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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.15893 |
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| _version_ | 1866913490354044928 |
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| author | Moutand, Mohammed |
| author_facet | Moutand, Mohammed |
| contents | Let $X$ be a smooth variety over a field $K$ with function field $K(X)$. Using the interpretation of the torsion part of the étale cohomology group $H_{\text{ét}}^2(K(X), \mathbb{G}_m)$ in terms of Milnor-Quillen algebraic $K$-group $K_2(K(X))$, we prove that under mild conditions on the norm maps along unramified extensions of $K(X)$ over $X$, there exist cohomological Brauer classes in $H_{\text{ét}}^2(X, \mathbb{G}_m)$ that are representable by Azumaya algebras on $X$. Theses conditions are almost satisfied in the case of number fields, providing then, a partial answer on a question of Grothendieck. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15893 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Azumaya algebras over unramified extensions of function fields Moutand, Mohammed Algebraic Geometry Let $X$ be a smooth variety over a field $K$ with function field $K(X)$. Using the interpretation of the torsion part of the étale cohomology group $H_{\text{ét}}^2(K(X), \mathbb{G}_m)$ in terms of Milnor-Quillen algebraic $K$-group $K_2(K(X))$, we prove that under mild conditions on the norm maps along unramified extensions of $K(X)$ over $X$, there exist cohomological Brauer classes in $H_{\text{ét}}^2(X, \mathbb{G}_m)$ that are representable by Azumaya algebras on $X$. Theses conditions are almost satisfied in the case of number fields, providing then, a partial answer on a question of Grothendieck. |
| title | Azumaya algebras over unramified extensions of function fields |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2408.15893 |