mapPiLocalization_bijective — Mathlib

English

If the PiLocalization map is surjective, then the index set is finite.

Русский

Если отображение PiLocalization сюръективно, то множество индексов конечно.

LaTeX

$$$\text{If } \mathrm{toPiLocalization}((R_i)_{i\in I}) \ \text{ is surjective, then } |I| < \infty$$$

Lean4

theorem mapPiLocalization_bijective : Function.Bijective (mapPiLocalization f hf) :=
  by
  let f := RingEquiv.ofBijective f hf
  let e := RingEquiv.ofRingHom (mapPiLocalization f hf) (mapPiLocalization (f.symm : S →+* R) f.symm.bijective) ?_ ?_
  · exact e.bijective
  · rw [← mapPiLocalization_comp]
    simp_rw [RingEquiv.comp_symm, mapPiLocalization_id]
  · rw [← mapPiLocalization_comp]
    simp_rw [RingEquiv.symm_comp, mapPiLocalization_id]

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