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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:
 It equals \(\operatorname {Spec} (S/IS)\).
 It equals \(\operatorname {Spec} (S/IS)\).
Table of Contents
Proofs: 1
 Lemma: Fiber of Prime Ideals
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References
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck