English
There is a natural compatibility between linear maps, tensoring, and quotients expressed via a sequence of canonical isomorphisms of tensor-quotient constructions.
Русский
Существует естественная совместимость линейных отображений, тензорирования и факторизации через последовательность канонических изоморфизмов конструкций тензор-фактор.
LaTeX
$$$$ \text{There are canonical isomorphisms } (M \otimes_R N)/ (M \otimes_R n) \cong M \otimes_R (N/n). $$$$
Lean4
theorem rTensor_range : range (rTensor Q g) = range (rTensor Q (Submodule.subtype (range g))) :=
by
have : g = (Submodule.subtype _).comp g.rangeRestrict := rfl
nth_rewrite 1 [this]
rw [rTensor_comp]
apply range_comp_of_range_eq_top
rw [range_eq_top]
apply rTensor_surjective
rw [← range_eq_top, range_rangeRestrict]
