The geometric concentration theorem
Olivier Haution
TL;DR
The paper proves a geometric concentration theorem for actions of linearly reductive groups on affine schemes, establishing the existence of a representation V with no trivial summand whose pullback to X has a G-equivariant section vanishing exactly on the fixed locus X^G. It then translates this geometric concentration into a motivic and cohomological framework, showing that after inverting Euler classes E_G associated to representations in \\mathcal{V}_G, the equivariant motivic stable homotopy category SH^G(X) reduces to SH^G(X^G), with analogous statements for Borel-type theories via a pseudo-orientation. The work develops a robust theory of equivariant cohomology, Euler classes, and localization, and applies these ideas to obtain a motivic Smith-type result for finite p-group actions, expressing fixed-point cohomology in terms of equivariant data and the Steenrod algebra action. Collectively, these results generalize classical concentration theorems to broader group types and cohomology theories, and provide powerful tools for fixed-point and localization calculations in algebraic geometry and motivic homotopy theory.
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.