# The Stacks Project

## Tag 08WR

Lemma 34.4.10. For a ring $R$, the functor $C : \text{Mod}_R \to \text{Mod}_R$ is exact and reflects injections and surjections.

Proof. Exactness is More on Algebra, Lemma 15.51.6 and the other properties follow from this, see Remark 34.4.4. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 956–960 (see updates for more information).

\begin{lemma}
\label{lemma-C-is-faithful}
For a ring $R$, the functor $C : \text{Mod}_R \to \text{Mod}_R$ is
exact and reflects injections and surjections.
\end{lemma}

\begin{proof}
Exactness is More on Algebra, Lemma \ref{more-algebra-lemma-vee-exact}
and the other properties follow from this, see
Remark \ref{remark-reflects}.
\end{proof}

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