59.87 Base change for pushforward
This section is preliminary and should be skipped on a first reading. In this section we discuss for what morphisms $f : X \to S$ we have $f^{-1}g_* = h_*e^{-1}$ on all sheaves (of sets) for every cartesian diagram
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
with $g$ quasi-compact and quasi-separated.
Lemma 59.87.1. Consider the cartesian diagram of schemes
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
Assume that $f$ is flat and every object $U$ of $X_{\acute{e}tale}$ has a covering $\{ U_ i \to U\} $ such that $U_ i \to S$ factors as $U_ i \to V_ i \to S$ with $V_ i \to S$ étale and $U_ i \to V_ i$ quasi-compact with geometrically connected fibres. Then for any sheaf $\mathcal{F}$ of sets on $T_{\acute{e}tale}$ we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.
Proof.
Let $U \to X$ be an étale morphism such that $U \to S$ factors as $U \to V \to S$ with $V \to S$ étale and $U \to V$ quasi-compact with geometrically connected fibres. Observe that $U \to V$ is flat (More on Flatness, Lemma 38.2.3). We claim that
\begin{align*} f^{-1}g_*\mathcal{F}(U) & = g_*\mathcal{F}(V) \\ & = \mathcal{F}(V \times _ S T) \\ & = e^{-1}\mathcal{F}(U \times _ X Y) \\ & = h_*e^{-1}\mathcal{F}(U) \end{align*}
Namely, thinking of $U$ as an object of $X_{\acute{e}tale}$ and $V$ as an object of $S_{\acute{e}tale}$ we see that the first equality follows from Lemma 59.39.31. Thinking of $V \times _ S T$ as an object of $T_{\acute{e}tale}$ the second equality follows from the definition of $g_*$. Observe that $U \times _ X Y = U \times _ S T$ (because $Y = X \times _ S T$) and hence $U \times _ X Y \to V \times _ S T$ has geometrically connected fibres as a base change of $U \to V$. Thinking of $U \times _ X Y$ as an object of $Y_{\acute{e}tale}$, we see that the third equality follows from Lemma 59.39.3 as before. Finally, the fourth equality follows from the definition of $h_*$.
Since by assumption every object of $X_{\acute{e}tale}$ has an étale covering to which the argument of the previous paragraph applies we see that the lemma is true.
$\square$
Lemma 59.87.2. Consider a cartesian diagram of schemes
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
where $f$ is flat and locally of finite presentation with geometrically reduced fibres. Then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ on $T_{\acute{e}tale}$.
Proof.
Combine Lemma 59.87.1 with More on Morphisms, Lemma 37.46.3.
$\square$
Lemma 59.87.3. Consider the cartesian diagrams of schemes
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
Assume that $S$ is the spectrum of a separably closed field. Then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ on $T_{\acute{e}tale}$.
Proof.
We may work locally on $X$. Hence we may assume $X$ is affine. Then we can write $X$ as a cofiltered limit of affine schemes of finite type over $S$. By Lemma 59.86.3 we may assume that $X$ is of finite type over $S$. Then Lemma 59.87.1 applies because any scheme of finite type over a separably closed field is a finite disjoint union of connected and geometrically connected schemes (see Varieties, Lemma 33.7.6).
$\square$
Lemma 59.87.4. Consider a cartesian diagram of schemes
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
Assume that
$f$ is flat and open,
the residue fields of $S$ are separably algebraically closed,
given an étale morphism $U \to X$ with $U$ affine we can write $U$ as a finite disjoint union of open subschemes of $X$ (for example if $X$ is a normal integral scheme with separably closed function field),
any nonempty open of a fibre $X_ s$ of $f$ is connected (for example if $X_ s$ is irreducible or empty).
Then for any sheaf $\mathcal{F}$ of sets on $T_{\acute{e}tale}$ we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.
Proof.
Omitted. Hint: the assumptions almost trivially imply the condition of Lemma 59.87.1. The for example in part (3) follows from Lemma 59.80.4.
$\square$
The following lemma doesn't really belong here but there does not seem to be a good place for it anywhere.
Lemma 59.87.5. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation with geometrically reduced fibres. Then $f^{-1} : \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ commutes with products.
Proof.
Let $I$ be a set and let $\mathcal{G}_ i$ be a sheaf on $S_{\acute{e}tale}$ for $i \in I$. Let $U \to X$ be an étale morphism such that $U \to S$ factors as $U \to V \to S$ with $V \to S$ étale and $U \to V$ flat of finite presentation with geometrically connected fibres. Then we have
\begin{align*} f^{-1}(\prod \mathcal{G}_ i)(U) & = (\prod \mathcal{G}_ i)(V) \\ & = \prod \mathcal{G}_ i(V) \\ & = \prod f^{-1}\mathcal{G}_ i(U) \\ & = (\prod f^{-1}\mathcal{G}_ i)(U) \end{align*}
where we have used Lemma 59.39.3 in the first and third equality (we are also using that the restriction of $f^{-1}\mathcal{G}$ to $U_{\acute{e}tale}$ is the pullback via $U \to V$ of the restriction of $\mathcal{G}$ to $V_{\acute{e}tale}$, see Sites, Lemma 7.28.2). By More on Morphisms, Lemma 37.46.3 every object $U$ of $X_{\acute{e}tale}$ has an étale covering $\{ U_ i \to U\} $ such that the discussion in the previous paragraph applies to $U_ i$. The lemma follows.
$\square$
Lemma 59.87.6. Let $f : X \to S$ be a flat morphism of schemes such that for every geometric point $\overline{x}$ of $X$ the map
\[ \mathcal{O}_{S, f(\overline{x})}^{sh} \longrightarrow \mathcal{O}_{X, \overline{x}}^{sh} \]
has geometrically connected fibres. Then for every cartesian diagram of schemes
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
with $g$ quasi-compact and quasi-separated we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ of sets on $T_{\acute{e}tale}$.
Proof.
It suffices to check equality on stalks, see Theorem 59.29.10. By Theorem 59.53.1 we have
\[ (h_*e^{-1}\mathcal{F})_{\overline{x}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \times _ X Y, e^{-1}\mathcal{F}) \]
and we have similarly
\[ (f^{-1}g_*^{-1}\mathcal{F})_{\overline{x}} = (g_*^{-1}\mathcal{F})_{f(\overline{x})} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh}) \times _ S T, \mathcal{F}) \]
These sets are equal by an application of Lemma 59.39.3 to the morphism
\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \times _ X Y \longrightarrow \mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh}) \times _ S T \]
which is a base change of $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh})$ because $Y = X \times _ S T$.
$\square$
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