Chromatic Purity in Hermitian K-Theory at $p=2$

TL;DR

This paper explores the chromatic purity of L-theory at prime 2, demonstrating that L-theory does not exhibit chromatic redshift and establishing new connections between GW, L-theory, and chromatic localization.

Contribution

It introduces chromatic purity results for L-theory at p=2 using Poincaré categories and Hermitian trace methods, and proves a chromatic analogue of the homotopy limit problem for GW-theory.

Findings

L-theory does not exhibit chromatic redshift.

Higher chromatic vanishing of quadratic L-theory for idempotent complete categories.

Dependence of T(n+1)-local GW-theory on T(n+1)-local K-theory for rings with involution.

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 Poincar\'e categories as laid out in the series of papers by Calm\`es 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.