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The Stacks project

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

f(\text{Ass}_ S(M)) \subset \text{Ass}_ R(M) \subset \text{WeakAss}_ R(M) \subset f(\text{WeakAss}_ S(M))

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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