Accessing Properties of the Cohomology Module#

The functions described in this section merely return data used to define the cohomology module. In each case, the argument \({\rm CM}\) must be a cohomology module returned by a call to CohomologyModule.

Module(CM): ModCoho -> ModGrp#: The \(K[G]\)-module used to define the cohomology module \({\rm CM}\). An error occurs if \({\rm CM}\) was defined by an action on a finitely generated abelian group.

Invariants(CM): ModCoho -> SeqEnum#: Given a cohomology module \({\rm CM}\) that was defined by an action on a finitely generated abelian group \(A\), return the invariants of \(A\). If \({\rm CM}\) was not defined by an action on an abelian group, an error results.

Dimension(CM): ModCoho -> RngIntElt#: Let \({\rm CM}\) be a cohomology module. If \({\rm CM}\) was defined by the action of a group on an \(R\)-module \(M\), return the dimension of \(M\). In the case in which \({\rm CM}\) was defined by the action of a group on a finitely generated abelian group \(A\), the rank of \(A\) is returned.

Ring(CM): ModCoho -> ModGrp#: The ring over which the module used to define the cohomology module \({\rm CM}\) is defined. If \({\rm CM}\) is defined in terms of an action on a finitely generated abelian group \(A\), then the ring will be the integers if \(A\) is infinite, and the integers modulo the exponent of \(A\) if \(A\) is finite.

Group(CM): ModCoho -> Grp#: The group used to define action on the cohomology module \({\rm CM}\).

FPGroup(CM): ModCoho -> Grp, HomGrp#: Given a cohomology module \({\rm CM}\) with associated group \(G\), return a finitely presented group \(F\) isomorphic to \(G\) and the isomorphism from \(F\) to \(G\). This presentation is on a strong generating set if \(G\) is a permutation or matrix group. It is used in the construction of presentations of extensions returned by the function Extension.

MatrixOfElement(CM, g): ModCoho, GrpElt -> AlgMatElt#: The matrix representing the action of the element \(g\) in the group of \({\rm CM}\) on the module of \({\rm CM}\).

Example: Coho Module2cont (ex-fdbfad)#

Following on from example Example: Coho Module2 above:

> G:=CyclicGroup(4);
> mats := [ Matrix(Integers(),2,2,[1,2,1,3]) ];
> invar := [2,4];
> CM := CohomologyModule(G,invar,mats);
> Invariants(CM);
[ 2, 4 ]
> FPGroup(CM);
Finitely presented group on 1 generator
Relations
    $.1^4 = Id($)
Mapping from: GrpFP to GrpPerm: G
> MatrixOfElement(CM, G.1^2);
[1 0]
[0 3]

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