# Simple examples of colimits of affine schemes (evaluated in the presheaf category) which are not affine schemes

Notation and Setting: let $$\operatorname{Aff}$$ denote the category of affine schemes whose objects are covariant representable functors $$\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{Set}$$ and $$\operatorname{Spec}:\operatorname{Ring^{op}}\rightarrow\operatorname{Func(Ring, Set)}$$ be the contravariant Yoneda embedding of $$\operatorname{Ring^{op}}$$ in its category of presheaves so that $$\operatorname{Aff}\simeq\operatorname{Ring^{op}}$$.

In addition, let $$\mathcal{O}:\operatorname{Func(Ring, Set)}\rightarrow\operatorname{Ring^{op}}$$ be the functor that sends a functor $$\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{Set}$$ to the ring of maps $$\operatorname{X}\rightarrow \mathbb{A}^1$$ (where $$\mathbb{A}^1$$ is the forgetful functor) so that $$\operatorname{Spec}$$ and $$\mathcal{O}$$ are inverse of one another.

Let $$\widehat{\operatorname{Aff}}$$ be the indization of $$\operatorname{Aff}$$, i.e. the category whose objects are functors $$\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{Set}$$ that are small filtered colimits of affine schemes.

My question: I am looking for (simple) examples of functors which are objects of $$\widehat{\operatorname{Aff}}$$ but are not affine schemes.

I am particularly interested in examples of the following form: Let $$\operatorname{X}$$ be an affine scheme, $$I\subseteq\mathcal{O}_{X}$$ an ideal and consider the following diagram in $$\operatorname{Func(Ring, Set)}$$ over $$\mathbb{Z_{\geq0}}$$

$$0=\operatorname{Spec(\mathcal{O}_{X}/I^{0})}\hookrightarrow\ldots\hookrightarrow\operatorname{Spec(\mathcal{O}_{X}/I^{n-1})}\hookrightarrow\operatorname{Spec(\mathcal{O}_{X}/I^{n})}\hookrightarrow\ldots$$

Since $$\operatorname{Func(Ring, Set)}$$ admits small colimits, $$\mathrm{colimit}_{n\in\mathbb{Z}_{\geq0}}\operatorname{Spec(\mathcal{O}_{X}/I^{n})}$$ exists. Thus I am looking for examples of affine schemes $$\operatorname{X}$$ and ideals $$I\subseteq\mathcal{O}_{X}$$ for which $$(\mathrm{colimit}_{n\in\mathbb{Z}_{\geq0}}\operatorname{Spec(\mathcal{O}_{X}/I^{n})})\neq \mathrm{colimit}_{n\in\mathbb{Z}_{\geq0}}(\operatorname{Spec(\mathcal{O}_{X}/I^{n})})$$

The only example that I could find so far was that of $$\operatorname{Spec(\mathbb{Z}[x])}$$ and the ideal $$(x)$$ which give the functor $$\operatorname{Nil}\simeq\mathrm{colimit}_{n\in\mathbb{Z}_{\geq0}}\operatorname{Spec(\mathbb{Z}[x]/(x)^{n})}\neq(\mathrm{colimit}_{n\in\mathbb{Z}_{\geq0}}\operatorname{Spec(\mathbb{Z}[x]/(x)^{n})})\simeq\operatorname{Spec}(\mathbb{Z}[\![ x ]\!])$$.

To see that you can either show that $$\operatorname{Nil}$$ is not representable1 or check, for example, that $$\operatorname{Spec}(\mathbb{Z}[\![ x ]\!])(\mathbb{Z})\neq\operatorname{Nil}(\mathbb{Z})$$.

• It looks like you are asking about formal schemes. Your example Nil is the formal spectrum $\operatorname{Spf}\mathbb{Z}[[x]]$. They are discussed at the end of EGA1. Jan 24, 2020 at 12:59
• I don't think that the question is really about formal schemes (since here no "global space" is given). A better keyword is "ind-schemes". Jan 24, 2020 at 20:33

A basic standard example is the colimit of $$\mathbb{A}^0 \to \mathbb{A}^1 \to \mathbb{A}^2 \to \cdots$$ with transition maps $$x \mapsto (x,0)$$. The $$R$$-valued points are finite sequences in $$R$$. This functor is not representable.
More generally, let $$A$$ be a commutative ring with a sequence of ideals $$I_0 \supseteq I_1 \supseteq I_2 \supseteq \cdots$$. (In the mentioned example, $$A = \mathbf{Z}[x_0,x_1,\dotsc]$$ and $$I_n = \langle x_n,x_{n+1},\dotsc\rangle$$.) Then the colimit $$X$$ of $$\mathrm{Spec}(A/I_0) \to \mathrm{Spec}(A/I_1) \to \cdots$$ is the subfunctor of $$\mathrm{Spec}(A)$$ whose $$R$$-valued points are those $$A \to R$$ whose kernel contains some $$I_n$$. We have $$\mathcal{O}(X) = \lim_n A/I_n =: \widehat{A}$$.
Then $$X$$ is representable aka affine iff the canonical morphism $$X \to \mathrm{Spec}(\widehat{A})$$ is an isomorphism. It is injective anyway, and it is surjective on $$R$$-valued points iff every homomorphism $$\widehat{A} \to R$$ factors through some projection $$\widehat{A} \to A/I_n$$. So $$X$$ is representable iff the identity $$\widehat{A} \to \widehat{A}$$ factors through some projection $$\widehat{A} \to A/I_n$$. But this clearly implies $$I_n = I_{n+1} = \dotsc$$ and the sequence is stationary.
Conversely this means that for every non-stationary sequence $$I_0 \supseteq I_1 \supseteq I_2 \supseteq \cdots$$ the functor $$X$$ is not representable.