§ Hilbert polynomial and dimension
- Think of non Cohen Macaulay ring (plane with line perpendicular to it). Here the dimension varies per point.
- Let be a graded ring. Let be noetherian. is finitely generated as an algebra over . This implies by hilbert basis theorem that is noetherian (finitely generated as a module over ).
- Suppose is a graded module over , and is finitely generated as a module over .
- How fast does grow? We need some notion of size.
- Define the size of as .Suppose is a field. Then is a vector space. We define to be the dimension of as a vector space over .
- 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 , we have dimension two (we expect dimension one)