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One of the exercises in Ravi Vakil's algebraic geometry notes, Ex. $5.4.$I(b), is to show that $$ \operatorname{Spec}\left(k[x_1, \ldots, x_n]/(x_1^2 + \cdots + x_m^2)\right) $$ is normal, where $k$ is any field of $\operatorname{char}(k)\neq 2$, and $n \geq m \geq 3$.

I have absolutely no idea how to get started on this. Is there anyone that could give a hint as to how one would approach this problem?

Samuel Forster
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  • One way to show this is using the Jacobian criterion. If I remember correctly, then there is another way to see that using non-degeneracy of a matrix associated to $\sum x_i^2$. – Youngsu Dec 07 '13 at 10:43
  • This is also an exercise in Hartshorne, Chapter II, Section 6. He gives some hints there. – Fredrik Meyer Dec 07 '13 at 17:04
  • As Fredrik mentioned, use induction on $m$ and exercise 6.4 of chapter II in Hartshorne. – Ehsan M. Kermani Dec 07 '13 at 20:00
  • @Youngsu: The jacobian criterion only works outside of the singularity $(0,\dotsc,0)$. – Martin Brandenburg Dec 08 '13 at 00:10
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    @MartinBrandenburg: I'm a bit confused by your comment. Can you explain? – Youngsu Dec 08 '13 at 00:18
  • And I'm confused by your comment ;). What do you mean exactly by "using the Jacobian criterion"? – Martin Brandenburg Dec 08 '13 at 00:28
  • @MartinBrandenburg: Hi. Let me be more specific. I thought the Jacobian ideal is $(x_1, \dots, x_m)$ which has codimension at least $2$. Isn't this enough to conclude that the ring is normal? I was confused by the part "only works outside the origin." Let me know if my question makes sense to you this time. – Youngsu Dec 11 '13 at 12:18

1 Answers1

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The following steps lead to a solution:

Step 1: Suppose the characteristic of a field $k$ is not $2$. If $f \in k[x_1,\ldots,x_n]$ is square-free and non-constant, then $A = k[x_1,\ldots,x_n][z]/(z^2 - f)$ is integrally closed.

Hint for step 1: Follow the method that I employ in my answer here.

Step 2: Show that the ring in your question above is a domain and then show it is integrally closed using step 1. Hint for showing that it is a domain: Dehomogenize with respect to the last variable and apply Eisenstein's Criterion.

<p><strong>Step 3:</strong> Conclude your result.</p>
Community
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  • In fact, isn't this the exercise (in Vakil) before the OP's question? – Alex Youcis Dec 07 '13 at 23:57
  • @AlexYoucis Indeed it is. I hadn't looked in Vakil before posting this, I was just posting how I did an identical exercise in Hartshorne chapter II, section 6 (Divisors). –  Dec 08 '13 at 00:48
  • @Benja, I thought Eisenstein's criterion only applied to polynomials in one variable. How do you want to apply it to $x_1^2+...+x_{m-1}^2+1$ to show that it is irreducible? – Rodrigo Jan 25 '14 at 20:13
  • @Rodrigo Apply it to the ring in the variable $x_{m-1}$ with coefficients in $k[x_1,\ldots,x_{m-2}]$ –  Jan 26 '14 at 13:36