AboutHap

About HAP: Tor and Ext of modules over a mod p group ring

Let FG be the group algebra of a finite group over the field F of p elements, and let M be an FG-module with G a p-group. The abelian groups

Tor_n^FG(M,F)

and

Extⁿ_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=Syl₂(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:=SylowSubgroup(GL(3,2),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

Tor_n^FG(DM,F) = Tor_n+1^FG(M,F)

and

Extⁿ_FG(DM,F) = Extⁿ⁺¹_FG(M,F)

gap> DM:=DesuspensionMtxModule(M);;

The following commands now compute the 6-dimensional vector spaces

Tor₅^FG(M,F) = Tor₄^FG(DM,F) = F⁶

and

Ext⁵_FG(M,F) = Ext⁴_FG(DM,F) = F⁶ .

gap> R:=ResolutionFpGModule(DM,5);;
gap> p:=2;;
gap> Homology(TensorWithIntegersModP(R,p),4);
6
gap> Cohomology(HomToIntegersModP(R,p),4);
6

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