The geometric concentration theorem

TL;DR

This paper proves a geometric version of the concentration theorem for reductive group actions on schemes, showing equivalences in equivariant motivic homotopy theory and extending classical results beyond diagonalizable groups.

Contribution

It introduces a purely geometric form of the concentration theorem applicable to linearly reductive groups, generalizing known algebraic geometry results and connecting to motivic homotopy theory.

Findings

Establishes a geometric concentration theorem for reductive group actions.

Shows equivalence of equivariant and fixed locus motivic homotopy theories after inverting Euler classes.

Derives a motivic Smith theory analogous to topological results.

Abstract

We establish a purely geometric form of the concentration theorem (also called localization theorem) for actions of a linearly reductive group $G$ on an affine scheme $X$ over an affine base scheme $S$. It asserts the existence of a $G$-representation without trivial summand over $S$, which acquires over $X$ an equivariant section vanishing precisely at the fixed locus of $X$. As a consequence, we show that the equivariant stable motivic homotopy theory of a scheme with an action of a linearly reductive group is equivalent to that of the fixed locus, upon inverting appropriate maps, namely the Euler classes of representations without trivial summands. We also discuss consequences for equivariant cohomology theories obtained using Borel's construction. This recovers most known forms of the concentration theorem in algebraic geometry, and yields generalizations valid beyond the setting of actions of diagonalizable groups on one hand, and that of oriented cohomology theories on the other hand. Finally, we derive a version of Smith theory for motivic cohomology, following the approach of Dwyer--Wilkerson in topology.