Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2016
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/1606.04175 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916136161902592 |
|---|---|
| author | Russell, Jeremy Dean, Samuel |
| author_facet | Russell, Jeremy Dean, Samuel |
| contents | For any ring $R$, the Auslander-Gruson-Jensen functor is the exact contravariant functor $$\textsf{D}_A:\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})\longrightarrow(\textsf{mod}(R^{op}),\textsf{Ab})$$ sending representable functors $(X,\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} )$ to tensor functors $X\otimes\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} $. We show that this functor admits a fully faithful left adjoint $\textsf{D}_L$ and a fully faithful right adjoint $\textsf{D}_R$. The left adjoint $$\textsf{D}_L\:(\textsf{mod}(R^{op}),\textsf{Ab})\longrightarrow \textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$$ induces an equivalence of categories $$\frac{\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})}{\{F\ |\ \textsf{D}_A F=0\}}\cong(\textsf{mod}(R^{op}),\textsf{Ab})^{op}$$ where $\{F \ |\ \textsf{D}_A F=0\}$ is the Serre subcategory of $\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$ consisting of all functors $F$ arising from pure exact sequences. As a result, the functor $\textsf{D}_A$ is seen to be a Serre localization functor. The right adjoint $$\textsf{D}_R:(\textsf{mod}(R^{op}),\textsf{Ab})\longrightarrow \textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$$ together with $\textsf{D}_A$ restricts to the well known Auslander-Gruson-Jensen duality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1606_04175 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | The Auslander-Gruson-Jensen Recollement Russell, Jeremy Dean, Samuel Representation Theory For any ring $R$, the Auslander-Gruson-Jensen functor is the exact contravariant functor $$\textsf{D}_A:\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})\longrightarrow(\textsf{mod}(R^{op}),\textsf{Ab})$$ sending representable functors $(X,\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} )$ to tensor functors $X\otimes\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} $. We show that this functor admits a fully faithful left adjoint $\textsf{D}_L$ and a fully faithful right adjoint $\textsf{D}_R$. The left adjoint $$\textsf{D}_L\:(\textsf{mod}(R^{op}),\textsf{Ab})\longrightarrow \textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$$ induces an equivalence of categories $$\frac{\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})}{\{F\ |\ \textsf{D}_A F=0\}}\cong(\textsf{mod}(R^{op}),\textsf{Ab})^{op}$$ where $\{F \ |\ \textsf{D}_A F=0\}$ is the Serre subcategory of $\textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$ consisting of all functors $F$ arising from pure exact sequences. As a result, the functor $\textsf{D}_A$ is seen to be a Serre localization functor. The right adjoint $$\textsf{D}_R:(\textsf{mod}(R^{op}),\textsf{Ab})\longrightarrow \textsf{fp}(\textsf{Mod}(R),\textsf{Ab})$$ together with $\textsf{D}_A$ restricts to the well known Auslander-Gruson-Jensen duality. |
| title | The Auslander-Gruson-Jensen Recollement |
| topic | Representation Theory |
| url | https://arxiv.org/abs/1606.04175 |