Page 101 - Textos de Matemática Vol. 45

7.7. Special Schreier extensions with trivial kernel 93
Theorem 7.7.3. Given any special Schreier extension with trivial kernel A en- dowed with its canonical structure of Y -bimodule, and any surjective homo- morphism h : A A′ of Y -bimodules, there is a special Schreier extension with trivial kernel A′ and direction given by the Y -bimodule structure on A′, called direct image of the special Schreier extension along h, together with a morphism h¯ which makes the following diagram commutative:
A//kf //Xf////Y
h A′ //
kf′
h¯
//X′ ////Y
f′
Given any morphism of special Schreier extensions:
A//kf //Xf////Y
K(l)

morphism, there is a unique morphism of special Schreier extensions:
l

//X′′ ////Y
f′′
with a factorization mh = K(l), where m: A′ → A′′ is a Y -bimodule homo-
A′′ //
kf′′
A′//kf′ //X′ f′ ////Y
mμ
A′′ //
kf′′
//X′′
f′′
////Y
such that μh¯ = l.
Proof. The Y -bimodule structure on A is given by a · y = qf (xR[f ](a · x)) and y·a=qf(xR[f](x·a))foranyx∈X suchthatf(x)=y.ThefactthathisaY- bimodules homomorphism means that we have hqf (xR[f ](a · x)) = h(a) · f (x) and hqf (xR[f ](x · a)) = f (x) · h(a). Consider now the following pullback in SRng:
j
S f∗//// K[h] Y kh 1Y

R[f] f¯ A Y
// //