The author gives an upper bound for the degrees of the generators of an homogeneous ideal defining a (generically reduced) subscheme \(X\subset {\mathbb{P}}^n\), by using an invariant which depends on the length of the cohomology \(H^i({\mathbb{P}}^n,{\mathcal J}_X(n))\), \(n>-\dim X\), where \({\mathcal J}_X\) is the ideal sheaf of \(X\) in \({\mathbb{P}}^n\).