Page 61 - Textos de Matemática Vol. 45

5.3. The homogeneous split extension classifier 53 For that, let us consider the following pullback of split epimorphisms, where i
is the inclusion Aut(X) → End(X):
X // ¯ιX
// Hol(X) oo ρ¯X // Aut(X) θ¯X
//
Since Aut(X) is a group, the split epimorphism is homogeneous.
X
ιX
Hol(X) // End(X) θX
¯i i oo ρX
//
The upper split epimorphism is a Schreier one, because so is the lower one.
Theorem 5.3.1. For any homogeneous split epimorphism with kernel X: Aoo s //B
f
there exists a unique morphism φ: B → Aut(X) such that the following diagram is commutative and its right hand side part is a pullback of split epimorphisms:
X oo X
qf // A oo s // B kf
φ¯ φ // ooρ¯X
¯ιX
Hol(X)
θ¯X
// Aut(X).
The explicit definition of φ is φ(b)(x) = qf (s(b) · x), where qf is the Schreier retraction. The pair (k, s) being jointly strongly epimorphic, the homomorphism φ¯ is completely determined by the commutativity of this diagram and we have: φ¯(a) = (φf (a), qf (a)).
Proof. A homogeneous split epimorphism being a Schreier one, it is enough to prove that a Schreier split epimorphism (A,B,f,s) is a homogeneous one if and only if its classifying homomorphism φ: B → End(X) factors through Aut(X). If φ factors through Aut(X), as a pullback of a homogeneous split epimorphism, (A, B, f, s) is homogeneous. Conversely, suppose that (A, B, f, s) is homogeneous. Let us show that any endomorphism φ(b) is an automorphism. Suppose qf (s(b) · x) = qf (s(b) · x′), then we have
s(b)·x = qf (s(b)·x)·sf(s(b)·x) = qf (s(b)·x)·s(b) = qf (s(b)·x′)·s(b) = s(b)·x′.
Since the split epimorphism is left homogeneous, we have x = x′ and φ(b) is injective. Let z be in X. Since the split epimorphism is left homogeneous, there existst∈X suchthats(b)·t=z·s(b).Then
φ(b)(t) = qf (s(b) · t) = qf (z · s(b)) = z
since z ∈ X = K[f], and consequently φ(b) is surjective.