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Tag 09ZS

Chapter 44: Étale Cohomology > Section 44.74: Affine analog of proper base change

Lemma 44.74.11. Let $(A, I)$ be a henselian pair. Set $X = \mathop{\rm Spec}(A)$ and $Z = \mathop{\rm Spec}(A/I)$. The functor $$ U \longmapsto U \times_X Z $$ is an equivalence of categories between finite étale schemes over $X$ and finite étale schemes over $Z$.

Proof. This is a translation of More on Algebra, Lemma 15.7.9. $\square$

    The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 11722–11731 (see updates for more information).

    \begin{lemma}
    \label{lemma-gabber}
    Let $(A, I)$ be a henselian pair. Set $X = \Spec(A)$ and $Z = \Spec(A/I)$.
    The functor
    $$
    U \longmapsto U \times_X Z
    $$
    is an equivalence of categories between finite \'etale schemes over $X$
    and finite \'etale schemes over $Z$.
    \end{lemma}
    
    \begin{proof}
    This is a translation of
    More on Algebra, Lemma \ref{more-algebra-lemma-finite-etale-equivalence}.
    \end{proof}

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