<!-- HAPPRIME - gradedalgebra.xml.in Function documentation template file for HAPprime Paul Smith Copyright (C) 2007-2008 Paul Smith National University of Ireland Galway $Id: functions.xml.in 200 2008-02-05 14:29:47Z pas $ --> <!-- ********************************************************** --> <Chapter Label="GradedAlgebraPresentationDatatype"> <Heading>Presentations of graded algebras</Heading> A graded algebra <M>A</M> is a an algebra that has additional structure, called a grading (see Section <Ref Sect="GradedAlgebra"/>). Graded algebras of the type found in &HAPprime; have a presentation as a quotient of a polynomial ring <Alt Only="LaTeX"> <Display> H^*(G, \mathbb{F}) = \frac{\mathbb{F}[x_1, x_2, \ldots, x_n]}{\langle I_1, I_2, \ldots, I_m \rangle} </Display> </Alt> <Alt Not="LaTeX"> <Display><![CDATA[ H^*(G, F) = F[x_1, x_2, ..., x_n] / <I_1, I_2, ..., I_m> ]]></Display> </Alt> where the polynomial ring indeterminates <M>x_i</M> each have an associated degree <M>d_i</M> and the <M>I_j</M> are relations which together generate an ideal in the ring. <Section> <Heading>The <K>GradedAlgebraPresentation</K> datatype</Heading> For algebras that have a presentation as a quotient of a polynomial ring, the <K>GradedAlgebraPresentation</K> datatype stores a quotient <M>R/I</M> where: <List> <Item><M>R</M> is a polynomial ring</Item> <Item><M>I</M> is a set of relations in <M>R</M> that generate an ideal</Item> </List> and it also stores a grading in the form of <List> <Item>the degree of each indeterminate of <M>R</M></Item> </List> </Section> <Section> <Heading>Construction function</Heading> <!-- GAPDocSourceSuffix="_DTmanGradedAlgebra_Con" --> </Section> <Section> <Heading>Data access functions</Heading> <!-- GAPDocSourceSuffix="_DTmanGradedAlgebra_Dat" --> <Subsection Label="GradedAlgebraExample1"> <Heading>Example: Constructing and accessing data of a <K>GradedAlgebraPresentation</K></Heading> We demonstrate creating a <K>GradedAlgebraPresentation</K> object and reading back its data by creating the graded algebra <M>A</M> with presentation <M>\mathbb{F}_2[x_1, x_2, x_3] / (x_1x_2, x_1^3+x_2^3)</M> where <M>x_1</M> and <M>x_2</M> have degree 1 and <M>x_3</M> 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]); gap> BaseRing(A); # REMOVED BECAUSE RESCLASSES CHANGES THE OUTPUT OF THIS gap> CoefficientsRing(A); gap> IndeterminatesOfGradedAlgebraPresentation(A); gap> PresentationIdeal(A); # REMOVED BECAUSE RESCLASSES CHANGES THE OUTPUT OF THIS gap> GeneratorsOfPresentationIdeal(A); gap> IndeterminateDegrees(A); --> <Example><![CDATA[ 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 ] ]]></Example> </Subsection> </Section> <Section> <Heading>Other functions</Heading> <!-- GAPDocSourceSuffix="_DTmanGradedAlgebra_Func" --> </Section> <Section Label="GradedAlgebraExample2"> <Heading>Example: Computing the Lyndon-Hoschild-Serre spectral sequence and mod-&p; cohomology ring for a small &p;-group</Heading> The Lyndon-Hoschild-Serre spectral sequence is relates the cohomologies of a normal subgroup <M>N</M> and a quotient group <M>G/N</M> to the cohomology of the total group <M>G</M>: the limiting sheet of the sequence is an associated graded ring of the cohomology of <M>G</M>. <P/> In this example we calculate the Lyndon-Hoschild-Serre spectral sequence for a group of order 16 using the centre of <M>G</M> 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 (<M>E_2</M>) and last (<M>E_\infty</M>) 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); 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]); gap> SubspaceDimensionDegree(SS[3], [1..10]); 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); gap> DegreeOfRepresentative(SS[3], B[1]*B[2]); gap> # gap> # And find the Poincare series gap> HilbertPoincareSeries(SS[3]); --> <Example><![CDATA[ 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) ]]></Example> 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, <M>n</M>, 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)); gap> # gap> # Find the maximum degree gap> n := MaximumDegreeForPresentation(Einf); gap> # gap> # And calculate the cohomology ring gap> H := ModPCohomologyRingPresentation(G, n); gap> # gap> # Check for an associated graded ring, and isomorphism gap> IsAssociatedGradedRing(H, Einf); gap> IsIsomorphicGradedAlgebra(H, Einf); --> <Example><![CDATA[ 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 ]]></Example> </Section> </Chapter>