§ Hilbert polynomial and dimension

• 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)