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<p><a id="X7CF4153B7903F639" name="X7CF4153B7903F639"></a></p>
<div class="ChapSects"><a href="chap4.html#X7CF4153B7903F639">4 <span class="Heading">Presentations of graded algebras</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7F7903AE8392D54A">4.1 <span class="Heading">The <code class="keyw">GradedAlgebraPresentation</code> datatype</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X8589CD117D1ECD29">4.2 <span class="Heading">Construction function</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84A14CBF7C499665">4.2-1 <span class="Heading">GradedAlgebraPresentation construction functions</span></a>
</span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7DE3278D7E5DEE03">4.3 <span class="Heading">Data access functions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X877889A5792202F7">4.3-1 BaseRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8235D10781BE8003">4.3-2 CoefficientsRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X844BB80282EA3EBA">4.3-3 IndeterminatesOfGradedAlgebraPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84299C5A7ACD6F71">4.3-4 GeneratorsOfPresentationIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7DABCC1D85B33DAC">4.3-5 PresentationIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BFB8AC97FE6BDD6">4.3-6 IndeterminateDegrees</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7DB98FC5869F4CBC">4.3-7 <span class="Heading">Example: Constructing and accessing data of a
<code class="keyw">GradedAlgebraPresentation</code></span></a>
</span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X87C3D1B984960984">4.4 <span class="Heading">Other functions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87EB0B4A852CF4C6">4.4-1 <span class="Heading">TensorProduct</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8306E03281AF1CED">4.4-2 IsIsomorphicGradedAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87B04D4580DE8E10">4.4-3 IsAssociatedGradedRing</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85D95BF07C8E5DF7">4.4-4 DegreeOfRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7CE381787AEC5CAC">4.4-5 MaximumDegreeForPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EF085B67B846215">4.4-6 <span class="Heading">SubspaceDimensionDegree</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7834EE487CAA02D4">4.4-7 <span class="Heading">SubspaceBasisRepsByDegree</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84D5111984AFDD31">4.4-8 CoefficientsOfPoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B93B7D082A50E61">4.4-9 HilbertPoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F5D00C97A46D686">4.4-10 LHSSpectralSequence</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X87DD66E67D0D5485">4.5 <span class="Heading">Example: Computing the Lyndon-Hoschild-Serre spectral sequence
and mod-p cohomology ring for a small p-group</span></a>
</div>
</div>
<h3>4 <span class="Heading">Presentations of graded algebras</span></h3>
<p>A graded algebra A is a an algebra that has additional structure, called a grading (see Section <a href="chap3.html#X8074AAF07A3E7D2C"><b>3</b></a>). Graded algebras of the type found in <strong class="pkg">HAPprime</strong> have a presentation as a quotient of a polynomial ring</p>
<p class="pcenter">
H^*(G, F) = F[x_1, x_2, ..., x_n] / <I_1, I_2, ..., I_m>
</p>
<p>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.</p>
<p><a id="X7F7903AE8392D54A" name="X7F7903AE8392D54A"></a></p>
<h4>4.1 <span class="Heading">The <code class="keyw">GradedAlgebraPresentation</code> datatype</span></h4>
<p>For algebras that have a presentation as a quotient of a polynomial ring, the <code class="keyw">GradedAlgebraPresentation</code> datatype stores a quotient R/I where:</p>
<ul>
<li><p>R is a polynomial ring</p>
</li>
<li><p>I is a set of relations in R that generate an ideal</p>
</li>
</ul>
<p>and it also stores a grading in the form of</p>
<ul>
<li><p>the degree of each indeterminate of R</p>
</li>
</ul>
<p><a id="X8589CD117D1ECD29" name="X8589CD117D1ECD29"></a></p>
<h4>4.2 <span class="Heading">Construction function</span></h4>
<p><a id="X84A14CBF7C499665" name="X84A14CBF7C499665"></a></p>
<h5>4.2-1 <span class="Heading">GradedAlgebraPresentation construction functions</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> GradedAlgebraPresentation</code>( <var class="Arg">R, I, degs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> GradedAlgebraPresentationNC</code>( <var class="Arg">R, I, degs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">GradedAlgebraPresentation</code></p>
