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DeclareRepresentation("IsCObjectCompRep",
IsComponentObjectRep, ["G","K","M","e","d","L","R","B","H","P"]);
CObject:=NewType(NewFamily("Cohomology Object"),
IsCObject and IsCObjectCompRep);
# The components of a CObject are the following.
# G the group
# p the PrimePGroup of G
# K the field
# M the KG-module M in Ext(M,N)
# e the augmentation P_0 -> K
# d the boundary maps P_i -> P_{i-1}
# -- these could arguably be thrown away
# L the full and minimal lists of matrices
# giving the action of G on KG
# R the boundary maps in "non-lifted" form
# B the Betti numbers, that is, the dimensions
# of the groups Ext^i(K,K)=H^i(G)
# H the chain maps corresponding with the cohomology
# generators -- these could arguably be thrown away
# P for internal usage in calculating generators
InstallMethod(CohomologyObject,
"Creates a cohomology object",[IsGroup,IsRecord],
function(G,M)
local K,p;
if not MTX.IsMTXModule(M) then
Error("M should be a MeatAxe module\n");fi;
if not IsFinite(G) or not IsPGroup(G) or IsTrivial(G) then
Error("G should be a finte, nontrivial p-group\n");fi;
if IsEmpty(MTX.Generators(M)) then
Error("M should be a module over some nontrivial group\n");fi;
K:=MTX.Field(M);
p:=PrimePGroup(G);
if not p=Characteristic(K) then
Error("PrimePGroup(G) should equal Characteristic(K)\n");fi;
return Objectify(CObject, rec(G:=G,p:=p,K:=K,M:=M));
end
);
InstallOtherMethod(CohomologyObject,
"Creates a cohomology object",[IsGroup],
function(G)
local p,K;
if not IsFinite(G) or not IsPGroup(G) or IsTrivial(G) then
Error("G should be a finte, nontrivial p-group\n");fi;
p:=PrimePGroup(G);
K:=GF(p);
return CohomologyObject(G,
GModuleByMats(List(Pcgs(G),x->[[One(K)]]),K)
);
# Have to call GModuleByMats because TrivialGModule
# uses GeneratorsOfGroup instead of Pcgs.
end
);
# Called as ProjectiveResolution(C,n), the following function computes
# P_n -> P_{n-1} -> ... -> P_1 -> P_0 -> M -> 0.
InstallMethod(ProjectiveResolution,"Computes a minimal projective resolution",
[IsCObject,IsPosInt],
function(C,n)
local p,d;
if not IsBound(C!.e) then C!.e:=FirstStep(C!.M,C!.p);fi;
if not IsBound(C!.B) then C!.B:=[];fi;
if not IsBound(C!.B[1]) then C!.B[1]:=Size(C!.e)/Size(C!.G);fi;
if not IsBound(C!.L) then C!.L:=PCGens(C!.G,C!.K);fi;
if not IsBound(C!.R) then C!.R:=[];fi;
if not IsBound(C!.d) then C!.d:=[];fi;
if not IsBound(C!.R[1]) then
C!.R[1]:=GeneratorsOfRadical(C!.e,C!.L[2]);
C!.d[1]:=LiftHom(C!.L[1],C!.R[1],C!.p);
C!.B[2]:=Size(C!.R[1]);
fi;
for d in [Size(C!.R)+1..n] do
C!.R[d]:=GeneratorsOfRadical(C!.d[d-1],C!.L[2]);
C!.d[d]:=LiftHom(C!.L[1],C!.R[d],C!.p);
C!.B[d+1]:=Size(C!.R[d]);
od;
return C!.B;
end
);
InstallMethod(BoundaryMap,"Returns a boundary map",
[IsCObject,IsInt],
function(C,n)
if n<0 then Error("n should be nonnegative\n");
elif n=0 then ProjectiveResolution(C,1); return C!.e;
else ProjectiveResolution(C,n); return C!.R[n];
fi;
end
);
[ Dauer der Verarbeitung: 0.24 Sekunden
(vorverarbeitet)
]
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