Lemma 10.14.4. Let $R \to S$ be a ring map. The functors $\text{Mod}_ S \to \text{Mod}_ R$, $N \mapsto N_ R$ (restriction) and $\text{Mod}_ R \to \text{Mod}_ S$, $M \mapsto \mathop{\mathrm{Hom}}\nolimits _ R(S, M)$ are adjoint functors. In a formula
Proof. If $\alpha : N_ R \to M$ is an $R$-module map, then we define $\alpha ' : N \to \mathop{\mathrm{Hom}}\nolimits _ R(S, M)$ by the rule $\alpha '(n) = (s \mapsto \alpha (sn))$. If $\beta : N \to \mathop{\mathrm{Hom}}\nolimits _ R(S, M)$ is an $S$-module map, we define $\beta ' : N_ R \to M$ by the rule $\beta '(n) = \beta (n)(1)$. We omit the verification that these constructions are mutually inverse. $\square$
Your email address will not be published.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 toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)