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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. |
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gap>
G:=GL(3,2);; gap> M:=GModuleByMats(GeneratorsOfGroup(G),GF(2));; |
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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)
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gap>
DM:=DesuspensionMtxModule(M);; |
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The
following commands now compute the 2-dimensional vector spaces Tor5FG(M,F) = Tor4FG(DM,F) = F2 Ext5FG(M,F)
= Ext4FG(DM,F) = F2 .
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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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