<!-- 
  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>
  </Section>

  <Section>
    <Heading>Data access functions</Heading>
  
    <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);
gap> CoefficientsRing(A);
gap> IndeterminatesOfGradedAlgebraPresentation(A);
gap> PresentationIdeal(A);
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> BaseRing(A);
GF(2)[x_1,x_2,x_3]
gap> CoefficientsRing(A);
GF(2)
gap> IndeterminatesOfGradedAlgebraPresentation(A);
[ x_1, x_2, x_3 ]
gap> PresentationIdeal(A);
<two-sided ideal in GF(2)[x_1,x_2,x_3], (2 generators)>
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>
  
  </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 := Mod2CohomologyRingPresentation(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 := Mod2CohomologyRingPresentation(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>

