The direct summand conjecture of Hochster asserts that if \(R\) is a regular subring of a commutative ring \(S\), then \(R\) is a module direct summand. This paper deals with a generalisation in which the regularity of \(R\) is replaced by the weaker condition that \(S\) has finite projective dimension as \(R\)-module. The authors use methods of algebraic geometry to show that the strengthened conjecture is false for rings of both prime and mixed characteristic. They also find several conditions under which the conjecture holds.