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Groups which do not admit ghosts. (English) Zbl 1190.20008

Proc. Am. Math. Soc. 136, No. 4, 1171-1179 (2008); corrigendum ibid. 136, No. 10, 3727 (2008).
Let \(G\) be a \(p\)-group and let \(k\) be a field of characteristic \(p\). A map \(f\colon M\to N\) is said to be a ghost if \(f\) is non zero in the stable module category of all (possibly infinite dimensional) \(kG\)-modules, but \(f\) induces the zero map on Tate cohomology.
The main result of the paper is the statement that \(G\) does not admit ghosts if and only if \(G\) is either of order \(2\) or of order \(3\). The reason why these two groups are special is that every non projective indecomposable module is a syzygy of the trivial module.
The proof of the theorem is to first abstractly construct for any module a “universal ghost”, which is a map through which every ghost will factor. Then, it is shown that all ghosts out of \(M\) are trivial if and only if the universal ghost out of \(M\) is trivial and this is then equivalent to saying that the module \(M\) is a retract of a coproduct of suspensions of the trivial module. The statement then follows by applying a result of Jennings on the nilpotency index of the radical of the group ring of a \(p\)-group.

MSC:

20C20 Modular representations and characters
20J06 Cohomology of groups
55P42 Stable homotopy theory, spectra
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16D90 Module categories in associative algebras
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References:

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