Proposition: Spectrum Function of Commutative Rings

Let \(\varphi \colon R\longrightarrow S\,\) be a ring homomorphism between two commutative rings. The following function, which maps the spectra of both rings is well-defined and called the spectrum function (given the homomorphism \(\varphi\)). \[\varphi ^{ * }\colon \cases{\operatorname {Spec} \left(S\right)\longrightarrow \operatorname {Spec} \left(R\right),\cr I \longmapsto \varphi ^{ * }(I):=\varphi ^{-1}(I)}\]

The spectrum function fulfills the following properties: * \(\varphi\) is continuous. * For every ideal \(I\lhd R\), we have \((\varphi ^{ * })^{-1}(D(I))=D(IS)\), where \(D\) denotes open sets of the Zariski topology on \(R\). * For any another ring homomorphism \(\psi \colon S\longrightarrow T\,\) it follows \((\psi \circ \varphi )^{ * }=\varphi ^{ * }\circ \psi ^{ * }.\)

Proofs: 1

Lemmas: 1
Proofs: 2
Propositions: 3

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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