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Tag 08WN

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

Example 34.4.7. For a ring $R$ and $f_1, \ldots, f_n \in R$ generating the unit ideal, the morphism $R \to R_{f_1} \oplus \ldots \oplus R_{f_n}$ is universally injective. Although this is immediate from Lemma 34.4.8, it is instructive to check it directly: we immediately reduce to the case where $R$ is local, in which case some $f_i$ must be a unit and so the map $R \to R_{f_i}$ is an isomorphism.

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 916–925 (see updates for more information).

    \begin{example}
    \label{example-cover-universally-injective}
    For a ring $R$ and $f_1, \ldots, f_n \in R$ generating the unit
    ideal, the morphism $R \to R_{f_1} \oplus \ldots \oplus R_{f_n}$ is
    universally injective. Although this is immediate from
    Lemma \ref{lemma-faithfully-flat-universally-injective},
    it is instructive to check it directly: we immediately reduce to the case
    where $R$ is local, in which case some $f_i$ must be a unit and so the map
    $R \to R_{f_i}$ is an isomorphism.
    \end{example}

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