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Let $X$ be an affine $k$-variety, and let $f$ be an element of $\mathcal{O}(X)$. The subvariety $D(f)$ of $X$ is a quasi-affine $k$-variety. Is $\mathcal{O}(X)_f$ the ring of regular functions of $D(f)$?

yannickvda
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1 Answers1

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Yes, this is Proposition 2.2 in Chapter II of Hartshorne's "Algebraic Geometry": for every affine scheme $X=\operatorname{Spec}(A)$, with $A$ a commutative ring, and every $f \in A$, one has $$ \mathcal{O}(D(f)) \cong A_f \cong \mathcal{O}(X)_f $$

Edit: a more elementary proof is given in Fulton's wonderful little book on algebraic curves, Proposition 6.3.5. A few comments are in order:

  1. Fulton assumes varieties to be irreducible, while you seem to allow reducible varieties. As far as I can see, however, this makes no difference in the proof of the above-mentioned proposition.
  2. Fulton's definition of regular function might look slightly different than the one you give, but they are really the same as in Hartshorne.
Nils Matthes
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  • Thanks, but is there a formulation/approach without the use of schemes? I am taking only an introductory algebraic geometry course. – yannickvda Aug 23 '13 at 21:02
  • @yannickvda: Yes, of course. Could you tell me then what is your definition of a $k$-variety, and what is a regular function for you? – Nils Matthes Aug 24 '13 at 06:31
  • (multi-comment for readability) A regular function on an affine $k$-set is a morphism of affine $k$-sets $X \to k$ where a morphism is understood in the sense of definition 2 @ http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_algebraist_2010;task=show_msg;msg=1703.0001 – yannickvda Aug 24 '13 at 20:01
  • There is also a second approach which says: $f$ is regular at a point if $f$ can written as a quotient of polynomials $f=h/g$ on some open neighborhood $U$ of $x$. It is also proven that in the case of affine $k$-varieties "regular at every point" = "regular in the first sense." – yannickvda Aug 24 '13 at 20:02
  • Finally, an affine $k$-set is a pair ($k^n,X)$ where $X$ is an algebraic set of $k^n$. We call an affine $k$-set an affine $k$-variety if $k$ is algebraically closed. – yannickvda Aug 24 '13 at 20:03
  • Dear @yannickvda, I'm away from home at the Moment, and don't have computer in front of me, and since texing with a smartphone is a pain in the behind, I won't answer your question before my return. Sorry, but I just wanted to let you know, I did not forget about your question. – Nils Matthes Aug 27 '13 at 10:51
  • Dear @yannickvda: I have given you an alternative reference, one which avoids the language of schemes, for the result you ask about. – Nils Matthes Sep 03 '13 at 09:26
  • the "ask an algebraist" link appears to be no longer working, but I think this is what it pointed to: http://at.yorku.ca/b/ask-an-algebraist/2010/1703.htm#1703 – ziggurism Aug 28 '23 at 23:05