The Stacks project

Proof. We will use Lemma 35.15.2. First we note that $\mathcal{P}$ is local in the Zariski topology. Next, we show that if $T \to S$ is a flat, finitely presented morphism of affines and $S$ has a finite number of irreducible components, then so does $T$. Namely, since $T \to S$ is flat, the generic points of $T$ map to the generic points of $S$, see Morphisms, Lemma 29.25.9. Hence it suffices to show that for $s \in S$ the fibre $T_ s$ has a finite number of generic points. Note that $T_ s$ is an affine scheme of finite type over $\kappa (s)$, see Morphisms, Lemma 29.15.4. Hence $T_ s$ is Noetherian and has a finite number of irreducible components (Morphisms, Lemma 29.15.6 and Properties, Lemma 28.5.7). Finally, we have to show that if $T \to S$ is a surjective flat, finitely presented morphism of affines and $T$ has a finite number of irreducible components, then so does $S$. This follows from Topology, Lemma 5.8.5. Thus (1), (2) and (3) of Lemma 35.15.2 hold and we win. $\square$