What is the definition of "local equation(s)" for a subscheme? 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?
 A: 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.
