In this section we prove the smooth base change theorem.

**First proof of smooth base change.**
This proof is very long but more direct (using less general theory) than the second proof given below.

The theorem is local on $X_{\acute{e}tale}$. More precisely, suppose we have $U \to X$ étale such that $U \to S$ factors as $U \to V \to S$ with $V \to S$ étale. Then we can consider the cartesian square

\[ \xymatrix{ U \ar[d]_{f'} & U \times _ X Y \ar[l]^{h'} \ar[d]^{e'} \\ V & V \times _ S T \ar[l]_{g'} } \]

and setting $\mathcal{F}' = \mathcal{F}|_{V \times _ S T}$ we have $f^{-1}R^ qg_*\mathcal{F}|_ U = (f')^{-1}R^ qg'_*\mathcal{F}'$ and $R^ qh_*e^{-1}\mathcal{F}|_ U = R^ qh'_*(e')^{-1}\mathcal{F}'$ (as follows from the compatibility of localization with morphisms of sites, see Sites, Lemma 7.28.2 and and Cohomology on Sites, Lemma 21.20.4). Thus it suffices to produce an étale covering of $X$ by $U \to X$ and factorizations $U \to V \to S$ as above such that the theorem holds for the diagram with $f'$, $h'$, $g'$, $e'$.

By the local structure of smooth morphisms, see Morphisms, Lemma 29.36.20, we may assume $X$ and $S$ are affine and $X \to S$ factors through an étale morphism $X \to \mathbf{A}^ d_ S$. If we have a tower of cartesian diagrams

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

and the theorem holds for the bottom and top squares, then the theorem holds for the outer rectangle; this is formal. Writing $X \to S$ as the composition

\[ X \to \mathbf{A}^{d - 1}_ S \to \mathbf{A}^{d - 2}_ S \to \ldots \to \mathbf{A}^1_ S \to S \]

we conclude that it suffices to prove the theorem when $X$ and $S$ are affine and $X \to S$ has relative dimension $1$.

For every $n \geq 1$ invertible on $S$, let $\mathcal{F}[n]$ be the subsheaf of sections of $\mathcal{F}$ annihilated by $n$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}[n]$ by our assumption on the stalks of $\mathcal{F}$. The functors $e^{-1}$ and $f^{-1}$ commute with colimits as they are left adjoints. The functors $R^ qh_*$ and $R^ qg_*$ commute with filtered colimits by Lemma 59.51.7. Thus it suffices to prove the theorem for $\mathcal{F}[n]$. From now on we fix an integer $n$, we work with sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules and we assume $S$ is a scheme over $\mathop{\mathrm{Spec}}(\mathbf{Z}[1/n])$.

Next, we reduce to the case where $T$ is affine. Since $g$ is quasi-compact and quasi-separate and $S$ is affine, the scheme $T$ is quasi-compact and quasi-separated. Thus we can use the induction principle of Cohomology of Schemes, Lemma 30.4.1. Hence it suffices to show that if $T = W \cup W'$ is an open covering and the theorem holds for the squares

\[ \xymatrix{ X \ar[d] & e^{-1}(W) \ar[l]^ i \ar[d] \\ S & W \ar[l]_ a } \quad \xymatrix{ X \ar[d] & e^{-1}(W') \ar[l]^ j \ar[d] \\ S & W' \ar[l]_ b } \quad \xymatrix{ X \ar[d] & e^{-1}(W \cap W') \ar[l]^-k \ar[d] \\ S & W \cap W' \ar[l]_ c } \]

then the theorem holds for the original diagram. To see this we consider the diagram

\[ \xymatrix{ f^{-1}R^{q - 1}c_*\mathcal{F}|_{W \cap W'} \ar[d]^{\cong } \ar[r] & f^{-1}R^ qg_*\mathcal{F} \ar[d] \ar[r] & f^{-1}R^ qa_*\mathcal{F}|_ W \oplus f^{-1}R^ qb_*\mathcal{F}|_{W'} \ar[d]_{\cong } \\ R^ qk_*e^{-1}\mathcal{F}|_{e^{-1}(W \cap W')} \ar[r] & R^ qh_*e^{-1}\mathcal{F} \ar[r] & R^ qi_*e^{-1}\mathcal{F}|_{e^{-1}(W)} \oplus R^ qj_*e^{-1}\mathcal{F}|_{e^{-1}(W')} } \]

whose rows are the long exact sequences of Lemma 59.50.2. Thus the $5$-lemma gives the desired conclusion.

