Page 100 - Textos de Matemática Vol. 45

92 Chapter 7. Special Schreier and special homogeneous surjections
Proof. According to Lemma 7.6.2, given any pair xR[f]x′ and a ∈ A, we have qf (a · xR[f]a · x′) = a · qf (xR[f]x′) = 0 since the ring A is trivial. Hence a·x=a·x′,sincethekernelofqf (inMon)isthemorphisms0.
Given a special Schreier surjection f, the kernel K[f] is trivial if and only if the kernel inclusion kf : K[f] X cooperates with itself, and, according to Proposition 6.4.5, this happens if and only if the Schreier equivalence relation R[f] is such that [R[f],R[f]] = 0. When it is the case, consider the associated double centralizing relation and its levelwise horizontal quotients:
R[f]×X R[f]oo p1 ////R[f] f¯ ////Q
OO (d0p0,p) OO (p,d1p1) d0 s0 d1
OO ψ σ
p0
R[f] //XfY
oo d1 // d0
¯¯
(ψ,σ) such that ψf = fd0 = fd1 and σf = fs0. Moreover, since any of the
left hand side square is a pullback, by the Barr-Kock Theorem (Theorem 9.2.2 in the Appendix) the right hand side one is a pullback as well, and the split epimorphism is a special Schreier one since so is (d0,s0).
Actually the ring Q can be described in the following way: we noticed that A is canonically endowed with a Y -bimodule structure; the split epimor- phism (ψ, σ) is nothing but the canonical split epimorphism associated with this structure. Namely, the semiring Q is the semidirect product (A Y ), where the multiplication is defined by (a,y) · (a′,y′) = (x · a′ + a · x′,y · y′) for any xandx′ suchthatf(x)=yandf(x′)=y′ (a·a′ =0,sinceAistrivial).We call this split special Schreier epimorphism the direction of the special Schreier epimorphism f.
As in Mon, the previous construction of the direction gives rise to a functorial construction:
D : T rSExtY (SRng) → AbSP tY (SRng)
from the category of special Schreier extensions with codomain Y and triv- ial kernel to the abelian category AbSPtY (SRng) of internal abelian groups in the fiber SPtY (SRng) (which are nothing but Schreier split epimorphisms whose kernel is a trivial ring). This direction functor D preserves and reflects monomorphisms, surjective homomorphims and consequently isomorphisms. Again as in Mon, the direction functor is cofibrant above the surjective homo- morphims:
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The universal property of the quotients produces a unique split epimorphism