Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma

Benutzer


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

Quelle aboutTorAndExt.html

Sprache: HTML

products/Sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/aboutTorAndExt.html

<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
  <meta http-equiv="content-type"
content="text/html; charset=ISO-8859-1">
  <title>AboutHap</title>
</head>
<body
style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);"
alink="#000066" link="#000066" vlink="#000066">
<br>
<table
style="text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, 102);"
border="0" cellpadding="20" cellspacing="10">
  <tbody>
    <tr align="center">
      <th style="vertical-align: top;">
      <table style="width: 100%; text-align: left;" cellpadding="2"
cellspacing="2">
        <tbody>
          <tr>
            <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>
            </td>
          </tr>
        </tbody>
      </table>
      <big><span style="font-weight: bold;"></span></big><br>
      </th>
    </tr>
    <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>
    </tr>
    <tr>
      <td style="vertical-align: top;">
      <table
style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
        <tbody>
          <tr>
            <td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutPoincareSeries.html">Previous
Page</a><br>
            </td>
            <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html"><span style="color: rgb(0, 0, 102);">Contents</span></a><br>
            </td>
            <td style="text-align: right; vertical-align: top;"><a
href="aboutTwistedCoefficients.html"><span
style="color: rgb(0, 0, 102);">Next
page</span><br>
            </a> </td>
          </tr>
        </tbody>
      </table>
      <a href="aboutTopology.html"><br>
      </a> </td>
    </tr>
  </tbody>
</table>
<br>
<br>
</body>
</html>

Messung V0.5 in Prozent

¤ Dauer der Verarbeitung: 0.15 Sekunden (vorverarbeitet am 2026-06-18) ¤

Wurzel

Suchen

PVS Prover

Isabelle Prover

NIST Cobol Testsuite

Cephes Mathematical Library

Vienna Development Method

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.