Equivariant Decomposition Theorem

Updated 20 May 2026

The Equivariant Decomposition Theorem is a structural result that splits equivariant (co)homology, K-theory, and related theories based on finite group actions.
It employs a rational Burnside ring and orthogonal idempotents to index decompositions by conjugacy classes, unifying classical decomposition results.
The theorem simplifies complex computations in equivariant frameworks and facilitates applications in algebraic geometry, topology, and representation theory.

The equivariant decomposition theorem is a general structural result describing the splitting of equivariant (co)homology, K-theory, and related categories for spaces or algebraic objects equipped with a finite group action. These decompositions reflect, in a precise algebraic sense, the stratification of the object according to the orbits, subgroups, or fixed-point loci of the group action, and are indexed by conjugacy classes of subgroups or related data. Such theorems unify and generalize classical results, including the "Mackey decomposition" in representation theory, the idempotent splitting of the Burnside ring, and, in algebraic geometry, Vistoli’s decomposition for rational equivariant K-theory. A comprehensive formalism was established in (Sala, 20 May 2025), extending these ideas to generalized (co)homology theories with the Mackey property and localization.

1. Algebraic (Co)homology Theories, the Mackey Property, and Localization

Let $A = \operatorname{Spec} R$ be a base scheme and $G$ a finite constant group over $A$ . Consider an algebraic (co)homology theory $E_*(–, G)$ (or $E^*(–, G)$ ) assigning to each $G$ -scheme $X$ a graded $\mathbb Q$ -vector space that is functorial for equivariant morphisms. The foundational axioms necessary for the theory are:

Functoriality: Equivariant morphisms $f$ induce compatible pullback and pushforward maps.
Mackey property: For any inclusion $H \leq G$ , the induction and restriction functors correspond via the projection and Mackey formulas, encoding double coset decompositions and induction-restriction identities.
Localization: For any closed immersion $G$ 0 with open complement, there is a long exact sequence relating the equivariant theory of $G$ 1, $G$ 2, and $G$ 3.
Stack dependence: The theory depends only on the quotient stack $G$ 4, and if $G$ 5 with $G$ 6, then $G$ 7.

These properties collectively ensure compatibility with the geometry and representation theory of the group action and enable the construction of decomposition theorems leveraging the structure of the Burnside ring.

2. The Rational Burnside Ring and Idempotent Splitting

The integral Burnside ring $G$ 8 is the Grothendieck ring of finite $G$ 9-sets, encoding the isomorphism classes of such sets under disjoint union and Cartesian product. Its rationalization $A$ 0 admits a canonical splitting, due to Conlon, into orthogonal idempotent summands indexed by conjugacy classes of subgroups of $A$ 1:

$A$ 2

where each summand is generated by an explicit central idempotent $A$ 3 constructed inductively: $A$ 4, and for $A$ 5, $A$ 6, and $A$ 7.

These idempotents are mutually orthogonal and sum to the identity, acting as projectors onto pieces reflecting the fixed-point structure of $A$ 8-actions.

3. The Equivariant Decomposition Theorem: Precise Formulation and Proof Structure

Main Theorem ((Sala, 20 May 2025), Thm. 1.5):

Let $A$ 9 be an equivariant homology theory with $E_*(–, G)$ 0-coefficients satisfying the above axioms. Then:

(a) The action of $E_*(–, G)$ 1 yields a direct-sum decomposition

$E_*(–, G)$ 2

(b) For each $E_*(–, G)$ 3, the map induced by the inclusion $E_*(–, G)$ 4, providing "induction from the fixed-point locus," is an isomorphism:

$E_*(–, G)$ 5

Or, in compact notation,

$E_*(–, G)$ 6

Proof Outline:

A finite $E_*(–, G)$ 7-invariant stratification of $E_*(–, G)$ 8 by open sets yields Deligne–Mumford gerbe strata, facilitating induction on the number of strata (five-lemma).
In the trivial-gerbe case, explicit induction-restriction arguments show invertibility (up to normalization) over $E_*(–, G)$ 9.
For gerbes banded by normal subgroups, Morita equivalence and module-theoretic arguments reduce to the previous cases.
The axioms ensure functoriality and compatibility throughout, especially in the handling of the Burnside ring and Mackey property.

