Lemma 58.87.1. Let $(A, I)$ be a henselian pair. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of schemes. Let $Z = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I)$. For any sheaf $\mathcal{F}$ on the topological space associated to $X$ we have $\Gamma (X, \mathcal{F}) = \Gamma (Z, \mathcal{F}|_ Z)$.

## 58.87 The proper base change theorem

The proper base change theorem is stated and proved in this section. Our approach follows roughly the proof in [XII, Theorem 5.1, SGA4] using Gabber's ideas (from the affine case) to slightly simplify the arguments.

**Proof.**
We will use Lemma 58.79.4 to prove this. First observe that the underlying topological space of $X$ is spectral by Properties, Lemma 28.2.4. Let $Y \subset X$ be an irreducible closed subscheme. To finish the proof we show that $Y \cap Z = Y \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I)$ is connected. Replacing $X$ by $Y$ we may assume that $X$ is irreducible and we have to show that $Z$ is connected. Let $X \to \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ be the Stein factorization of $f$ (More on Morphisms, Theorem 37.49.5). Then $A \to B$ is integral and $(B, IB)$ is a henselian pair (More on Algebra, Lemma 15.11.8). Thus we may assume the fibres of $X \to \mathop{\mathrm{Spec}}(A)$ are geometrically connected. On the other hand, the image $T \subset \mathop{\mathrm{Spec}}(A)$ of $f$ is irreducible and closed as $X$ is proper over $A$. Hence $T \cap V(I)$ is connected by More on Algebra, Lemma 15.11.16. Now $Y \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I) \to T \cap V(I)$ is a surjective closed map with connected fibres. The result now follows from Topology, Lemma 5.7.5.
$\square$

Lemma 58.87.2. Let $(A, I)$ be a henselian pair. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of schemes. Let $i : Z \to X$ be the closed immersion of $X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I)$ into $X$. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (Z, i_{small}^{-1}\mathcal{F})$.

**Proof.**
This follows from Lemma 58.79.2 and 58.87.1 and the fact that any scheme finite over $X$ is proper over $\mathop{\mathrm{Spec}}(A)$.
$\square$

Lemma 58.87.3. Let $A$ be a henselian local ring. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of schemes. Let $X_0 \subset X$ be the fibre of $f$ over the closed point. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $\Gamma (X, \mathcal{F}) = \Gamma (X_0, \mathcal{F}|_{X_0})$.

**Proof.**
This is a special case of Lemma 58.87.2.
$\square$

Let $f : X \to S$ be a morphism of schemes. Let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point. The fibre of $f$ at $\overline{s}$ is the scheme $X_{\overline{s}} = \mathop{\mathrm{Spec}}(k) \times _{\overline{s}, S} X$ viewed as a scheme over $\mathop{\mathrm{Spec}}(k)$. If $\mathcal{F}$ is a sheaf on $X_{\acute{e}tale}$, then denote $\mathcal{F}_{\overline{s}} = p_{small}^{-1}\mathcal{F}$ the pullback of $\mathcal{F}$ to $(X_{\overline{s}})_{\acute{e}tale}$. In the following we will consider the set

Let $s \in S$ be the image point of $\overline{s}$. Let $\kappa (s)^{sep}$ be the separable algebraic closure of $\kappa (s)$ in $k$ as in Definition 58.55.1. By Lemma 58.39.5 pullback defines a bijection

where $p_{sep} : X_{\kappa (s)^{sep}} = \mathop{\mathrm{Spec}}(\kappa (s)^{sep}) \times _ S X \to X$ is the projection.

Lemma 58.87.4. Let $f : X \to S$ be a proper morphism of schemes. Let $\overline{s} \to S$ be a geometric point. For any sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ the canonical map

is bijective.

**Proof.**
By Theorem 58.52.1 (for sheaves of sets) we have

where $p : X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, \overline{s}}^{sh}) \to X$ is the projection. Since the residue field of the strictly henselian local ring $\mathcal{O}_{S, \overline{s}}^{sh}$ is $\kappa (s)^{sep}$ we conclude from the discussion above the lemma and Lemma 58.87.3. $\square$

Lemma 58.87.5. Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times _ Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$. Let $\mathcal{F}$ be any sheaf on $X_{\acute{e}tale}$. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.

