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What is a general technique for finding the ring of functions on an open subset of affine space over an algebraically closed field? I.e. "Find the ring of regular functions on $ \mathbb{A}_k^n- \{(0,...,0)\}$." Or find the ring of regular functions on $U_f$ a distinguished open subset?

Martin Brandenburg
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bijection
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1 Answers1

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I think that there is no general method. But the following two statements are fundamental and useful. I will state them in the scheme language (of course they hold, in particular, for varieties):

1) If $X=\mathrm{Spec}(A)$ is an affine scheme and $f \in A$, then $X_f$ is isomorphic to the affine scheme $\mathrm{Spec}(A_f)$. This also explains the notation $X_f$. In particular, the ring of regular functions on $X_f$ is $A_f$.

2) (Algebraic Hartogs Lemma) If $X$ is a noetherian normal scheme and $U \subseteq X$ is an open subset such that $X \setminus U \hookrightarrow X$ codimension at least $2$, then the restriction morphism $\Gamma(X,\mathcal{O}_X) \to \Gamma(U,\mathcal{O}_X)$ is an isomorphism.

In particular, for $n \geq 2$, the ring of regular functions on $\mathbb{A}^n \setminus \{0\}$ is the same as for $\mathbb{A}^n$, i.e. $k[x_1,\dotsc,x_n]$. A direct computation is also possible, use the affine cover given by the basic open subsets $\mathbb{A}^n_{x_i}$ and observe $\cap_i k[x_1,\dotsc,x_n]_{x_i} = k[x_1,\dotsc,x_n]$.

Martin Brandenburg
  • 181,922
  • thank you for your answer! I'm not familiar with scheme theory but does $2)$ say that the ring of regular functions on $U \subset X$ is the ring of regular functions on $X$ if U has codimension $ \geq 2$? – bijection Feb 20 '13 at 02:28
  • It is $X\setminus U$ that should be of codimension at least $2$, not $U$. – Georges Elencwajg Feb 20 '13 at 08:47
  • Thanks Georges, of course. I have corrected it. @Seth: Yes. – Martin Brandenburg Feb 20 '13 at 11:28
  • By the same reasoning given an affine set, $U \subset \mathbb{A}_k^n$, if we have a finite cover of basic open sets, can't we reason saying that a function $\phi$ is regular on $U$ iff it is regular on $U_f$ for all the sets in the cover; and then we can take the intersection of the localizations?? – bijection Feb 21 '13 at 17:29