In the book algebraice geometry by R.Harshorne we always say :an open affine subset U=SpecA of a sheme X.
I was always wondering that whether there existed an open but not affine subset. I want to have a good example.
Thank you all very much!
In the book algebraice geometry by R.Harshorne we always say :an open affine subset U=SpecA of a sheme X. I was always wondering that whether there existed an open but not affine subset. I want to have a good example. Thank you all very much! 

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The weirdest example of affineness / nonaffineness I know is the following. There is a smooth projective scheme $X$ over $\mathbb{Z}[1/7]$ and a closed subscheme $Z$ (in fact, $Z$ is a smooth irreducible divisor) such that $U = X \setminus Z$ is not affine. However, after tensoring with $\mathbb{Z}/p$ , $$U_p := U \otimes_{\mathbb{Z}} \mathbb{Z}/p$$ is affine for infinitely many $p > 0$ and also nonaffine for infinitely many $p > 0$. For details, see Remark 4.8 in MR2220102, Brenner and Katzman, J. Amer. Math. Soc. 19 (2006), no. 3. EDIT: As far as I know, it is unknown whether similar examples exist in the equal characteristic setting. 


The standard example is to let $X$ be the affine plane over a field, and $U=X\{(0,0)\}$. 


1) Obviously if $X$ is not affine, then $U=X$ is an open subset which is not affine (see Kevin's comment above) but this might feel cheating. 2) A more general example along the lines of Robin's example is the following: Let $X$ be an arbitrary $S_2$ affine scheme of dimension at least $2$. (If you don't know what $S_2$ is, take normal, or smooth). Let $Z\subset X$ be a nonempty closed subscheme of codimension at least $2$. Then $U=X\setminus Z\subset X$ is a proper open subset which is not affine. Here is why it's not affine: Assume that $X={\rm Spec} A$ is affine and let $\iota:U\hookrightarrow X$ be the natural embedding. By the $S_2$ property it follows that $\iota_*\mathcal O_U\simeq \mathcal O_X$, and consequently the induced morphism on global sections $\Gamma(U, \mathcal O_U)\to A$ is an isomorphism, but then if $U$ were affine, then this would imply that the embedding $\iota$ is an isomorphism, which is not true by construction. 