Summarizing, we may assume $S$, $X$, $T$, and $Y$ affine, $\mathcal{F}$ is $n$ torsion, $X \to S$ is smooth of relative dimension $1$, and $S$ is a scheme over $\mathbf{Z}[1/n]$. We will prove the theorem by induction on $q$. The base case $q = 0$ is handled by Lemma 59.87.2. Assume $q > 0$ and the theorem holds for all smaller degrees. Choose a short exact sequence $0 \to \mathcal{F} \to \mathcal{I} \to \mathcal{Q} \to 0$ where $\mathcal{I}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules. Consider the induced diagram

\[ \xymatrix{ f^{-1}R^{q - 1}g_*\mathcal{I} \ar[d]_{\cong } \ar[r] & f^{-1}R^{q - 1}g_*\mathcal{Q} \ar[d]_{\cong } \ar[r] & f^{-1}R^ qg_*\mathcal{F} \ar[d] \ar[r] & 0 \ar[d] \\ R^{q - 1}h_*e^{-1}\mathcal{I} \ar[r] & R^{q - 1}h_*e^{-1}\mathcal{Q} \ar[r] & R^ qh_*e^{-1}\mathcal{F} \ar[r] & R^ qh_*e^{-1}\mathcal{I} } \]

with exact rows. We have the zero in the right upper corner as $\mathcal{I}$ is injective. The left two vertical arrows are isomorphisms by induction hypothesis. Thus it suffices to prove that $R^ qh_*e^{-1}\mathcal{I} = 0$.

Write $S = \mathop{\mathrm{Spec}}(A)$ and $T = \mathop{\mathrm{Spec}}(B)$ and say the morphism $T \to S$ is given by the ring map $A \to B$. We can write $A \to B = \mathop{\mathrm{colim}}\nolimits _{i \in I} (A_ i \to B_ i)$ as a filtered colimit of maps of rings of finite type over $\mathbf{Z}[1/n]$ (see Algebra, Lemma 10.127.14). For $i \in I$ we set $S_ i = \mathop{\mathrm{Spec}}(A_ i)$ and $T_ i = \mathop{\mathrm{Spec}}(B_ i)$. For $i$ large enough we can find a smooth morphism $X_ i \to S_ i$ of relative dimension $1$ such that $X = X_ i \times _{S_ i} S$, see Limits, Lemmas 32.10.1, 32.8.9, and 32.18.4. Set $Y_ i = X_ i \times _{S_ i} T_ i$ to get squares

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

Observe that $\mathcal{I}_ i = (T \to T_ i)_*\mathcal{I}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules on $T_ i$, see Cohomology on Sites, Lemma 21.14.2. We have $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits (T \to T_ i)^{-1}\mathcal{I}_ i$ by Lemma 59.51.9. Pulling back by $e$ we get $e^{-1}\mathcal{I} = \mathop{\mathrm{colim}}\nolimits (Y \to Y_ i)^{-1}e_ i^{-1}\mathcal{I}_ i$. By Lemma 59.51.8 applied to the system of morphisms $Y_ i \to X_ i$ with limit $Y \to X$ we have

\[ R^ qh_*e^{-1}\mathcal{I} = \mathop{\mathrm{colim}}\nolimits (X \to X_ i)^{-1} R^ qh_{i, *} e_ i^{-1}\mathcal{I}_ i \]

This reduces us to the case where $T$ and $S$ are affine of finite type over $\mathbf{Z}[1/n]$.

