Lemma 27.4.2. In Situation 27.3.1. Let $F$ be the functor associated to $(S, \mathcal{A})$ above. If $S$ is affine, then $F$ is representable by the affine scheme $\mathop{\mathrm{Spec}}(\Gamma (S, \mathcal{A}))$.

Proof. Write $S = \mathop{\mathrm{Spec}}(R)$ and $A = \Gamma (S, \mathcal{A})$. Then $A$ is an $R$-algebra and $\mathcal{A} = \widetilde A$. The ring map $R \to A$ gives rise to a canonical map

$f_{univ} : \mathop{\mathrm{Spec}}(A) \longrightarrow S = \mathop{\mathrm{Spec}}(R).$

We have $f_{univ}^*\mathcal{A} = \widetilde{A \otimes _ R A}$ by Schemes, Lemma 26.7.3. Hence there is a canonical map

$\varphi _{univ} : f_{univ}^*\mathcal{A} = \widetilde{A \otimes _ R A} \longrightarrow \widetilde A = \mathcal{O}_{\mathop{\mathrm{Spec}}(A)}$

coming from the $A$-module map $A \otimes _ R A \to A$, $a \otimes a' \mapsto aa'$. We claim that the pair $(f_{univ}, \varphi _{univ})$ represents $F$ in this case. In other words we claim that for any scheme $T$ the map

$\mathop{\mathrm{Mor}}\nolimits (T, \mathop{\mathrm{Spec}}(A)) \longrightarrow \{ \text{pairs } (f, \varphi )\} ,\quad a \longmapsto (f_{univ} \circ a, a^*\varphi _{univ})$

is bijective.

Let us construct the inverse map. For any pair $(f : T \to S, \varphi )$ we get the induced ring map

$\xymatrix{ A = \Gamma (S, \mathcal{A}) \ar[r]^{f^*} & \Gamma (T, f^*\mathcal{A}) \ar[r]^{\varphi } & \Gamma (T, \mathcal{O}_ T) }$

This induces a morphism of schemes $T \to \mathop{\mathrm{Spec}}(A)$ by Schemes, Lemma 26.6.4.

The verification that this map is inverse to the map displayed above is omitted. $\square$

Comment #7016 by on

Just before the construction of the inverse map : is that $\varphi_{\mathrm{univ}}$ instead of $\varphi$ ?

Comment #7121 by Elías Guisado on

Who is exactly the map $A = \Gamma (S, \mathcal{A}) \xrightarrow{f^*} \Gamma (T, f^*\mathcal{A})$ in the description of the inverse?

Comment #7122 by Elías Guisado on

Is it $\mathcal{A}\to f_*f^*\mathcal{A}$, the unit of the pushforward-pullback adjunction of sheaves of modules?

Comment #7236 by on

@#7016. Thanks and fixed here.

@#7121 and #7122. Yes. Discussion. Say we have a continuous map $f : X \to Y$ of topological spaces and we have a sheaf $\mathcal{G}$ on $Y$, then we have a map $f^{-1} : \Gamma(Y, \mathcal{G}) \to \Gamma(X, f^{-1}\mathcal{G})$ often called a pullback map. Now it is indeed true that you can write $\Gamma(X, f^{-1}\mathcal{G}) = \Gamma(Y, f_*f^{-1}\mathcal{G})$ and then you can get the map by appying the adjunction map $\mathcal{G} \to f_*f^{-1}\mathcal{G}$. Another method, is to say that $\Gamma(Y, \mathcal{G}) = \text{Mor}(*, \mathcal{G})$, use the fact that $f^{-1}$ is a functor, and use that $f^{-1}* = *$. Here $*$ is the singleton sheaf AKA the final object of the category of sheaves of sets. These constructions give the same thing.

Of course, if we have a map of ringed spaces and $\mathcal{G}$ is a sheaf of modules, then we also have a pullback map $\Gamma(Y, \mathcal{G}) \to \Gamma(X, f^*\mathcal{G})$. It can be constructed by either method discussed above or its existence can be deduced from the construction of the map for sheaves of sets.

There are also:

• 9 comment(s) on Section 27.4: Relative spectrum as a functor

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).