**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 representable](https://math.stackexchange.com/questions/3055234/functor-textttnil-ring-longrightarrow-set-is-not-representable?noredirect=1&lq=1)[1](https://math.stackexchange.com/questions/3055234/functor-textttnil-ring-longrightarrow-set-is-not-representable?noredirect=1&lq=1) or check, for example, that $\operatorname{Spec}(\mathbb{Z}[\![ x ]\!])(\mathbb{Z})\neq\operatorname{Nil}(\mathbb{Z})$.



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


