Page 27 - Textos de Matemática Vol. 45

2.3. First properties of Schreier and homogeneous split epimorphisms 19 Proposition 2.3.4. Schreier (resp. homogeneous) split epimorphisms are stable
under pullbacks along any morphism.
Proof. Consider the following diagram, where the lower row is a Schreier split
sequence and the right-hand side square is a pullback:
oo q′
πY
≃ πA h
ooq oos K[f] // A // // B.
kf
The map q′ defined by q′(a, y) = (q(a), 1) satisfies the conditions of Proposition 2.1.4 since:
(a, y) = (q(a), 1) · (sf (a), y) = (q(a), 1) · (sh(y), y)
for any (a, y) ∈ A×B Y . Moreover, the elements of K[πY ] are of the form (α, 1),
with α ∈ K[f], and then:
q′((α, 1) · (sh(y), y)) = q′(α · sh(y), y) = (q(α · sh(y)), 1) = (α, 1)
Accordingly the upper row is a Schreier split epimorphism. So the right ho- mogeneous split epimorphisms are stable under pullbacks. Dually, the same holds for the left homogeneous split epimorphisms, and consequently for the homogeneous ones.
Proposition 2.3.5. Consider the lower right hand side vertical morphism of split epimorphisms:
R[K(g)] //oo qR // R[g] oo R(s′) // R[h] OO R(kf′) OO R(f′) OO
K[πY ]
// A ×B Y
⟨oosh,1Y ⟩
// // Y
⟨k,0⟩
p0 p1 oo
K[f′] //
p0 p1
qf′ oo s′
//A′
kf′ f′
p0 p1 //B′
g
K[f] A B
K(g)
h // oos //
//kf
UU f
qf
Complete it with the horizontal kernels and the vertical kernel equivalence rela- tions. Commutativity of limits makes the upper row a kernel diagram. Suppose that g is a regular epimorphism (= a surjective homomorphism). Suppose that moreover K(g) is a regular epimorphism. If the two upper split epimorphisms are Schreier (resp. homogeneous) ones, so is the lower one.