On very effective hermitian $K$-theory

TL;DR

The paper demonstrates that the very effective cover of hermitian K-theory in motivic homotopy theory effectively generalizes connective real topological K-theory, facilitating calculations and aligning with expected algebraic structures.

Contribution

It establishes that the very effective cover of hermitian K-theory serves as a suitable algebro-geometric analogue of connective real topological K-theory, with correct Betti realization and computable spectral sequences.

Findings

Aligns motivic cohomology with the motivic Steenrod algebra

Ensures Betti realization matches topological K-theory

Spectral sequences become more computationally accessible

Abstract

We argue that the very effective cover of hermitian $K$-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological $K$-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.