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Let $\mathcal X$ be a scheme of finite type over $\mathbb Z$. We know that the number of geometrically irreducible components (of geometric fibers) is the constant at almost everywhere, by EGA. It is not true for the number of irreducible components. For example $\mathbb Z[x]/(x^2+1)$ is split and nonsplit for infinitely many primes.

However, is it true that the number of irreducible components of $\mathcal X_{\mathbb F_p}$ is the same as $\mathcal X_{\mathbb Q_p}$ for almost all $p$? For this example, factorizing mod $p$ is the same as factorizing over $\mathbb Q_p$ for almost all prime (excluding the bad primes) due to Hensel's lemma. What about higher dimensional case?

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    $\begingroup$ Yes, you can reduce to the 0-dimensional case using a construction of Romagny link.springer.com/article/10.1007/s00229-010-0424-7. $\endgroup$ Commented Nov 25 at 10:41
  • $\begingroup$ @TimSantens Thank you very much for the reference. My french is not very good. Could you tell me very briefly which part is the most relevant? $\endgroup$ Commented Nov 25 at 10:58
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    $\begingroup$ Given a scheme $X$ of finite presentation over a scheme $S$ he constructs an étale sheaf $\mathrm{Irr}(X/S)$ such that for any geometric point $s \in S$ the stalk $\mathrm{Irr}(X/S)_s$ consists of the irreducible components of $X_s$. If the maximal points are dense in $S$, such as for $\mathrm{Spec} \mathbb{Z}$, then he shows in Lemma 2.1.3 that this sheaf is represented by a quasi-compact étale algebraic space. This algebraic space is quasi-separated by Lemma 2.1.2 and thus has a dense open subscheme by stacks.math.columbia.edu/tag/03JI. This is a 0-dimensional family. $\endgroup$ Commented Nov 25 at 11:26
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    $\begingroup$ More generally the paper really has a lot of results about connected or irreducible components in families of varieties. $\endgroup$ Commented Nov 25 at 11:32

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