Remark 10.66.12. Let $\varphi : R \to S$ be a ring map. Let $M$ be an $S$-module. Denote $f : \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ the associated map on spectra. Then we have

see Lemmas 10.63.11, 10.66.11, and 10.66.6. In general all of the inclusions may be strict, see Remarks 10.63.12 and 10.66.10. If $S$ is Noetherian, then all the inclusions are equalities as the outer two are equal by Lemma 10.66.9.

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