Lemma 10.47.1. Let R \to S be a ring map. Assume
\mathop{\mathrm{Spec}}(R) is irreducible,
R \to S is flat,
R \to S is of finite presentation,
the fibre rings S \otimes _ R \kappa (\mathfrak p) have irreducible spectra for a dense collection of primes \mathfrak p of R.
Then \mathop{\mathrm{Spec}}(S) is irreducible. This is true more generally with (b) + (c) replaced by “the map \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R) is open”.
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