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I came up with an example I wanted to try to work through. Let $k$ be an algebraically closed field with characteristic $0$ and let $A=k[x,y]/(x^3-y^3)$. Let $X=\operatorname{Spec}(A)$ and let $\mathcal{O}_X$ be the structure sheaf. If $\omega$ is a primitive third root of unity, then the minimal primes of $A$ are $(x-y)$, $(x-\omega y)$, and $(x-\omega^2 y)$. Every other prime ideal is maximal, so takes the form $(x-a,y-a)$, $(x-\omega a,y-a)$, or $(x-\omega^2 a,y-a)$ for some $a\in k$. If $m$ is a maximal ideal other than $(x,y)$, then $U=X\setminus\{m\}$ is an open set which is not a basic open set under the Zariski topology, so I wanted to work out $\mathcal{O}_X(U)$. I understand the definition of $\mathcal{O}(U)$ as an inverse limit and was able to write down a nontrivial element, but I can't quite characterize this ring as a whole. Is there a simple(ish) way to express $\mathcal{O}(U)$ in terms of $A$? I would be satisfied with an answer in the case $m=(x-a,y-a)$ or even just $m=(x-1,y-1)$.

Edit: I have found a solution thanks to Daniel's comment.

Anonymous
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    Say $m = (x - a, y - b)$. Then, $X \setminus {m} = D(x - a) \cup D(y - b)$ so you can represent a regular function on $U$ by a pair of regular function $(f,g)$ on $ D(x - a)$ and $D(y - b)$ respectively, subject to the condition that $f$ and $g$ agree on $D(x - a) \cap D(x - b) = D((x - a)(x - b))$. In your case, these can be translated to some concrete statements about polynomials and localization. – Daniel Feb 16 '22 at 03:50
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    @Daniel Thanks! Your comment was a great help. I was able to solve it and might write a full solution here if I have time. – Anonymous Feb 18 '22 at 23:50
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    Great! I'm glad to have been able to help. The idea behind my comment was the Cech complex which is used to compute the cohomology of sheaves. This is the computation for the 0th cohomology, which is equal to the global sections. – Daniel Feb 19 '22 at 02:54

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