**Proof.**
There is a canonical map $g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$. Namely, it is adjoint to the map

which is $f_*$ applied to the canonical map $\mathcal{F} \to g'_*(g')^{-1}\mathcal{F}$. To check this map is an isomorphism we can compute what happens on stalks. Let $y' : \mathop{\mathrm{Spec}}(k) \to Y'$ be a geometric point with image $y$ in $Y$. By Lemma 58.87.4 the stalks are $\Gamma (X'_{y'}, \mathcal{F}_{y'})$ and $\Gamma (X_ y, \mathcal{F}_ y)$ respectively. Here the sheaves $\mathcal{F}_ y$ and $\mathcal{F}_{y'}$ are the pullbacks of $\mathcal{F}$ by the projections $X_ y \to X$ and $X'_{y'} \to X$. Thus we see that the groups agree by Lemma 58.39.5. We omit the verification that this isomorphism is compatible with our map. $\square$

At this point we start discussing the proper base change theorem. To do so we introduce some notation. consider a commutative diagram

of morphisms of schemes. Then we obtain a commutative diagram of sites

For any object $E$ of $D(X_{\acute{e}tale})$ we obtain a canonical base change map

in $D(Y'_{\acute{e}tale})$. See Cohomology on Sites, Remark 21.19.3 where we use the constant sheaf $\mathbf{Z}$ as our sheaf of rings. We will usually omit the subscripts ${}_{small}$ in this formula. For example, if $E = \mathcal{F}[0]$ where $\mathcal{F}$ is an abelian sheaf on $X_{\acute{e}tale}$, the base change map is a map

in $D(Y'_{\acute{e}tale})$.

The map (58.87.5.2) has no chance of being an isomorphism in the generality given above. The goal is to show it is an isomorphism if the diagram (58.87.5.1) is cartesian, $f : X \to Y$ proper, the cohomology sheaves of $E$ are torsion, and $E$ is bounded below. To study this question we introduce the following terminology. Let us say that *cohomology commutes with base change for $f : X \to Y$* if (58.87.5.3) is an isomorphism for every diagram (58.87.5.1) where $X' = Y' \times _ Y X$ and every torsion abelian sheaf $\mathcal{F}$.

Lemma 58.87.6. Let $f : X \to Y$ be a proper morphism of schemes. The following are equivalent

cohomology commutes with base change for $f$ (see above),

for every prime number $\ell $ and every injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules $\mathcal{I}$ on $X_{\acute{e}tale}$ and every diagram (58.87.5.1) where $X' = Y' \times _ Y X$ the sheaves $R^ qf'_*(g')^{-1}\mathcal{I}$ are zero for $q > 0$.

**Proof.**
It is clear that (1) implies (2). Conversely, assume (2) and let $\mathcal{F}$ be a torsion abelian sheaf on $X_{\acute{e}tale}$. Let $Y' \to Y$ be a morphism of schemes and let $X' = Y' \times _ Y X$ with projections $g' : X' \to X$ and $f' : X' \to Y'$ as in diagram (58.87.5.1). We want to show the maps of sheaves

are isomorphisms for all $q \geq 0$.

For every $n \geq 1$, let $\mathcal{F}[n]$ be the subsheaf of sections of $\mathcal{F}$ annihilated by $n$. Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}[n]$. The functors $g^{-1}$ and $(g')^{-1}$ commute with arbitrary colimits (as left adjoints). Taking higher direct images along $f$ or $f'$ commutes with filtered colimits by Lemma 58.51.7. Hence we see that

Thus it suffices to prove the result in case $\mathcal{F}$ is annihilated by a positive integer $n$.

If $n = \ell n'$ for some prime number $\ell $, then we obtain a short exact sequence

Observe that $\mathcal{F}/\mathcal{F}[\ell ]$ is annihilated by $n'$. Moreover, if the result holds for both $\mathcal{F}[\ell ]$ and $\mathcal{F}/\mathcal{F}[\ell ]$, then the result holds by the long exact sequence of higher direct images (and the $5$ lemma). In this way we reduce to the case that $\mathcal{F}$ is annihilated by a prime number $\ell $.

Assume $\mathcal{F}$ is annihilated by a prime number $\ell $. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ in $D(X_{\acute{e}tale}, \mathbf{Z}/\ell \mathbf{Z})$. Applying assumption (2) and Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) we see that

computes $Rf'_*(g')^{-1}\mathcal{F}$. We conclude by applying Lemma 58.87.5. $\square$

Lemma 58.87.7. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of schemes. Assume

cohomology commutes with base change for $f$,

cohomology commutes with base change for $g \circ f$, and

$f$ is surjective.

Then cohomology commutes with base change for $g$.

**Proof.**
We will use the equivalence of Lemma 58.87.6 without further mention. Let $\ell $ be a prime number. Let $\mathcal{I}$ be an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $Y_{\acute{e}tale}$. Choose an injective map of sheaves $f^{-1}\mathcal{I} \to \mathcal{J}$ where $\mathcal{J}$ is an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $Z_{\acute{e}tale}$. Since $f$ is surjective the map $\mathcal{I} \to f_*\mathcal{J}$ is injective (look at stalks in geometric points). Since $\mathcal{I}$ is injective we see that $\mathcal{I}$ is a direct summand of $f_*\mathcal{J}$. Thus it suffices to prove the desired vanishing for $f_*\mathcal{J}$.

