Let $M$ be a variety and $Y\subset M$ a subscheme. Is the scheme-theoretic complement $M\setminus Y$ of $Y$ in $M$ equal to the scheme-theoretic complement $M\setminus Y_{\text{red}}$, where $Y_{\text{red}}$ denotes the underlying reduced scheme upon which $Y$ is supported? I'm thinking the answer should be "yes", since complements if I remember correctly are defined in terms of localizing at a prime, and if the subscheme $Y$ is non-reduced I don't think it's ideal can be prime. But if not, then how would one describe the scheme $M\setminus Y$, say when $M=\mathbb{A}_k^2$ and $Y$ is a fat point, e.g. when $Y$ is the subscheme corresponding to the ideal $(x,y)^2$?

## 1 Answer

If $Y\subseteq X$ is a closed subscheme, then the topological space underlying $Y$ is a closed subset of $X$. Its complement $X\setminus Y$ is an open subset of $X$ and the restriction of $\mathcal{O}_X$ to it makes it into a scheme. It has the universal property Allen Knutson described in his comment. It is clear from this definition that $X\setminus Y = X\setminus Y_{red}$.

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