Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings
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arXiv
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| Auteurs principaux: | , , , , , , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866913063333003264 |
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| author | Calmès, Baptiste Dotto, Emanuele Harpaz, Yonatan Hebestreit, Fabian Land, Markus Moi, Kristian Nardin, Denis Nikolaus, Thomas Steimle, Wolfgang |
| author_facet | Calmès, Baptiste Dotto, Emanuele Harpaz, Yonatan Hebestreit, Fabian Land, Markus Moi, Kristian Nardin, Denis Nikolaus, Thomas Steimle, Wolfgang |
| contents | We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-dévissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_07225 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings Calmès, Baptiste Dotto, Emanuele Harpaz, Yonatan Hebestreit, Fabian Land, Markus Moi, Kristian Nardin, Denis Nikolaus, Thomas Steimle, Wolfgang K-Theory and Homology Algebraic Topology 11E70, 18F25, 19G38 (Primary), 11E39, 11E81, 19D25 (Secondary) We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated, and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of Noetherian rings of global dimension $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen's localisation-dévissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi. |
| title | Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings |
| topic | K-Theory and Homology Algebraic Topology 11E70, 18F25, 19G38 (Primary), 11E39, 11E81, 19D25 (Secondary) |
| url | https://arxiv.org/abs/2009.07225 |