Constructing endotrivial modules. (English) Zbl 1098.20004
Let \(G\) be a finite \(p\)-group and let \(k\) be an algebraically closed field of characteristic \(p\). A \(kG\)-module \(M\) is called endotrivial if its endomorphism ring over \(k\) is isomorphic to a direct sum of the trivial \(kG\)-module \(k\) and a projective module. Endotrivial modules form a finitely generated Abelian group \(T(G)\) whose structure was determined very recently. Puig showed that the torsion part of \(T(G)\) is finitely generated, Carlson-Thévenaz gave the precise structure, Bouc-Thévenaz and Alperin gave a subgroup of finite index in the torsion free part of \(T(G)\), and Carlson-Thévenaz showed that the subgroup described by Alperin is actually all of the torsion free part of \(T(G)\).
In the paper under review the author gives explicit generators of \(T(G)\) by methods coming from the cohomology variety of the group. The modules are constructed by this well developed theory by modules corresponding to irreducible components of this variety.
In the paper under review the author gives explicit generators of \(T(G)\) by methods coming from the cohomology variety of the group. The modules are constructed by this well developed theory by modules corresponding to irreducible components of this variety.
Reviewer: Alexander Zimmermann (Amiens)
MSC:
| 20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
| 20J06 | Cohomology of groups |
| 20C20 | Modular representations and characters |
Keywords:
endotrivial modules; Dade groups; support varieties; cohomology of groups; finite \(p\)-groups; endomorphism ringsReferences:
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