<p>Construct a <code class="keyw">GradedAlgebraPresentation</code> object representing a presentation of a graded algebra as the quotient of a polynomial ring <var class="Arg">R</var> by the ideal <var class="Arg">I</var> (as a list of relations in <var class="Arg">R</var>) where the indeterminates of <var class="Arg">R</var> (as returned by <code class="func">IndeterminatesOfGradedAlgebraPresentation</code> (<a href="chap4.html#X844BB80282EA3EBA"><b>4.3-3</b></a>) have degrees <var class="Arg">degs</var> respectively.</p>
<p>The function <code class="keyw">GradedAlgebraPresentation</code> checks that the arguments are compatible, while the <code class="code">NC</code> method performs no checks.</p>
<p><a id="X7DE3278D7E5DEE03" name="X7DE3278D7E5DEE03"></a></p>
<h4>4.3 <span class="Heading">Data access functions</span></h4>
<p><a id="X877889A5792202F7" name="X877889A5792202F7"></a></p>
<h5>4.3-1 BaseRing</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> BaseRing</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>Polynomial ring</p>
<p>Returns the base ring of the graded algebra presentation <var class="Arg">A</var>.</p>
<p><a id="X8235D10781BE8003" name="X8235D10781BE8003"></a></p>
<h5>4.3-2 CoefficientsRing</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CoefficientsRing</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>Ring</p>
<p>Returns the ring of coefficients of the graded algebra presentation <var class="Arg">A</var>.</p>
<p><a id="X844BB80282EA3EBA" name="X844BB80282EA3EBA"></a></p>
<h5>4.3-3 IndeterminatesOfGradedAlgebraPresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IndeterminatesOfGradedAlgebraPresentation</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>List</p>
<p>Returns the indeterminates used in the graded algebra presentation <var class="Arg">A</var>.</p>
<p><a id="X84299C5A7ACD6F71" name="X84299C5A7ACD6F71"></a></p>
<h5>4.3-4 GeneratorsOfPresentationIdeal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> GeneratorsOfPresentationIdeal</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>List</p>
<p>Returns the relations in the ring presentation for the graded algebra <var class="Arg">A</var>. The relations are returned sorted in order of increasing degree, and by indeterminate within each degree.</p>
<p><a id="X7DABCC1D85B33DAC" name="X7DABCC1D85B33DAC"></a></p>
<h5>4.3-5 PresentationIdeal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> PresentationIdeal</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>Ideal</p>
<p>Returns the ideal in the graded algebra presentation <var class="Arg">A</var> as a <strong class="pkg">GAP</strong> ideal <a href="../../../../../gap4r4/doc/htm/ref/CHAP054.htm#SECT002"><b>Reference: Ideal</b></a>.</p>
<p><a id="X7BFB8AC97FE6BDD6" name="X7BFB8AC97FE6BDD6"></a></p>
<h5>4.3-6 IndeterminateDegrees</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IndeterminateDegrees</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>List</p>
<p>Returns the degrees of the polynomial ring indeterminates in the graded algebra presentation <var class="Arg">A</var>. The ordering corresponds to the order of the ring indeterminates returned by <code class="func">IndeterminatesOfGradedAlgebraPresentation</code> (<a href="chap4.html#X844BB80282EA3EBA"><b>4.3-3</b></a>).</p>
<p><a id="X7DB98FC5869F4CBC" name="X7DB98FC5869F4CBC"></a></p>
<h5>4.3-7 <span class="Heading">Example: Constructing and accessing data of a
<code class="keyw">GradedAlgebraPresentation</code></span></h5>
<p>We demonstrate creating a <code class="keyw">GradedAlgebraPresentation</code> object and reading back its data by creating the graded algebra A with presentation 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</p>
<table class="example">
<tr><td><pre>
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 ]
</pre></td></tr></table>
<p><a id="X87C3D1B984960984" name="X87C3D1B984960984"></a></p>
<h4>4.4 <span class="Heading">Other functions</span></h4>
<p><a id="X87EB0B4A852CF4C6" name="X87EB0B4A852CF4C6"></a></p>
<h5>4.4-1 <span class="Heading">TensorProduct</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> TensorProduct</code>( <var class="Arg">A, B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> TensorProduct</code>( <var class="Arg">coll</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>GradedAlgebraPresentation</p>
