The Stacks project

Lemma 10.26.3. Let $R$ be a ring. Let $\mathfrak p \subset R$ be a prime.

  1. the set of irreducible closed subsets of $\mathop{\mathrm{Spec}}(R)$ passing through $\mathfrak p$ is in one-to-one correspondence with primes $\mathfrak q \subset R_{\mathfrak p}$.

  2. The set of irreducible components of $\mathop{\mathrm{Spec}}(R)$ passing through $\mathfrak p$ is in one-to-one correspondence with minimal primes $\mathfrak q \subset R_{\mathfrak p}$.

Proof. Follows from Lemma 10.26.1 and the description of $\mathop{\mathrm{Spec}}(R_\mathfrak p)$ in Lemma 10.17.5 which shows that $\mathop{\mathrm{Spec}}(R_\mathfrak p)$ corresponds to primes $\mathfrak q$ in $R$ with $\mathfrak q \subset \mathfrak p$. $\square$

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