Cohomological Mackey Theory

• Mackey theory with cohomological techniques is a framework that combines induction, restriction, and conjugation with homological algebra to compute invariants in group theory.
• It introduces novel constructions such as the OGIN inflation functor and explicit extension algebra presentations, enhancing the analysis of permutation and simple modules.
• This synthesis bridges representation theory, algebraic topology, and geometric methods, enabling derived equivalences and categorical decompositions in complex algebraic structures.

Mackey theory with cohomological techniques refers to a sustained synthesis between the foundational structure of Mackey functors—encoding induction, restriction, and conjugation for group representations—and the sophisticated machinery of homological algebra, cohomology, and modern categorical frameworks. This interplay enables the articulation of deep structural, computational, and categorical results in group theory, representation theory, algebraic topology, algebraic geometry, and beyond. Cohomological techniques sharpen Mackey theory by providing invariants, new functorial constructions, explicit extension algebras, and links to geometric and higher-categorical contexts.

1. Fundamentals of Cohomological Mackey Functors

A Mackey functor for a finite group $G$ consists of systems of abelian groups equipped with restriction, induction (transfer), and conjugation maps between various subgroups, subject to the “double coset formula” and compatibility relations. A cohomological Mackey functor refines this by imposing the cohomologicality condition: $$\operatorname{Tr}_K^H \circ \operatorname{Res}_K^H = |H:K| \cdot \operatorname{id}_{M(H)}$$ for all $K \leq H \leq G$, reflecting the behaviour of transfer and restriction in group cohomology. The category of cohomological Mackey functors (e.g., %%%%2%%%% over a ring $R$) is abelian and supports a rich homological structure, including projective and injective objects, extension groups, and derived categories.

Yoshida’s theorem states that cohomological Mackey functors correspond (via idempotent completion) to contravariant $k$-linear functors on the category of permutation $kG$-modules (or $p$-permutation modules, depending on context), making the analysis of such functors amenable to the machinery of module categories and homological algebra.

2. Functorial Constructions: Inflation, Restriction, and Explicit Presentations

The introduction of new functorial constructions is a central application of cohomological techniques within Mackey theory. Standard inflation from $G/N$ to $G$ fails to preserve the cohomological property when $N$ is a $p$-group. To address this, a new “OGIN” inflation functor is constructed as follows (Bouc et al., 2010):

• For a permutation $kG$-module $U$, define the $N$-trace submodule:

$$\operatorname{tr}^{(N)}(U) = \{u\in U \mid u = \sum_{n\in N} n\cdot v\ \text{for some}\ v\in U\}$$

• For any finite $G$-set $X$, $\operatorname{tr}^{(N)}(kX)$ is a direct summand of a permutation module for $G/N$.
• Precomposition with $X \mapsto \operatorname{tr}^{(N)}(kX)$, followed by passage to functor categories under Yoshida equivalence, defines an exact functor:

$$\mathrm{OGIN}:\ \mathrm{ME}(G/N) \to \mathrm{ME}(G)$$

• For $V$ a simple $k(G/N)$-module,

$$\mathrm{OGIN}(S_{1,V}) \cong S_{1, \mathrm{Inf}_G V}$$

with further splitting properties when $G = N \rtimes H$.

This construction enables explicit comparison of extension groups, supports induction techniques for computation, and preserves the essential features of cohomological functors across group extensions.

For elementary abelian $p$-groups $G$, there is an explicit presentation of the graded extension algebra

$$\mathcal{E} = \operatorname{Ext}^*_{\mathrm{ME}(G)}(S_1, S_1)$$

where $S_1$ is the simple cohomological Mackey functor concentrated at $1$. Generators $T_y$ (degree 1) parametrize group homomorphisms $y: G \to k^+$ and generators $Y_X$ (degree 2) correspond to order $p$ subgroups $X \leq G$. Relations (see Theorem 1.2 and 1.3 in (Bouc et al., 2010)) include:

• Additivity: $T_{y+z} = T_y + T_z$,
• Quadratic: $T_y^2 = 0$,
• Commutator: $[T_{y}, Y_X] = 0$ if $X \leq \ker y$,
• Further $[Y_{X}, Y_{Y}]$ relations depending on the orderings and $p$.

A sharp description of the Poincaré series is given: $$P(t) = \prod_{i=0}^{r-1} \frac{1}{1 - t - (p^{i} - 1) t^2}$$ where $r = \mathrm{rank}(G)$.

These explicit presentations allow for computations of decomposition series, complexity invariants, and support inductive proofs.

3. Applications to Representation Theory and Homology

Cohomological Mackey functors underpin several structural results in modular representation theory and cohomological algebra:

• They detect permutation modules—e.g., for cyclic $p$-groups, a lattice module $M$ is permutation if and only if $H^1(U, \mathrm{Res}^G_U M) = 0$ for all $U \leq G$ (Torrecillas et al., 2012).
• The global dimension of the category of cohomological Mackey functors for a cyclic $p$-group is 3; the (projective) lattice subcategory has dimension 2.
• Projective dimensions in the full Mackey functor category are rigid: only projective functors have finite such dimension, while cohomological Mackey functors may be Gorenstein under group-theoretic conditions (Sylow subgroups cyclic or dihedral) and have finite global dimension precisely when $|G|$ is invertible or Sylow subgroups are cyclic of order 2 (Bouc et al., 2015).

