The Kessar–Linckelmann refinement of Broué's abelian defect group conjecture over arbitrary p-modular systems

Broué's abelian defect group conjecture predicts that a block $b$ of a group algebra with abelian defect group $P$ is derived equivalent (indeed splendidly Rickard equivalent) to its Brauer correspondent in $N_G(P)$ . Kessar and Linckelmann refined this: the predicted splendid Rickard equivalence should exist over any p-modular system $(K,\mathcal{O},k)$ , in particular over $(\mathbb{Q}_p,\mathbb{Z}_p,\mathbb{F}_p)$ , so that the equivalence is independent of the choice of coefficient field. The conjecture is verified only case-by-case (cyclic, Klein four, symmetric and alternating groups, $SL(2,q)$ / $GL(2,q)$ , unipotent blocks of $GL_n(q)$ , p-nilpotent groups, RoCK blocks of double covers, and — by Miller (2026), via a new characterization of endosplit p-permutation resolutions and their Galois descent — the condition- $(\ast)$ blocks of p-solvable groups with abelian Sylow subgroup). Whether the refinement holds for all blocks of all finite groups with abelian defect groups — or even for all p-solvable groups — remains open; Miller (2026) documents the concrete obstruction (Harris–Linckelmann's Lemma 3.3 fails over fields not large enough for $G$ ).

Background

Let $p$ be a prime. A p-modular system $(K,\mathcal{O},k)$ consists of a complete discrete valuation ring $\mathcal{O}$ with residue field $k=\mathcal{O}/J(\mathcal{O})$ of characteristic $p$ and fraction field $K$ ; the minimal choice is $(\mathbb{Q}_p,\mathbb{Z}_p,\mathbb{F}_p)$ . A block of $\mathcal{O}G$ is a primitive idempotent $b$ of the center $Z(\mathcal{O}G)$ ; the corresponding block algebra is $\mathcal{O}Gb$ . A defect group $P$ of $b$ is a maximal p-subgroup of $G$ on which the Brauer homomorphism (the map, defined on $P$ -fixed elements of $\mathcal{O}G$ , that retains precisely the terms indexed by the centralizer $C_G(P)$ , with values in $kC_G(P)$ ) does not annihilate $b$ ; Brauer's first main theorem gives a canonical bijection between blocks of $\mathcal{O}G$ with defect group $P$ and blocks of $\mathcal{O}N_G(P)$ with defect group $P$ , and the block $c$ corresponding to $b$ is its Brauer correspondent. Broué's abelian defect group conjecture (1990) asserts that when $P$ is abelian, $\mathcal{O}Gb$ and $\mathcal{O}N_G(P)c$ are derived equivalent; more precisely, Rickard conjectured a splendid Rickard equivalence: a bounded complex $X$ of $(\mathcal{O}Gb,\mathcal{O}N_G(P)c)$ -bimodules whose terms are p-permutation modules (modules that restrict to permutation modules on every p-subgroup), which is a Rickard tilting complex ( $X\otimes^{\mathbb{L}}_{\mathcal{O}N_G(P)c}X^{\ast}$ has homology $\mathcal{O}Gb$ concentrated in degree zero) and is splendid (every indecomposable direct summand of each term of $X$ is, regarded as a module over $G\times N_G(P)$ , a p-permutation module with twisted-diagonal vertex). Most verifications construct such a complex over a splitting field for $G$ (a field $k$ over which every simple module of every subgroup of $G$ is absolutely simple; such fields are called large enough for $G$ ), and it is not automatic that the construction descends to a smaller field such as $\mathbb{F}_p$ : for a field extension $k'/k$ , a $k'G$ -module $M'$ descends to $k$ if $M'\cong k'\otimes_k M$ for some $kG$ -module $M$ , and small fields can genuinely change the block-theoretic picture (over $\mathbb{F}_4$ , a block of $\mathbb{F}_2G$ can split into several blocks permuted by $\mathrm{Gal}(\mathbb{F}_4/\mathbb{F}_2)$ ). Kessar and Linckelmann (2018) therefore proposed the refined conjecture that Broué's conjecture should hold over any p-modular system $(K,\mathcal{O},k)$ , not just over splitting systems: for any block $b$ of $\mathcal{O}G$ with abelian defect group $P$ and Brauer correspondent $c$ , there is a splendid Rickard equivalence between $\mathcal{O}Gb$ and $\mathcal{O}N_G(P)c$ . This refinement connects to Galois-invariance refinements of the McKay and Alperin–McKay conjectures (Isaacs–Navarro, Navarro, Turull). Key tools are endosplit p-permutation resolutions: bounded complexes of p-permutation modules with homology concentrated in degree zero whose endomorphism complex is split, introduced by Rickard (1996); they resolve endo-p-permutation modules (modules $M$ for which $M^{\ast}\otimes_k M$ is a p-permutation module). Harris–Linckelmann (2000) verified Broué's conjecture for all blocks with abelian defect groups of p-solvable groups (finite groups built from p-groups and p'-groups in composition series) by reducing to groups with $G=O_{p',p,p'}(G)$ (the largest normal subgroup in the series $O_{p'}(G)\leq O_{p',p}(G)\leq O_{p',p,p'}(G)$ , where $O_{p'}(G)$ is the largest normal subgroup of $G$ of order prime to $p$ and each later term is obtained by alternating the largest-normal-p-subgroup operator with $O_{p'}$ ) and building the tilting complex from endosplit resolutions of endopermutation modules (the classical p-group analogue of endo-p-permutation modules). The refinement has been verified for: blocks with cyclic defect groups (Kessar–Linckelmann 2018), blocks of symmetric groups (Chuang–Rouquier 2008, field-independent), blocks with Klein four defect groups (Huang 2023), blocks of $SL(2,p^n)$ and $GL(2,p^n)$ (Huang–Li–Zhang 2023), blocks of alternating groups (Huang 2024), unipotent blocks of $GL_n(q)$ (Huang–Li–Zhang 2024), blocks of p-nilpotent groups (Boltje–Yılmaz 2022 for p-permutation equivalences; Miller 2025 for splendid Rickard equivalences), RoCK blocks (a combinatorially controlled family of blocks with abelian defect groups introduced by Rouquier and generalized by Chuang–Kessar) of double covers of symmetric and alternating groups (Du–Huang 2025), and, by Miller (2026), the blocks of p-solvable groups with $G=O_{p',p,p'}(G)$ and abelian Sylow p-subgroup that satisfy condition $(\ast)$ : the block idempotent $b$ of $\mathbb{F}_pG$ satisfies $bb'\neq 0$ for some G-stable block $b'$ of $\overline{\mathbb{F}}_pO_{p'}(G)$ . Miller's proof gives a new characterization of all endosplit p-permutation resolutions (via endotrivial complexes, the invertible objects of the bounded homotopy category of p-permutation modules) and proves that Galois descent of such a resolution is equivalent to Galois descent of the module it resolves. What no construction achieves is a uniform, group-independent argument: every known verification exploits the specific block theory of a class of groups, and the descent step to small fields is not automatic — Miller (2026) shows that Harris–Linckelmann's Lemma 3.3 fails over fields not large enough for $G$ , so its reduction step does not directly extend even to all p-solvable groups.

