A Noetherian affine scheme has finitely many generic points.

Lemma 10.30.6. If $R$ is a Noetherian ring then $\mathop{\mathrm{Spec}}(R)$ has finitely many irreducible components. In other words $R$ has finitely many minimal primes.

Proof. By Lemma 10.30.5 and Topology, Lemma 5.9.2 we see there are finitely many irreducible components. By Lemma 10.25.1 these correspond to minimal primes of $R$. $\square$

Comment #1549 by Bhargav Bhatt on

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