Start with a graded ring \(\CC[x_0,\ldots,x_n]\) with \(\deg x_i = 1\). Consider a graded module \(M = \bigoplus_d M_d\) over \(R\). Hilbert looked at the map \(d\mapsto H_M (d)= \dim_{\CC} M_d\). For example we can take \(R\) to be the homogeneous coordinate ring of \(\PP^n\) and \(V(I)\subset \PP^n\) a subvariety where \(I\) is a homogeneous ideal. We then take \(M = R/I\), if \(V\) is a curve \(C\) then \(H_{R/I}(d) = \deg(V)\cdot d + (1 -g(C))\). Hilbert showed that the function \(H_M(d)\) is eventually polynomial. We can compute this function easily if \(M\) is free so we try to replace \(M\) by free modules. First we take \[K_0\to F_0 \to M\] where \(K_0\) is the kernel of the surjective map from \(F_0\) to \(M\). We can continue this getting \begin{gather*} K_1\to F_1 \to K_0\\ K_2\to F_2 \to K_1\\ \vdots \end{gather*} we can then write \[\cdots \to F_2\to F_1\to F_0 \to M \to 0,\] this is a free resolution of \(M\). We also have the following.