Spec_map_tensorInl_fromSpecResidueField

English

For any Triplet f g, the tensor fiber product T.tensor is nontrivial.

Русский

Для любой троики f g тензорное произведение ненулево.

LaTeX

$$$\forall {X Y S : Scheme}, {f : X ⟶ S}, {g : Y ⟶ S}, (T : Triplet f g) \Rightarrow \text{Nontrivial}(T.tensor)$$

Lean4

theorem Spec_map_tensorInl_fromSpecResidueField :
    (Spec.map T.tensorInl ≫ X.fromSpecResidueField T.x) ≫ f = (Spec.map T.tensorInr ≫ Y.fromSpecResidueField T.y) ≫ g :=
  by
  simp only [residueFieldCongr_inv, Category.assoc, tensorInl, tensorInr,
    ← Hom.Spec_map_residueFieldMap_fromSpecResidueField]
  rw [← residueFieldCongr_fromSpecResidueField T.hx.symm, ← residueFieldCongr_fromSpecResidueField T.hy.symm]
  simp only [← Category.assoc, ← Spec.map_comp, pushout.condition]

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