- #1

disregardthat

Science Advisor

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## Main Question or Discussion Point

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

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 [tex]x^{d-r-1}yz^r[/tex] for r < d, and [tex]x^{d-r}z^r[/tex] 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?

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 [tex]x^{d-r-1}yz^r[/tex] for r < d, and [tex]x^{d-r}z^r[/tex] 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?

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