MAGMA Computational Algebra System

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 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 CM. An error occurs if CM was defined by an action on a finitely generated abelian group.

Invariants(CM) : ModCoho -> SeqEnum

Given a cohomology module CM that was defined by an action on a finitely generated abelian group A, return the invariants of A. If CM was not defined by an action on an abelian group, an error results.

Dimension(CM) : ModCoho -> RngIntElt

Let CM be a cohomology module. If CM was defined by the action of a group on an R-module M, return the dimension of M. In the case in which 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 CM is defined. If 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 CM.

FPGroup(CM) : ModCoho -> Grp, HomGrp

Given a cohomology module 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 CM on the module of CM.