Summarizing, we have an integer $q \geq 1$ such that the theorem holds in degrees $< q$, the schemes $S$ and $T$ affine of finite type type over $\mathbf{Z}[1/n]$, we have $X \to S$ smooth of relative dimension $1$ with $X$ affine, and $\mathcal{I}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules and we have to show that $R^ qh_*e^{-1}\mathcal{I} = 0$. We will do this by induction on $\dim (T)$.

The base case is $T = \emptyset $, i.e., $\dim (T) < 0$. If you don't like this, you can take as your base case the case $\dim (T) = 0$. In this case $T \to S$ is finite (in fact even $T \to \mathop{\mathrm{Spec}}(\mathbf{Z}[1/n])$ is finite as the target is Jacobson; details omitted), so $h$ is finite too and hence has vanishing higher direct images (see references below).

Assume $\dim (T) = d \geq 0$ and we know the result for all situations where $T$ has lower dimension. Pick $U$ affine and étale over $X$ and a section $\xi $ of $R^ qh_*q^{-1}\mathcal{I}$ over $U$. We have to show that $\xi $ is zero. Of course, we may replace $X$ by $U$ (and correspondingly $Y$ by $U \times _ X Y$) and assume $\xi \in H^0(X, R^ qh_*e^{-1}\mathcal{I})$. Moreover, since $R^ qh_*e^{-1}\mathcal{I}$ is a sheaf, it suffices to prove that $\xi $ is zero locally on $X$. Hence we may replace $X$ by the members of an étale covering. In particular, using Lemma 59.51.6 we may assume that $\xi $ is the image of an element $\tilde\xi \in H^ q(Y, e^{-1}\mathcal{I})$. In terms of $\tilde\xi $ our task is to show that $\tilde\xi $ dies in $H^ q(U_ i \times _ X Y, e^{-1}\mathcal{I})$ for some étale covering $\{ U_ i \to X\} $.

By More on Morphisms, Lemma 37.38.8 we may assume that $X \to S$ factors as $X \to V \to S$ where $V \to S$ is étale and $X \to V$ is a smooth morphism of affine schemes of relative dimension $1$, has a section, and has geometrically connected fibres. Observe that $\dim (V \times _ S T) \leq \dim (T) = d$ for example by More on Algebra, Lemma 15.44.2. Hence we may then replace $S$ by $V$ and $T$ by $V \times _ S T$ (exactly as in the discussion in the first paragraph of the proof). Thus we may assume $X \to S$ is smooth of relative dimension $1$, geometrically connected fibres, and has a section $\sigma : S \to X$.

Let $\pi : T' \to T$ be a finite surjective morphism. We will use below that $\dim (T') \leq \dim (T) = d$, see Algebra, Lemma 10.112.3. Choose an injective map $\pi ^{-1}\mathcal{I} \to \mathcal{I}'$ into an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules. Then $\mathcal{I} \to \pi _*\mathcal{I}'$ is injective and hence has a splitting (as $\mathcal{I}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules). Denote $\pi ' : Y' = Y \times _ T T' \to Y$ the base change of $\pi $ and $e' : Y' \to T'$ the base change of $e$. Picture

\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e & Y' \ar[l]^{\pi '} \ar[d]^{e'} \\ S & T \ar[l]_ g & T' \ar[l]_\pi } \]

By Proposition 59.55.2 and Lemma 59.55.3 we have $R\pi '_*(e')^{-1}\mathcal{I}' = e^{-1}\pi _*\mathcal{I}'$. Thus by the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.5) we have

\[ H^ q(Y', (e')^{-1}\mathcal{I}') = H^ q(Y, e^{-1}\pi _*\mathcal{I}') \supset H^ q(Y, e^{-1}\mathcal{I}) \]

and this remains true after base change by any $U \to X$ étale. Thus we may replace $T$ by $T'$, $\mathcal{I}$ by $\mathcal{I}'$ and $\tilde\xi $ by its image in $H^ q(Y', (e')^{-1}\mathcal{I}')$.

