The algebra of quasi-symmetric functions is free over the integers

Let L denote the Leibniz-Hopf algebra, which also turns up as the Solomon descent algebra and the algebra of noncommutative symmetric functions. As an algebra L = Z<Z(1), Z(2), ...>, the free associative algebra over the integers in countably many indeterminates. The coalgebra structure is given by mu(Z(n)) = Sigma(i=0)(n) Z(i) circle times Z(n-i), Z(0) = 1. Let M be the graded dual of L. This is the algebra of quasi-symmetric functions. The Ditters conjecture says that this algebra is a free commutative algebra over the integers. In this paper the Ditters conjecture is proved. (C) 2001 Elsevier Science.