Let $Z' \to Z$ be a morphism of schemes and set $Y' = Z' \times _ Z Y$ and $X' = Z' \times _ Z X = Y' \times _ Y X$. Denote $a : X' \to X$, $b : Y' \to Y$, and $c : Z' \to Z$ the projections. Similarly for $f' : X' \to Y'$ and $g' : Y' \to Z'$. By Lemma 58.87.5 we have $b^{-1}f_*\mathcal{J} = f'_*a^{-1}\mathcal{J}$. On the other hand, we know that $R^ qf'_*a^{-1}\mathcal{J}$ and $R^ q(g' \circ f')_*a^{-1}\mathcal{J}$ are zero for $q > 0$. Using the spectral sequence (Cohomology on Sites, Lemma 21.14.7)

we conclude that $ R^ pg'_*(b^{-1}f_*\mathcal{J}) = R^ pg'_*(f'_*a^{-1}\mathcal{J}) = 0$ for $p > 0$ as desired. $\square$

Lemma 58.87.8. Let $f : X \to Y$ and $g : Y \to Z$ be proper morphisms of schemes. Assume

cohomology commutes with base change for $f$, and

cohomology commutes with base change for $g$.

Then cohomology commutes with base change for $g \circ f$.

**Proof.**
We will use the equivalence of Lemma 58.87.6 without further mention. Let $\ell $ be a prime number. Let $\mathcal{I}$ be an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $X_{\acute{e}tale}$. Then $f_*\mathcal{I}$ is an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $Y_{\acute{e}tale}$ (Cohomology on Sites, Lemma 21.14.2). The result follows formally from this, but we will also spell it out.

Let $Z' \to Z$ be a morphism of schemes and set $Y' = Z' \times _ Z Y$ and $X' = Z' \times _ Z X = Y' \times _ Y X$. Denote $a : X' \to X$, $b : Y' \to Y$, and $c : Z' \to Z$ the projections. Similarly for $f' : X' \to Y'$ and $g' : Y' \to Z'$. By Lemma 58.87.5 we have $b^{-1}f_*\mathcal{I} = f'_*a^{-1}\mathcal{I}$. On the other hand, we know that $R^ qf'_*a^{-1}\mathcal{I}$ and $R^ q(g')_*b^{-1}f_*\mathcal{I}$ are zero for $q > 0$. Using the spectral sequence (Cohomology on Sites, Lemma 21.14.7)

we conclude that $R^ p(g' \circ f')_*a^{-1}\mathcal{I} = 0$ for $p > 0$ as desired. $\square$

Lemma 58.87.9. Let $f : X \to Y$ be a finite morphism of schemes. Then cohomology commutes with base change for $f$.

**Proof.**
Observe that a finite morphism is proper, see Morphisms, Lemma 29.44.11. Moreover, the base change of a finite morphism is finite, see Morphisms, Lemma 29.44.6. Thus the result follows from Lemma 58.87.6 combined with Proposition 58.54.2.
$\square$

Lemma 58.87.10. To prove that cohomology commutes with base change for every proper morphism of schemes it suffices to prove it holds for the morphism $\mathbf{P}^1_ S \to S$ for every scheme $S$.

**Proof.**
Let $f : X \to Y$ be a proper morphism of schemes. Let $Y = \bigcup Y_ i$ be an affine open covering and set $X_ i = f^{-1}(Y_ i)$. If we can prove cohomology commutes with base change for $X_ i \to Y_ i$, then cohomology commutes with base change for $f$. Namely, the formation of the higher direct images commutes with Zariski (and even étale) localization on the base, see Lemma 58.51.6. Thus we may assume $Y$ is affine.

Let $Y$ be an affine scheme and let $X \to Y$ be a proper morphism. By Chow's lemma there exists a commutative diagram

where $X' \to \mathbf{P}^ n_ Y$ is an immersion, and $\pi : X' \to X$ is proper and surjective, see Limits, Lemma 32.12.1. Since $X \to Y$ is proper, we find that $X' \to Y$ is proper (Morphisms, Lemma 29.41.4). Hence $X' \to \mathbf{P}^ n_ Y$ is a closed immersion (Morphisms, Lemma 29.41.7). It follows that $X' \to X \times _ Y \mathbf{P}^ n_ Y = \mathbf{P}^ n_ X$ is a closed immersion (as an immersion with closed image).

By Lemma 58.87.7 it suffices to prove cohomology commutes with base change for $\pi $ and $X' \to Y$. These morphisms both factor as a closed immersion followed by a projection $\mathbf{P}^ n_ S \to S$ (for some $S$). By Lemma 58.87.9 the result holds for closed immersions (as closed immersions are finite). By Lemma 58.87.8 it suffices to prove the result for projections $\mathbf{P}^ n_ S \to S$.

