English
There is a canonical correspondence sending a point of the tensor Spec back to the corresponding pullback data; this realizes a homeomorphism between the pullback space and the fiber of the Spec tensor.
Русский
Существует каноническое соответствие, отправляющее точку Spec в соответствующие данные вытяжки; оно реализует гомеоморфизм между пространством вытяжки и волокном тензора Spec.
LaTeX
$$$\text{SpecOfPoint}(t) = (T, p)$$$
Lean4
theorem Spec_ofPointTensor_SpecTensorTo (T : Triplet f g) (p : Spec T.tensor) :
Spec.map (Hom.residueFieldMap T.SpecTensorTo p) ≫
Spec.map (ofPointTensor (T.SpecTensorTo.base p)) ≫ Spec.map (tensorCongr (T.ofPoint_SpecTensorTo p).symm).hom =
(Spec T.tensor).fromSpecResidueField p :=
by
apply T.Spec_map_tensor_isPullback.hom_ext
· rw [← cancel_mono <| X.fromSpecResidueField T.x]
simp_rw [Category.assoc, ← T.specTensorTo_fst, tensorCongr_SpecTensorTo_assoc]
rw [← Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc, ofPointTensor_SpecTensorTo_assoc]
· rw [← cancel_mono <| Y.fromSpecResidueField T.y]
simp_rw [Category.assoc, ← T.specTensorTo_snd, tensorCongr_SpecTensorTo_assoc]
rw [← Hom.Spec_map_residueFieldMap_fromSpecResidueField_assoc, ofPointTensor_SpecTensorTo_assoc]
