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Fix typo and add internal reference
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aisejohan committed May 24, 2024
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Expand Up @@ -1305,14 +1305,14 @@ \subsection{Descent for modules and their morphisms}
Hence $f^*$ is not fully faithful.

\medskip\noindent
We finally check that (c) implies (b). By Lemma \ref{lemma-descent-lemma}, for
We finally check that (c) implies (b). By Lemmas
\ref{lemma-equalizer-M} and \ref{lemma-descent-lemma}, for
$(M, \theta) \in DD_{S/R}$,
the natural map $f^* f_*(M,\theta) \to M$ is an isomorphism of $S$-modules. On
the other hand, for $M_0 \in \text{Mod}_R$,
we may tensor (\ref{equation-equalizer-S}) with $M_0$ over $R$ to obtain an
equalizer sequence,
so $M_0 \to f_* f^* M$ is an isomorphism. Consequently, $f_*$ and $f^*$ are
quasi-inverse functors, proving the claim.
equalizer sequence, so $M_0 \to f_* f^* M_0$ is an isomorphism.
Consequently, $f_*$ and $f^*$ are quasi-inverse functors, proving the claim.
\end{proof}

\subsection{Descent for properties of modules}
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