Restriction to finite-index subgroups as \'etale extensions in topology, KK-theory and geometry

TL;DR

This paper demonstrates that restricting to a finite-index subgroup in equivariant homotopy, KK-theory, and derived categories produces a finite separable extension, similar to finite étale extensions in algebraic geometry.

Contribution

It establishes a new analogy between subgroup restriction in equivariant theories and finite étale extensions in algebraic geometry.

Findings

Restriction to finite-index subgroups yields finite separable extensions.

Analogies are drawn between topological and algebraic geometric structures.

Provides a unified perspective across multiple equivariant theories.

Abstract

For equivariant stable homotopy theory, equivariant KK-theory and equivariant derived categories, we show how restriction to a subgroup of finite index yields a finite commutative separable extension, analogous to finite \'etale extensions in algebraic geometry.