The Stacks project

Lemma 59.97.7. Let $K$ be a field. Let $n \geq 1$ be invertible in $K$. Consider a commutative diagram

\[ \xymatrix{ X \ar[d] & X' \ar[l]^ p \ar[d]_{f'} & Y \ar[l]^ h \ar[d]^ e \\ \mathop{\mathrm{Spec}}(K) & S' \ar[l] & T \ar[l]_ g } \]

of schemes with $X' = X \times _{\mathop{\mathrm{Spec}}(K)} S'$ and $Y = X' \times _{S'} T$ and $g$ quasi-compact and quasi-separated. The canonical map

\[ p^{-1}E \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} (f')^{-1}Rg_*F \longrightarrow Rh_*(h^{-1}p^{-1}E \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} e^{-1}F) \]

is an isomorphism if $E$ in $D^+(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$ has tor amplitude in $[a, \infty ]$ for some $a \in \mathbf{Z}$ and $F$ in $D^+(T_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$.

Proof. This lemma is a generalization of Lemma 59.97.6 to objects of the derived category; the assertion of our lemma is true because in Lemma 59.97.6 the scheme $X$ over $K$ is arbitrary. We strongly urge the reader to skip the laborious proof (alternative: read only the last paragraph).

We may represent $E$ by a bounded below K-flat complex $\mathcal{E}^\bullet $ consisting of flat $\mathbf{Z}/n\mathbf{Z}$-modules. See Cohomology on Sites, Lemma 21.46.4. Choose an integer $b$ such that $H^ i(F) = 0$ for $i < b$. Choose a large integer $N$ and consider the short exact sequence

\[ 0 \to \sigma _{\geq N + 1}\mathcal{E}^\bullet \to \mathcal{E}^\bullet \to \sigma _{\leq N}\mathcal{E}^\bullet \to 0 \]

of stupid truncations. This produces a distinguished triangle $E'' \to E \to E' \to E''[1]$ in $D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$. For fixed $F$ both sides of the arrow in the statement of the lemma are exact functors in $E$. Observe that

\[ p^{-1}E'' \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} (f')^{-1}Rg_*F \quad \text{and}\quad Rh_*(h^{-1}p^{-1}E'' \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} e^{-1}F) \]

are sitting in degrees $\geq N + b$. Hence, if we can prove the lemma for the object $E'$, then we see that the lemma holds in degrees $\leq N + b$ and we will conclude. Some details omitted. Thus we may assume $E$ is represented by a bounded complex of flat $\mathbf{Z}/n\mathbf{Z}$-modules. Doing another argument of the same nature, we may assume $E$ is given by a single flat $\mathbf{Z}/n\mathbf{Z}$-module $\mathcal{E}$.

Next, we use the same arguments for the variable $F$ to reduce to the case where $F$ is given by a single sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules $\mathcal{F}$. Say $\mathcal{F}$ is annihilated by an integer $m | n$. If $\ell $ is a prime number dividing $m$ and $m > \ell $, then we can look at the short exact sequence $0 \to \mathcal{F}[\ell ] \to \mathcal{F} \to \mathcal{F}/\mathcal{F}[\ell ] \to 0$ and reduce to smaller $m$. This finally reduces us to the case where $\mathcal{F}$ is annihilated by a prime number $\ell $ dividing $n$. In this case observe that

\[ p^{-1}\mathcal{E} \otimes _{\mathbf{Z}/n\mathbf{Z}}^\mathbf {L} (f')^{-1}Rg_*\mathcal{F} = p^{-1}(\mathcal{E}/\ell \mathcal{E}) \otimes _{\mathbf{F}_\ell }^\mathbf {L} (f')^{-1}Rg_*\mathcal{F} \]

by the flatness of $\mathcal{E}$. Similarly for the other term. This reduces us to the case where we are working with sheaves of $\mathbf{F}_\ell $-vector spaces which is discussed

