Krdytkyu

Let $X$ be a scheme and $Y$ and $Z$ closed subschemes. What does it mean for $Y$ to be contained in $Z$?

This question adresses the same question, but it covers only the affine case and it uses a different definition of a closed subscheme. So for the non-affine case, let $(Y, mathcal{O}_Y)$ and $(Z, mathcal{O}_Z)$ be two closed subschemas of a scheme $(X, mathcal{O}_X)$. Then $mathcal{O}_Y$ and $mathcal{O}_Z$ are quotients of the structure sheaf $mathcal{O}_X$ by a quasi-coherent sheaf of ideals, that is,
$$mathcal{O}_Y = mathcal{O}_X/mathcal{I}$$
and
$$mathcal{O}_Z = mathcal{O}_X/mathcal{J} , .$$
Then, to say that $Y subset Z$, we would like to say that $|Y| subset |X|$ as topological spaces, and that $mathcal{O}_Y$ is a sub sheaf of $mathcal{O}_Z$. I believe we can sharpen it a bit more by saying that the quasi-coherent sheaf $mathcal{J}$ is a subsheaf of $mathcal{I}$. Would this suffice?

For some mysterious reason the notion of closed subscheme is handled in a rather confusing way in much of the literature (egregiously and very untypically in Hartshorne).

Be that as it may, just remember that there is a bijective order reversing corrrespondence between closed subschemes $Ysubset X$ and quasi-coherent ideal sheaves $mathcal Jsubset mathcal O_X$.

The corresponding items are written $Y=V(mathcal J)$, resp. $mathcal J=mathcal I_Y$.

And then the answer to your question is given by the biblical simple equivalence:

$$Ysubset Ziff mathcal I_Y supset mathcal I_Z$$

Edit: a simple but edifying example.

Let $(A,mathfrak m)$ be a discrete valuation ring, $X=operatorname {Spec}A={M,eta}$ its associated affine scheme and $U={eta}$ its non trivial open subset.

a) The only non zero ideals of $A$ are the powers $mathfrak m^n$ and the corresponding quasi-coherent sheaves of ideals $mathcal I_n$ define the closed subschemes $Y_n=V(mathcal I_n)subset X$.

These $Y_n$'s are the infinitesimal neighbourhoods of the closed reduced point $Y_1={M}$ and they constitute the collection of all the closed subschemes $subsetneq X$.

Notice that $Y_n$ is non reduced for $ngt 1$ and that $Y_nsubsetneq Y_m$ for $n lt m$.

b) However $mathcal O_X$ has another sheaf of ideals $mathcal J$ defined by $mathcal J(X)=A$ and $mathcal J(U)={0}$.

This sheaf is not quasi-coherent and thus does not correspond to a closed subscheme of $X$.