<p>Returns a presentation for the graded algebra that is the tensor product of two graded algebras presented by <var class="Arg">A</var> and <var class="Arg">B</var>, or of a list of graded algebras.</p>
<p><a id="X8306E03281AF1CED" name="X8306E03281AF1CED"></a></p>
<h5>4.4-2 IsIsomorphicGradedAlgebra</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsIsomorphicGradedAlgebra</code>( <var class="Arg">A, B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns <code class="keyw">true</code> if the graded algebras <var class="Arg">A</var> and <var class="Arg">B</var> are isomorphic, or <code class="keyw">false</code> otherwise. This function tries all possible ring isomorphisms, so may take a considerable length of time for graded algebras with a large number of dimensions in each degree.</p>
<p><a id="X87B04D4580DE8E10" name="X87B04D4580DE8E10"></a></p>
<h5>4.4-3 IsAssociatedGradedRing</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsAssociatedGradedRing</code>( <var class="Arg">A, B</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns <code class="keyw">true</code> if the algebra <var class="Arg">A</var> is an associated graded ring of the algebra <var class="Arg">B</var>. This is the case if the additive structure is the same (i.e. the Hilbert-Poincaré series is the same), and the generators for <var class="Arg">A</var> (and their degrees) are included in the generators for <var class="Arg">B</var>.</p>
<p><a id="X85D95BF07C8E5DF7" name="X85D95BF07C8E5DF7"></a></p>
<h5>4.4-4 DegreeOfRepresentative</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> DegreeOfRepresentative</code>( <var class="Arg">A, p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>Integer</p>
<p>Returns the degree of a polynomial representative <var class="Arg">p</var> from the graded ring presentation <var class="Arg">A</var>.</p>
<p><a id="X7CE381787AEC5CAC" name="X7CE381787AEC5CAC"></a></p>
<h5>4.4-5 MaximumDegreeForPresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MaximumDegreeForPresentation</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>Integer</p>
<p>Returns the maximum degree in generators or relations that is needed to generate the graded algebra presentation <var class="Arg">A</var>. This is not necessarily the same as the largest degree in any of the relations and generators - some relations may be redundant (for example due to being a Groebner basis), so this routine checks for the largest degree of a required generator, and returns the maximum of this and the generator degrees.</p>
<p><a id="X7EF085B67B846215" name="X7EF085B67B846215"></a></p>
<h5>4.4-6 <span class="Heading">SubspaceDimensionDegree</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SubspaceDimensionDegree</code>( <var class="Arg">A, d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SubspaceDimensionDegree</code>( <var class="Arg">A, degs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>Integer or list</p>
<p>Returns the dimension of degree <var class="Arg">d</var> of the graded algebra <var class="Arg">A</var>, or a list of dimensions corresponding to the list of degrees <var class="Arg">degs</var>.</p>
<p><a id="X7834EE487CAA02D4" name="X7834EE487CAA02D4"></a></p>
<h5>4.4-7 <span class="Heading">SubspaceBasisRepsByDegree</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SubspaceBasisRepsByDegree</code>( <var class="Arg">A, d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SubspaceBasisRepsByDegree</code>( <var class="Arg">A, degs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>List or list of lists</p>
<p>Returns a basis for degree <var class="Arg">d</var> of the graded algebra <var class="Arg">A</var>, or a list of bases for the list of degrees <var class="Arg">degs</var>. Each basis is returned as a list of representatives.</p>
<p><a id="X84D5111984AFDD31" name="X84D5111984AFDD31"></a></p>
<h5>4.4-8 CoefficientsOfPoincareSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CoefficientsOfPoincareSeries</code>( <var class="Arg">A, n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>List</p>
<p>Returns the first <var class="Arg">n</var> coefficients of the Poincaré series for the graded algebra with <var class="Arg">A</var>. These are equal to the dimensions of degrees 0 to n-1 of the algebra (a fact that is used in the function <code class="func">SubspaceDimensionDegree</code> (<a href="chap4.html#X7EF085B67B846215"><b>4.4-6</b></a>)).</p>
<p><em>This function uses the <strong class="pkg">singular</strong> package.</em></p>
