Solitons: Differential equations, symmetries, and infinite-dimensional algebras
《Solitons: Differential equations, symmetries, and infinite-dimensional algebras》 · 第 484–485 页 · Definition 14.138；Theorem 14.139
Let $X$ be an $S$-scheme. The Hilbert functor $\operatorname{Hilb}_{X/S}$ is the functor $(\operatorname{Sch}/S)^{\mathrm{opp}}\to(\operatorname{Sets})$, $T\mapsto\operatorname{Hilb}(X\times_S T/T)$, where $\operatorname{Hilb}(X/S)$ consists of the closed subschemes $Z\subseteq X$ such that $Z\to S$ is proper, flat, and of finite presentation. If $S$ is noetherian and $X\to S$ is (quasi-)projective, then the Hilbert functor is representable; the representing functor is called the Hilbert scheme.