Galois cohomology revisited
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arXiv
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| Format: | Preprint |
| Published: |
2017
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| _version_ | 1866913915588313088 |
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| author | Nikolaev, Igor V. |
| author_facet | Nikolaev, Igor V. |
| contents | We recast the Galois cohomology of the variety $V$ over a number field $k$ in terms of the K-theory of a $C^*$-algebra $\mathscr{A}_V$ connected to $V$. It is proved that $V$ is isomorphic to $V'$ over $k$ (algebraic closure of $k$, resp.) if and only if $\mathscr{A}_V$ is isomorphic (Morita equivalent, resp.) to $\mathscr{A}_{V'}$. In particular, the Morita equivalent $C^*$-algebras $\mathscr{A}_V$ parametrize twists of the variety $V$. The case of rational elliptic curves is considered in detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1712_07516 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Galois cohomology revisited Nikolaev, Igor V. Number Theory Algebraic Geometry Operator Algebras 11G35, 14A22, 46L85 We recast the Galois cohomology of the variety $V$ over a number field $k$ in terms of the K-theory of a $C^*$-algebra $\mathscr{A}_V$ connected to $V$. It is proved that $V$ is isomorphic to $V'$ over $k$ (algebraic closure of $k$, resp.) if and only if $\mathscr{A}_V$ is isomorphic (Morita equivalent, resp.) to $\mathscr{A}_{V'}$. In particular, the Morita equivalent $C^*$-algebras $\mathscr{A}_V$ parametrize twists of the variety $V$. The case of rational elliptic curves is considered in detail. |
| title | Galois cohomology revisited |
| topic | Number Theory Algebraic Geometry Operator Algebras 11G35, 14A22, 46L85 |
| url | https://arxiv.org/abs/1712.07516 |