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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2406.12261 |
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| _version_ | 1866911921962221568 |
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| author | Friedlander, Eric M. |
| author_facet | Friedlander, Eric M. |
| contents | We investigate infinite dimensional modules for a linear algebraic group $\mathbb G$ over a field of positive characteristic $p$. For any subcoalgebra $C \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we consider the abelian subcategory $CoMod(C) \subset Mod(\mathbb G)$ and the left exact functor $(-)_C: Mod(\mathbb G) \to CoMod(C)$ that is right adjoint to the inclusion functor. The class of cofinite $\mathbb G$-modules is formulated using finite dimensional subcoalgebras of $\mathcal O(\mathbb G)$ and the new invariant of "cofinite type" is introduced.
We are particularly interested in mock injective $\mathbb G$-modules, $\mathbb G$-modules which are not seen by earlier support theories. Various properties of these ghostly $\mathbb G$-modules are established. The stable category $StMock(\mathbb G)$ is introduced, enabling mock injective $\mathbb G$-modules to fit into the framework of tensor triangulated categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12261 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinite dimensional modules for linear algebraic groups Friedlander, Eric M. Representation Theory We investigate infinite dimensional modules for a linear algebraic group $\mathbb G$ over a field of positive characteristic $p$. For any subcoalgebra $C \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we consider the abelian subcategory $CoMod(C) \subset Mod(\mathbb G)$ and the left exact functor $(-)_C: Mod(\mathbb G) \to CoMod(C)$ that is right adjoint to the inclusion functor. The class of cofinite $\mathbb G$-modules is formulated using finite dimensional subcoalgebras of $\mathcal O(\mathbb G)$ and the new invariant of "cofinite type" is introduced. We are particularly interested in mock injective $\mathbb G$-modules, $\mathbb G$-modules which are not seen by earlier support theories. Various properties of these ghostly $\mathbb G$-modules are established. The stable category $StMock(\mathbb G)$ is introduced, enabling mock injective $\mathbb G$-modules to fit into the framework of tensor triangulated categories. |
| title | Infinite dimensional modules for linear algebraic groups |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2406.12261 |