Authors:
========
Shigeo Koshitani
Department of Mathematics, Graduate School of Science,
Chiba University, Chiba, 263-8522, Japan


J{\"u}rgen M{\"u}ller
Lehrstuhl D f{\"u}r Mathematik, RWTH Aachen,
52062 Aachen, Germany


Felix Noeske
Lehrstuhl D f{\"u}r Mathematik, RWTH Aachen,
52062 Aachen, Germany


Title:
======
Brou{\'e}'s abelian defect group conjecture holds
for the sporadic simple Conway group {\sf Co}$_3$}

Abstract:
=========
In the representation theory of finite groups, there is a well-known and
important conjecture due to M.~Brou{\'e}. He conjectures that, for
any prime $p$, if a $p$-block $A$ of a finite group $G$ has an abelian
defect group $P$, then $A$ and its Brauer corresponding block $A_N$
of the normaliser $N_G(P)$ of $P$ in $G$ are derived equivalent
(Rickard equivalent).
This conjecture is called
{\sl Strong Version of \sl Brou{\'e}'s Abelian Defect Group Conjecture}.
In this paper, we prove that the strong version of
Brou{\'e}'s abelian defect group conjecture
is true for the non-principal $2$-block $A$ with an elementary abelian
defect group $P$ of order $8$ of the sporadic simple Conway
group {\sf Co}$_3$. This result completes the verification of the strong
version of Brou{\'e}'s abelian defect group conjecture for all primes $p$
and for all $p$-blocks
of {\sf Co}$_3$.

Status:
=======
Submitted