Lemma 10.31.6. 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.

** A Noetherian affine scheme has finitely many generic points. **

**Proof.**
By Lemma 10.31.5 and Topology, Lemma 5.9.2 we see there are finitely many irreducible components. By Lemma 10.26.1 these correspond to minimal primes of $R$.
$\square$

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## Comments (1)

Comment #1549 by Bhargav Bhatt on