Suppose we have a factorization $T \to S' \to S$ where $\pi : S' \to S$ is finite. Setting $X' = S' \times _ S X$ we can consider the induced diagram

\[ \xymatrix{ X \ar[d]_ f & X' \ar[l]^{\pi '} \ar[d]^{f'} & Y \ar[l]^{h'} \ar[d]^ e \\ S & S' \ar[l]_\pi & T \ar[l]_ g } \]

Since $\pi '$ has vanishing higher direct images we see that $R^ qh_*e^{-1}\mathcal{I} = \pi '_*R^ qh'_*e^{-1}\mathcal{I}$ by the Leray spectral sequence. Hence $H^0(X, R^ qh_*e^{-1}\mathcal{I}) = H^0(X', R^ qh'_*e^{-1}\mathcal{I})$. Thus $\xi $ is zero if and only if the corresponding section of $R^ qh'_*e^{-1}\mathcal{I}$ is zero^{1}. Thus we may replace $S$ by $S'$ and $X$ by $X'$. Observe that $\sigma : S \to X$ base changes to $\sigma ' : S' \to X'$ and hence after this replacement it is still true that $X \to S$ has a section $\sigma $ and geometrically connected fibres.

We will use that $S$ and $T$ are Nagata schemes, see Algebra, Proposition 10.162.16 which will guarantee that various normalizations are finite, see Morphisms, Lemmas 29.53.15 and 29.54.11. In particular, we may first replace $T$ by its normalization and then replace $S$ by the normalization of $S$ in $T$. Then $T \to S$ is a disjoint union of dominant morphisms of integral normal schemes, see Morphisms, Lemma 29.53.13. Clearly we may argue one connnected component at a time, hence we may assume $T \to S$ is a dominant morphism of integral normal schemes.

Let $s \in S$ and $t \in T$ be the generic points. By Lemma 59.89.1 there exist finite field extensions $K/\kappa (t)$ and $k/\kappa (s)$ such that $k$ is contained in $K$ and a finite étale Galois covering $Z \to X_ k$ with Galois group $G$ of order dividing a power of $n$ split over $\sigma (\mathop{\mathrm{Spec}}(k))$ such that $\tilde\xi $ maps to zero in $H^ q(Z_ K, e^{-1}\mathcal{I}|_{Z_ K})$. Let $T' \to T$ be the normalization of $T$ in $\mathop{\mathrm{Spec}}(K)$ and let $S' \to S$ be the normalization of $S$ in $\mathop{\mathrm{Spec}}(k)$. Then we obtain a commutative diagram

\[ \xymatrix{ S' \ar[d] & T' \ar[l] \ar[d] \\ S & T \ar[l] } \]

whose vertical arrows are finite. By the arguments given above we may and do replace $S$ and $T$ by $S'$ and $T'$ (and correspondingly $X$ by $X \times _ S S'$ and $Y$ by $Y \times _ T T'$). After this replacement we conclude we have a finite étale Galois covering $Z \to X_ s$ of the generic fibre of $X \to S$ with Galois group $G$ of order dividing a power of $n$ split over $\sigma (s)$ such that $\tilde\xi $ maps to zero in $H^ q(Z_ t, (Z_ t \to Y)^{-1}e^{-1}\mathcal{I})$. Here $Z_ t = Z \times _ S t = Z \times _ s t = Z \times _{X_ s} Y_ t$. Since $n$ is invertible on $S$, by Fundamental Groups, Lemma 58.31.8 we can find a finite étale morphism $U \to X$ whose restriction to $X_ s$ is $Z$.

At this point we replace $X$ by $U$ and $Y$ by $U \times _ X Y$. After this replacement it may no longer be the case that the fibres of $X \to S$ are geometrically connected (there still is a section but we won't use this), but what we gain is that after this replacement $\tilde\xi $ maps to zero in $H^ q(Y_ t, e^{-1}\mathcal{I})$, i.e., $\tilde\xi $ restricts to zero on the generic fibre of $Y \to T$.

