Lannes' $T$-functor and mod-$p$ cohomology of profinite groups
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911409611210752 |
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| author | Boggi, Marco |
| author_facet | Boggi, Marco |
| contents | The Lannes-Quillen theorem relates the mod-$p$ cohomology of a finite group $G$ with the mod-$p$ cohomology of centralizers of abelian elementary $p$-subgroups of $G$, for $p>0$ a prime number. This theorem was extended to profinite groups whose mod-$p$ cohomology algebra is finitely generated by Henn. In a weaker form, the Lannes-Quillen theorem was then extended by Symonds to arbitrary profinite groups. Building on Symonds' result, we formulate and prove a full version of this theorem for all profinite groups. For this purpose, we develop a theory of products for families of discrete torsion modules, parameterized by a profinite space, which is dual, in a very precise sense, to the theory of coproducts for families of profinite modules, parameterized by a profinite space, developed by Haran, Melnikov and Ribes. In the last section, we give applications to the problem of conjugacy separability of $p$-torsion elements and finite $p$-subgroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lannes' $T$-functor and mod-$p$ cohomology of profinite groups Boggi, Marco Group Theory Geometric Topology 20F65 (Primary) 20J05, 20E18 (Secondary) The Lannes-Quillen theorem relates the mod-$p$ cohomology of a finite group $G$ with the mod-$p$ cohomology of centralizers of abelian elementary $p$-subgroups of $G$, for $p>0$ a prime number. This theorem was extended to profinite groups whose mod-$p$ cohomology algebra is finitely generated by Henn. In a weaker form, the Lannes-Quillen theorem was then extended by Symonds to arbitrary profinite groups. Building on Symonds' result, we formulate and prove a full version of this theorem for all profinite groups. For this purpose, we develop a theory of products for families of discrete torsion modules, parameterized by a profinite space, which is dual, in a very precise sense, to the theory of coproducts for families of profinite modules, parameterized by a profinite space, developed by Haran, Melnikov and Ribes. In the last section, we give applications to the problem of conjugacy separability of $p$-torsion elements and finite $p$-subgroups. |
| title | Lannes' $T$-functor and mod-$p$ cohomology of profinite groups |
| topic | Group Theory Geometric Topology 20F65 (Primary) 20J05, 20E18 (Secondary) |
| url | https://arxiv.org/abs/2408.12488 |