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The Stacks project

Lemma 10.47.1. Let R \to S be a ring map. Assume

  1. \mathop{\mathrm{Spec}}(R) is irreducible,

  2. R \to S is flat,

  3. R \to S is of finite presentation,

  4. 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”.

Proof. The assumptions (b) and (c) imply that the map on spectra is open, see Proposition 10.41.8. Hence the lemma follows from Topology, Lemma 5.8.14. \square


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