Recall that $t$ is the spectrum of the function field of $T$, i.e., as a scheme $t$ is the limit of the nonempty affine open subschemes of $T$. By Lemma 59.51.5 we conclude there exists a nonempty open subscheme $V \subset T$ such that $\tilde\xi $ maps to zero in $H^ q(Y \times _ T V, e^{-1}\mathcal{I}|_{Y \times _ T V})$.

Denote $Z = T \setminus V$. Consider the diagram

\[ \xymatrix{ Y \times _ T Z \ar[d]_{e_ Z} \ar[r]_{i'} & Y \ar[d]_ e & Y \times _ T V \ar[l]^{j'} \ar[d]^{e_ V} \\ Z \ar[r]^ i & T & V \ar[l]_ j } \]

Choose an injection $i^{-1}\mathcal{I} \to \mathcal{I}'$ into an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules on $Z$. Looking at stalks we see that the map

\[ \mathcal{I} \to j_*\mathcal{I}|_ V \oplus i_*\mathcal{I}' \]

is injective and hence splits as $\mathcal{I}$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules. Thus it suffices to show that $\tilde\xi $ maps to zero in

\[ H^ q(Y, e^{-1}j_*\mathcal{I}|_ V) \oplus H^ q(Y, e^{-1}i_*\mathcal{I}') \]

at least after replacing $X$ by the members of an étale covering. Observe that

\[ e^{-1}j_*\mathcal{I}|_ V = j'_*e_ V^{-1}\mathcal{I}|_ V,\quad e^{-1}i_*\mathcal{I}' = i'_*e_ Z^{-1}\mathcal{I}' \]

By induction hypothesis on $q$ we see that

\[ R^ aj'_*e_ V^{-1}\mathcal{I}|_ V = 0, \quad a = 1, \ldots , q - 1 \]

By the Leray spectral sequence for $j'$ and the vanishing above it follows that

\[ H^ q(Y, j'_*(e_ V^{-1}\mathcal{I}|_ V)) \longrightarrow H^ q(Y \times _ T V, e_ V^{-1}\mathcal{I}_ V) = H^ q(Y \times _ T V, e^{-1}\mathcal{I}|_{Y \times _ T V}) \]

is injective. Thus the vanishing of the image of $\tilde\xi $ in the first summand above because we know $\tilde\xi $ vanishes in $H^ q(Y \times _ T V, e^{-1}\mathcal{I}|_{Y \times _ T V})$. Since $\dim (Z) < \dim (T) = d$ by induction the image of $\tilde\xi $ in the second summand

\[ H^ q(Y, e^{-1}i_*\mathcal{I}') = H^ q(Y, i'_*e_ Z^{-1}\mathcal{I}') = H^ q(Y \times _ T Z, e_ Z^{-1}\mathcal{I}') \]

dies after replacing $X$ by the members of a suitable étale covering. This finishes the proof of the smooth base change theorem.
$\square$

**Second proof of smooth base change.**
This proof is the same as the longer first proof; it is shorter only in that we have split out the arguments used in a number of lemmas.

The case of $q = 0$ is Lemma 59.87.2. Thus we may assume $q > 0$ and the result is true for all smaller degrees.

For every $n \geq 1$ invertible on $S$, let $\mathcal{F}[n]$ be the subsheaf of sections of $\mathcal{F}$ annihilated by $n$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}[n]$ by our assumption on the stalks of $\mathcal{F}$. The functors $e^{-1}$ and $f^{-1}$ commute with colimits as they are left adjoints. The functors $R^ qh_*$ and $R^ qg_*$ commute with filtered colimits by Lemma 59.51.7. Thus it suffices to prove the theorem for $\mathcal{F}[n]$. From now on we fix an integer $n$ invertible on $S$ and we work with sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules.

