residueFieldMap_comp — Mathlib

English

For morphisms f: X → Y and g: Y → Z, the residue field map along the composite equals the composition of the residue maps along g and f.

Русский

Для гомоморфизмов f: X → Y и g: Y → Z резидуума-перевод по композиции равен композиции резидуума по g и f.

LaTeX

$$$ \\mathrm{residueFieldMap}(f \\circ g)\\; x = \\mathrm{residueFieldMap}(g)(f.base x) \\circ \\mathrm{residueFieldMap}(f)\\; x $$$

Lean4

@[simp]
theorem residueFieldMap_comp {Z : LocallyRingedSpace.{u}} (g : Y ⟶ Z) (x : X) :
    residueFieldMap (f ≫ g) x = residueFieldMap g (f.base x) ≫ residueFieldMap f x :=
  by
  ext : 1
  simp only [comp_toShHom, SheafedSpace.comp_base, residueFieldMap, CommRingCat.hom_comp, TopCat.comp_app]
  simp_rw [stalkMap_comp]
  apply IsLocalRing.ResidueField.map_comp (Hom.stalkMap g (f.base x)).hom (Hom.stalkMap f x).hom

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