# The Stacks Project

## Tag 08X9

Remark 34.4.21. If $f$ is a split injection in $\text{Mod}_R$, one can simplify the argument by splitting $f$ directly, without using $C$. Things are even simpler if $f$ is faithfully flat; in this case, the conclusion of Lemma 34.4.20 is immediate because tensoring over $R$ with $S$ preserves all equalizers.

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1274–1282 (see updates for more information).

\begin{remark}
\label{remark-descent-lemma}
If $f$ is a split injection in $\text{Mod}_R$, one can simplify the argument by
splitting $f$ directly,
without using $C$. Things are even simpler if $f$ is faithfully flat; in this
case,
the conclusion of Lemma \ref{lemma-descent-lemma}
is immediate because tensoring over $R$ with $S$ preserves all equalizers.
\end{remark}

There are no comments yet for this tag.

There are also 4 comments on Section 34.4: Descent.

## Add a comment on tag 08X9

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 lower-right corner).