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Hartshorne mentions "local equations" a few times without (so far as I can tell) actually defining them anywhere. As best as I can guess, the definition would be something like this:

If $Y \subseteq X$ is a closed subscheme, then "local equations for $Y$" consist of an open affine set $U \subseteq X$ and a finite set of generators $f_1, \ldots, f_n \in \mathcal{O}_X(U)$ of the ideal sheaf $\mathscr{I}_Y(U)$ considered as an $\mathcal{O}_X(U)$-module.

(Here I've attempted to adapt the definition of local equations for a subvariety given in Shafarevich.) Is this the accepted definition? Or should the assumption that $U$ is affine or that $Y$ is a closed subscheme be weakened? Or is there something else wrong with it?

What should be done if $X$ is non-noetherian? Might a closed subscheme simply not have local equations in that case?

Or is "local equation" defined somewhere in Hartshorne?

Daniel McLaury
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  • Can you give an example of where Hartshorne's usage confuses you? I can't recall Hartshorne using the terminology "local equations" except for defining Cartier divisors, where there are sufficient conditions on $X$ for this definition to make sense. – bzc May 18 '13 at 05:16
  • Oh I'm definitely not asking anything particularly sophisticated here; it's just that in order to do a lot of exercises here it's necessary to fix some definition, and I want to make sure I'm using the right one. – Daniel McLaury May 18 '13 at 05:57
  • (I also don't have a lot of experience at this sort of thing and am prone to making mistakes, leading me to want to run things by someone before I get too far down a dead end.) – Daniel McLaury May 18 '13 at 06:04

1 Answers1

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Let $X$ be an arbitrary scheme and $i : Y \hookrightarrow X$ be a closed immersion corresponding to the quasi-coherent ideal $I \subseteq \mathcal{O}_X$ . If $U \subseteq X$ is an open subset, the "local equations of $i$ on $U$" actually mean the ideal $I|_U \subseteq \mathcal{O}_U$ (in particular they exist). Often it also means any set of its global generators, i.e. a family of elements $(f_s)_{s \in S}$ of $\Gamma(U,I)$ such that $\oplus_{s \in S} \mathcal{O}_U \to I$ is an epimorphism. If $U$ is affine, this means that $\Gamma(U,I)$ is generated by the $f_s$. If $X$ is noetherian, one can choose $S$ to be finite.

When $X$ is projective, we often use another definition. Let's say $X=\mathrm{Proj}(S)$ with a nice graded algebra $S$. Then closed subschemes $Y$ of $X$ come from graded ideals $I$ of $S$ via $Y=\mathrm{Proj}(S/I)$. Then generators of $I$ are also called the (homogeneous) equations for $Y \hookrightarrow X$. We have the obvious notion of local (homogeneous) equation when $S$ is a sheaf of graded algebras on a base scheme.

Martin Brandenburg
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  • Just to make sure I'm 100% clear on this, your $\mathcal{O}_U$ and $I|_U$ are what Hartshorne would write $\mathcal{O}_X|_U$ and $\mathscr{I}_Y|_U$, right? – Daniel McLaury May 18 '13 at 10:27