We create the Basic algebra for the principal block of the sporadic simple
group M
11 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 ]
We apply the same procedure in the case of 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
The ext-algebra of an algebra B is the algebra ExtB * (S, S) for
S = S1 direct-sum ... direct-sum Sn where S1, ..., Sn are all of
the simple B-modules. In the event that the algebra B had finite
global dimension, this is a finite dimensional algebra and we can form
its basic algebra.
The function computes the information on the
ext-algebra B of the basic algebra A
where the projective resolutions and cohomology have been computed to
degree n. The function returns a record carrying the data:
- (i)
- A free algebra F,
- (ii)
- A list of relations in the elements of F, such that the ext-algebra
is the quotient F/I where I is the ideal generated by the relations,
- (iii)
- The sequence of chain maps of the generators. Each chain map goes
from the projective resolution of one simple module to that of another.
- (iv)
- The sequence of degrees of the generators.
- (v)
- A sequence of sequences of sequences of basis element such the
jth element of the ith sequence is a basis of
in Ext * A(Si, Sj) where Si is the ith simple module.
- (vi)
- A basis of the entire ext-algebra, the concatenation of the
previous sequences.
- (vii)
- The number of steps of the cohomology that were computed.
- (viii)
- The global dimension that has been computed. This number is
smaller than the number of steps only in the case that the global
dimension of A is less than n, meaning that the nth step
in the projective resolution of any simple module is the zero module.
The function creates the basic algebra from a computed ext-algebra. The
input is the output of the ExtAlgebra function. If the ext-algebra
is not verified to be finite dimensional by the computation, then an
error is returned.
The function forms the basic algebra from a computed ext-algebra of the
basic algebra A. If no ext-algebra for A has been computed or if the
ext-algebra is not verified to be finite dimensional then an error is
returned.
The function creates the basic algebra for the ext-algebra of A computed
to n steps. If no ext-algebra for A to n steps has been computed then
it computes one. If the ext-algebra is not verified to be finite
dimensional by the computation, then an error is returned.
This function computes the Betti numbers of all of simple A-modules out to
degree n. This is the dimension of the ext-algebra of A computed to
degree n.
We construct the basic algebra of the algebra B of lower triangular
matrices over the field with 5 elements. Note that because the identity
element of B is the sum of primitive idempotents of rank 1, B is
actually isomorphic to its basic algebra.
> A := MatrixAlgebra(GF(5),10);
> U := A!0;
> ElementaryMatrix := function(i,j);
> W := U;
> W[i,j] := 1;
> return W;
> end function;
> S := &cat[[ElementaryMatrix(i,j): i in [j .. 10]]:j in [1 .. 10]];
> B := sub<A|S>;
> B;
Matrix Algebra of degree 10 with 55 generators over GF(5)
> C := BasicAlgebra(B);
> C;
Basic algebra of dimension 55 over GF(5)
Number of projective modules: 10
Number of generators: 19
Note that C has the same dimension as B. The two are isomorphic,
though they have different types.
> SumOfBettiNumbersOfSimpleModules(C,9);
19
> SumOfBettiNumbersOfSimpleModules(C,10);
19
From the above we see that C has global dimension at most 9. Consequently,
we can compute the basic algebra of its ext-algebra.
> D:= ExtAlgebra(C,10);
> E := BasicAlgebraOfExtAlgebra(D);
> E;
Basic algebra of dimension 19 over GF(5)
Number of projective modules: 10
Number of generators: 19
> SumOfBettiNumbersOfSimpleModules(E,8);
54
> SumOfBettiNumbersOfSimpleModules(E,9);
55
> SumOfBettiNumbersOfSimpleModules(E,10);
55
Here we see that E has global dimension 9. So we compute the basic
algebra of its ext-algebra.
> F := BasicAlgebraOfExtAlgebra(E,10);
> F;
Basic algebra of dimension 55 over GF(5)
Number of projective modules: 10
Number of generators: 19
> G := BasicAlgebraOfExtAlgebra(F,10);
> G;
Basic algebra of dimension 19 over GF(5)
Number of projective modules: 10
Number of generators: 19
So it would appear that C and F are isomorphic as well as E and G.
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