厄米K理论、导出的等价关系和Karoubi基本定理
Hermitian K-theory, derived equivalences and Karoubi's Fundamental Theorem
摘要：
在具有弱等价和对偶的dg范畴的框架内，我们证明了高阶Grothendieck-Witt群（aka. Hermitian K-群）在导出等价下不变，Morita正合序列诱导Grothendieck-Witt群的长正合序列。这意味着一个代数Bott序列和Karoubi基本定理的一个新的证明和推广。对于向量丛的高阶Grothendieck-Witt群（可能是奇异的）具有大量的线丛使得2在正则函数的环中可逆，我们得到了Nisnevich覆盖和沿正则嵌入中心爆破的Mayer-Vietoris长精确序列，射影丛公式和Bass基本定理.对于凝聚Grothendieck-Witt群，我们得到了一个类似于Quillen的K '-局部化定理的局部化定理.
英文摘要：
Within the framework of dg categories with weak equivalences and duality that have uniquely 2-divisible mapping complexes, we show that higher Grothendieck-Witt groups (aka. hermitian K-groups) are invariant under derived equivalences and that Morita exact sequences induce long exact sequences of Grothendieck-Witt groups. This implies an algebraic Bott sequence and a new proof and generalization of Karoubi's Fundamental Theorem. For the higher Grothendieck-Witt groups of vector bundles of (possibly singular) schemes with an ample family of line-bundles such that 2 is invertible in the ring of regular functions, we obtain Mayer-Vietoris long exact sequences for Nisnevich coverings and blow-ups along regularly embedded centers, projective bundle formulas, and a Bass fundamental theorem. For coherent Grothendieck-Witt groups, we obtain a localization theorem analogous to Quillen's K'-localization theorem.
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