论文标题
Hermitian K理论稳定$ \ infty $ - 类别III：Grothendieck-Witt戒指
Hermitian K-theory for stable $\infty$-categories III: Grothendieck-Witt groups of rings
论文作者
论文摘要
我们建立了一个光纤序列，将环$ r $的经典色我们使用此光纤序列来删除以下假设，即从Grothendieck-Witt组中的各种结果中是$ r $的单位。 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 $是$ \ geq d+3 $的等效性。作为一个重要的工具，我们建立了Quillen Quillen destisation-dévissage序列的Hermitian类似物，用于Dedekind Rings，并使用它来解决Berrick-Karoubi的猜想。
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.
