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

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.