# The Stacks Project

## Tag 08WK

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}

There are no comments yet for this tag.

There are also 3 comments on Section 34.4: Descent.

## Add a comment on tag 08WK

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).