Equivariant localizing invariants of simple varieties

TL;DR

This paper introduces a class of simple varieties over fields, demonstrating how to analyze their equivariant invariants and applying results to important examples like Schubert varieties.

Contribution

It defines simple varieties and shows how to control their equivariant truncating invariants, including $K$-theory and trace maps, with applications to Schubert varieties.

Findings

$p$-adic cyclotomic trace is an equivalence for algebraically closed fields of characteristic $p$.

Goodwillie-Jones trace is an isomorphism in degree zero over $Q$.

Homotopy invariant $K$-theory $KH$ is equivariantly formal and topologically determined.

Abstract

We define a certain class of simple varieties over a field $k$ by a constructive recipe and show how to control their (equivariant) truncating invariants. Consequently, we prove that on simple varieties: (i) if $k=\overline{k}$ and $\mathrm{char} \ k = p$, the $p$-adic cyclotomic trace is an equivalence; (ii) if $k = \mathbb{Q}$, the Goodwillie-Jones trace is an isomorphism in degree zero; (iii) we can control homotopy invariant $K$-theory $KH$, which is equivariantly formal and determined by its topological counterparts. Simple varieties are quite special, but encompass important singular examples appearing in geometric representation theory. We in particular show that both finite and affine Schubert varieties for $GL_n$ lie in this class, so all the above results hold for them.