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products/Sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 19.6.2025 mit Größe 4 kB

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            <td style="vertical-align: top;"><a
href="aboutPoincareSeries.html"><small style="color: rgb(0, 0, 102);">Previous</small></a><br>
            </td>
            <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big><span
style="font-weight: bold;">About HAP: Tor and Ext of modules over a
mod p group ring<br>
            </span></big></td>
            <td style="text-align: right; vertical-align: top;"><a
href="aboutTwistedCoefficients.html"><small
style="color: rgb(0, 0, 102);">next</small></a><br>
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    <tr>
      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Let
FG be the group algebra of a finite group over the field F of p
elements, and let  M be an FG-module with G a p-group.  The abelian groups<br>
      <br>
      <div style="text-align: center;">Tor<sub>n</sub><sup>FG</sup>(M,F)
      <br>
      </div>
and <br>
      <div style="text-align: center;">Ext<sup>n</sup><sub>FG</sub>(M,F)<br>
      <br>
      </div>
can be calculated from a free resolution of M.  <br>
      <br>
We illustrate this for the module M arising from the canonical action
of the group G=Syl<sub>2</sub>(GL<sub>3</sub>(2)) on the 3-dimensional <span
style="font-weight: bold;">column</span> vector space over
GF(2). The module M can be entered as a meat-axe module using the
following standard GAP commands.<br>
      </td>
    </tr>
    <tr>
      <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
G:=SylowSubgroup(GL(3,2),2);;<br>
gap> M:=GModuleByMats(GeneratorsOfGroup(G),GF(2));;<br>
      </td>
    </tr>
    <tr>
      <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
module can be converted to an FpG-module DM using the following
command. The "desuspended module" DM is mathematically related to M via
a short exact sequence<br>
      <br>
      <div style="text-align: center;">0 → DM → PM → M → 0<br>
      </div>
      <br>
where PM is a free module. Thus<br>
      <br>
      <div style="text-align: center;">Tor<sub>n</sub><sup>FG</sup>(DM,F)
=  Tor<sub>n+1</sub><sup>FG</sup>(M,F) <br>
      </div>
and <br>
      <div style="text-align: center;">Ext<sup>n</sup><sub>FG</sub>(DM,F)
=  Ext<sup>n+1</sup><sub>FG</sub>(M,F)  <br>
      </div>
      </td>
    </tr>
    <tr>
      <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
DM:=DesuspensionMtxModule(M);;<br>
      </td>
    </tr>
    <tr>
      <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands now compute the 6-dimensional vector spaces<br>
      <div style="text-align: center;"><br>
Tor<sub>5</sub><sup>FG</sup>(M,F)  =  Tor<sub>4</sub><sup>FG</sup>(DM,F)
= F<sup>6</sup><br>
      </div>
and <br>
      <div style="text-align: center;">Ext<sup>5</sup><sub>FG</sub>(M,F)
=  Ext<sup>4</sup><sub>FG</sub>(DM,F)  = F<sup>6</sup> .<br>
      </div>
      <br>
      </td>
    </tr>
    <tr>
      <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=ResolutionFpGModule(DM,5);;<br>
gap>
p:=2;;<br>
gap> Homology(TensorWithIntegersModP(R,p),4);<br>
6<br>
gap> Cohomology(HomToIntegersModP(R,p),4);<br>
6<br>
      </td>
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style="color: rgb(0, 0, 102);" href="aboutPoincareSeries.html">Previous
Page</a><br>
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            <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html"><span style="color: rgb(0, 0, 102);">Contents</span></a><br>
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