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How do I show that $\frac{k[x,y,z]}{\langle xz-y^2\rangle}$ is not isomorphic to $k[x,y,z]$? (I want to understand the coordinate ring of this surface) Is there a 'natural' evaluation map that realizes $\langle xz-y^2\rangle$ as its kernel?

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How much do you know about dimension? –  Dylan Moreland May 30 '12 at 19:45
@DylanMoreland, What a luck! I was also going to ask about dimension of ${A^n}_k$, the affine space of dimension $n$. It s not hard to show that dim $\geq n$. But my question is that in the proofs I see in books, to prove the upper bound, it is first assumed that the there is a maximal finite chain of ascending irreducible closed subsets. But I could not figure out why there must exist a finite chain. I thought it was because $A^n$ is Noetherian, but it seems I need more. –  Herband May 30 '12 at 19:54
I changed $< xz-y^2>$ to $\langle xz-y^2\rangle$. the code is \langle xz-y^2\rangle. That is standard $\TeX$ usage. –  Michael Hardy May 30 '12 at 19:56
@Herband: The dimension of $\mathbb{A}^n$ is a different question. Not every noetherian ring has finite dimension. But I don't think that the proofs for $\mathbb{A}^n$ need that the dimension is bounded a priori. Read them again. –  Martin Brandenburg May 30 '12 at 19:56
@MartinBrandenburg, For instance, here is a reference where in the proof, a longest chain is assumed to exist (see page 53, first line of first proof)(here is the link: mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf) –  Herband May 30 '12 at 20:15
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2 Answers

up vote 3 down vote accepted

The dimension of $k[x,y,z]/(xz-y^2)$ is $2$. In geometric terms, its spectrum is an algebraic surface. But the dimension of $k[x,y,z]$ is $3$.

This computation relies on the following general result (see Qing Liu, Algebraic Geometry and Arithmetic Curves, Prop. 2.5.23): If $A$ is a finitely generated $k$-algebra, where $k$ is a field and $A$ is a domain, and $\mathfrak{p}$ is a prime ideal of $A$, then $\dim(A/\mathfrak{p})+\operatorname{ht}(\mathfrak{p})=\dim(A)$. A corollary of this (2.5.26 loc. cit.) is that the dimension of $V(f) \subseteq X$ is $n-1$, where $X$ is an irreducible variety of dimension $n$ and $f$ is a regular function which is not nilpotent.

After your edit, the following might be of interest:

The homomorphism $k[x,y,z]/(xz-y^2) \to k[s,t]$, $x \mapsto s^2$, $y \mapsto st$, $z \mapsto t^2$ is injective, and the image identifies with $k[s^2,t^2,st]$.

For the sake of completeness (and the hope that someone will explain me how to improve this), here is the proof for injectivity: Since $y^2-xz \in k[x,z][y]$ is monic, we see that $k[x,y,z]/(y^2-xz)$ is free over $k[x,z]$ with basis $1,y$. Hence, it's free over $k$ with basis $x^{\mathbb{N}} z^{\mathbb{N}}$, $x^{\mathbb{N}} y z^{\mathbb{N}}$, which we can rewrite as $1,x^{\mathbb{N}^+} z^{\mathbb{N}^+}$, $x^{\mathbb{N}} y z^{\mathbb{N}}$. A typical element has the form $a + \sum_{i,j > 0} a_{ij} x^i z^j + \sum_{i,j > 0} b_{ij} x^i y z^j$. If it gets mapped to $0$, this means $a + \sum_{i,j > 0} a_{ij} s^{2i} t^{2j} + \sum_{i,j > 0} b_{ij} s^{2i+1} t^{2j+1}=0$. The three sums here don't share any monomial (consider the parities of the exponents). Thus they vanish individually. But also every monomial appears only once. Thus all coefficients vanish.

Actually this enables us to find a more ad hoc invariant (which coincides with the dimension, but one doesn't have to know that) which distinguishes the two rings, namely the transzendence degree of the field of fractions over $k$. For $k[x,y,z]$ it is $3$. For $k[x,y,z]/(xz-y^2)$ the field $k(s^2,st,t^2)$ contains $a := s/t$ and $b := s^2$; one checks that $s,t$ are algebraic over $k(a,b)$, and the latter is purely transcendent. Thus $\{a,b\}$ is a transcendence basis and the transcendence degree is $2$.

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At Martin, thanks for both approaches. What do you think of my other question in the comments? Or should I ask it separately? –  Herband May 30 '12 at 20:00
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Here's a way to avoid using Krull dimension. Given a $k$-algebra $A$ and a morphism $e_p : A \to k$ (which corresponds to evaluation at a point $p$ of $\text{Spec } A$), a derivation at $p$ is a linear map $D : A \to k$ such that $$D(ab) = D(a) e_p(b) + e_p(a) D(b).$$

The space of all such $D$ is a vector space over $k$ which I'll denote by $T_p(A)$; geometrically it is the Zariski tangent space at $p$ ($D$ represents the directional derivative with respect to a tangent vector). More algebraically, a derivation at $p$ is a morphism $$\phi = e_p + \epsilon D : A \to k[\epsilon]/\epsilon^2$$

such that composition with the natural quotient map $k[\epsilon]/\epsilon^2 \to k$ gives $e_p$.

This definition allows us to use the universal property of polynomial rings to conclude that a derivation at any point for $k[x, y, z]$ is freely determined by $D(x), D(y)$, and $D(z)$; consequently $\dim T_p(k[x, y, z]) = 3$ for all $p$. On the other hand, a derivation $D : k[x, y, z]/(xz - y^2)$ at any point $p = (x_0, y_0, z_0)$ with $y_0 \neq 0$ satisfies $$D(y^2) = 2y_0 D(y) = D(xz) = z_0 D(x) + x_0 D(z)$$

hence $D(y)$ is determined by $D(x)$ and $D(z)$ (in characteristic not equal to $2$). Consequently $\dim T_p(k[x, y, z]/(xz - y^2)) = 2$.

(Note that when $p = (0, 0, 0)$ we instead have $\dim T_p(k[x, y, z]/(xz - y^2)) = 3$. This is a singular point of the variety $V(xz - y^2)$, so $k[x, y, z]/(xz - y^2)$ also cannot be isomorphic to $k[x, y]$ even though they have the same dimension.)

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At Qiaochu, thanks for this third method, which also teaches me more. [In fact, I arrived at this coordinate ring while calculating the coordinate ring of the image of $P^1$ under the Veronese map $v_2$.] –  Herband May 30 '12 at 20:20
At Qiaochu, I am afraid I should accept Martin's answer since he answered first. Thanks a lot. –  Herband May 30 '12 at 20:43
You should except the answer which helped you most. Not the first one ... –  Martin Brandenburg May 31 '12 at 15:45
@MartinBrandenburg, Laurent's comment to your answer here: mathoverflow.net/questions/98549/… was the piece that was needed to answer my question in the comments, I guess. That is " the Dependent Choice Axiom is needed to prove that ``every increasing sequence terminates'' implies the existence of maximal elements". Am I right? I did not find this explicitly in any textbook of Algebraic Geometry that I have seen so far. –  Herband Jun 2 '12 at 20:41
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