In a first course on algebraic geometry (following Chapter 1 of Hartshorne), I learned there was a one-to-one correspondence that is inclusion-reversing between the set of algebraic sets in projective space $\mathbf{P}^n_k$ and homogeneous radical ideals of $S=k[x_0, \ldots, x_n]$ not containing the "irrelevant" ideal $S_+$.
Moving on to study schemes, I found myself with the following questions unanswered:
- Does there still exist such a correspondance between the set of closed subschemes of a given scheme of the form $Proj(A)$, and the set of homogeneous ideals in $A$?
- If yes, then is this correspondence still inclusion-reversing?
There is a bijective correspondence between closed subschemes $Z \subset X$ and coherent ideal sheaves $\mathcal{I}_Z \subset \mathcal{O}_X$ by taking the kernel of the surjective map $\mathcal{O}_X \to \mathcal{O}_Z$. This correspondence is order reversing. Indeed, if $Z \subset W \subset X$ is a sequence of closed subschemes, then the quotient map $\mathcal{O}_X \to \mathcal{O}_Z$ factors as a sequence of quotient maps $\mathcal{O}_X \to \mathcal{O}_W \to \mathcal{O}_Z$, so the kernel of the first map is contained the kernel of the composition.
If $X = \mathbb{P}^n_k = \operatorname{Proj} S$, the exercise mentioned in the comments gives a bijective correspondence between coherent ideal sheaves $\mathcal{I}$ and saturated homogeneous ideals $I \subset S$ not containing the irrelevant ideal, explicitly by taking $I = \bigoplus_{d \geq 0} \Gamma(\operatorname{Proj} S, \mathcal{I}(d))$. Since the correspondence between closed subschemes and ideal sheaves is order-reversing it suffices to show this correspondence is order-preserving.
Since $\mathcal{O}(d)$ are locally free, an inclusion $\mathcal{I} \subset \mathcal{I}'$ yield inclusions $\mathcal{I}(d) \subset \mathcal{I}'(d)$. Hence, the order is preserved by the left exactness of $\Gamma(\operatorname{Proj} S, -).$