This question is from Mumford's The Red Book of Varieties and Schemes (Section I.4).

Let $X\subseteq k^n$ be an irreducible algebraic set, $R$ its affine coordinate ring. Since $X$ is irreducible, $I(X)$ is prime and $R$ is an integral domain. Let $K$ be its field of fractions.

Let $\underline{o}_x=\{f/g\mid f, g\in R, g(x)\neq 0\}\subseteq K$. Now for $U$ open in $X$, let $$\underline{o}_X(U)=\bigcap_{x\in U}\underline{o}_x.$$

Let $X_f=\{x\in X\mid f(x)\neq 0\}$.

The author said: Since the sets $X_f$ are a basis of the Zaiski topology of $X$, we have $$\varinjlim_{x\in U}\underline{o}_X(U)=\varinjlim_{x\in X_f}\underline{o}_X(X_f).$$

My Question: Why? I think the first part I have to prove is that $$\bigcup_{x\in U}\underline{o}_X(U)=\bigcup_{x\in X_f}\underline{o}_X(X_f).$$ But I cannot prove it.

  • $\begingroup$ If you know some complex analyis compare $\mathcal O_x$ with germs of meromorphic functions at a point. Might be insightful. $\endgroup$ – Ignorant Mathematician Apr 14 at 7:00
  • $\begingroup$ @Ghosh Thx. But I am looking for a purely algebraic proof. I will come back to see your hint after learning complex analysis. $\endgroup$ – bfhaha Apr 14 at 7:06

The sets $X_f$ are cofinal in the poset of open sets of $X$ containing $x$ (this follows from the $X_f$ being a basis for the topology of $X$). Since stalks are calculated as colimits, it suffices to calculate along any cofinal collection - which is exactly the claim the author makes.

  • $\begingroup$ Hello. Thanks for you answer. Which book where I can find the theorem you mentioned? "Since stalks are calculated as colimits, it suffices to calculate along any cofinal collection" $\endgroup$ – bfhaha Apr 14 at 19:22
  • $\begingroup$ I don't have a reference offhand - just look at the definition of cofinal: a subcollection $A$ of a poset $B$ is cofinal iff for any $b\in B$ there is an $a\in A$ so that $b\leq a$. In particular, this means that the morphisms from any $b\in B$ to the colimit factor through the cofinal collection, so the colimit of the original collection and the cofinal collection satisfy the same universal property and must be isomorphic. $\endgroup$ – KReiser Apr 14 at 20:34

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