English
For morphisms f: X → Y and a point x ∈ X, the residue map through Y followed by the residue field map equals the stalk map composed with the residue map at x in X.
Русский
Для гомоморфизмов f: X → Y и точки x ∈ X, композиция остаточных гомоморфизмов через Y и резидуума равна композиции stalkMap и residuе в X.
LaTeX
$$$ \\operatorname{ofHom}(\\ IsLocalRing.\\residue}(Y.presheaf.stalk(f. base x))) \\;\\succ \\; residueFieldMap f x = f.stalkMap x \\;\\succ \\; \\operatorname{ofHom}(\\ IsLocalRing.\\residue}(X.presheaf.stalk x)) $$$
Lean4
theorem residue_comp_residueFieldMap_eq_stalkMap_comp_residue (x : X) :
CommRingCat.ofHom (IsLocalRing.residue (Y.presheaf.stalk (f.base x))) ≫ residueFieldMap f x =
f.stalkMap x ≫ CommRingCat.ofHom (IsLocalRing.residue (X.presheaf.stalk x)) :=
by
simp [residueFieldMap]
rfl
