Presentations of graded algebras

A graded algebra A is a an algebra that has additional structure, called a grading (see Section ). Graded algebras of the type found in &HAPprime; have a presentation as a quotient of a polynomial ring H^*(G, \mathbb{F}) = \frac{\mathbb{F}[x_1, x_2, \ldots, x_n]}{\langle I_1, I_2, \ldots, I_m \rangle} H^*(G, F) = F[x_1, x_2, ..., x_n] / <I_1, I_2, ..., I_m> where the polynomial ring indeterminates x_i each have an associated degree d_i and the I_j are relations which together generate an ideal in the ring.

The GradedAlgebraPresentation datatype

For algebras that have a presentation as a quotient of a polynomial ring, the GradedAlgebraPresentation datatype stores a quotient R/I where: R is a polynomial ring I is a set of relations in R that generate an ideal and it also stores a grading in the form of the degree of each indeterminate of R
Construction function
<#Include Label="GradedAlgebraPresentation_DTmanGradedAlgebra_Con">
Data access functions
<#Include Label="BaseRing_DTmanGradedAlgebra_Dat"> <#Include Label="CoefficientsRing_DTmanGradedAlgebra_Dat"> <#Include Label="IndeterminatesOfGradedAlgebraPresentation_DTmanGradedAlgebra_Dat"> <#Include Label="GeneratorsOfPresentationIdeal_DTmanGradedAlgebra_Dat"> <#Include Label="PresentationIdeal_DTmanGradedAlgebra_Dat"> <#Include Label="IndeterminateDegrees_DTmanGradedAlgebra_Dat">

Example: Constructing and accessing data of a GradedAlgebraPresentation

We demonstrate creating a GradedAlgebraPresentation object and reading back its data by creating the graded algebra A with presentation \mathbb{F}_2[x_1, x_2, x_3] / (x_1x_2, x_1^3+x_2^3) where x_1 and x_2 have degree 1 and x_3 has degree 4 gap> R := PolynomialRing(GF(2), 3);; gap> A := GradedAlgebraPresentation(R, [R.1*R.2, R.1^3+R.2^3], [1,1,4]); Graded algebra GF(2)[ x_1, x_2, x_3 ] / [ x_1*x_2, x_1^3+x_2^3 ] with indeterminate degrees [ 1, 1, 4 ] gap> CoefficientsRing(A); GF(2) gap> IndeterminatesOfGradedAlgebraPresentation(A); [ x_1, x_2, x_3 ] gap> GeneratorsOfPresentationIdeal(A); [ x_1*x_2, x_1^3+x_2^3 ] gap> IndeterminateDegrees(A); [ 1, 1, 4 ]
Other functions
<#Include Label="TensorProduct_DTmanGradedAlgebra_Func"> <#Include Label="IsIsomorphicGradedAlgebra_DTmanGradedAlgebra_Func"> <#Include Label="IsAssociatedGradedRing_DTmanGradedAlgebra_Func"> <#Include Label="DegreeOfRepresentative_DTmanGradedAlgebra_Func"> <#Include Label="MaximumDegreeForPresentation_DTmanGradedAlgebra_Func"> <#Include Label="SubspaceDimensionDegree_DTmanGradedAlgebra_Func"> <#Include Label="SubspaceBasisRepsByDegree_DTmanGradedAlgebra_Func"> <#Include Label="CoefficientsOfPoincareSeries_DTmanGradedAlgebra_Func"> <#Include Label="HilbertPoincareSeries_DTmanGradedAlgebra_Func"> <#Include Label="LHSSpectralSequence_DTmanGradedAlgebra_Func">

Example: Computing the Lyndon-Hoschild-Serre spectral sequence and mod-&p; cohomology ring for a small &p;-group

The Lyndon-Hoschild-Serre spectral sequence is relates the cohomologies of a normal subgroup N and a quotient group G/N to the cohomology of the total group G: the limiting sheet of the sequence is an associated graded ring of the cohomology of G.

In this example we calculate the Lyndon-Hoschild-Serre spectral sequence for a group of order 16 using the centre of G as our normal subgroup. By asking for an infinite number of terms, this function calculates enough terms to be sure that the sequence has converged. We compare the dimensions in the first (E_2) and last (E_\infty) sheet, we demonstrate that the limiting sheet (the last in the list) is a graded algebra by multiplying some elements, and we calculate the Poincaré series of the last sheet. gap> G := SmallGroup(16, 4);; gap> SS := LHSSpectralSequence(G, Centre(G), infinity); [ , Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ ] with indeterminate degrees [ 1, 1, 1, 1 ], Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ x_2^2, x_1^2+x_1*x_2 ] with indeterminate degrees [ 1, 1, 2, 2 ], Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ x_2^2, x_1^2+x_1*x_2 ] with indeterminate degrees [ 1, 1, 2, 2 ] ] gap> # i.e. we identify convergence after 3 terms gap> # gap> # Compare the dimensions of the first and last sheet gap> SubspaceDimensionDegree(SS[2], [1..10]); [ 4, 10, 20, 35, 56, 84, 120, 165, 220, 286 ] gap> SubspaceDimensionDegree(SS[3], [1..10]); [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] gap> # gap> # Take the two basis elements from degree 1 and check that the gap> # product is in degree two gap> B := SubspaceBasisRepsByDegree(SS[3], 1); [ x_1, x_2 ] gap> DegreeOfRepresentative(SS[3], B[1]*B[2]); 2 gap> # gap> # And find the Poincare series gap> HilbertPoincareSeries(SS[3]); (1)/(x_1^2-2*x_1+1)

The largest degree in the presentation for the limiting sheet in the Lyndon-Hoschild-Serre spectral sequence for &G; is the same as the largest degree in the presentation for the mod-&p; cohomology ring of &G;. We continue this example by calculating this maximum degree, n, for our group &G; and then computing the mod-&p; cohomology ring. We confirm that the cohomology ring is an associated graded ring of the limiting sheet of the spectral sequence, and check whether in this case it is in fact also isomorphic. gap> G := SmallGroup(16, 4);; gap> Einf := LHSSpectralSequenceLastSheet(G, Centre(G)); Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ x_2^2, x_1^2+x_1*x_2 ] with indeterminate degrees [ 1, 1, 2, 2 ] gap> # gap> # Find the maximum degree gap> n := MaximumDegreeForPresentation(Einf); 2 gap> # gap> # And calculate the cohomology ring gap> H := ModPCohomologyRingPresentation(G, n); Graded algebra GF(2)[ x_1, x_2, x_3, x_4 ] / [ x_1*x_2+x_2^2, x_1^2 ] with indeterminate degrees [ 1, 1, 2, 2 ] gap> # gap> # Check for an associated graded ring, and isomorphism gap> IsAssociatedGradedRing(H, Einf); true gap> IsIsomorphicGradedAlgebra(H, Einf); true