A Universe of Sorts

Think of non Cohen Macaulay ring (plane with line perpendicular to it). Here the dimension varies per point.
Let $R$ be a graded ring. Let $R^0$ be noetherian. $R$ is finitely generated as an algebra over $R^0$ . This implies by hilbert basis theorem that $R$ is noetherian (finitely generated as a module over $R^0$ ).
Suppose $M$ is a graded module over $R$ , and $M$ is finitely generated as a module over $R$ .
How fast does $M_n$ grow? We need some notion of size.
Define the size of $M_n$ as $\lambda(M_n)$ .Suppose $R$ is a field. Then $M_n$ is a vector space. We define $\lambda(M_n)$ to be the dimension of $M_n$ as a vector space over $R$ .
What about taking dimension of tangent space? Doesn't work for cusps! (singular points). Can be used to define singular points.
TODO: show that at $y^2 = x^3$ , we have dimension two (we expect dimension one)

§ Hilbert Polynomial and Dimension