For every $n \geq 1$ there is a finite surjective morphism

given on coordinates by

where $F_0, \ldots , F_ n$ in $x_1, \ldots , y_ n$ are the polynomials with integer coefficients such that

Applying Lemmas 58.87.7, 58.87.9, and 58.87.8 one more time we conclude that the lemma is true. $\square$

Theorem 58.87.11. Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times _ Y X$ and consider the cartesian diagram

Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$. Then the base change map

is an isomorphism.

**Proof.**
In the terminology introduced above, this means that cohomology commutes with base change for every proper morphism of schemes. By Lemma 58.87.10 it suffices to prove that cohomology commutes with base change for the morphism $\mathbf{P}^1_ S \to S$ for every scheme $S$.

Let $S$ be the spectrum of a strictly henselian local ring with closed point $s$. Set $X = \mathbf{P}^1_ S$ and $X_0 = X_ s = \mathbf{P}^1_ s$. Let $\mathcal{F}$ be a sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $X_{\acute{e}tale}$. The key to our proof is that

Namely, choose a resolution $\mathcal{F} \to \mathcal{I}^\bullet $ by injective sheaves of $\mathbf{Z}/\ell \mathbf{Z}$-modules. Then $\mathcal{I}^\bullet |_{X_0}$ is a resolution of $\mathcal{F}|_{X_0}$ by right $H^0_{\acute{e}tale}(X_0, -)$-acyclic objects, see Lemma 58.81.2. Leray's acyclicity lemma tells us the right hand side is computed by the complex $H^0_{\acute{e}tale}(X_0, \mathcal{I}^\bullet |_{X_0})$ which is equal to $H^0_{\acute{e}tale}(X, \mathcal{I}^\bullet )$ by Lemma 58.87.3. This complex computes the left hand side.

Assume $S$ is general and $\mathcal{F}$ is a sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules on $X_{\acute{e}tale}$. Let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point of $S$ lying over $s \in S$. We have

where $\kappa (s)^{sep}$ is the residue field of $\mathcal{O}_{S, \overline{s}}^{sh}$, i.e., the separable algebraic closure of $\kappa (s)$ in $k$. The first equality by Theorem 58.52.1 and the second equality by the displayed formula in the previous paragraph.

Finally, consider any morphism of schemes $g : T \to S$ where $S$ and $\mathcal{F}$ are as above. Set $f' : \mathbf{P}^1_ T \to T$ the projection and let $g' : \mathbf{P}^1_ T \to \mathbf{P}^1_ S$ the morphism induced by $g$. Consider the base change map

Let $\overline{t}$ be a geometric point of $T$ with image $\overline{s} = g(\overline{t})$. By our discussion above the map on stalks at $\overline{t}$ is the map

Since $\kappa (s)^{sep} \subset \kappa (t)^{sep}$ this map is an isomorphism by Lemma 58.80.11.

This proves cohomology commutes with base change for $\mathbf{P}^1_ S \to S$ and sheaves of $\mathbf{Z}/\ell \mathbf{Z}$-modules. In particular, for an injective sheaf of $\mathbf{Z}/\ell \mathbf{Z}$-modules the higher direct images of any base change are zero. In other words, condition (2) of Lemma 58.87.6 holds and the proof is complete. $\square$

Lemma 58.87.12. Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times _ Y X$ and denote $f' : X' \to Y'$ and $g' : X' \to X$ the projections. Let $E \in D^+(X_{\acute{e}tale})$ have torsion cohomology sheaves. Then the base change map (58.87.5.2) $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism.

**Proof.**
This is a simple consequence of the proper base change theorem (Theorem 58.87.11) using the spectral sequences

converging to $R^ nf_*E$ and $R^ nf'_*(g')^{-1}E$. The spectral sequences are constructed in Derived Categories, Lemma 13.21.3. Some details omitted. $\square$

Lemma 58.87.13. Let $f : X \to Y$ be a proper morphism of schemes. Let $\overline{y} \to Y$ be a geometric point.

For a torsion abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $(R^ nf_*\mathcal{F})_{\overline{y}} = H^ n_{\acute{e}tale}(X_{\overline{y}}, \mathcal{F}_{\overline{y}})$.

For $E \in D^+(X_{\acute{e}tale})$ with torsion cohomology sheaves we have $(R^ nf_*E)_{\overline{y}} = H^ n_{\acute{e}tale}(X_{\overline{y}}, E_{\overline{y}})$.

**Proof.**
In the statement, $\mathcal{F}_{\overline{y}}$ denotes the pullback of $\mathcal{F}$ to the scheme theoretic fibre $X_{\overline{y}} = \overline{y} \times _ Y X$. Since pulling back by $\overline{y} \to Y$ produces the stalk of $\mathcal{F}$, the first statement of the lemma is a special case of Theorem 58.87.11. The second one is a special case of Lemma 58.87.12.
$\square$

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