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Tag 02KH

Chapter 29: Cohomology of Schemes > Section 29.5: Cohomology and base change, I

Lemma 29.5.2 (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 20.18.1 is an isomorphism $$ g^*R^if_*\mathcal{F} \longrightarrow R^if'_*\mathcal{F}', $$
  2. if $S = \mathop{\rm Spec}(A)$ and $S' = \mathop{\rm Spec}(B)$, then $H^i(X, \mathcal{F}) \otimes_A B = H^i(X', \mathcal{F}')$.

Proof. We claim that 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 $S = \mathop{\rm Spec}(A)$ and $S' = \mathop{\rm Spec}(B)$ as in (2). Then since $R^if_*\mathcal{F}$ is quasi-coherent (Lemma 29.4.5), 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 29.4.6). 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 25.7.3) 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 $X_B = X \times_{\mathop{\rm Spec}(A)} \mathop{\rm Spec}(B)$ 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 where $X_B = \mathop{\rm Spec}(B) \times_{\mathop{\rm Spec}(A)} X$ and $\mathcal{F}_B$ is the pullback of $\mathcal{F}$ to $X_B$.

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 29.2.6. 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 29.5.1. 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 20.12.5. The reader who wishes to avoid this spectral sequence can use Mayer-Vietoris and induction on $t$ as in the proof of Lemma 29.4.5. 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. $\square$

    The code snippet corresponding to this tag is a part of the file coherent.tex and is located in lines 907–930 (see updates for more information).

    \begin{lemma}[Flat base change]
    \label{lemma-flat-base-change-cohomology}
    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 $i \geq 0$
    \begin{enumerate}
    \item the base change map of
    Cohomology, Lemma \ref{cohomology-lemma-base-change-map-flat-case}
    is an isomorphism
    $$
    g^*R^if_*\mathcal{F} \longrightarrow R^if'_*\mathcal{F}',
    $$
    \item if $S = \Spec(A)$ and $S' = \Spec(B)$, then
    $H^i(X, \mathcal{F}) \otimes_A B = H^i(X', \mathcal{F}')$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    We claim that 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 $S = \Spec(A)$
    and $S' = \Spec(B)$ as in (2).
    Then since $R^if_*\mathcal{F}$ is quasi-coherent
    (Lemma \ref{lemma-quasi-coherence-higher-direct-images}),
    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 \ref{lemma-quasi-coherence-higher-direct-images-application}).
    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 \ref{schemes-lemma-widetilde-pullback})
    we see that it suffices to prove (2).
    
    \medskip\noindent
    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 $X_B = X \times_{\Spec(A)} \Spec(B)$ 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 where $X_B = \Spec(B) \times_{\Spec(A)} X$ and
    $\mathcal{F}_B$ is the pullback of $\mathcal{F}$ to $X_B$.
    
    \medskip\noindent
    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 \ref{lemma-cech-cohomology-quasi-coherent}.
    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 {\v C}ech 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 \ref{lemma-affine-base-change}.
    Since $A \to B$ is flat, the same thing remains true on taking cohomology.
    
    \medskip\noindent
    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
    {\v C}ech-to-cohomology spectral sequence
    Cohomology, Lemma \ref{cohomology-lemma-cech-spectral-sequence}.
    The reader who wishes to avoid this spectral sequence
    can use Mayer-Vietoris and induction on $t$ as in the proof of
    Lemma \ref{lemma-quasi-coherence-higher-direct-images}.
    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.
    \end{proof}

    Comments (4)

    Comment #936 by correction_bot on August 22, 2014 a 8:03 pm UTC

    "Similarly, we have a spectral sequence with $E_2$-page $E_2^{p, q} = \check{H}^p(\mathcal{U}_B, \underline{H}^q(\mathcal{F}_B))$ and converges to $H^{p + q}(X_B, \mathcal{F}_B)$." Write instead "which converges to".

    Comment #957 by Johan (site) on August 28, 2014 a 12:54 pm UTC

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

    Comment #2343 by Daniel on January 4, 2017 a 2:30 pm UTC

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

    Comment #2412 by Johan (site) on February 17, 2017 a 1:24 pm UTC

    Thanks Daniel. Fixed here.

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