Chromatic Purity in Hermitian K-Theory at $p=2$
Jordan Levin
TL;DR
This work analyzes chromatic behavior of L-theory at the prime $p=2$ by embedding additive Hermitian $K$-theory into the framework of Poincaré categories. It proves a chromatic purity result: for an $\mathbf{E}_1$-ring with anti-involution $(A,\sigma)$, the maps to $L_n^f A$ and to $L_T A$ induce $T(n)$-local equivalences on quadratic $L$-theory, implying no chromatic redshift and enabling higher chromatic vanishing results in idempotent complete Poincaré categories. The paper also develops descent/ fracture-square techniques for Hermitian $K$-theory, deriving chromatic descent properties for $GW$-theory and establishing a chromatic analogue of the homotopy limit problem in this Hermitian setting. Leveraging additive $L$-theory and trace methods, the results integrate with existing redshift and vanishing frameworks and set the stage for further applications of Hermitian trace methods to chromatic questions in $GW$ and $L$-theory.
Abstract
In this article we investigate the question of chromatic purity of L-theory. To do so, we utilize the theory of additive GW and L-theory in the language of Poincaré categories as laid out in the series of papers by Calmès et al. We apply this theory to chromatically localised L-theory at the prime $p=2$ and recover the L-theoretic analogues of chromatic purity for $E_1$-rings with involution. From this, we deduce that L-theory does not exhibit chromatic redshift. We deduce the higher chromatic vanishing of quadratic L-theory of arbitrary idempotent complete categories, thereby allowing the use of Hermitian trace methods to probe chromatic behaviour of GW and L-theory. Finally, we show that for $T(n+1)$-acyclic rings with involution, $T(n+1)$-local GW-theory depends only on $T(n+1)$-local K-theory and the associated duality, thereby proving a chromatic analogue of the homotopy limit problem for GW-theory.