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Tag 08WK

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

Remark 34.4.4. Any functor $F : \mathcal{A} \to \mathcal{B}$ of abelian categories which is exact and takes nonzero objects to nonzero objects reflects injections and surjections. Namely, exactness implies that $F$ preserves kernels and cokernels (compare with Homology, Section 12.7). For example, if $f : R \to S$ is a faithfully flat ring homomorphism, then $\bullet \otimes_R S: \text{Mod}_R \to \text{Mod}_S$ has these properties.

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 886–896 (see updates for more information).

    \begin{remark}
    \label{remark-reflects}
    Any functor $F : \mathcal{A} \to \mathcal{B}$ of abelian categories
    which is exact and takes nonzero objects to nonzero objects reflects
    injections and surjections. Namely, exactness implies that
    $F$ preserves kernels and cokernels (compare with
    Homology, Section \ref{homology-section-functors}).
    For example, if $f : R \to S$ is a 
    faithfully flat ring homomorphism, then
    $\bullet \otimes_R S: \text{Mod}_R \to \text{Mod}_S$ has these properties.
    \end{remark}

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