| Ghosts in modular representation theory (2006) | |||||||||
Abstract | |||||||||
| Let $G$ be a finite $p$-group and let $k$ be a field of characteristic $p$. Recall that the \emph{stable module category} $\StMod(kG)$ is the following tensor triangulated category. The objects are left $kG$-modules and the space of morphisms between $kG$-modules $M$ and $N$, denoted $\uHom_{kG}(M,N)$, is the $k$-vector space of $kG$-module homomorphisms modulo those maps that factor through a projective module. The category $\stmod(kG)$ is the full subcategory of finite-dimensional left $kG$-modules. A \emph{ghost} in the stable module category is a map between $kG$-modules that is trivial in Tate cohomology. In \cite{CCM3}, we formulated the \emph{generating hypothesis} (GH) for $kG$ as the statement that all ghosts between finite-dimensional $kG$-modules are trivial in the stable module category, i.e., they factor through a projective. (This formulation was motivated by the famous classical generating hypothesis of Peter Freyd \cite{freydGH} in the stable homotopy category which is the conjecture that there no non-trivial maps between finite spectra that are trivial in stable homotopy groups.) We have shown in \cite{CCM3} that the GH holds for $kG$, where $G$ is a non-trivial finite $p$-group and $k$ is a field of characteristic $p$, if and only if $G$ is either $C_2$ or $C_3$.. Comment: 15 pages, final version, to appear in Advances in Mathematics | |||||||||
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