Math Conjectures

Broué’s Abelian Defect Group Conjecture

OPENMajorConjectureProposed 1990 · Standard version

Canonical statement

Let pp be a prime, let (K,O,k)(K,\mathcal O,k) be a splitting pp-modular system for a finite group GG: O\mathcal O is a complete discrete valuation ring with characteristic-00 fraction field KK and algebraically closed residue field kk of characteristic pp, and both fields split every subgroup of GG. Let BB be a block algebra of OG\mathcal O G with abelian defect group DD. If bb is the Brauer-correspondent block of ONG(D)\mathcal O N_G(D), then
Db(B-modfg)≃Db(b-modfg) D^b(B\text{-}\mathrm{mod}_{\mathrm{fg}}) \simeq D^b(b\text{-}\mathrm{mod}_{\mathrm{fg}})
as triangulated categories, where these are bounded derived categories of finitely generated left modules.
View source LaTeX
Let \(p\) be a prime, let \((K,\mathcal O,k)\) be a splitting \(p\)-modular system for a finite group \(G\): \(\mathcal O\) is a complete discrete valuation ring with characteristic-\(0\) fraction field \(K\) and algebraically closed residue field \(k\) of characteristic \(p\), and both fields split every subgroup of \(G\). Let \(B\) be a block algebra of \(\mathcal O G\) with abelian defect group \(D\). If \(b\) is the Brauer-correspondent block of \(\mathcal O N_G(D)\), then
\[
  D^b(B\text{-}\mathrm{mod}_{\mathrm{fg}})
    \simeq D^b(b\text{-}\mathrm{mod}_{\mathrm{fg}})
\] as triangulated categories, where these are bounded derived categories of finitely generated left modules.
ALG-MODREP-011:[Broue1990][Rickard1996Splendid][Rouquier2001BlockTheory]
Algebra and representation theory

Notes

Broué's conjecture predicts a precise structural form of the local–global principle in modular representation theory. Let BB be a block of OG\mathcal O G for a finite group GG, taken over a splitting pp-modular system, and suppose its defect group DD is abelian. The conjecture, formulated by Broué in 1990, asserts that BB and its Brauer-correspondent block bb of ONG(D)\mathcal O N_G(D) have equivalent bounded derived categories [Broue1990]. This would explain numerical coincidences between a block and its local counterpart by an actual equivalence of triangulated categories.

Rickard proposed the finer notion of a splendid equivalence, built from complexes of permutation modules, as the expected shape of such equivalences [Rickard1996Splendid], and Rouquier developed block-theoretic machinery based on stable and Rickard equivalences [Rouquier2001BlockTheory]. Derived equivalences of the predicted kind have been constructed for many families of blocks and for small or highly structured defect groups.

No uniform construction, however, treats every finite group and every block with abelian defect, and the conjecture remains open in general.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.