## 59.86 Preliminaries on base change

If you are interested in either the smooth base change theorem or the proper base change theorem, you should skip directly to the corresponding sections. In this section and the next few sections we consider commutative diagrams

$\xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g }$

of schemes; we usually assume this diagram is cartesian, i.e., $Y = X \times _ S T$. A commutative diagram as above gives rise to a commutative diagram

$\xymatrix{ X_{\acute{e}tale}\ar[d]_{f_{small}} & Y_{\acute{e}tale}\ar[d]^{e_{small}} \ar[l]^{h_{small}} \\ S_{\acute{e}tale}& T_{\acute{e}tale}\ar[l]_{g_{small}} }$

of small étale sites. Let us use the notation

$f^{-1} = f_{small}^{-1}, \quad g_* = g_{small, *}, \quad e^{-1} = e_{small}^{-1}, \text{ and}\quad h_* = h_{small, *}.$

By Sites, Section 7.45 we get a base change or pullback map

$f^{-1}g_*\mathcal{F} \longrightarrow h_*e^{-1}\mathcal{F}$

for a sheaf $\mathcal{F}$ on $T_{\acute{e}tale}$. If $\mathcal{F}$ is an abelian sheaf on $T_{\acute{e}tale}$, then we get a derived base change map

$f^{-1}Rg_*\mathcal{F} \longrightarrow Rh_*e^{-1}\mathcal{F}$

see Cohomology on Sites, Lemma 21.15.1. Finally, if $K$ is an arbitrary object of $D(T_{\acute{e}tale})$ there is a base change map

$f^{-1}Rg_*K \longrightarrow Rh_*e^{-1}K$

see Cohomology on Sites, Remark 21.19.3.

Lemma 59.86.1. Consider a cartesian diagram of schemes

$\xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g }$

Let $\{ U_ i \to X\}$ be an étale covering such that $U_ i \to S$ factors as $U_ i \to V_ i \to S$ with $V_ i \to S$ étale and consider the cartesian diagrams

$\xymatrix{ U_ i \ar[d]_{f_ i} & U_ i \times _ X Y \ar[l]^{h_ i} \ar[d]^{e_ i} \\ V_ i & V_ i \times _ S T \ar[l]_{g_ i} }$

Let $\mathcal{F}$ be a sheaf on $T_{\acute{e}tale}$. Let $K$ in $D(T_{\acute{e}tale})$. Set $K_ i = K|_{V_ i \times _ S T}$ and $\mathcal{F}_ i = \mathcal{F}|_{V_ i \times _ S T}$.

1. If $f_ i^{-1}g_{i, *}\mathcal{F}_ i = h_{i, *}e_ i^{-1}\mathcal{F}_ i$ for all $i$, then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.

2. If $f_ i^{-1}Rg_{i, *}K_ i = Rh_{i, *}e_ i^{-1}K_ i$ for all $i$, then $f^{-1}Rg_*K = Rh_*e^{-1}K$.

3. If $\mathcal{F}$ is an abelian sheaf and $f_ i^{-1}R^ qg_{i, *}\mathcal{F}_ i = R^ qh_{i, *}e_ i^{-1}\mathcal{F}_ i$ for all $i$, then $f^{-1}R^ qg_*\mathcal{F} = R^ qh_*e^{-1}\mathcal{F}$.

Proof. Proof of (1). First we observe that

$(f^{-1}g_*\mathcal{F})|_{U_ i} = f_ i^{-1}(g_*\mathcal{F}|_{V_ i}) = f_ i^{-1}g_{i, *}\mathcal{F}_ i$

The first equality because $U_ i \to X \to S$ is equal to $U_ i \to V_ i \to S$ and the second equality because $g_*\mathcal{F}|_{V_ i} = g_{i, *}\mathcal{F}_ i$ by Sites, Lemma 7.28.2. Similarly we have

$(h_*e^{-1}\mathcal{F})|_{U_ i} = h_{i, *}(e^{-1}\mathcal{F}|_{U_ i \times _ X Y}) = h_{i, *}e_ i^{-1}\mathcal{F}_ i$

Thus if the base change maps $f_ i^{-1}g_{i, *}\mathcal{F}_ i \to h_{i, *}e_ i^{-1}\mathcal{F}_ i$ are isomorphisms for all $i$, then the base change map $f^{-1}g_*\mathcal{F} \to h_*e^{-1}\mathcal{F}$ restricts to an isomorphism over $U_ i$ for all $i$ and we conclude it is an isomorphism as $\{ U_ i \to X\}$ is an étale covering.

