Hilbert nullstellensatz for schemes

Question

I know that there is a bijection between the irreducible closed sets and the prime ideals of a ring (via functions $V()$ and $I()$).

Is this true not only for affine schemes but also for general schemes? And Why?

Answer 1

Response to the comment:

The correspondence is a point $p$ to $\overline{\{p\}}$, the closure of $p$. It says two things: existence (that each irreducible closed set has a generic point), and uniqueness (that each irreducible closed set has a unique generic point)

Let's say our scheme is $X$, an irreducible closed set being $C$.

Existence: We want to find a generic point of $C$.

Intersect $C$ with an affine open subset $U$. This intersection is dense in $C$, while by the affine case $C \cap U$ has a generic point $p$ (considered as a subset of $U$) in $U$. Try to show that the closure of $p$ in $X$ is $C$.

Uniqueness: Suppose $p$ and $q$ has the same closure $C$. We want to show that $p = q$.

Take an affine open subset $U$ that contains $p$, and try to show that $p$ is the same as the generic point of $C \cap U$ in $U$.

Now take an affine open subset $V$ that contains $q$. Irreducibility of $C$ forces $U \cap V$ to be nonempty. The last paragraph then shows that both $p$ and $q$ are the generic point of $C \cap U \cap V$ in $U \cap V$, which means that they are the same.

Answer 2

This is the whole point of scheme theory. Hilbert's Nullstellensatz only gives a bijection between radical ideals in $k[x_1,\cdots,x_n]$ and closed subsets of $\mathbb{A}^n$, for $k$ algebraically closed.

Now for any scheme, there is 1-1 correspondence between quasi-coherent ideal sheaves (subsheaves of the structure sheaf) and closed subschemes of X. (this is the content of proposition 5.9 in Harthorne)

The correspondence is given as follows: If $Y$ is a closed subscheme of $X$, then the inclusion morphism $i:Y \to X$ gives a morphism of sheaves: $i^\#:\mathcal{O}_X \to i_*\mathcal{O}_Y$. The ideal sheaf is the kernel of this map.

Conversely, given a scheme $X$ and a quasi-coherent shraf of ideals $\mathcal{J}$, let $Y$ be the support of the quotient sheaf $\mathcal{O}_X/\mathcal{J}$. Then $Y$ is a closed subscheme with structure sheaf $\mathcal{O}_X/\mathcal{J}$.