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Tag 08XF

Chapter 34: Descent > Section 34.4: Descent for universally injective morphisms

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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