基于定向Gorenstein代数的Hermitian K-理论
Hermitian K-theory via oriented Gorenstein algebras
摘要：
证明了交换环R的Hermitian K-理论空间在A^1-同伦下可以与定向有限Gorenstein R-代数的广群的群完备化相一致，即，具有平凡化对偶层的有限局部自由R-代数我们推出，埃尔米特K-理论是普遍的广义motivic上同调理论与转移沿定向有限Gorenstein态射。作为应用，我们得到了作为运动空间的Hermitian K-理论的Hilbert格式模型。我们也给出了一个应用程序的计算复杂性：我们证明了1-通用的最小边界秩张量退化到大Coppersmith-Winograd张量。
英文摘要：
We show that the hermitian K-theory space of a commutative ring R can be identified, up to A^1-homotopy, with the group completion of the groupoid of oriented finite Gorenstein R-algebras, i.e., finite locally free R-algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith-Winograd tensor.
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