By Lemma 59.86.1 the question is étale local on $X$ and $S$. By the local structure of smooth morphisms, see Morphisms, Lemma 29.36.20, we may assume $X$ and $S$ are affine and $X \to S$ factors through an étale morphism $X \to \mathbf{A}^ d_ S$. Writing $X \to S$ as the composition

\[ X \to \mathbf{A}^{d - 1}_ S \to \mathbf{A}^{d - 2}_ S \to \ldots \to \mathbf{A}^1_ S \to S \]

we conclude from Lemma 59.86.2 that it suffices to prove the theorem when $X$ and $S$ are affine and $X \to S$ has relative dimension $1$.

By Lemma 59.88.7 it suffices to show that $R^ qh_*\mathbf{Z}/d\mathbf{Z} = 0$ for $d | n$ whenever we have a cartesian diagram

\[ \xymatrix{ X \ar[d] & Y \ar[d] \ar[l]^ h \\ S & \mathop{\mathrm{Spec}}(K) \ar[l] } \]

where $X \to S$ is affine and smooth of relative dimension $1$, $S$ is the spectrum of a normal domain $A$ with algebraically closed fraction field $L$, and $K/L$ is an extension of algebraically closed fields.

Recall that $R^ qh_*\mathbf{Z}/d\mathbf{Z}$ is the sheaf associated to the presheaf

\[ U \longmapsto H^ q(U \times _ X Y, \mathbf{Z}/d\mathbf{Z}) = H^ q(U \times _ S \mathop{\mathrm{Spec}}(K), \mathbf{Z}/d\mathbf{Z}) \]

on $X_{\acute{e}tale}$ (Lemma 59.51.6). Thus it suffices to show: given $U$ and $\xi \in H^ q(U \times _ S \mathop{\mathrm{Spec}}(K), \mathbf{Z}/d\mathbf{Z})$ there exists an étale covering $\{ U_ i \to U\} $ such that $\xi $ dies in $H^ q(U_ i \times _ S \mathop{\mathrm{Spec}}(K), \mathbf{Z}/d\mathbf{Z})$.

Of course we may take $U$ affine. Then $U \times _ S \mathop{\mathrm{Spec}}(K)$ is a (smooth) affine curve over $K$ and hence we have vanishing for $q > 1$ by Theorem 59.83.10.

Final case: $q = 1$. We may replace $U$ by the members of an étale covering as in More on Morphisms, Lemma 37.38.8. Then $U \to S$ factors as $U \to V \to S$ where $U \to V$ has geometrically connected fibres, $U$, $V$ are affine, $V \to S$ is étale, and there is a section $\sigma : V \to U$. By Lemma 59.80.4 we see that $V$ is isomorphic to a (finite) disjoint union of (affine) open subschemes of $S$. Clearly we may replace $S$ by one of these and $X$ by the corresponding component of $U$. Thus we may assume $X \to S$ has geometrically connected fibres, has a section $\sigma $, and $\xi \in H^1(Y, \mathbf{Z}/d\mathbf{Z})$. Since $K$ and $L$ are algebraically closed we have

\[ H^1(X_ L, \mathbf{Z}/d\mathbf{Z}) = H^1(Y, \mathbf{Z}/d\mathbf{Z}) \]

See Lemma 59.83.12. Thus there is a finite étale Galois covering $Z \to X_ L$ with Galois group $G \subset \mathbf{Z}/d\mathbf{Z}$ which annihilates $\xi $. You can either see this by looking at the statement or proof of Lemma 59.89.1 or by using directly that $\xi $ corresponds to a $\mathbf{Z}/d\mathbf{Z}$-torsor over $X_ L$. Finally, by Fundamental Groups, Lemma 58.31.9 we find a (necessarily surjective) finite étale morphism $X' \to X$ whose restriction to $X_ L$ is $Z \to X_ L$. Since $\xi $ dies in $X'_ K$ this finishes the proof.
$\square$

The following immediate consequence of the smooth base change theorem is what is often used in practice.

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