The Norm Functor over Schemes
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910738335924224 |
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| author | Gille, Philippe Neher, Erhard Ruether, Cameron |
| author_facet | Gille, Philippe Neher, Erhard Ruether, Cameron |
| contents | We construct a globalization of Ferrand's norm functor over rings which generalizes it to the setting of a finite locally free morphism of schemes $T\to S$ of constant rank. It sends quasi-coherent modules over $T$ to quasi-coherent modules over $S$. These functors restrict to the category of quasi-coherent algebras. We also assemble these functors into a norm morphism from the stack of quasi-coherent modules over a finite locally free of constant rank extension of the base scheme into the stack of quasi-coherent modules. This morphism also restricts to the analogous stacks of algebras. Restricting our attention to finite étale covers, we give a cohomological description of the norm morphism in terms of the Segre embedding. Using this cohomological description, we show that the norm gives an equivalence of stacks of algebras $A_1^2 \equiv D_2$, akin to the result shown in The Book of Involutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Norm Functor over Schemes Gille, Philippe Neher, Erhard Ruether, Cameron Algebraic Geometry 16H05, 14F20, 20G10, 20G35 We construct a globalization of Ferrand's norm functor over rings which generalizes it to the setting of a finite locally free morphism of schemes $T\to S$ of constant rank. It sends quasi-coherent modules over $T$ to quasi-coherent modules over $S$. These functors restrict to the category of quasi-coherent algebras. We also assemble these functors into a norm morphism from the stack of quasi-coherent modules over a finite locally free of constant rank extension of the base scheme into the stack of quasi-coherent modules. This morphism also restricts to the analogous stacks of algebras. Restricting our attention to finite étale covers, we give a cohomological description of the norm morphism in terms of the Segre embedding. Using this cohomological description, we show that the norm gives an equivalence of stacks of algebras $A_1^2 \equiv D_2$, akin to the result shown in The Book of Involutions. |
| title | The Norm Functor over Schemes |
| topic | Algebraic Geometry 16H05, 14F20, 20G10, 20G35 |
| url | https://arxiv.org/abs/2401.15051 |