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Given a compact projective resolution P for a
simple module S over a basic
algebra A, the function returns the chain maps
in compact form of a minimal set of
generators for the cohomology ExtA^ * (S, S),
as well as some other information. The record that is
returned has the following fields:
- (a)
- The list of maps in compact form for the chain map
of the generators
(field name ChainMapRecord).
- (b)
- The sequence of degrees of cohomology generators
(field name ChainDegrees).
- (c)
- The tops of the chain maps (maps on modules modulo
radicals) for the purposes of computing products (field name
TopsOfCohomologyGenerators).
- (d)
- The tops of the chain maps representing monomials
in the generators (field name
TopsOfCohomologyChainMaps).
- (e)
- The original compact projective resolution (field name
ProjectiveResolution).
Given projective resolutions P and Q for simple modules
S and T over a basic algebra A and the cohomology
generators CQ for T associated to the resolution
Q, the function returns the chain maps
in compact form of the minimal generators for the
cohomology ExtA^ * (S, T) as a right module over the cohomology ring
ExtA^ * (T, T). The function returns a record consisting of
the following fields.
- (a)
- The list of maps in compact form for the chain map of
each cohomology generator
(field name ChainMapRecord).
- (b)
- The sequence of degrees of cohomology generators
(field name ChainDegrees).
- (c)
- The tops of the chain maps (maps on modules module
radicals) for the purposes of computing products (field name
TopsOfCohomologyGenerators).
Given projective resolutions P and Q for simple
modules S and T over a basic
algebra A and the cohomology generators CP for T
associated to the resolution
Q, the function returns the chain maps
in compact form of the minimal generators for the
cohomology ExtA^ * (S, T) as a left module over
the cohomology ring ExtA^ * (S, S).The
function returns a record consisting of the following fields.
- (a)
- The list of maps in compact form for the chain map
of each cohomology generator
(field name ChainMapRecord).
- (b)
- The sequence of degrees of cohomology generators
(field name ChainDegrees).
- (c)
- The tops of the chain maps (maps on modules module
radicals) for the purposes of computing products (field name
TopsOfCohomologyGenerators).
Given the generators C for cohomology, as either module generators or
as ring generators, the function returns the list
of degrees of the minimal generators.
Given the projective resolutions P and Q of two modules M and N and
the cohomology generators C of the cohomology module, ( Ext)B^ * (M, N), the
function returns the chain map from P to Q that lifts the nth generator
of the cohomology module and has degree equal to the degree of that
generator.
Given the projective resolution P of a module and the cohomology
generators C of the cohomology ring of that module, the function returns
the chain map from P to P that lifts the nth generator of the
cohomology ring and has degree equal to the degree of that generator.
We create the Basic algebra for the principal block of the sporadic simple
group M11 in characteristic 2. The block algebra has three simple modules
of dimension 1, 44, and 10. The basic algebra has dimension 22.
> ff := GF(2);
> VV8 := VectorSpace(ff,8);
> BB8 := Basis(VV8);
> MM8 := MatrixAlgebra(ff,8);
> e11 := MM8!0;
> e12 := MM8!0;
> e13 := MM8!0;
> e11[1] := BB8[1];
> e11[4] := BB8[4];
> e11[5] := BB8[5];
> e11[8] := BB8[8];
> e12[2] := BB8[2];
> e12[7] := BB8[7];
> e13[3] := BB8[3];
> e13[6] := BB8[6];
> a1 := MM8!0;
> b1 := MM8!0;
> c1 := MM8!0;
> d1 := MM8!0;
> e1 := MM8!0;
> f1 := MM8!0;
> a1[1] := BB8[2];
> a1[5] := BB8[7];
> b1[1] := BB8[3];
> b1[4] := BB8[6];
> c1[2] := BB8[4];
> c1[7] := BB8[8];
> e1[3] := BB8[5];
> e1[6] := BB8[8];
> f1[3] := BB8[6];
> A1 := sub< MM8 | [e11, e12, e13, a1, b1, c1, d1, e1, f1] >;
> T1 := [ <1,1>,<1,4>,<1,5>,<2,6>,<3,8>,<4,5>,<5,4>,<6,8>];
> VV6 := VectorSpace(ff,6);
> BB6 := Basis(VV6);
> MM6 := MatrixAlgebra(ff,6);
