Hermitian K-theory via oriented Gorenstein algebras

Abstract

AbstractWe show that the hermitian K-theory space of a commutative ringRcan be identified, up to𝐀1{\mathbf{A}^{1}}-homotopy, with the group completion of the groupoid of oriented finite GorensteinR-algebras, i.e., finite locally freeR-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.