Past exam of the introduction to schemes course of the University of Oxford / 2023 / 1 / c

For an idempotent $e\in R$, define the principal open subset $D(e)\subseteq \mathrm{Spec}R$. Since $e(1-e)=0$, a prime ideal contains at least one of $e,1-e$, and it cannot contain both. ThereforeThus $D(e)$ is a clopen set.

Conversely, let $C$ be a clopen set in $T=\mathrm{Spec}R$. The structure sheaf glues the section $1$ on $C$ and the section $0$ on its complement to a unique global section $e\in \Gamma (T,{\mathcal{O}}_T)=R$. Checking on these two open subsets gives $e^2=e$, and $D(e)=C$. Conversely, the restriction of an idempotent $e$ to $D(e)$ is $1$ because $e$ is a unit there, and its restriction to $D(1-e)$ is $0$. Thus the two constructions are inverse, including for the zero ring:

Label the two charts of the affine line with doubled origin by $A,B$, and use $t$ for their common coordinate. Let $u:T=\mathrm{Spec}R\to X$ be an $R$-valued element of the functor of points. Its composite with $\varphi$ corresponds to a unique $r\in R$ under the affine-target adjunction for schemes:The closed subscheme $Z=V(r)=\mathrm{Spec}(R/(r))$ maps into the disjoint union of the two zero sections,The inverse image $Z_A$ of $A_0$ is a clopen set in $Z$. By the idempotent–clopen correspondence, it determines a unique idempotent $e\in R/(r)$, equal to $1$ on $Z_A$ and $0$ on the inverse image $Z_B$ of $B_0$. HenceOver $\mathbb{Z}$, the two “origins” are entire zero sections, rather than single closed points; this construction uses their full scheme-theoretic fibers.

Let $T=\mathrm{Spec}R$ and $Z=\mathrm{Spec}(R/(r))$. The idempotent $e$ determines the disjoint clopen sets $Z_A=D_Z(e)$ and $Z_B=D_Z(1-e)$. Each is closed in $T$. Define the open subsetsThey cover $T$ and satisfy $T_A\cap T_B=T\setminus Z=D(r)$. The global section $r$ defines morphisms of schemes $T_A\to A$ and $T_B\to B$, both sending the coordinate $t$ to $r$. On their intersection $r$ is a unit, so both maps land in the common open subscheme $A\cap B=\mathrm{Spec}\mathbb{Z}[t,t^{-1}]$, where they agree. They therefore glue uniquely:For an entirely explicit description, choose any lift $a\in R$ of $e$. Then $a(1-a)\in (r)$ andThe equality $a+(1-a)=1$ proves that these open subsets cover. The inclusion $D(a(1-a))\subseteq D(r)$ proves that their intersection is exactly $D(r)$. Their description as complements of $Z_A,Z_B$ proves independence of the chosen lift. In particular, no lift of $e$ to an idempotent of $R$ is required.

Starting from $(r,e)$, the construction in (iii) preserves the composite to the affine line and sends $Z_A$ to the $A$ zero section and $Z_B$ to the $B$ zero section. Part (ii) therefore recovers $(r,e)$.

Conversely, start with $u:T\to X$ and construct $(r,e)$. The only points of $X$ outside $A$ lie on $B_0$, and the only points outside $B$ lie on $A_0$. Consequently $u^{-1}(A)=T\setminus Z_B=T_A$ and $u^{-1}(B)=T\setminus Z_A=T_B$. On each of these open subsets, $u$ is the unique morphism of schemes to the appropriate affine line with coordinate $r$. Thus part (iii) recovers $u$.

Finally, a ring homomorphism $R\to R^{\prime }$ sends $r$ to $r^{\prime }$ and induces $R/(r)\to R^{\prime }/(r^{\prime })$, sending $e$ to an idempotent $e^{\prime }$. Inverse images of the clopen sets $Z_A,Z_B$ are precisely those determined by $e^{\prime }$. The constructions therefore commute with this base change. We have a natural bijectionso $F$ is a functor represented by a scheme, represented by the affine line with doubled origin. The representing scheme is not required to be an affine scheme.