<p><a id="X7B93B7D082A50E61" name="X7B93B7D082A50E61"></a></p>
<h5>4.4-9 HilbertPoincareSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> HilbertPoincareSeries</code>( <var class="Arg">A</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>Rational function</p>
<p>Returns the Poincaré series for the graded algebra <var class="Arg">A</var>. This is a rational function P(t)/Q(t) which is a is a polynomial whose coefficients are the dimensions of each degree of the algebra.</p>
<p><em>This function uses the <strong class="pkg">singular</strong> package.</em></p>
<p><a id="X7F5D00C97A46D686" name="X7F5D00C97A46D686"></a></p>
<h5>4.4-10 LHSSpectralSequence</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> LHSSpectralSequence</code>( <var class="Arg">G[, N], n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> LHSSpectralSequenceLastSheet</code>( <var class="Arg">G[, N]</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">GradedAlgebraPresentation</code> or list</p>
<p>Computes the Lyndon-Hoschild-Serre spectral sequence for the group extension N -> G -> G/N. If a normal suggroup <var class="Arg">N</var> is not provided, then the largest central subgroup of G is used, or (if the order of the centre is larger than sqrt|G|) then the central subgroup that leads to the smallest initial sheet size is chosen.</p>
<p>The function <code class="func">LHSSpectralSequence</code> returns the first <var class="Arg">n</var> sheets of the spectral sequence, or all of the sequence up to convergence, if that occurs before the (n+1)th sheet. The Lyndon-Hoschild-Serre spectral sequence starts at the E_2 sheet, so the first element in returned list will always be empty. If <var class="Arg">n</var> is set to <code class="keyw">infinity</code> then the length of the returned list equals the number of sheets for convergence, and the last sheet in the list is the limiting sheet.</p>
<p>The function <code class="func">LHSSpectralSequenceLastSheet</code> returns only the limiting sheet of the spectral sequence. This ring is an associated graded algebra of the mod-p cohomology ring of G, with the same additive structure while not necessarily being isomorphic to it.</p>
<p>There are four options <a href="../../../../../gap4r4/doc/htm/ref/CHAP008.htm"><b>Reference: Options Stack</b></a> which can be used to guide this algorithm:</p>
<ul>
<li><p><code class="code">InitialLHSBicomplexSize</code> can be used to specify the initial size of the bicomplex (the default is 5). If, in the process of computing the spectral sequence, this is found to be too small then the algorithm restarts with a larger value. Specifying a larger initial value in these cases can save time.</p>
</li>
<li><p><code class="code">LargerLHSBicomplexBreak</code> if set to <code class="keyw">true</code> will force the calculation to enter a break loop before restarting with a larger bicomplex, should the bicomplex be found to be too small. The user user is prompted to type <code class="code">return;</code> before continuing. The default behaviour is <code class="keyw">false</code>, i.e. no prompt.</p>
</li>
<li><p><code class="code">LargerLHSBicomplexFail</code> if set to <code class="keyw">true</code> will return <code class="keyw">fail</code> should the bicomplex be found to be too small. The default behaviour is <code class="keyw">false</code>, i.e. to either restart or prompt, depending on the setting of the previous option.</p>
</li>
<li><p><code class="code">NoInductiveProof</code> if set to <code class="keyw">true</code> will not check that the cohomology rings for N and G/N are correct. Instead, it will compute the cohomology rings only up to the degree needed for the bicomplex size (5 by default, or specified by the <code class="code">InitialLHSBicomplexSize</code> option).</p>
</li>
</ul>
<p><a id="X87DD66E67D0D5485" name="X87DD66E67D0D5485"></a></p>
<h4>4.5 <span class="Heading">Example: Computing the Lyndon-Hoschild-Serre spectral sequence
and mod-p cohomology ring for a small p-group</span></h4>
<p>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.</p>
<p>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.</p>
<table class="example">
<tr><td><pre>
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)
</pre></td></tr></table>
<p>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.</p>
<table class="example">
<tr><td><pre>
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
</pre></td></tr></table>
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