Page 21 - Textos de Matemática Vol. 45

2.1. Definitions and first properties 13
(b) qs = 0;
(c) q(1) = 1;
(d) if b∈B and α∈K[f], then q(s(b)·α)·s(b)=s(b)·α; (e) for every a,a′ ∈A q(a·a′)=q(a)·q(sf(a)·q(a′)).
Proof. (a) it is a straighforward consequence of the second identity in Propo- sition 2.1.4.
(b) forb∈Bwehave:
s(b) = 1 · sf(s(b))
and the uniqueness of q gives that qs(b) = 1 for every b ∈ B.
(c) obviously we have 1 = 1 · sf (1).
(d) foranyb∈Bandanyα∈K[f]wehave:
s(b) · α = q(s(b) · α) · sf(s(b) · α) =
= q(s(b) · α) · sfs(b) · sf(α) = q(s(b) · α) · s(b).
(e) q(a · a′) is the unique element of K[f] such that
a·a′ =q(a·a′)·sf(a·a′)=q(a·a′)·sf(a)·sf(a′),
so it suffices to prove that
q(a) · q(sf(a) · q(a′)) · sf(a) · sf(a′) = a · a′.
By point (d), we have that
q(sf(a) · q(a′)) · sf(a) = sf(a) · q(a′)
and hence
q(a) · q(sf(a) · q(a′)) · sf(a) · sf(a′) = q(a) · sf(a) · q(a′) · sf(a′) = a · a′.

We end this section with two important observations:
Lemma 2.1.6. A Schreier split epimorphism (and, a fortiori, a homogeneous split epimorphism) is a strongly split epimorphism.
Proof. Given a Schreier split epimorphism, the formula a = q(a) · sf (a) proves that A is the supremum of the subobjects k: K[f] A and s: B A.