For the other two statements we replace the appeal to Sites, Lemma 7.28.2 by an appeal to Cohomology on Sites, Lemma 21.20.4. $\square$

Lemma 59.86.2. Consider a tower of cartesian diagrams of schemes

$\xymatrix{ W \ar[d]_ i & Z \ar[l]^ j \ar[d]^ k \\ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g }$

Let $K$ in $D(T_{\acute{e}tale})$. If

$f^{-1}Rg_*K \to Rh_*e^{-1}K \quad \text{and}\quad i^{-1}Rh_*e^{-1}K \to Rj_*k^{-1}e^{-1}K$

are isomorphisms, then $(f \circ i)^{-1}Rg_*K \to Rj_*(e \circ k)^{-1}K$ is an isomorphism. Similarly, if $\mathcal{F}$ is an abelian sheaf on $T_{\acute{e}tale}$ and if

$f^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F} \quad \text{and}\quad i^{-1}R^ qh_*e^{-1}\mathcal{F} \to R^ qj_*k^{-1}e^{-1}\mathcal{F}$

are isomorphisms, then $(f \circ i)^{-1}R^ qg_*\mathcal{F} \to R^ qj_*(e \circ k)^{-1}\mathcal{F}$ is an isomorphism.

Proof. This is formal, provided one checks that the composition of these base change maps is the base change maps for the outer rectangle, see Cohomology on Sites, Remark 21.19.5. $\square$

Lemma 59.86.3. Let $I$ be a directed set. Consider an inverse system of cartesian diagrams of schemes

$\xymatrix{ X_ i \ar[d]_{f_ i} & Y_ i \ar[l]^{h_ i} \ar[d]^{e_ i} \\ S_ i & T_ i \ar[l]_{g_ i} }$

with affine transition morphisms and with $g_ i$ quasi-compact and quasi-separated. Set $X = \mathop{\mathrm{lim}}\nolimits X_ i$, $S = \mathop{\mathrm{lim}}\nolimits S_ i$, $T = \mathop{\mathrm{lim}}\nolimits T_ i$ and $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ to obtain the cartesian diagram

$\xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g }$

Let $(\mathcal{F}_ i, \varphi _{i'i})$ be a system of sheaves on $(T_ i)$ as in Definition 59.51.1. Set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits p_ i^{-1}\mathcal{F}_ i$ on $T$ where $p_ i : T \to T_ i$ is the projection. Then we have the following

1. If $f_ i^{-1}g_{i, *}\mathcal{F}_ i = h_{i, *}e_ i^{-1}\mathcal{F}_ i$ for all $i$, then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.

2. If $\mathcal{F}_ i$ is an abelian sheaf for all $i$ and $f_ i^{-1}R^ qg_{i, *}\mathcal{F}_ i = R^ qh_{i, *}e_ i^{-1}\mathcal{F}_ i$ for all $i$, then $f^{-1}R^ qg_*\mathcal{F} = R^ qh_*e^{-1}\mathcal{F}$.

Proof. We prove (2) and we omit the proof of (1). We will use without further mention that pullback of sheaves commutes with colimits as it is a left adjoint. Observe that $h_ i$ is quasi-compact and quasi-separated as a base change of $g_ i$. Denoting $q_ i : Y \to Y_ i$ the projections, observe that $e^{-1}\mathcal{F} = \mathop{\mathrm{colim}}\nolimits e^{-1}p_ i^{-1}\mathcal{F}_ i = \mathop{\mathrm{colim}}\nolimits q_ i^{-1}e_ i^{-1}\mathcal{F}_ i$. By Lemma 59.51.8 this gives

