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Tag 08XT

Chapter 45: Dualizing Complexes > Section 45.3: Injective modules

Example 45.3.6. Let $R$ be a reduced ring. Let $\mathfrak p \subset R$ be a minimal prime so that $K = R_\mathfrak p$ is a field (Algebra, Lemma 10.24.1). Then $K$ is an injective $R$-module. Namely, we have $\mathop{\rm Hom}\nolimits_R(M, K) = \mathop{\rm Hom}\nolimits_K(M_\mathfrak p, K)$ for any $R$-module $M$. Since localization is an exact functor and taking duals is an exact functor on $K$-vector spaces we conclude $\mathop{\rm Hom}\nolimits_R(-, K)$ is an exact functor, i.e., $K$ is an injective $R$-module.

    The code snippet corresponding to this tag is a part of the file dualizing.tex and is located in lines 279–289 (see updates for more information).

    \begin{example}
    \label{example-reduced-ring-injective}
    Let $R$ be a reduced ring. Let $\mathfrak p \subset R$ be a minimal prime
    so that $K = R_\mathfrak p$ is a field
    (Algebra, Lemma \ref{algebra-lemma-minimal-prime-reduced-ring}).
    Then $K$ is an injective $R$-module. Namely, we have
    $\Hom_R(M, K) = \Hom_K(M_\mathfrak p, K)$ for any $R$-module
    $M$. Since localization is an exact functor and taking duals is
    an exact functor on $K$-vector spaces we conclude $\Hom_R(-, K)$
    is an exact functor, i.e., $K$ is an injective $R$-module.
    \end{example}

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