> e21 := MM6!0;
> e22 := MM6!0;
> e23 := MM6!0;
> e22[1] := BB6[1];
> e22[5] := BB6[5];
> e22[6] := BB6[6];
> e21[2] := BB6[2];
> e21[4] := BB6[4];
> e23[3] := BB6[3];
> a2 := MM6!0;
> b2 := MM6!0;
> c2 := MM6!0;
> d2 := MM6!0;
> e2 := MM6!0;
> f2 := MM6!0;
> a2[4] := BB6[6];
> b2[2] := BB6[3];
> c2[1] := BB6[2];
> d2[1] := BB6[5];
> d2[5] := BB6[6];
> e2[3] := BB6[4];
> A2 := sub< MM6 | [e21, e22, e23, a2, b2, c2, d2, e2, f2]>;
> T2 := [ <1,2>, <1,6>, <2,5>, <3,8>, <1,7>, <5,7> ];
> VV8 := VectorSpace(ff,8);
> BB8 := Basis(VV8);
> MM8 := MatrixAlgebra(ff,8);
> e31 := MM8!0;
> e32 := MM8!0;
> e33 := MM8!0;
> e31[2] := BB8[2];
> e31[6] := BB8[6];
> e32[4] := BB8[4];
> e33[1] := BB8[1];
> e33[3] := BB8[3];
> e33[5] := BB8[5];
> e33[7] := BB8[7];
> e33[8] := BB8[8];
> a3 := MM8!0;
> b3 := MM8!0;
> c3 := MM8!0;
> d3 := MM8!0;
> e3 := MM8!0;
> f3 := MM8!0;a3[2] := BB8[4];
> b3[6] := BB8[8];
> b3[2] := BB8[7];
> c3[4] := BB8[6];
> e3[1] := BB8[2];
> e3[3] := BB8[6];
> f3[1] := BB8[3];
> f3[3] := BB8[5];
> f3[5] := BB8[7];
> f3[7] := BB8[8];
> A3 := sub< MM8 | [e31, e32, e33, a3, b3, c3, d3, e3, f3] >;
> T3 := [ <1,3>,<1,8>,<1,9>,<2,4>,<3,9>,<4,6>,<5,9>,<6,5>];
>
> m11 := BasicAlgebra( [<A1, T1>, <A2, T2>, <A3, T3>] );
> m11;
Basic algebra of dimension 22 over GF(2)
Number of projective modules: 3
Number of generators: 9
> s1 := SimpleModule(m11,1);
> s2 := SimpleModule(m11,2);
Now we compute the projective resolutions of the first and second simple
modules. Then we find the degrees of their cohomology ring generators.
> prj1 := CompactProjectiveResolution(s1,20);
> prj2 := CompactProjectiveResolution(s2,20);
> CR1 := CohomologyRingGenerators(prj1);
> CR2 := CohomologyRingGenerators(prj2);
> DegreesOfCohomologyGenerators(CR1);
[ 3, 4, 5 ]
> DegreesOfCohomologyGenerators(CR2);
[ 1, 2 ]
Finally we look at the cohomology Ext(cs(s2), cs(s1)) as a
left module over
the cohomology ring of cs{s1} and as a right module over the cohomology
ring of cs{s2}.
> CR12 := CohomologyLeftModuleGenerators(prj1,CR1,prj2);
> DegreesOfCohomologyGenerators(CR12);
[ 1, 2, 3, 4 ]
> CR12 := CohomologyRightModuleGenerators(prj1,prj2,CR2);
> DegreesOfCohomologyGenerators(CR12);
[ 1 ]
So as a module over the cohomology ring of cs{s1} it is generated by 4
elements. But as a module over the cohomology ring of cs{s2} it is generated
by a single element.
Next we get the chain complex for the projective resolution of the first
simple module and the chain map for the third generator of the cohomology ring
of the first simple module.
> pj1 := ProjectiveResolution(s1,20);
> pj1;
Basic algebra complex with terms of degree 20 down to 0
Dimensions of terms: 74 66 68 68 60 54 54 54 48 40 40 42 34 26 28 28 20 14 14
14 8
> gen113 := CohomologyGeneratorToChainMap(pj1,CR1,3);
> gen113;
Basic algebra chain map of degree -5
We can compose this with itself.
> gen113*gen113;
Basic algebra chain map of degree -10
Now compute the kernel and the dimensions of the homology of the kernel.
> Ker, phi := Kernel(gen113);
> Ker, phi;
Basic algebra complex with terms of degree 20 down to 0
Dimensions of terms: 20 15 19 20 20 17 17 20 22 15 17 22 20 15 19 20 20 14 14
14 8
Basic algebra chain map of degree 0
> DimensionsOfHomology(Ker);
[ 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0 ]
Same for the cokernel.
> Cok, mu := Cokernel(gen113);
> Cok, mu;
Basic algebra complex with terms of degree 20 down to 0
Dimensions of terms: 74 66 68 68 60 0 3 5 0 0 3 5 0 0 3 5 0 0 3 5 0
Basic algebra chain map of degree 0
> DimensionsOfHomology(Cok);
[ 0, 0, 0, 27, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2 ]
We can also check the image.
> Imm, theta, gamma := Image(gen113);
> Imm;
Basic algebra complex with terms of degree 20 down to 0
Dimensions of terms: 0 0 0 0 0 54 51 49 48 40 37 37 34 26 25 23 20 14 11 9 8
> DimensionsOfHomology(Imm);
[ 0, 0, 0, 0, 27, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0 ]
We can check that certain things make sense.
> IsChainMap(theta);
true
> IsChainMap(gamma);
true
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