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تسجيل مستخدم جديدHermitian K-theory for stable $infty$-categories III: Grothendieck-Witt groups of rings
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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 Ranickis 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 Quillens localisation-devissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.
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This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable $infty$-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings an
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