The Stacks project

Proof. For $q > 1$ we know that $\xi $ dies in $H^ q(X_{\overline{K}}, (X_{\overline{K}} \to \mathop{\mathrm{Spec}}(K))^{-1}\mathcal{F})$ (Theorem 59.83.10). By Lemma 59.51.5 we see that this means there is a finite extension $K'/K$ such that $\xi $ dies in $H^ q(X_{K'}, (X_{K'} \to \mathop{\mathrm{Spec}}(K))^{-1}\mathcal{F})$. Thus we can take $k' = k$ and $Z = X$ in this case.

Assume $q = 1$. Recall that $\mathcal{F}$ corresponds to a discrete module $M$ with continuous $\text{Gal}_ K$-action, see Lemma 59.59.1. Since $M$ is $n$-torsion, it is the uninon of finite $\text{Gal}_ K$-stable subgroups. Thus we reduce to the case where $M$ is a finite abelian group annihilated by $n$, see Lemma 59.51.4. After replacing $K$ by a finite extension we may assume that the action of $\text{Gal}_ K$ on $M$ is trivial. Thus we may assume $\mathcal{F} = \underline{M}$ is the constant sheaf with value a finite abelian group $M$ annihilated by $n$.

We can write $M$ as a direct sum of cyclic groups. Any two finite étale Galois coverings whose Galois groups have order invertible in $k$, can be dominated by a third one whose Galois group has order invertible in $k$ (Fundamental Groups, Section 58.7). Thus it suffices to prove the lemma when $M = \mathbf{Z}/d\mathbf{Z}$ where $d | n$.

Assume $M = \mathbf{Z}/d\mathbf{Z}$ where $d | n$. In this case $\overline{\xi } = \xi |_{X_{\overline{K}}}$ is an element of

See Theorem 59.83.10. This group classifies $\mathbf{Z}/d\mathbf{Z}$-torsors, see Cohomology on Sites, Lemma 21.4.3. The torsor corresponding to $\overline{\xi }$ (viewed as a sheaf on $X_{\overline{k}, {\acute{e}tale}}$) in turn gives rise to a finite étale morphism $T \to X_{\overline{k}}$ endowed an action of $\mathbf{Z}/d\mathbf{Z}$ transitive on the fibre of $T$ over $x_{\overline{k}}$, see Lemma 59.64.4. Choose a connected component $T' \subset T$ (if $\overline{\xi }$ has order $d$, then $T$ is already connected). Then $T' \to X_{\overline{k}}$ is a finite étale Galois cover whose Galois group is a subgroup $G \subset \mathbf{Z}/d\mathbf{Z}$ (small detail omitted). Moreover the element $\overline{\xi }$ maps to zero under the map $H^1(X_{\overline{k}}, \mathbf{Z}/d\mathbf{Z}) \to H^1(T', \mathbf{Z}/d\mathbf{Z})$ as this is one of the defining properties of $T$.

Next, we use a limit argument to choose a finite extension $k'/k$ contained in $\overline{k}$ such that $T' \to X_{\overline{k}}$ descends to a finite étale Galois cover $Z \to X_{k'}$ with group $G$. See Limits, Lemmas 32.10.1, 32.8.3, and 32.8.10. After increasing $k'$ we may assume that $Z$ splits over $x_{k'}$. The image of $\xi $ in $H^1(Z_{\overline{K}}, \mathbf{Z}/d\mathbf{Z})$ is zero by construction. Thus by Lemma 59.51.5 we can find a finite subextension $\overline{K}/K'/K$ containing $k'$ such that $\xi $ dies in $H^1(Z_{K'}, \mathbf{Z}/d\mathbf{Z})$ and this finishes the proof. $\square$

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

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

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

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

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

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

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

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

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

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

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

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 zero¹. 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 29.54.10. 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

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

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

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

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

By induction hypothesis on $q$ we see that

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

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

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

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

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

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

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. $\square$

Proof. Consider the spectral sequences

converging to $R^ nf_*E$ and $R^ nf'_*(g')^{-1}E$. These spectral sequences are constructed in Derived Categories, Lemma 13.21.3. Combining the smooth base change theorem (Theorem 59.89.2) with Lemma 59.86.3 we see that

Combining all of the above we get the lemma. $\square$