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Tag 0959

Chapter 53: Étale Cohomology > Section 53.54: Vanishing of finite higher direct images

Lemma 53.54.3. Consider a cartesian square $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ of schemes with $f$ a finite morphism. For any sheaf of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $f'_*(g')^{-1}\mathcal{F} = g^{-1}f_*\mathcal{F}$.

Proof. In great generality there is a pullback map $g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$, see Sites, Section 7.44. To check this map is an isomorphism it suffices to check on stalks (Theorem 53.29.10). This is clear from the description of stalks in Proposition 53.54.2 and Lemma 53.36.2. $\square$

    The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 7563–7575 (see updates for more information).

    \begin{lemma}
    \label{lemma-finite-pushforward-commutes-with-base-change}
    Consider a cartesian square
    $$
    \xymatrix{
    X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\
    Y' \ar[r]^g & Y
    }
    $$
    of schemes with $f$ a finite morphism. For any sheaf of sets
    $\mathcal{F}$ on $X_\etale$ we have
    $f'_*(g')^{-1}\mathcal{F} = g^{-1}f_*\mathcal{F}$.
    \end{lemma}
    
    \begin{proof}
    In great generality there is a pullback map
    $g^{-1}f_*\mathcal{F} \to f'_*(g')^{-1}\mathcal{F}$, see
    Sites, Section \ref{sites-section-pullback}.
    To check this map is an isomorphism it suffices to check
    on stalks (Theorem \ref{theorem-exactness-stalks}).
    This is clear from the description of stalks
    in Proposition \ref{proposition-finite-higher-direct-image-zero} and
    Lemma \ref{lemma-stalk-pullback}.
    \end{proof}

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