{small, *}\mathcal{F} & = \epsilon _{Y, *}a_ Y^{-1}f_{small, *}\mathcal{F} \\ & \to \epsilon _{Y, *}f_{big, fppf, *}a_ X^{-1}\mathcal{F} \\ & = f_{big, {\acute{e}tale}, *} \epsilon _{X, *}a_ X^{-1}\mathcal{F} \\ & = f_{big, {\acute{e}tale}, *}\pi _ X^{-1}\mathcal{F} \end{align*}

is an isomorphism. This is part (1) of Lemma 82.5.6.

To see (2) we use that

\begin{align*} R\epsilon _{Y, *}Rf_{big, fppf, *}a_ X^{-1}K & = Rf_{big, {\acute{e}tale}, *}R\epsilon _{X, *}a_ X^{-1}K \\ & = Rf_{big, {\acute{e}tale}, *}\pi _ X^{-1}K \\ & = \pi _ Y^{-1}Rf_{small, *}K \\ & = R\epsilon _{Y, *} a_ Y^{-1}Rf_{small, *}K \end{align*}

The first equality by the commutative diagram in Lemma 82.6.4 and Cohomology on Sites, Lemma 21.19.2. Then second equality is Lemma 82.6.2. The third is Lemma 82.5.6 part (2). The fourth is Lemma 82.6.2 again. Thus the base change map $a_ Y^{-1}(Rf_{small, *}K) \to Rf_{big, fppf, *}(a_ X^{-1}K)$ induces an isomorphism

\[ R\epsilon _{Y, *}a_ Y^{-1}Rf_{small, *}K \to R\epsilon _{Y, *}Rf_{big, fppf, *}a_ X^{-1}K \]

The proof is finished by the following remark: a map $\alpha : a_ Y^{-1}L \to M$ with $L$ in $D^+(Y_{\acute{e}tale})$ and $M$ in $D^+((\textit{Spaces}/Y)_{fppf})$ such that $R\epsilon _{Y, *}\alpha $ is an isomorphism, is an isomorphism. Namely, we show by induction on $i$ that $H^ i(\alpha )$ is an isomorphism. This is true for all sufficiently small $i$. If it holds for $i \leq i_0$, then we see that $R^ j\epsilon _{Y, *}H^ i(M) = 0$ for $j > 0$ and $i \leq i_0$ by Lemma 82.6.1 because $H^ i(M) = a_ Y^{-1}H^ i(L)$ in this range. Hence $\epsilon _{Y, *}H^{i_0 + 1}(M) = H^{i_0 + 1}(R\epsilon _{Y, *}M)$ by a spectral sequence argument. Thus $\epsilon _{Y, *}H^{i_0 + 1}(M) = \pi _ Y^{-1}H^{i_0 + 1}(L) = \epsilon _{Y, *}a_ Y^{-1}H^{i_0 + 1}(L)$. This implies $H^{i_0 + 1}(\alpha )$ is an isomorphism (because $\epsilon _{Y, *}$ reflects isomorphisms as it is the identity on underlying presheaves) as desired. $\square$


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