On the equations which are needed to define a closed subscheme of the projective space (Q1346819)

Let \(R = k[X_0, \ldots, X_n]\) be the polynomial algebra in \(n + 1\) variables over an infinite field \(k\), and let \(I\) be a proper homogeneous ideal of \(R\). The main result of this paper shows that there exists a minimal system of homogeneous generators \(g_1, \ldots, g_s\) of \(I\) such that \(g_1, \ldots, g_t\) define the subscheme \(X = \text{Proj} (R/I)\) of \(\mathbb{P}^n_k\) scheme-theoretically, where the integer \(t\) \((t \leq s)\) is the minimal number of homogeneous equations needed to define \(X\) scheme-theoretically in \(\mathbb{P}^n_k\).