Problem Statement

Prove or refute the Kessar–Linckelmann refinement of Broué's abelian defect group conjecture: for every prime $p$ , every p-modular system $(K,\mathcal{O},k)$ — in particular the minimal system $(\mathbb{Q}_p,\mathbb{Z}_p,\mathbb{F}_p)$ — every finite group $G$ , and every block $b$ of $\mathcal{O}G$ whose defect group $P$ is abelian with Brauer correspondent $c$ in $\mathcal{O}N_G(P)$ , there exists a splendid Rickard equivalence between $\mathcal{O}Gb$ and $\mathcal{O}N_G(P)c$ , i.e. a bounded complex of p-permutation $(\mathcal{O}Gb,\mathcal{O}N_G(P)c)$ -bimodules that is a Rickard tilting complex and is splendid (every indecomposable direct summand of each of its terms is, regarded as a module over $G\times N_G(P)$ , a p-permutation module with twisted-diagonal vertex), defined over the given $\mathcal{O}$ . As a concrete intermediate target identified in Miller (2026) — but not a substitute for the general statement — the refinement is open already for all p-solvable groups with abelian defect groups, where Harris–Linckelmann's reduction step is currently available over $\mathbb{F}_p$ only for blocks satisfying condition $(\ast)$ , and the documented obstruction is the failure of their Lemma 3.3 over fields not large enough for $G$ .

References

Sam K. Miller, On endosplit p-permutation resolutions and Broué's conjecture for p-solvable groups, Representation Theory (2026), DOI 10.1090/ert/712, arXiv:2408.04094, https://arxiv.org/abs/2408.04094
Radha Kessar and Markus Linckelmann, Descent of Equivalences and Character Bijections, Springer Proceedings in Mathematics & Statistics (2018), DOI 10.1007/978-3-319-94033-5_7, https://doi.org/10.1007/978-3-319-94033-5_7
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