Abstract
Abstract We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel Φ : M → End ( A ) Inn ( A ) {\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}} . If an abstract kernel factors through SEnd ( A ) Inn ( A ) {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} , where SEnd ( A ) {\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group U ( Z ( A ) ) {U(Z(A))} of invertible elements of the center Z ( A ) {Z(A)} of A, on which M acts via Φ. An abstract kernel Φ : M → SEnd ( A ) Inn ( A ) {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ( A ) Inn ( A ) {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}} ) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel Φ : M → SEnd ( A ) Inn ( A ) {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. Φ : M → Aut ( A ) Inn ( A ) {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}} ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U ( Z ( A ) ) {U(Z(A))} .
Highlights
The classification of extensions is a classical problem in group theory
We show that any regular Schreier extension of a monoid by a monoid induces an abstract kernel
Called abstract kernel of the extension, and he determined conditions on such a homomorphism Φ in order to get the existence of extensions having it as abstract kernel. (The notation for group extensions is borrowed from Mac Lane’s book [16], and it is justified by the fact that we will use the multiplicative notation for the group G and the additive one for the other groups.) Later, Eilenberg and Mac Lane [10] gave an interpretation of such results in terms of cohomology: to an abstract kernel Φ can be associated an element Obs(Φ), called obstruction of the abstract kernel, of the third cohomology group H3(G, Z(A)), where Z(A) is the center of A and the left G-module structure on Z(A) is induced by Φ
Summary
The classification of extensions is a classical problem in group theory. It is well known that extensions with abelian kernel inducing the same action are classified by the 2-dimensional cohomology group. In [27] the Schreier extensions of a monoid M by an M -module A were classified by H2(M, A), the classical second cohomology group of M with coefficients in the M -module A. Homogeneous) Schreier extensions of M by A (see Definition 3.11) which induce the same abstract not empty, is in bijection with the second cohomology group H2(M, U (Z(A))) of M with coefficients in the M -module U (Z(A)). This is done, as for the classical case of extensions of groups, by showing that there is a transitive action of the abelian group H2(M, U (Z(A))) on the set Ext(M, A, Φ).
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