Explicit decompositions and recollements of categories of cohomological Mackey functors can be obtained by analyzing associated source algebras and their endomorphism rings, yielding robust invariants for blocks of finite group algebras and facilitating the paper of derived and Morita equivalences at the level of Mackey categories (Linckelmann, 2015, Linckelmann et al., 2016).

4. Categorical and 2-Categorical Generalizations

Cohomological Mackey functors admit higher-categorical enhancements:

• The bicategory of Mackey 2-motives and its cohomological quotient (modding out 2-cells corresponding to the cohomological relation $E_i \circ T_{n_i} = [G:H] \cdot id$) categorifies the classical Yoshida theorem (Balmer et al., 2021).
• The decategorification process recovers endomorphism rings isomorphic to the center of the group algebra, linking block theory and the motivic decomposition governed by the crossed Burnside ring and its image in $Z(kG)$ (Oda et al., 2022).

These categorical settings provide a robust context to generalize and unify derived equivalences, block decompositions, and the structure of module and functor categories. They streamline computations and conceptual understanding of how local and global invariants interact.

5. Integration of Mackey Theory with Cohomological and Geometric Techniques

Cohomological techniques enable the extension of Mackey theory into geometrically and homotopically rich environments:

• In the context of equivariant cohomology and stable homotopy, spectral Mackey functors—presheaves enriched over spectra on suitable spectral Burnside or multicategorical frameworks—allow for the algebraic classification of genuine $G$-objects, equivariant cohomology theories, and stable model categories (Johnson et al., 2022).
• For critical cohomology of quotient stacks, induction and restriction systems can be constructed to endow such cohomology theories with a localized induction-restriction system, satisfying a cohomological Mackey formula involving Euler class normalizations and double coset summations (Hennecart, 14 May 2025). Such results have direct implications for the structure of cohomological Hall algebras and enumerative geometry.
• In other analytic and geometric settings, such as the solution theory for twisted cohomological equations over partially hyperbolic flows, Mackey theory provides the framework to decompose unitary representations and analyze obstructions and Sobolev estimates (Wang, 2018).

These advanced implementations show how Mackey theory, enhanced with cohomological techniques, provides computational and conceptual tools for modern geometric representation theory, algebraic geometry, and equivariant topology.

6. Cohomological Mackey Functors in Class Field Theory and Higher Arithmetic

Cohomological Mackey functors are used to generalize class field theory. Through the work on Fesenko–Neukirch reciprocity and ramification theory (Thiel, 2011):

• Cohomological Mackey functors model class field theories by assigning to subgroups of the Galois/absolute Galois group values such as Milnor–Paršin $K$-groups.
• Reciprocity maps in this context generalize the Frobenius morphism construction and allow for uniform descriptions of abelian extensions in both rank-one and higher rank local field settings, with applications to the paper of higher-dimensional local fields, ramification, and the Paršin topology.

This generalizes both the classical local theory and allows adaptation to settings where invariants are not discrete modules over Galois groups but more general Mackey-structured coefficient systems.

7. Unification and Thematic Directions

Across these contexts, several unifying themes emerge:

• Structure Theorems and Homological Invariants: Explicit presentations of extension algebras, projective/injective dimensions, Gorenstein/global dimensions, and their direct connection to group-theoretic properties (e.g., Sylow subgroup structure) provide fine control in representation theory and cohomological classification.
• Derived and Block Equivalences: Morita and derived equivalences among blocks of group algebras and their associated categories of cohomological Mackey functors are closely linked through source algebra and endomorphism realizations, and can be characterized at the categorical and 2-categorical level.
• Geometric and Topological Applications: The framework supports the algebraic modelling of equivariant phenomena, the computation of invariants in algebraic topology (Bredon cohomology, $RO(G)$-graded cohomology), and the geometric organization of cohomological theories on quotient stacks and Landau–Ginzburg models.
• Obstruction and Classification via Cohomology: Cohomological obstruction maps classify the possibility of extending modules over graded algebras, linking graded Artin–Wedderburn decompositions and projective representation theory to $H^2$-classes.
• Finiteness and Dimension Criteria: Criteria for finiteness properties (FP$_n$ type, cohomological dimension) for Mackey and cohomological Mackey functors coincide with analogous conditions in Bredon cohomology and can be checked on families of subgroups (notably $p$-subgroups), reducing the complexity of verification and computation (John-Green, 2013).

Mackey theory, when augmented with cohomological and homological methods, is thus a powerful and flexible language for organizing and analyzing rich algebraic, geometric, and categorical invariants across mathematics.