Dimension of A(Y) and Height of p in Ring Let A = k[x,y,z]

[disregardthat]

Let A = k[x,y,z] and Y = \{(t,t^2,t^3)|t \in k\}, which is irreducible. It corresponds to the prime ideal p=(y-x^2,z-x^3).

A(Y) is generated by x,y,z of degree 1 as a k-algebra in its graded ring structure. Each group corresponding to the degree d is spanned by the linearly independent monomials x^{d-r-1}yz^r for r < d, and x^{d-r}z^r for r <= d.

For each group there are 2d+1 such monomials. This polynomial has degree 1, so doesn't this imply the dimension of A(Y) is 2 and hence the height of p is 1? But the height of p is obviously 2, so what is wrong here?

 

[mathwonk]

i'm not very algebraic minded, but i forget how A(Y) is graded, when the ideal defining it is not homogeneous?

 

[disregardthat]

i'm not very algebraic minded, but i forget how A(Y) is graded, when the ideal defining it is not homogeneous?

Oh, maybe you are right, I may have been too quick in verifying that A(Y) is graded. Basically I assumed S_d mod I(Y) would constitute the groups. Certainly S_d \mod I(Y) * S_k \mod I(Y) \subseteq S_{d+k} \mod I(Y), but I suppose that \oplus S_d \mod I(Y) is not a grading of A(Y).

EDIT: Of course, I see it now. The S_d \mod I(Y)'s are intersecting non-trivially, so the direct sum is not equal to A(Y). Reading on homogeneous ideals it makes sense now that only homogeneous ideals have a grading in this sense.