Descent and finite permutation resolutions for discrete groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913173827747840 |
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| author | Gómez, Juan Omar Pol, Luca |
| author_facet | Gómez, Juan Omar Pol, Luca |
| contents | Let $G$ be a discrete group with a finite-dimensional model for the classifying space for proper actions, and let $k$ be a commutative Noetherian ring of finite global dimension. In this setting, we prove that the homotopy category of projective $kG$-modules, the stable module category of $kG$-modules as defined by Mazza-Symonds, and the derived category of permutation $kG$-modules with finite isotropy, admit descent to finite subgroups. As an application, we show that any $kG$-module of type $FP_\infty$ is a retract of a module that admits a finite resolution by finitely generated $\natural$-permutation modules with finite isotropy, generalizing a result of Balmer-Gallauer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31384 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Descent and finite permutation resolutions for discrete groups Gómez, Juan Omar Pol, Luca Representation Theory 18F99, 20C07, 20C12, 16E05, 55U35 Let $G$ be a discrete group with a finite-dimensional model for the classifying space for proper actions, and let $k$ be a commutative Noetherian ring of finite global dimension. In this setting, we prove that the homotopy category of projective $kG$-modules, the stable module category of $kG$-modules as defined by Mazza-Symonds, and the derived category of permutation $kG$-modules with finite isotropy, admit descent to finite subgroups. As an application, we show that any $kG$-module of type $FP_\infty$ is a retract of a module that admits a finite resolution by finitely generated $\natural$-permutation modules with finite isotropy, generalizing a result of Balmer-Gallauer. |
| title | Descent and finite permutation resolutions for discrete groups |
| topic | Representation Theory 18F99, 20C07, 20C12, 16E05, 55U35 |
| url | https://arxiv.org/abs/2605.31384 |