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<div class="ChapSects"><a href="chap3.html#X7EE4339B83BFDA2D">3 <span class="Heading">Functions for Homological Algebra</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7C0B125E7D5415B4">3.1 <span class="Heading">Resolutions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86934BE9858F7199">3.1-1 <span class="Heading">ResolutionPrimePowerGroup</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B435C307F28D44F">3.1-2 <span class="Heading">ExtendResolutionPrimePowerGroup</span></a>
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<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7FF2605B79D7B5F8">3.2 <span class="Heading">Poincaré Series</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E1A4C8781A02CD0">3.2-1 PoincareSeriesLHS</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7A9561E47A4994F5">3.3 <span class="Heading">Cohomology Ring structure</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X85CFF2AB7A7A99D2">3.3-1 ModPCohomologyRingPresentation</a></span>
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<h3>3 <span class="Heading">Functions for Homological Algebra</span></h3>
<p><a id="X7C0B125E7D5415B4" name="X7C0B125E7D5415B4"></a></p>
<h4>3.1 <span class="Heading">Resolutions</span></h4>
<p><a id="X86934BE9858F7199" name="X86934BE9858F7199"></a></p>
<h5>3.1-1 <span class="Heading">ResolutionPrimePowerGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ResolutionPrimePowerGroupRadical</code>( <var class="Arg">G, 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">> ResolutionPrimePowerGroupGF</code>( <var class="Arg">G, 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">> ResolutionPrimePowerGroupAutoMem</code>( <var class="Arg">G, 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">> ResolutionPrimePowerGroupGF2</code>( <var class="Arg">G, 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">> ResolutionPrimePowerGroupRadical</code>( <var class="Arg">M, 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">> ResolutionPrimePowerGroupGF</code>( <var class="Arg">M, 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">> ResolutionPrimePowerGroupAutoMem</code>( <var class="Arg">M, 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">> ResolutionPrimePowerGroupGF2</code>( <var class="Arg">M, n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">HAPResolution</code></p>
<p>Returns <var class="Arg">n</var> terms of a minimal free FG-resolution for either the ground ring of a prime power group <var class="Arg">G</var> or of a module <var class="Arg">M</var>. For the module version, <var class="Arg">M</var> must be passed as an <code class="keyw">FpGModuleGF</code> object - see <a href="/home/pas/GAP/pkg/happrime-0.3.2/doc/datatypes/chap5.html#X820435E87D83DF34"><b>HAPprime Datatypes: FG-modules</b></a> in the <strong class="pkg">HAPprime</strong> datatypes reference manual.</p>
<p>Three versions of this function are provided:</p>
<dl>
<dt><strong class="Mark"><code class="keyw">ResolutionPrimePowerGroupRadical</code></strong></dt>
<dd><p>uses the same resolution-building method as the <strong class="pkg">HAP</strong> function <code class="func">ResolutionPrimePowerGroup</code>, but stores the resolution in a different format that takes only about half the memory of the <strong class="pkg">HAP</strong> version.</p>
</dd>
<dt><strong class="Mark"><code class="keyw">ResolutionPrimePowerGroupGF</code></strong></dt>
<dd><p>calculates the resolution using <strong class="pkg">HAPprime</strong>'s G-generator form of modules, which reduces memory use by around a factor of two over <code class="keyw">ResolutionPrimePowerGroupRadical</code>, but is slower by an order of magnitude.</p>
</dd>
<dt><strong class="Mark"><code class="keyw">ResolutionPrimePowerGroupAutoMem</code></strong></dt>
<dd><p>automatically switches between the two previous versions based on the available memory. It uses the <code class="code">Radical</code> version until it gets close to the limit of the available memory, and then switches to the <code class="code">GF</code> version.</p>
</dd>
<dt><strong class="Mark"><code class="keyw">ResolutionPrimePowerGroupGF2</code></strong></dt>
<dd><p>calculates the resolution by FG-matrix partitioning. The amount of partitioning is governed by the <a href="../../../../../gap4r4/doc/htm/ref/CHAP008.htm"><b>Reference: Options Stack</b></a> option <code class="code">MaxFGExpansionSize</code>. The default value means that until the boundary map takes about 128Mb, the method is equivalent to <code class="keyw">ResolutionPrimePowerGroupRadical</code>, and then it tends towards <code class="keyw">ResolutionPrimePowerGroupGF</code> in terms of time, but saves less memory.</p>
</dd>
</dl>
<p>See the <strong class="pkg">HAPprime</strong> datatypes reference manual for details of the different algorithms, in particular the chapters on the G-generator form of FG-modules <a href="/home/pas/GAP/pkg/happrime-0.3.2/doc/datatypes/chap5.html#X820435E87D83DF34"><b>HAPprime Datatypes: FG-modules</b></a> and FG-module homomorphisms <a href="/home/pas/GAP/pkg/happrime-0.3.2/doc/datatypes/chap6.html#X82F28552819A6542"><b>HAPprime Datatypes: FG-module homomorphisms</b></a> and on resolutions <a href="/home/pas/GAP/pkg/happrime-0.3.2/doc/datatypes/chap2.html#X7C0B125E7D5415B4"><b>HAPprime Datatypes: Resolutions</b></a>.</p>