$R^ qh_*e^{-1}\mathcal{F} = \mathop{\mathrm{colim}}\nolimits r_ i^{-1}R^ qh_{i, *}e_ i^{-1}\mathcal{F}_ i$

where $r_ i : X \to X_ i$ is the projection. Similarly, we have

$f^{-1}Rg_*\mathcal{F} = f^{-1}\mathop{\mathrm{colim}}\nolimits s_ i^{-1}R^ qg_{i, *}\mathcal{F}_ i = \mathop{\mathrm{colim}}\nolimits r_ i^{-1}f_ i^{-1}R^ qg_{i, *}\mathcal{F}_ i$

where $s_ i : S \to S_ i$ is the projection. The lemma follows. $\square$

Lemma 59.86.4. Let $I$, $X_ i$, $Y_ i$, $S_ i$, $T_ i$, $f_ i$, $h_ i$, $e_ i$, $g_ i$, $X$, $Y$, $S$, $T$, $f$, $h$, $e$, $g$ be as in the statement of Lemma 59.86.3. Let $0 \in I$ and let $K_0 \in D^+(T_{0, {\acute{e}tale}})$. For $i \in I$, $i \geq 0$ denote $K_ i$ the pullback of $K_0$ to $T_ i$. Denote $K$ the pullback of $K_0$ to $T$. If $f_ i^{-1}Rg_{i, *}K_ i = Rh_{i, *}e_ i^{-1}K_ i$ for all $i \geq 0$, then $f^{-1}Rg_*K = Rh_*e^{-1}K$.

Proof. It suffices to show that the base change map $f^{-1}Rg_*K \to Rh_*e^{-1}K$ induces an isomorphism on cohomology sheaves. In other words, we have to show that $f^{-1}R^ pg_*K \to R^ ph_*e^{-1}K$ is an isomorphism for all $p \in \mathbf{Z}$ if we are given that $f_ i^{-1}R^ pg_{i, *}K_ i \to R^ ph_{i, *}e_ i^{-1}K_ i$ is an isomorphism for all $i \geq 0$ and $p \in \mathbf{Z}$. At this point we can argue exactly as in the proof of Lemma 59.86.3 replacing reference to Lemma 59.51.8 by a reference to Lemma 59.52.4. $\square$

Lemma 59.86.5. Consider a cartesian diagram of schemes

$\xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g }$

where $g : T \to S$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be an abelian sheaf on $T_{\acute{e}tale}$. Let $q \geq 0$. The following are equivalent

1. For every geometric point $\overline{x}$ of $X$ with image $\overline{s} = f(\overline{x})$ we have

$H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S T, \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S T, \mathcal{F})$
2. $f^{-1}R^ qg_*\mathcal{F} \to R^ qh_*e^{-1}\mathcal{F}$ is an isomorphism.

Proof. Since $Y = X \times _ S T$ we have $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ X Y = \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S T$. Thus the map in (1) is the map of stalks at $\overline{x}$ for the map in (2) by Theorem 59.53.1 (and Lemma 59.36.2). Thus the result by Theorem 59.29.10. $\square$

Lemma 59.86.6. Let $f : X \to S$ be a morphism of schemes. Let $\overline{x}$ be a geometric point of $X$ with image $\overline{s}$ in $S$. Let $\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ be a morphism with $K$ a separably closed field. Let $\mathcal{F}$ be an abelian sheaf on $\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}$. Let $q \geq 0$. The following are equivalent

1. $H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S \mathop{\mathrm{Spec}}(K), \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S \mathop{\mathrm{Spec}}(K), \mathcal{F})$

2. $H^ q(\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _{\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})} \mathop{\mathrm{Spec}}(K), \mathcal{F}) = H^ q(\mathop{\mathrm{Spec}}(K), \mathcal{F})$

Proof. Observe that $\mathop{\mathrm{Spec}}(K) \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ is the spectrum of a filtered colimit of étale algebras over $K$. Since $K$ is separably closed, each étale $K$-algebra is a finite product of copies of $K$. Thus we can write

$\mathop{\mathrm{Spec}}(K) \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) = \mathop{\mathrm{lim}}\nolimits _{i \in I} \coprod \nolimits _{a \in A_ i} \mathop{\mathrm{Spec}}(K)$

as a cofiltered limit where each term is a disjoint union of copies of $\mathop{\mathrm{Spec}}(K)$ over a finite set $A_ i$. Note that $A_ i$ is nonempty as we are given $\mathop{\mathrm{Spec}}(K) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. It follows that

\begin{align*} \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _ S \mathop{\mathrm{Spec}}(K) & = \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _{\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})} \left( \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \times _ S \mathop{\mathrm{Spec}}(K)\right) \\ & = \mathop{\mathrm{lim}}\nolimits _{i \in I} \coprod \nolimits _{a \in A_ i} \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{X, \overline{x}}) \times _{\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})} \mathop{\mathrm{Spec}}(K) \end{align*}

Since taking cohomology in our setting commutes with limits of schemes (Theorem 59.51.3) we conclude. $\square$

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