Assume $\ell $ is a prime number invertible in $K$. Assume $\mathcal{E}$, $\mathcal{F}$ are sheaves of $\mathbf{F}_\ell $-vector spaces on $X_{\acute{e}tale}$ and $T_{\acute{e}tale}$. We want to show that

\[ p^{-1}\mathcal{E} \otimes _{\mathbf{F}_\ell } (f')^{-1}R^ qg_*\mathcal{F} \longrightarrow R^ qh_*(h^{-1}p^{-1}\mathcal{E} \otimes _{\mathbf{F}_\ell } e^{-1}\mathcal{F}) \]

is an isomorphism for every $q \geq 0$. This question is local on $X$ hence we may assume $X$ is affine. We can write $\mathcal{E}$ as a filtered colimit of constructible sheaves of $\mathbf{F}_\ell $-vector spaces on $X_{\acute{e}tale}$, see Lemma 59.73.2. Since tensor products commute with filtered colimits and since higher direct images do too (Lemma 59.51.7) we may assume $\mathcal{E}$ is a constructible sheaf of $\mathbf{F}_\ell $-vector spaces on $X_{\acute{e}tale}$. Then we can choose an integer $m$ and finite and finitely presented morphisms $\pi _ i : X_ i \to X$, $i = 1, \ldots , m$ such that there is an injective map

\[ \mathcal{E} \to \bigoplus \nolimits _{i = 1, \ldots , m} \pi _{i, *}\mathbf{F}_\ell \]

See Lemma 59.74.4. Observe that the direct sum is a constructible sheaf as well (Lemma 59.73.9). Thus the cokernel is constructible too (Lemma 59.71.6). By dimension shifting, i.e., induction on $q$, on the category of constructible sheaves of $\mathbf{F}_\ell $-vector spaces on $X_{\acute{e}tale}$, it suffices to prove the result for the sheaves $\pi _{i, *}\mathbf{F}_\ell $ (details omitted; hint: start with proving injectivity for $q = 0$ for all constructible $\mathcal{E}$). To prove this case we extend the diagram of the lemma to

\[ \xymatrix{ X_ i \ar[d]^{\pi _ i} & X'_ i \ar[l]^{p_ i} \ar[d]^{\pi '_ i} & Y_ i \ar[l]^{h_ i} \ar[d]^{\rho _ i} \\ X \ar[d] & X' \ar[l]^ p \ar[d]_{f'} & Y \ar[l]^ h \ar[d]^ e \\ \mathop{\mathrm{Spec}}(K) & S' \ar[l] & T \ar[l]_ g } \]

with all squares cartesian. In the equations below we are going to use that $R\pi _{i, *} = \pi _{i, *}$ and similarly for $\pi '_ i$, $\rho _ i$, we are going to use the Leray spectral sequence, we are going to use Lemma 59.55.3, and we are going to use Lemma 59.96.6 (although this lemma is almost trivial for finite morphisms) for $\pi _ i$, $\pi '_ i$, $\rho _ i$. Doing so we see that

\begin{align*} p^{-1}\pi _{i, *}\mathbf{F}_\ell \otimes _{\mathbf{F}_\ell } (f')^{-1}R^ qg_*\mathcal{F} & = \pi '_{i, *}\mathbf{F}_\ell \otimes _{\mathbf{F}_\ell } (f')^{-1}R^ qg_*\mathcal{F} \\ & = \pi '_{i, *}((\pi '_ i)^{-1} (f')^{-1}R^ qg_*\mathcal{F}) \end{align*}

Similarly, we have

\begin{align*} R^ qh_*(h^{-1}p^{-1} \pi _{i, *}\mathbf{F}_\ell \otimes _{\mathbf{F}_\ell } e^{-1}\mathcal{F}) & = R^ qh_*(\rho _{i, *}\mathbf{F}_\ell \otimes _{\mathbf{F}_\ell } e^{-1}\mathcal{F}) \\ & = R^ qh_*(\rho _ i^{-1}e^{-1}\mathcal{F}) \\ & = \pi '_{i, *}R^ qh_{i, *} \rho _ i^{-1}e^{-1}\mathcal{F}) \end{align*}

Simce $R^ qh_{i, *} \rho _ i^{-1}e^{-1}\mathcal{F} = (\pi '_ i)^{-1} (f')^{-1}R^ qg_*\mathcal{F}$ by Lemma 59.97.6 we conclude. $\square$

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