Smallness of the category of schemes of finite type

Question

Most sources about motivic homotopy theory mention that the category of (smooth) separated schemes of finite type over a (Noetherian of finite Krull dimension) base $S$ is essentially small, which is essential for the homotopy theory of its simplicial sheaves. Yet, none of the sources I came across proves or references that this category is not small, or indicates the small category that it is equivalent to. I guess the latter should be the category of (smooth) finitely generated algebras over $\Gamma(S,\mathcal{O}_S)$.

I would like to ask for a reference where such a statement is proven, preferably in more generality, if it exists.

Answer 1

You can prove essential smallness of the category of finite type $S$-schemes (without further assumptions), where $S$ is any scheme, as follows:

• If $S$ is affine and we only consider affine finite type $S$-schemes, these correspond to finite type $\Gamma(S)$-algebras. These are isomorphic to algebras of the form $\Gamma(S)[x_1,\dotsc,x_n]/I$ for some natural number $n$ and some ideal $I$. These algebras form a set.

• If $S$ is affine and we allow separated finite type $S$-schemes, these may be glued from affine finite type $S$-schemes $X_1,\dotsc,X_n$, $n \in \mathbb{N}$, and isomorphisms between the affine schemes $X_i \cap X_j \subseteq X_i$ and $X_j \cap X_i \subseteq X_j$. By the affine case, we can classify them by a set.

• If $S$ is affine and we allow arbitrary finite type $S$-schemes, these may be glued from finitely many affine finite type $S$-schemes $X_1,\dotsc,X_n$, $n \in \mathbb{N}$, and isomorphisms between the separated schemes $X_i \cap X_j \subseteq X_i$ and $X_j \cap X_i \subseteq X_j$. By the first two cases, we can classify them by a set.

• If $S$ is arbitrary, then finite type $S$-schemes may be glued from finite type $U$-schemes $X_U$, where $U$ runs through all open affines $U \subseteq S$, together with isomorphisms $X_U \times_U V \cong X_V$ for $V \subseteq U$. By the affine case, we may classify them by a set.

Remark that the category of locally finite type $S$-schemes is only essentially small when $S=\emptyset$: Consider $\coprod_{i \in I} S$ for sets $I$.

As Qiaochu points out, the category of (finite type) $S$-schemes is never small because it contains, for every set $X$, the scheme $(\emptyset,\mathcal{O})$ with the structure sheaf $\mathcal{O}(\emptyset)=(\{X\},+,\cdot)$.

Answer 2

Asking about smallness instead of essential smallness is a distraction; smallness is not invariant under equivalence of categories, so the answer potentially depends delicately on how you construct the category of schemes (e.g. as locally ringed spaces or as functors of points).

It's also a distraction in the sense that it almost never holds. The affine case already exhibits the relevant phenomenon so let me stick to it. The category of finitely generated algebras over a base ring $R$ is essentially small, but it is never small for dumb reasons: an algebra is in particular a set equipped with some extra data, and the set part alone can already vary over a proper class. This phenomenon is ubiquitous whenever discussing categories of sets with extra structure. For example, there are a proper class's worth of zero ring, because $\{ X \}$ is a set with one element for every set $X$, and so we can put the zero ring structure on it.

Given an essentially small category you can construct a small category it's equivalent to by finding a skeleton of it.