---
title: Cohomological Mackey Algebra Overview
url: https://www.emergentmind.com/topics/cohomological-mackey-algebra
type: topic
---

# Cohomological Mackey Algebra Overview

The cohomological Mackey algebra is a finite-dimensional associative algebra associated to a finite group $G$ and a commutative ring or field $k$, defined as a quotient of the Mackey algebra by imposing cohomological relations. It plays a central role in encoding the algebraic structure of cohomological Mackey functors—functors that model induction, restriction, and transfer in equivariant algebra and topology—and in providing a foundation for the homological algebra of induction-restriction systems in modular representation theory, equivariant cohomology, and algebraic topology.

## 1. Structural Definition and Fundamental Properties

The cohomological Mackey algebra, denoted $\operatorname{co}\mu_k(G)$ or similar (notation varies), is a quotient of the Mackey algebra $\mu_k(G)$. The Mackey algebra is generated by symbols $T_H^K$ (transfer/induction), $R_H^K$ (restriction), and $c_g$ (conjugation) for subgroups $H \leq K \leq G$ and $g \in G$, subject to relations encoding the axioms of Mackey functors. In the cohomological Mackey algebra, one adds the "cohomological relation":
$$
T_H^K R_H^K = [K : H] \cdot \mathrm{id}
$$
for all inclusions $H \leq K$, reflecting the property that composition of restriction and transfer for cohomological functors scales as the group index.

As a $k$-algebra, $\operatorname{co}\mu_k(G)$ is finite-dimensional and is Morita equivalent to certain Hecke algebras of permutation modules due to Yoshida’s equivalence. Its module category is equivalent to the category of cohomological Mackey functors for $G$ over $k$.

Yoshida’s theorem provides an important isomorphism:
$$
\operatorname{co}\mu_k(G) \cong \operatorname{End}_{kG}(k\Omega_G)
$$
where $k\Omega_G$ is the direct sum over all transitive $G$-sets (i.e., permutation modules), with $G$ acting diagonally.

## 2. Block Structure and Cartan Matrix Analysis

A key structural feature is the block decomposition of $\operatorname{co}\mu_k(G)$, paralleling that of the group algebra $kG$. There is a canonical one-to-one correspondence between the blocks of $kG$ and the central primitive idempotents (blocks) of $\operatorname{co}\mu_k(G)$. Explicitly, for central elements $z \in Z(kG)$ written as $z = \sum_{x \in G} \lambda_x x$, the corresponding central element in $\operatorname{co}\mu_k(G)$ is:
$$
L(z) = \sum_{H \leq G} \frac{1}{|H|} \sum_{x \in G} \lambda_x t_H c_{x,H}
$$
where $t_H$ and $c_{x,H}$ denote the algebra generators reflecting transfer and conjugation.

**Cartan matrix structure** is crucial for representation theory. For the Cartan matrix $C(\operatorname{co}\mu_k(G))$ (whose entries count composition factors of projective indecomposable modules), the rank admits an explicit formula [1001.0321]:
$$
\operatorname{rk} C(\operatorname{co}\mu_k(G)) = \sum_{R \in [C_p(G)]} |N_G(R) \backslash C_G(R)_{p'}|
$$
where $[C_p(G)]$ is a set of $G$-conjugacy class representatives of cyclic $p$-subgroups and $C_G(R)_{p'}$ denotes the $p'$-elements in the centralizer of $R$.

The Cartan matrix is **nonsingular if and only if** $G$ is $p$-nilpotent with cyclic Sylow $p$-subgroups. This equivalence is preserved when restricted to blocks: a block algebra of $\operatorname{co}\mu_k(G)$ is nonsingular precisely when the associated block of $kG$ is nilpotent with cyclic defect groups.

## 3. Homological and Finiteness Properties

The homological characteristics of $\operatorname{co}\mu_k(G)$ are governed by the structure of $G$'s $p$-Sylow subgroups.

- **Gorenstein property**: $\operatorname{co}\mu_k(G)$ is Gorenstein if and only if, for each prime divisor $p$ of $|G|$, the Sylow $p$-subgroups are cyclic or, in characteristic $2$, either cyclic or dihedral. This ensures that projective objects have finite injective dimension and vice versa [1503.03955].
- **Finite global dimension**: $\operatorname{co}\mu_k(G)$ has finite global dimension if and only if $|G|$ is invertible in $k$ or $k$ has characteristic $2$ and Sylow $2$-subgroups are cyclic of order $2$. Over the integers, finite global dimension occurs simultaneously with all Sylow $p$-subgroups being cyclic (for $p$ odd) or cyclic/dihedral for $p=2$ [1503.03955].
- **Dominant dimension**: For a block $B$ of $kG$ with a nontrivial defect group, the dominant dimension of $\operatorname{co}\mu_k(B)$ is $2$ [1703.07820].

The **finitistic dimension** of a block algebra of $\operatorname{co}\mu_k(G)$ is $1 + s(P)$, where $s(P)$ is the sectional $p$-rank of the defect group $P$ [1510.01627].

