Nonunique indecomposable projective decompositions over the localized group ring of PSL2(F11)

Abstract

Put \(O=\mathbb Z_{(11)}\) and \(G=\operatorname{PSL}_2(\mathbb F_{11})\). We construct nonzero finitely generated indecomposable projective \(O[G]\)-modules \(A,B,C,D\), each of \(O\)-rank \(44\), with \[A\oplus D\cong B\oplus C,\qquad A\not\cong B,\qquad A\not\cong C.\] Thus a projective module of rank \(88\) has two inequivalent decompositions into indecomposable projectives, giving a negative answer to Kourovka Problem 4.55. Three of the modules are summands of representations induced from subgroups of orders \(12\) and \(60\). Their \(11\)-adic completions split into pairs of projective covers. Irrational character values prevent either member of a pair from descending separately to \(O\), while approximation of homomorphisms descends the required complementary summand.