Page 99 - Textos de Matemática Vol. 45

7.7. Special Schreier extensions with trivial kernel 91 In this case the kernel K[f] of f, being isomorphic to the kernel of the projection
p0 : R[f] → X, is necessarily a ring.
Lemma 7.6.2. Given a special Schreier surjection f, the Schreier retraction qf associated with the Schreier split epimorphism d0 satisfies the following identity: given xR[f]x′ and zR[f]z′, we get
qf((x·zR[f]x′ ·z′) = qf(xR[f]x′)·qf(zR[f]z′)+qf(xR[f]x′)·z+x·qf(zR[f]z′). On the other hand, for any k ∈ K[f] and any pair xR[f]x′, we have
qf ((k · x)R[f](k · x′)) = k · qf (xR[f]x′) qf ((x · k)R[f](x′ · k)) = qf (xR[f]x′) · k.
and similarly
Proof. The first point is just a translation of the point (e) in Proposition 6.0.11.
The second one is obtained from:
k·qf (xR[f]x′)+k·x = k·(qf (xR[f]x′)+x) = k·x′ = qf ((k·x)R[f](k·x′))+k·x,
where the last two equalities come from Lemma 3.1.4. The proof of the second formula is completely analogous.
For the same reasons as in Mon, the following sequence K[f] // k //X f //// Y
is exact in SRng, namely f is the cokernel of its kernel k. In SRng, the special Schreier surjections are stable under products and pullbacks, the pullbacks along surjective homomorphisms reflect the special Schreier surjections, and we have a special Schreier short five lemma.
7.7 Special Schreier extensions with trivial kernel Definition 7.7.1. A special Schreier extension in SRng with trivial kernel is an
exact sequence
A=K[f]// k //X f ////Y
where f is a special Schreier surjection and K[f] is a trivial ring A.
Lemma 7.7.2. Let f : X Y be a special Schreier surjection with trivial kernel A. Given any pair xR[f]x′, we have a·x = a·x′ and x·a = x′ ·a for any a ∈ A. Accordingly the abelian group A is endowed with a canonical structure of Y -bimodule (see Section 6.7) by: a · y = a · x and y · a = x · a, for any x such that f(x) = y.