  In the central theorem of this article we prove the following: if $R$ is a
complete regular local ring and $B$ is the integral closure of $R$ in the
algebraic closure of the fraction field of $R$, then $\Hom_R(B, R) \neq 0$. Our
proof of this theorem does not involve almost mathematics and it works for all
characteristics. As consequences we derive the validity of a) descent of
flatness for integral extensions of Noetherian rings and b) direct summand
conjecture due to M. Hochster.
