nLab ultrascheme

Idea
Definition
Equivalence of categories with schemes
References

Idea

Ultraschemes are locally ringed spaces locally isomorphic to the maximal spectrum of a Jacobson ring. The category of ultraschemes is equivalent to a certain non-full subcategory of schemes, Namely to Jacobson schemes with the morphisms locally of finite type between them. One can understand this equivalence as a “scheme theory with only closed points.”

Ultraschemes were defined by Grothendieck in EGA IV₃ to show that the category of algebraic varieties in the sense of Serre‘s FAC (over an algebraically closed field $k$ ) is equivalent to that of reduced separated schemes of finite type over $k$ . Actually, this latter equivalence is a subequivalence of the former one.

Definition

Let $A$ be a commutative ring with unit. Its maximal spectrum, denoted $Spm A$ in EGA (and $Spec_m A$ or $MaxSpec A$ by some authors), is the set of maximal ideals of $A$ . As a subset of the topological space $Spec A$ , the maximal spectrum inherits a topology. The structure sheaf on $Spm A$ is $\mathcal{O}_{Spm A}\coloneqq i^{-1}\mathcal{O}_{Spec A}$ , where $i\colon Spm A\hookrightarrow Spec A$ is the inclusion. An affine ultra-scheme is a locally ringed space $X$ isomorphic to $(Spm A,\mathcal{O}_{Spm A})$ , for some Jacobson ring $A$ (one also says that $X$ is ultra-affine). An ultra-scheme is a locally ringed space that has a cover by ultra-affine open subsets. Every ultra-scheme is a T₁ topological space.

A morphism of ultra-schemes $f:X\to Y$ is a morphism of locally ringed spaces that is “locally of finite type” in the following sense: every point in $X$ has an ultra-affine open neighborhood $U\subset X$ for which there is an ultra-affine open neighborhood $V\subseteq Y$ such that $f(U)\subseteq V$ and $\Gamma(\mathcal{O}_Y,V)\to\Gamma(\mathcal{O}_X,U)$ is a morphism of rings of finite type.

Equivalence of categories with schemes

Theorem

(EGA IV, 10.9.6)

The category of ultra-schemes is equivalent to the category of Jacobson schemes with all morphisms that are locally of finite type between them.

The mutual quasi-inverse functors that give the equivalence are the following. The functor to the left sends a Jacobson scheme $X$ to its subset of closed points $X_0$ with the restriction topology and structure sheaf $\mathcal{O}_{X_0}\coloneqq i^{-1}\mathcal{O}_X$ , where $i\colon X_0\hookrightarrow X$ is the inclusion. The functor to the right sends an ultra-scheme $U$ to its sobrification $S U$ (see also (EGA I, 1971 ed., p. 68), or Tag 0A2N) with structure sheaf $\mathcal{O}_{S U}\coloneqq i_*\mathcal{O}_U$ , where $i:U\to S U$ is the canonical morphism. This latter functor, $U\maspto S U$ , is called the schematization functor.

Let $i:X'\to X$ be a morphism of locally ringed spaces that is isomorphic to a morphism of the form $X_0\hookrightarrow X$ , where $X$ is a scheme, or equivalently to $U\hookrightarrow S U$ , where $U$ is an ultra-scheme. The morphism $i:X'\to X$ satisfies the following universal property: Given a scheme $Y$ and a morphism of locally ringed spaces $f':X'\to Y$ , there exists a unique morphism of schemes $f:X\to Y$ such that $f\circ i=f'$ (Guisado Villalgordo, Lemma 2.28). In other words, we have a relative adjunction: the schematization functor is left adjoint to the forgetful $R\colon JacSch_{lft}\to LRS$ , relative to the forgetful $J\colon UltSch\to LRS$ . Here, $UltSch$ , $LRS$ and $JacSch_{flt}$ denote respectively the categories of ultra-schemes, that of locally ringed spaces and that of Jacobson schemes with locally of finite type morphisms. The morphism $X'\to X$ is then interpreted as the unit of this relative adjunction.

Given an algebraically closed field $k$ , a (pre)variety over $k$ in the sense of FAC (without the quasi-compactness restriction) is essentially the same as a (separated) reduced ultra-scheme over $Spec k=Spm k$ (an ultra-scheme $U$ is separated or reduced if the scheme $S U$ is). Morphisms of (pre)varieties are always over $Spm k$ . The equivalence from the theorem restricts to an equivalence between (pre)varieties in the sense of FAC and (separated) reduced schemes locally of finite type over $Spec k$ .

References

The original definition (with the name ultra-préschéma) appeared in section 10.9 of:

Alexander Grothendieck, Étude locale des schémas et des morphismes de schémas (Troisième partie) Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255. (numdam:PMIHES_1966__28__5_0/)

They were later renamed to ultra-schémas in the Appendix of:

Alexander Grothendieck, Éléments de Géométrie Algébrique I. (2nd ed.) Grundlehren der Mathematischen Wissenschaften (1971). 166. Springer-Verlag.

Ultra-schemes are considered in Chapter 5 of:

Yves Diers: Categories of Commutative Algebras, Oxford University Press (1992) [https://doi.org/10.1093/oso/9780198535867.001.0001]