The Stacks project

Lemma 37.36.1. Let $R = \mathop{\mathrm{colim}}\nolimits R_ i$ be colimit of a directed system of rings whose transition maps are faithfully flat. Then the number of minimal primes of $R$ taken as an element of $\{ 0, 1, 2, \ldots , \infty \} $ is the supremum of the numbers of minimal primes of the $R_ i$.

Proof. If $A \to B$ is a flat ring map, then $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ maps minimal primes to minimal primes by going down (Algebra, Lemma 10.39.19). If $A \to B$ is faithfully flat, then every minimal prime is the image of a minimal prime (by Algebra, Lemma 10.39.16 and 10.30.7). Hence the number of minimal primes of $R_ i$ is $\geq $ the number of minimal primes of $R_{i'}$ if $i \leq i'$. By Algebra, Lemma 10.39.20 each of the maps $R_ i \to R$ is faithfully flat and we also see that the number of minimal primes of $R$ is $\geq $ the number of minimal primes of $R_ i$. Finally, suppose that $\mathfrak q_1, \ldots , \mathfrak q_ n$ are pairwise distinct minimal primes of $R$. Then we can find an $i$ such that $R_ i \cap \mathfrak q_1, \ldots , R_ i \cap \mathfrak q_ n$ are pairwise distinct (as sets and hence as prime ideals). This implies the lemma. $\square$

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