functionField_isFractionRing_of_affine

English

For P : Fin 3 → R and u ∈ R, W'.dblZ (u • P) = u^4 · W'.dblZ P.

Русский

Для P : Fin 3 → R и u ∈ R выполняется W'.dblZ (u • P) = u^4 · W'.dblZ P.

LaTeX

$$W'.dblZ (u \cdot P) = u^4 \cdot W'.dblZ P$$

Lean4

instance functionField_isFractionRing_of_affine (R : CommRingCat.{u}) [IsDomain R] :
    IsFractionRing R (Spec R).functionField :=
  by
  convert StructureSheaf.IsLocalization.to_stalk R (genericPoint (Spec R))
  delta IsFractionRing IsLocalization.AtPrime
  apply Eq.to_iff
  congr 1
  rw [genericPoint_eq_bot_of_affine]
  ext
  exact mem_nonZeroDivisors_iff_ne_zero

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