The Stacks project

Lemma 30.5.2 (Flat base change). Consider a cartesian diagram of schemes

\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module with pullback $\mathcal{F}' = (g')^*\mathcal{F}$. Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated. For any $i \geq 0$

  1. the base change map of Cohomology, Lemma 20.17.1 is an isomorphism

    \[ g^*R^ if_*\mathcal{F} \longrightarrow R^ if'_*\mathcal{F}', \]
  2. if $S = \mathop{\mathrm{Spec}}(A)$ and $S' = \mathop{\mathrm{Spec}}(B)$, then $H^ i(X, \mathcal{F}) \otimes _ A B = H^ i(X', \mathcal{F}')$.

Proof. Using Cohomology, Lemma 20.17.1 in (1) is allowed since $g'$ is flat by Morphisms, Lemma 29.25.8. Having said this, part (1) follows from part (2). Namely, part (1) is local on $S'$ and hence we may assume $S$ and $S'$ are affine. In other words, we have $S = \mathop{\mathrm{Spec}}(A)$ and $S' = \mathop{\mathrm{Spec}}(B)$ as in (2). Then since $R^ if_*\mathcal{F}$ is quasi-coherent (Lemma 30.4.5), it is the quasi-coherent $\mathcal{O}_ S$-module associated to the $A$-module $H^0(S, R^ if_*\mathcal{F}) = H^ i(X, \mathcal{F})$ (equality by Lemma 30.4.6). Similarly, $R^ if'_*\mathcal{F}'$ is the quasi-coherent $\mathcal{O}_{S'}$-module associated to the $B$-module $H^ i(X', \mathcal{F}')$. Since pullback by $g$ corresponds to $- \otimes _ A B$ on modules (Schemes, Lemma 26.7.3) we see that it suffices to prove (2).

Let $A \to B$ be a flat ring homomorphism. Let $X$ be a quasi-compact and quasi-separated scheme over $A$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Set $X_ B = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)$ and denote $\mathcal{F}_ B$ the pullback of $\mathcal{F}$. We are trying to show that the map

\[ H^ i(X, \mathcal{F}) \otimes _ A B \longrightarrow H^ i(X_ B, \mathcal{F}_ B) \]

(given by the reference in the statement of the lemma) is an isomorphism.

In case $X$ is separated, choose an affine open covering $\mathcal{U} : X = U_1 \cup \ldots \cup U_ t$ and recall that

\[ \check{H}^ p(\mathcal{U}, \mathcal{F}) = H^ p(X, \mathcal{F}), \]

see Lemma 30.2.6. If $\mathcal{U}_ B : X_ B = (U_1)_ B \cup \ldots \cup (U_ t)_ B$ we obtain by base change, then it is still the case that each $(U_ i)_ B$ is affine and that $X_ B$ is separated. Thus we obtain

\[ \check{H}^ p(\mathcal{U}_ B, \mathcal{F}_ B) = H^ p(X_ B, \mathcal{F}_ B). \]

We have the following relation between the Čech complexes

\[ \check{\mathcal{C}}^\bullet (\mathcal{U}_ B, \mathcal{F}_ B) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A B \]

as follows from Lemma 30.5.1. Since $A \to B$ is flat, the same thing remains true on taking cohomology.

In case $X$ is quasi-separated, choose an affine open covering $\mathcal{U} : X = U_1 \cup \ldots \cup U_ t$. We will use the Čech-to-cohomology spectral sequence Cohomology, Lemma 20.11.5. The reader who wishes to avoid this spectral sequence can use Mayer-Vietoris and induction on $t$ as in the proof of Lemma 30.4.5. The spectral sequence has $E_2$-page $E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F}))$ and converges to $H^{p + q}(X, \mathcal{F})$. Similarly, we have a spectral sequence with $E_2$-page $E_2^{p, q} = \check{H}^ p(\mathcal{U}_ B, \underline{H}^ q(\mathcal{F}_ B))$ which converges to $H^{p + q}(X_ B, \mathcal{F}_ B)$. Since the intersections $U_{i_0 \ldots i_ p}$ are quasi-compact and separated, the result of the second paragraph of the proof gives $\check{H}^ p(\mathcal{U}_ B, \underline{H}^ q(\mathcal{F}_ B)) = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \otimes _ A B$. Using that $A \to B$ is flat we conclude that $H^ i(X, \mathcal{F}) \otimes _ A B \to H^ i(X_ B, \mathcal{F}_ B)$ is an isomorphism for all $i$ and we win. $\square$

Comments (8)

Comment #936 by correction_bot on

"Similarly, we have a spectral sequence with -page and converges to ." Write instead "which converges to".

Comment #957 by on

Fixed the typos pointed out in comments 931--936. Thanks! See here for changes.

Comment #2343 by Daniel on

Typo in the very end of the first paragraph of the proof: "It suffices to prove (2)"

Comment #4682 by Nicolas Müller on

The last part in the last sentence of the second paragraph of the proof ("where and is the pullback of to .") is redundant, as and are already defined in the sentence before.

Comment #5458 by Du on

Lemma 02N7 requirs to be flat. So this assumption should be added.

Comment #5676 by on

@#5458: Yes, we need to be flat in order to use Lemma 20.17.1. But we're saved by the fact that the base change of a flat morphism is flat (Lemma 29.25.8) and that is the base change of by . I have added a remark to the proof here.

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