# The Stacks Project

## Tag 08WU

Lemma 34.4.12. Let $R$ be a ring. A morphism $f: M \to N$ in $\text{Mod}_R$ is universally injective if and only if $C(f): C(N) \to C(M)$ is a split surjection.

Proof. By (34.4.11.1), for any $P \in \text{Mod}_R$ we have a commutative diagram $$\xymatrix@C=9pc{ \mathop{\rm Hom}\nolimits_R( P, C(N)) \ar[r]_{\mathop{\rm Hom}\nolimits_R(P,C(f))} \ar[d]^{\cong} & \mathop{\rm Hom}\nolimits_R(P,C(M)) \ar[d]^{\cong} \\ C(P \otimes_R N ) \ar[r]^{C(1_{P} \otimes f)} & C(P \otimes_R M ). }$$ If $f$ is universally injective, then $1_{C(M)} \otimes f: C(M) \otimes_R M \to C(M) \otimes_R N$ is injective, so both rows in the above diagram are surjective for $P = C(M)$. We may thus lift $1_{C(M)} \in \mathop{\rm Hom}\nolimits_R(C(M), C(M))$ to some $g \in \mathop{\rm Hom}\nolimits_R(C(N), C(M))$ splitting $C(f)$. Conversely, if $C(f)$ is a split surjection, then both rows in the above diagram are surjective, so by Lemma 34.4.10, $1_{P} \otimes f$ is injective. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 996–1000 (see updates for more information).

\begin{lemma}
\label{lemma-split-surjection}
Let $R$ be a ring. A morphism $f: M \to N$ in $\text{Mod}_R$ is universally
injective if and only if $C(f): C(N) \to C(M)$ is a split surjection.
\end{lemma}

\begin{proof}
By (\ref{equation-adjunction}), for any $P \in \text{Mod}_R$ we have a
commutative diagram
$$\xymatrix@C=9pc{ \Hom_R( P, C(N)) \ar[r]_{\Hom_R(P,C(f))} \ar[d]^{\cong} & \Hom_R(P,C(M)) \ar[d]^{\cong} \\ C(P \otimes_R N ) \ar[r]^{C(1_{P} \otimes f)} & C(P \otimes_R M ). }$$
If $f$ is universally injective, then $1_{C(M)} \otimes f: C(M) \otimes_R M \to C(M) \otimes_R N$ is injective,
so both rows in the above diagram are surjective for $P = C(M)$. We may thus
lift
$1_{C(M)} \in \Hom_R(C(M), C(M))$ to some $g \in \Hom_R(C(N), C(M))$ splitting
$C(f)$.
Conversely, if $C(f)$ is a split surjection, then
both rows in the above diagram are surjective,
so by Lemma \ref{lemma-C-is-faithful}, $1_{P} \otimes f$ is injective.
\end{proof}

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