


Let
FG be the group algebra of a finite group over the field F of p
elements, and let M be an FGmodule. The abelian groups Tor_{n}^{FG}(M,F)
and Ext^{n}_{FG}(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=GL_{3}(2) on the 3dimensional column vector space over GF(2). The module M can be entered as a meataxe 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 FpGmodule 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 Tor_{n}^{FG}(DM,F)
= Tor_{n+1}^{FG}(M,F)
and Ext^{n}_{FG}(DM,F)
= Ext^{n+1}_{FG}(M,F)


gap>
DM:=DesuspensionMtxModule(M);; 

The
following commands now compute the 2dimensional vector spaces Tor_{5}^{FG}(M,F) = Tor_{4}^{FG}(DM,F) = F^{2} Ext^{5}_{FG}(M,F)
= Ext^{4}_{FG}(DM,F) = F^{2} .


gap>
R:=ResolutionFpGModule(DM,5);; gap> p:=2;; gap> Homology(TensorWithIntegersModP(R,p),4); 2 gap> Cohomology(HomToIntegersModP(R,p),4); 2 