<p><a id="X7B435C307F28D44F" name="X7B435C307F28D44F"></a></p>
<h5>3.1-2 <span class="Heading">ExtendResolutionPrimePowerGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ExtendResolutionPrimePowerGroupRadical</code>( <var class="Arg">R</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">> ExtendResolutionPrimePowerGroupGF</code>( <var class="Arg">R</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">> ExtendResolutionPrimePowerGroupAutoMem</code>( <var class="Arg">R</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">> ExtendResolutionPrimePowerGroupGF2</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">HAPResolution</code></p>
<p>Returns the resolution <var class="Arg">R</var> extended by one term. The three variants offer a choice between memory and speed, and correspond to the different versions of <code class="keyw">ResolutionPrimePowerGroup</code> in <strong class="pkg">HAPprime</strong>. See the documentation (<a href="chap3.html#X86934BE9858F7199"><b>above</b></a>) for those functions for a description of the different variants.</p>
<p><a id="X7FF2605B79D7B5F8" name="X7FF2605B79D7B5F8"></a></p>
<h4>3.2 <span class="Heading">Poincaré Series</span></h4>
<p><a id="X7E1A4C8781A02CD0" name="X7E1A4C8781A02CD0"></a></p>
<h5>3.2-1 PoincareSeriesLHS</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> PoincareSeriesLHS</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>Rational function</p>
<p>For a finite p-group <var class="Arg">G</var>, this function calculates and returns a quotient of polynomials f(x) = P(x)/Q(x) (i.e. the Poincaré series) whose coefficient of x^k equals the rank of the vector space H_k(G, F_p) for all k in the range k=1 to k=n.</p>
<p>This function computes a Lyndon-Hoschild-Serre spectral sequence for the p-group G. The last sheet of this sequence will have the same additive structure as the mod-p group cohomology ring of G, and thus the same Poincaré series, which is returned by this function.</p>
<p>See Section <a href="chap2.html#X78F3639083A7DE62"><b>2.2-3</b></a> for an example and more description.</p>
<p><a id="X7A9561E47A4994F5" name="X7A9561E47A4994F5"></a></p>
<h4>3.3 <span class="Heading">Cohomology Ring structure</span></h4>
<p><a id="X85CFF2AB7A7A99D2" name="X85CFF2AB7A7A99D2"></a></p>
<h5>3.3-1 ModPCohomologyRingPresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ModPCohomologyRingPresentation</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ModPCohomologyRingPresentation</code>( <var class="Arg">G, 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">> ModPCohomologyRingPresentation</code>( <var class="Arg">R</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">> ModPCohomologyRingPresentation</code>( <var class="Arg">A</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">GradedAlgebraPresentation</code></p>
<p>Calculates and returns a cohomology ring presentation for the group G. See <a href="/home/pas/GAP/pkg/happrime-0.3.2/doc/datatypes/chap4.html#X7CF4153B7903F639"><b>HAPprime Datatypes: Presentations of graded algebras</b></a> in the datatypes reference manual for details of the <code class="keyw">GradedAlgebraPresentation</code> type.</p>
<p>If the only argument is a p-group <var class="Arg">G</var> then this function computes and returns the provably-correct cohomology ring presentation. This version first computes the Lyndon-Hoschild-Serre Spectral Sequence until convergence to find the additive structure of the cohomology ring, and then computes the cohomology ring up to and including the maximum necessary generator or relation, using the <code class="code">(G, n)</code> method described below. For certain groups, the cohomology ring is returned without computation: the known mod-p cohomology ring presentation for cyclic groups is returned without calculation, and for groups which can be expressed as a direct product, the cohomology ring is computed as a tensor product of its direct factors (thus the cohomology ring of all Abelian groups are also returned with minimal computation.)</p>
<p>When given a p-group <var class="Arg">G</var> and integer <var class="Arg">n</var>, this function computes the presentation modulo all elements of degree greater <var class="Arg">n</var>. Alternatively, a minimal resolution <var class="Arg">R</var> (with <var class="Arg">n</var> terms) can be input, or a structure constant algebra <var class="Arg">A</var> with embedded degrees (from <code class="func">ModPCohomologyRing</code> (<a href="/home/pas/GAP/pkg/Hap1.8/doc/chap8.html#X7A9561E47A4994F5"><b>HAP: ModPCohomologyRing</b></a>)).</p>
<p>See Section <a href="chap2.html#X81BCBAA18423E1C8"><b>2.2-1</b></a> and <a href="chap2.html#X825971A27C00C1B6"><b>2.2-2</b></a> for examples and more description. See also <code class="func">LHSSpectralSequence</code> (<a href="/home/pas/GAP/pkg/happrime-0.3.2/doc/datatypes/chap4.html#X7F5D00C97A46D686"><b>HAPprime Datatypes: LHSSpectralSequence</b></a>) for details of options that can be used to guide the spectral sequence computation.</p>
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