Page 79 - Textos de Matemática Vol. 45

6.5. Schreier accessibility 71
such that l = fk, rf = gp and fs = tg.
Schreier split extensions with fixed kernel X form a category, which we
will denote by SSplExtSRng(X), or simply by SSplExt(X).
Remark. Since the pair (k, s) is jointly strongly epimorphic, the morphism
f in (6.5.4) is uniquely determined by g.
Definition 6.5.1. An object in SSplExt(X) is said to be faithful if any object
in SSplExt(X) admits at most one morphism into it.
Definition 6.5.2. An object in SSplExt(X) is said to be accessible if it admits a morphism into a faithful object. We call that unique possible morphism an index of the Schreier split extension in question.
Proposition 6.5.3. Given a Schreier split extension
// oo q′ r // //
0 X //Coo D 0 lt
in SRng, the following conditions are equivalent: (i) the split extension is faithful;
(ii) if d1,d2 ∈ D are such that t(d1) · l(x) = t(d2) · l(x) and l(x) · t(d1) = l(x)·t(d2) for every x∈X, then d1 =d2.
Proof. Let us first prove that (ii) implies (i). Given another Schreier split ex- tension
0 //Xoo q //Aoo p //B //0,
k
s
suppose we have two morphisms (f,g) and (f′,g′) of Schreier split extensions, as in the following commutative diagram
0 // X oo q // A oo p // B // 0
s
k
1X ff′gg′
// oo q′ r //
0 X //Coo D 0.
lt
It suffices to prove that g = g′. For any b ∈ B and any x ∈ X, we have that s(b) · k(x) ∈ k(X), hence there exists y ∈ X such that s(b) · k(x) = k(y). So we have that
tg(b) · l(x) = fs(b) · fk(x) = f(s(b) · k(x)) = fk(y) = l(y) = f′k(y) = f′(s(b) · k(x)) = f′s(b) · f′k(x) = tg′(b) · l(x),
//