The Stacks Project


Tag 08WU

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

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}

    Comments (0)

    There are no comments yet for this tag.

    There are also 4 comments on Section 34.4: Descent.

    Add a comment on tag 08WU

    Your email address will not be published. Required fields are marked.

    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).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?