The Stacks project

Lemma 30.5.3 (Finite locally free base change). Consider a cartesian diagram of schemes

\[ \xymatrix{ Y \ar[d]_{g} \ar[r]_ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(B) \ar[r] & \mathop{\mathrm{Spec}}(A) } \]

Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module with pullback $\mathcal{G} = h^*\mathcal{F}$. If $B$ is a finite locally free $A$-module, then $H^ i(X, \mathcal{F}) \otimes _ A B = H^ i(Y, \mathcal{G})$.

Proof. In case $X$ is separated, choose an affine open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ and recall that

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

see Lemma 30.2.6. Let $\mathcal{V} : Y = \bigcup _{i \in I} g^{-1}(U_ i)$ be the corresponding affine open covering of $Y$. The opens $V_ i = g^{-1}(U_ i) = U_ i \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)$ are affine and $Y$ is separated. Thus we obtain

\[ \check{H}^ p(\mathcal{V}, \mathcal{G}) = H^ p(Y, \mathcal{G}). \]

We claim the map of Čech complexes

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

is an isomorphism. Namely, as $B$ is finitely presented as an $A$-module we see that tensoring with $B$ over $A$ commutes with products, see Algebra, Proposition 10.89.3. Thus it suffices to show that the maps $\Gamma (U_{i_0 \ldots i_ p}, \mathcal{F}) \otimes _ A B \to \Gamma (V_{i_0 \ldots i_ p}, \mathcal{G})$ are isomorphisms which follows from Lemma 30.5.1. Since $A \to B$ is flat, the same thing remains true on taking cohomology.

In the general case we argue in exactly the same way using affine open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ and the corresponding covering $\mathcal{V} : Y = \bigcup _{i \in I} V_ i$ with $V_ i = g^{-1}(U_ i)$ as above. We will use the Čech-to-cohomology spectral sequence Cohomology, Lemma 20.11.5. 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 $E_2^{p, q} = \check{H}^ p(\mathcal{V}, \underline{H}^ q(\mathcal{G}))$ which converges to $H^{p + q}(Y, \mathcal{G})$. Since the intersections $U_{i_0 \ldots i_ p}$ are separated, the result of the previous paragraph gives isomorphisms $\Gamma (U_{i_0 \ldots i_ p}, \underline{H}^ q(\mathcal{F})) \otimes _ A B \to \Gamma (V_{i_0 \ldots i_ p}, \underline{H}^ q(\mathcal{G}))$. Using that $- \otimes _ A B$ commutes with products and is exact, we conclude that $\check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \otimes _ A B \to \check{H}^ p(\mathcal{V}, \underline{H}^ q(\mathcal{G}))$ is an isomorphism. Using that $A \to B$ is flat we conclude that $H^ i(X, \mathcal{F}) \otimes _ A B \to H^ i(Y, \mathcal{G})$ is an isomorphism for all $i$ and we win. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CKW. Beware of the difference between the letter 'O' and the digit '0'.