Lemma: Fiber of Prime Ideals Under a Spectrum Function

Let \(\varphi \colon R\longrightarrow S\,\) be a ring homomorphism between two commutative rings and let \[\varphi ^{ * }\colon \cases{\operatorname {Spec} \left(S\right)\longrightarrow \operatorname {Spec} \left(R\right),\cr J\longmapsto \varphi ^{ * }(J)}\] be the corresponding spectrum function. Then the fiber of a prime ideal \(I\in \operatorname {Spec} \left(R\right)\) under the spectrum function fulfills the following properties:

  1. It equals \(\operatorname {Spec} (S/IS)\).
  2. It equals \(\operatorname {Spec} (S/IS)\).

Proofs: 1

  1. Lemma: Fiber of Prime Ideals

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück