# The Stacks Project

## Tag 08XF

Remark 34.4.27. It would make things easier to have a faithfully flat ring homomorphism $g: R \to T$ for which $T \to S \otimes_R T$ has some extra structure. For instance, if one could ensure that $T \to S \otimes_R T$ is split in $\textit{Rings}$, then it would follow that every property of a module or algebra which is stable under base extension and which descends along faithfully flat morphisms also descends along universally injective morphisms. An obvious guess would be to find $g$ for which $T$ is not only faithfully flat but also injective in $\text{Mod}_R$, but even for $R = \mathbf{Z}$ no such homomorphism can exist.

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1501–1515 (see updates for more information).

\begin{remark}
\label{remark-when-locally-split}
It would make things easier to have a faithfully
flat ring homomorphism $g: R \to T$ for which $T \to S \otimes_R T$ has some
extra structure.
For instance, if one could ensure that $T \to S \otimes_R T$ is split in
$\textit{Rings}$,
then it would follow that every property of a module or algebra which is stable
under base extension
and which descends along faithfully flat morphisms also descends along
universally injective morphisms.
An obvious guess would be to find $g$ for which $T$ is not only faithfully flat
but also injective in $\text{Mod}_R$,
but even for $R = \mathbf{Z}$ no such homomorphism can exist.
\end{remark}

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