4. Applications: Algebraic K-Theory, Modular K-Theory, and Representation-Theoretic Models

4.1 Rational Equivariant K-Theory (Vistoli’s Theorem)

For $E^*(–, G)$ 0, the decomposition specializes to a direct sum over cyclic subgroups $E^*(–, G)$ 1 of $E^*(–, G)$ 2:

$E^*(–, G)$ 3

reflecting classical fixed-point formulas for the equivariant $E^*(–, G)$ 4-theory of quotient stacks. Only cyclic subgroups contribute, per the structure of $E^*(–, G)$ 5.

4.2 Modular K-Theory (Borne)

Let $E^*(–, G)$ 6-mod be a full subcategory with finite generators, and $E^*(–, G)$ 7. The decomposition includes all $E^*(–, G)$ 8, including nonabelian subgroups, producing

$E^*(–, G)$ 9

In positive characteristic, this provides cases where nonabelian fixed-point data contribute summands to the equivariant $G$ 0-theory.

4.3 Equivariant K-Theory Decomposition in Topology

For finite groups and spaces with a normal subgroup $G$ 1 acting trivially, the equivariant $G$ 2-theory decomposes into a direct sum of twisted $G$ 3-equivariant $G$ 4-theories, indexed by $G$ 5-orbits of irreducible representations of $G$ 6; see (Gómez et al., 2016):

$G$ 7

where the twist $G$ 8 is a 2-cocycle obstructing the extension of $G$ 9-representations. This generalizes the algebraic decomposition theorem to topological and operator-algebraic contexts.

5. Relation to Burnside Ring Theory and Fixed Point Decompositions

The underlying mechanism is the action of the rational Burnside ring on equivariant (co)homology. The central idempotents $X$ 0 project to the parts controlled by $X$ 1-fixed points and their normalizer symmetries. This perspective is reflected in the fixed-point localization theorems for equivariant $X$ 2-theory and bordism—the decomposition of spectra or modules corresponds to summing over fixed-point data modulo symmetries.

This formalism is unified with the Mackey-classification perspective (induction/restriction across subgroup data), stable homotopy splittings in equivariant stable categories, and supports advanced computations in both algebraic and topological equivariant theories.

6. Extensions and Generalizations

The equivariant decomposition theorem framework extends to:

Twisted K-theory and projective representations, where obstructions from group cohomology appear as twisting cocycles (Gómez et al., 2020).
Proper actions of Lie groups with compact normal subgroups, using central extensions to produce twisted equivariant $X$ 3-theories indexed by orbits of the irreducible representations (Angel et al., 2020).
Semiorthogonal decompositions of equivariant and orbifold derived categories, where categorical blocks can be described by fixed-point/centralizer data (Polishchuk et al., 2015, Lim et al., 2017).
Homotopical and spectral contexts, e.g., splittings of $X$ 4-equivariant spectra indexed by conjugacy classes in the rational stable homotopy category.

The construction is robust under various modifications (e.g., base change in algebraic geometry, as for invariant divisors (Lim et al., 2017)) and is compatible with further refinements such as localization in representation theory and derived categories.

7. Significance, Computability, and Examples

The equivariant decomposition theorem dramatically simplifies calculations in equivariant (co)homology, $X$ 5-theory, and categorical settings by reducing global computations to local fixed-point contributions, computed modulo normalizers/centralizers and weighted by cohomological or representation-theoretic obstructions.

The decomposition is particularly powerful when:

The underlying theory (e.g., $X$ 6-theory, bordism) is semi-simple rationally (i.e., after tensoring with $X$ 7),
The Burnside ring action provides a complete set of orthogonal projectors,
The fixed-point loci and their symmetry are geometrically accessible.

Explicit examples, such as cyclic and dihedral group actions, modular $X$ 8-theory in positive characteristic, and direct calculations in the category of $X$ 9-equivariant coherent sheaves, demonstrate the breadth and computability of the decomposition (Sala, 20 May 2025, Angel et al., 2017, Gómez et al., 2016). In representation theory and topology, the corresponding splittings underpin deep connections between local symmetry, global algebro-geometric invariants, and categorical orthogonality.

Key Reference:

A. Vistoli’s decomposition and its generalizations: "On a decomposition theorem in equivariant generalized homology theories for finite group actions" (Sala, 20 May 2025). Further: (Gómez et al., 2016, Polishchuk et al., 2015, Angel et al., 2020, Angel et al., 2017).