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90 Chapter 7. Special Schreier and special homogeneous surjections
associated with the Y -module structure φ: Y → End(A). Similarly to what happens in the category Gp of groups we are going to show that SExtφ(Y,A) (resp. HExtφ(Y,A)) is endowed with a natural structure of abelian group.
For that, given a pair of classes of special Schreier extensions with abelian kernel A and same direction (A φ Y, Y, pφY , ιφY )
A // kf //X f //// Y A // kf′ //X′ f′ //// Y
we can define the Baer sum of the two previous classes as the class of the direct image of their product in AbSExtY (Mon) (given by their pullback above Y ) along the homomorphism + : A × A → A of Y -modules given by the abelian group operation in A:
kf×kf′ f×Y f′ A×A // //X×Y X′ //// Y
+ +˜
// // ′ ////
A X⊕YX ′Y kf′ f⊕Y f
This operation gives SExtφ(Y, A) (resp. HExtφ(Y, A)) an abelian group struc- ture. The zero of this group is just the isomorphic class of the split extension corresponding to the direction (A φ Y, Y, pφY , ιφY ). The inverse of a special
Schreier (resp. homogeneous) extension in M on: A // k // X f // // Y is given
by: A // −k //X f //// Y .TheproofofthecommutativityoftheBaersumis straightforward (since the pullback is commutative up to isomorphisms), while the associativity requires long and heavy calculations, which are similar to the ones classically done for groups (see, for example, [24] for more details). However, the result follows by general categorical arguments thanks to the properties of the direction functor described in the previous section. We will not describe here the details of these arguments, that can be found in Section VI in [9] and in the paper [11].
7.6 Special Schreier surjections in SRng
In this section, and in the following ones, we shall mimic in the category SRng
what we did in the category Mon.
Definition 7.6.1. A surjective homomorphism f : X Y in SRng will be said a special Schreier one when its kernel equivalence relation R[f] is a Schreier one.