Given a 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 Ext_A^ * (S, S).
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 Ext_A^ * (S, T) as a right module over the cohomology ring Ext_A^ * (T, T).
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 Ext_A^ * (S, T) as a left module over the cohomology ring Ext_A^ * (S, S).
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 n^(th) 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 n^(th) generator of the cohomology ring and has degree equal to the degree of that generator.
> gg := ExtraSpecialGroup(3,1);
> gg;
Permutation group gg acting on a set of cardinality 27
(1, 19, 10)(2, 20, 11)(3, 21, 12)(4, 22, 13)(5, 23, 14)(6, 24, 15)(7, 25,
16)(8, 26, 17)(9, 27, 18)
(1, 7, 4)(2, 8, 5)(3, 9, 6)(10, 18, 14)(11, 16, 15)(12, 17, 13)(19, 26,
24)(20, 27, 22)(21, 25, 23)
(1, 3, 2)(4, 6, 5)(7, 9, 8)(10, 12, 11)(13, 15, 14)(16, 18, 17)(19, 21,
20)(22, 24, 23)(25, 27, 26)
The group cs{gg} has order 27.
This will also be the dimension of its basic algebra.
> A := BasicAlgebra(gg,GF(3)); > A; Basic algebra of dimension 27 over GF(3) Number of projective modules: 1 Number of generators: 4 > s := SimpleModule(A,1); > prj := CompactProjectiveResolution(s,12); > SimpleHomologyDimensions(s); [ 16, 15, 14, 12, 10, 9, 8, 7, 6, 4, 2, 1 ]Here are the ranks as free modules over the group algebra of the terms of the projective resolution of the trivial module cs{s}.
> chm := CohomologyRingGenerators(prj); > DegreesOfCohomologyGenerators(chm); [ 1, 1, 2, 2, 2, 2, 3, 3, 6 ]These are the degrees of the minimal generators of the cohomology ring.
> 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 cohomolgy 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 5We can compose this with itself.
> gen113*gen113; Basic algebra chain map of degree 10Now 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 to be certain things make sense.
> IsChainMap(theta); true > IsChainMap(gamma); true[Next][Prev] [Right] [Left] [Up] [Index] [Root]