The Stacks Project


Tag 058I

Chapter 10: Commutative Algebra > Section 10.81: Universally injective module maps

Definition 10.81.1. Let $f: M \to N$ be a map of $R$-modules. Then $f$ is called universally injective if for every $R$-module $Q$, the map $f \otimes_R \text{id}_Q: M \otimes_R Q \to N \otimes_R Q$ is injective. A sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ of $R$-modules is called universally exact if it is exact and $M_1 \to M_2$ is universally injective.

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 18700–18708 (see updates for more information).

    \begin{definition}
    \label{definition-universally-injective}
    Let $f: M \to N$ be a map of $R$-modules.  Then $f$ is called
    {\it universally injective} if for every $R$-module $Q$, the map $f
    \otimes_R \text{id}_Q: M \otimes_R Q \to N \otimes_R Q$
    is injective.  A sequence $0 \to M_1 \to M_2 \to M_3
    \to 0$ of $R$-modules is called {\it universally exact} if it is exact
    and $M_1 \to M_2$ is universally injective.
    \end{definition}

    Comments (0)

    There are no comments yet for this tag.

    There are also 2 comments on Section 10.81: Commutative Algebra.

    Add a comment on tag 058I

    Your email address will not be published.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?