## 4. Derived and Wild Representation Type

The module and derived category of $\operatorname{co}\mu_k(G)$ can be wild or tame, depending on the group structure and field. Over an algebraically closed field $k$ of characteristic $p$:

- $\operatorname{co}\mu_k(G)$ is **derived wild** if and only if the Sylow $p$-subgroup of $G$ has order greater than $2$. In particular, for $G$ a nontrivial $p$-group other than $C_2$, the algebra is derived wild [2509.17604].
- The wildness transfers to the stable module category of compact $G$-equivariant $H\underline{k}$-modules in homotopy theory, rendering classification of compact objects infeasible whenever $G$ surjects onto a $p$-group of order $>2$.

Conversely, for $G = C_2$, the cohomological Mackey algebra is gentle and derived tame; yet even then, the full Mackey algebra can be wild [2509.17604].

## 5. Equivalences, Blocks, and Connections to Broué-type Conjectures

There are deep connections between structural equivalences of $\operatorname{co}\mu_k(G)$ and classical block theory:

- A **permeable derived equivalence** (induced by complexes of $p$-permutation bimodules that respect the subcategory of $p$-permutation modules) between two block algebras $RGb$ and $RHc$ (with $R$ a complete discrete valuation ring or field) induces a derived equivalence between the associated blocks of cohomological Mackey algebras [1406.6241].
- For group algebras associated to blocks $b$ and $b'$ that are splendidly derived equivalent (as in Broué’s abelian defect group conjecture), the corresponding blocks of $\operatorname{co}\mu_k(G)$ are also derived equivalent. Thus, invariants such as Cartan matrices are preserved under such derived equivalences.
- Conversely, an equivalence between categories of cohomological Mackey functors for blocks induces a permeable Morita equivalence between the block group algebras [1609.07870].

These results support using the cohomological Mackey algebra as a test for derived equivalences, and provide new invariants—such as the nonsingularity of Cartan matrices—that must be shared by splendidly equivalent blocks.

## 6. Explicit Presentation and Functorial Perspective

The functorial viewpoint informs the algebra’s structure:

- **Intrinsic description via source algebras**: For a block $b$ with source algebra $A$, the category of cohomological Mackey functors for $b$ is equivalent to $\mathrm{mod}(\mathrm{End}_A(N)^{\mathrm{op}})$, where $N = \bigoplus_{Q \leq P} A \otimes_{OQ} O$ sums over subgroups of a defect group $P$ [1510.01627]. This allows the construction of explicit two-sided tilting complexes realizing derived equivalences.
- **Idempotent recollement**: The module category for a block, the category of cohomological Mackey functors, and the subcategory vanishing at the trivial group form an idempotent recollement, reflecting a tight structural linkage [1510.01627].

## 7. Applications, Computations, and Broader Context

The cohomological Mackey algebra's module categories serve as a framework for:

- The calculation of extension algebras in modular representation theory, especially for elementary abelian $p$-groups and in computing the self-extension algebra of simple Mackey functors [1010.1346].
- Understanding the projective classification of $R[G]$-lattices and permutation modules via cohomological vanishing criteria in cyclic $p$-groups, including a new cohomological proof of the classical equivalence between vanishing of Tate cohomology and being a permutation lattice [1209.1932].
- Providing invariants for Bredon cohomology and refining closure under group extensions, subgroups, and finite quotients for the cohomological dimension of groups with respect to Mackey functors [1305.1227, 1310.6262].
- Functoriality under restriction, inflation, and induction functors, which is essential in the context of higher categorical generalizations (Mackey 2-motives) and their connection to the blocks of group algebras via categorical decompositions [2103.03974, 2201.04744].

## 8. Summary Table: Rank and Nonsingularity of the Cartan Matrix

| Object                   | Rank Formula                                                                                                                                           | Nonsingularity Condition                        |
|--------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------|
| $\operatorname{co}\mu_k(G)$     | $\displaystyle \sum_{R \in [C_p(G)]} |N_G(R)\backslash C_G(R)_{p'}|$                                                                                    | $G$ is $p$-nilpotent with cyclic Sylow $p$-subgroups   |
| Block $\operatorname{co}\mu_k(b)$ | $\displaystyle \sum_{(R, c)\in [C_p(b)]} |N_G(R)\backslash \mathrm{Irr}_k(k C_G(R)c)|$                                                               | $b$ nilpotent, cyclic defect                     |

## 9. Future Directions and Open Problems

Outstanding problems include:

- Refining invariants and equivalences at the level of derived or stable categories for both algebraic and topological versions of Mackey functors, especially when the algebra is wild.
- Explicit computation of Cartan matrices and their ranks for further classes of finite groups and blocks, especially beyond the nilpotent/cyclic case.
- Further development of the structure theory for cohomological Mackey 2-functors, and their connections to equivariant stable homotopy theory and localization phenomena [2103.03974, 2201.04744].

The cohomological Mackey algebra thus lies at the intersection of modular representation theory, algebraic topology, and higher category theory, serving as a unifying structure for functorial induction-restriction systems, block theory, and derived equivalence theory.

Source: https://www.emergentmind.com/topics/cohomological-mackey-algebra