English
There is a canonical homomorphism from RingAut R to RingAut (IsLocalRing.ResidueField R) given by mapping an automorphism to its induced residue-field automorphism; this defines a group homomorphism mapAut with toFun = mapEquiv, map_mul' = mapEquiv_trans, and map_one' = mapEquiv_refl.
Русский
Существует каноническое гомоморфизм from RingAut R к RingAut κ(R), задаваемый отображением авто на автоморфизм поля остатка; это образует гомоморфизм группы mapAut с toFun = mapEquiv, map_mul' = mapEquiv_trans, map_one' = mapEquiv_refl.
LaTeX
$$$ \\text{mapAut} : \\mathrm{RingAut}(R) \\to^* \\mathrm{RingAut}(\\kappa(R)), \\; toFun = \\operatorname{mapEquiv}, \\; map_mul' e_1 e_2 = mapEquiv\\_trans\\ e_2\\ e_1, \\; map\_one' = mapEquiv\\_refl $$$
Lean4
/-- The group homomorphism from `RingAut R` to `RingAut k` where `k`
is the residue field of `R`. -/
@[simps]
noncomputable def mapAut : RingAut R →* RingAut (IsLocalRing.ResidueField R)
where
toFun := mapEquiv
map_mul' e₁ e₂ := mapEquiv_trans e₂ e₁
map_one' := mapEquiv_refl
