English
Same as 37919: The map pointsPi is injective for quasi-separated X.
Русский
Повторение: pointsPi инъективно при квазиразделимой X.
LaTeX
$$Injectivity of pointsPi: If X is quasi-separated, then pointsPi R X is injective.$$
Lean4
theorem pointsPi_injective [QuasiSeparatedSpace X] : Function.Injective (pointsPi R X) :=
by
rintro f g e
have :=
isIso_of_comp_eq_sigmaSpec R (V := equalizer f g)
(equalizer.lift (sigmaSpec R) (by ext1 i; simpa using congr_fun e i)) (equalizer.ι f g) (by simp)
rw [← cancel_epi (equalizer.ι f g), equalizer.condition]