Compact objects in category of schemes over a base scheme?

Question

I'm wondering if the compact objects in $\textbf{Scheme}/X$ (schemes over $X$) are the quasi-compact morphisms with codomain $X$. By "compact object" I mean that the covariant representative functor $\textbf{Scheme}(X, -)$ commutes with filtered colimits.

If the compact objects aren't quasi-compact, what are they? Or, under what conditions does this statement hold?

Answer 1

It's definitely not true that quasicompact morphisms are compact objects in general. In fact, I suspect that the only compact object over any base $X$ is the empty scheme, and I can prove this in the case when $X$ is Spec of a field. Let's start with a lemma.

Lemma: Let $(A_i)$ be an inverse system of local rings, where the maps in the system are local homomorphisms. Let $A=\varprojlim A_i$ be the inverse limit. Then $\operatorname{Spec} A$ is the colimit of the direct system $(\operatorname{Spec} A_i)$ in the category of schemes.

Proof: Let us write $Y_i=\operatorname{Spec}A_i$, with closed point $p_i\in Y_i$. Let $S$ be any scheme and suppose we have compatible morphisms $f_i:Y_i\to S$ for each $i$. We may assume the index set of our inverse system has a least element $0$. Let $U=\operatorname{Spec}B\subseteq S$ be an affine open subscheme that contains $f_0(p_0)$. Since every other point of $Y_0$ is a generalization of $p_0$, the entire image of $f_0$ is contained in $U$, and so $f_0$ is induced by a homomorphism $g_0:B\to A_0$. Moreover, for any $i$, the map $Y_0\to Y_i$ in the inverse system sends $p_0$ to $p_i$, since it is induced by a local homomorphism. It follows that for any $i$, $f_i$ is also induced by a homomorphism $g_i:B\to A_i$. These homomorphisms $g_i$ then determine a unique homomorphism $g:B\to A$ and thus a morphism $f:\operatorname{Spec} A\to U\subseteq S$ compatible with the $f_i$. Moreover, $A$ is also a local ring and the maps $A\to A_i$ are local homomorphisms (this is easy to check: the maximal ideal of $A$ is the inverse limit of the maximal ideals of the $A_i$), and it follows easily that this $f$ is the unique morphism $\operatorname{Spec} Y\to S$ compatible with the $f_i$.

Using this lemma, you can then find lots of examples of affine schemes that are not compact objects in the category of schemes. Indeed, if $C$ is any ring with a homomorphism $A\to C$ which does not factor through $A_i$ for any $i$, then $\operatorname{Spec} C$ cannot be compact (it has a map to $\operatorname{Spec}A$ which does not factor through any $\operatorname{Spec} A_i$). For an explicit example, you could let $k$ be a field and $A_n=k[x]/(x^n)$, so $A=k[[x]]$. Then we conclude that for any ring $C$ with an injective homomorphism $k[[x]]\to C$, $\operatorname{Spec} C$ is not compact.

In particular, let's now consider schemes over a field $k$. Let $Z$ be any nonempty scheme over $X=\operatorname{Spec} k$, and let $T$ be a set of cardinality greater than that of the residue field of some point $p\in Z$. For each cofinite subset $I\subseteq T$, consider the field $k(I)$ of rational functions over $k$ with a variable for each element of $I$. These fields form an inverse system of subfields of $k(T)$, and their inverse limit (i.e., intersection) is just $k$. Thus by the Lemma, the schemes $\operatorname{Spec} k(I)$ form a directed system with colimit $X$. Now $Z$ has a morphism to $X$, but it does not have a morphism to $\operatorname{Spec} k(I)$ for any $I$, since the residue field of $Z$ at the point $p$ has cardinality smaller than that of $k(I)$. Thus $Z$ is not compact in the category of schemes over $X$.