The Stacks project

Proof. We will use Lemma 59.82.4 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 37.53.5). 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. $\square$

Proof. This follows from Lemma 59.82.2 and 59.91.1 and the fact that any scheme finite over $X$ is proper over $\mathop{\mathrm{Spec}}(A)$. $\square$

Proof. This is a special case of Lemma 59.91.2. $\square$

Proof. By Theorem 59.53.1 (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 59.91.3. $\square$

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 59.91.4 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 59.39.5. We omit the verification that this isomorphism is compatible with our map. $\square$

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 (59.91.5.1). 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 59.51.7. 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 59.91.5. $\square$

Proof. We will use the equivalence of Lemma 59.91.6 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 59.91.5 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. $\square$

Proof. We will use the equivalence of Lemma 59.91.6 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 59.91.5 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. $\square$

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 59.91.6 combined with Proposition 59.55.2. $\square$

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 59.51.6. 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 59.91.7 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 59.91.9 the result holds for closed immersions (as closed immersions are finite). By Lemma 59.91.8 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 59.91.7, 59.91.9, and 59.91.8 one more time we conclude that the lemma is true. $\square$

Proof. In the terminology introduced above, this means that cohomology commutes with base change for every proper morphism of schemes. By Lemma 59.91.10 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 59.85.2. 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 59.91.3. 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 59.53.1 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 59.83.12.

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 59.91.6 holds and the proof is complete. $\square$

Proof. This is a simple consequence of the proper base change theorem (Theorem 59.91.11) 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. $\square$

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 59.91.11. The second one is a special case of Lemma 59.91.12. $\square$