Hermitian K-theory via oriented Gorenstein algebras

TL;DR

This paper establishes a new geometric model for hermitian K-theory using oriented Gorenstein algebras, linking algebraic K-theory with motivic cohomology and providing computational insights.

Contribution

It introduces a novel identification of hermitian K-theory with the group completion of oriented Gorenstein algebra groupoids, and explores applications in motivic cohomology and tensor complexity.

Findings

Hermitian K-theory is equivalent to the group completion of oriented Gorenstein algebra groupoids.

Provides a motivic space model for hermitian K-theory via Hilbert schemes.

Shows minimal border rank tensors degenerate to the Coppersmith-Winograd tensor.

Abstract

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.