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Let FG be the group algebra of a finite group over the field F of p elements, and let M be an FG-module. The abelian groups TornFG(M,F) and ExtnFG(M,F) can be calculated from a free resolution of M. We illustrate this for the module M arising from the canonical action of the group G=GL3(2) on the 3-dimensional column vector space over GF(2). The module M can be entered as a meat-axe module using the following standard GAP commands.
gap> G:=GL(3,2);; gap> M:=GModuleByMats(GeneratorsOfGroup(G),GF(2));;
The module can be converted to an FpG-module DM using the following command. The "desuspended module" DM is mathematically related to M via a short exact sequence 0 → DM → PM → M → 0 where PM is a free module. Thus TornFG(DM,F) = Torn+1FG(M,F) and ExtnFG(DM,F) = Extn+1FG(M,F)
gap> DM:=DesuspensionMtxModule(M);;
The following commands now compute the 2-dimensional vector spaces Tor5FG(M,F) = Tor4FG(DM,F) = F2 and Ext5FG(M,F) = Ext4FG(DM,F) = F2 .
gap> R:=ResolutionFpGModule(DM,5);; gap> p:=2;; gap> Homology(TensorWithIntegersModP(R,p),4); 2 gap> Cohomology(HomToIntegersModP(R,p),4); 2
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