## 27.4 Relative spectrum as a functor

We place ourselves in Situation 27.3.1, i.e., $S$ is a scheme and $\mathcal{A}$ is a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras.

For any $f : T \to S$ the pullback $f^*\mathcal{A}$ is a quasi-coherent sheaf of $\mathcal{O}_ T$-algebras. We are going to consider pairs $(f : T \to S, \varphi )$ where $f$ is a morphism of schemes and $\varphi : f^*\mathcal{A} \to \mathcal{O}_ T$ is a morphism of $\mathcal{O}_ T$-algebras. Note that this is the same as giving a $f^{-1}\mathcal{O}_ S$-algebra homomorphism $\varphi : f^{-1}\mathcal{A} \to \mathcal{O}_ T$, see Sheaves, Lemma 6.20.2. This is also the same as giving an $\mathcal{O}_ S$-algebra map $\varphi : \mathcal{A} \to f_*\mathcal{O}_ T$, see Sheaves, Lemma 6.24.7. We will use all three ways of thinking about $\varphi$, without further mention.

Given such a pair $(f : T \to S, \varphi )$ and a morphism $a : T' \to T$ we get a second pair $(f' = f \circ a, \varphi ' = a^*\varphi )$ which we call the pullback of $(f, \varphi )$. One way to describe $\varphi ' = a^*\varphi$ is as the composition $\mathcal{A} \to f_*\mathcal{O}_ T \to f'_*\mathcal{O}_{T'}$ where the second map is $f_*a^\sharp$ with $a^\sharp : \mathcal{O}_ T \to a_*\mathcal{O}_{T'}$. In this way we have defined a functor

27.4.0.1
\begin{eqnarray} \label{constructions-equation-spec} F : \mathit{Sch}^{opp} & \longrightarrow & \textit{Sets} \\ T & \longmapsto & F(T) = \{ \text{pairs }(f, \varphi ) \text{ as above}\} \nonumber \end{eqnarray}

Lemma 27.4.1. In Situation 27.3.1. Let $F$ be the functor associated to $(S, \mathcal{A})$ above. Let $g : S' \to S$ be a morphism of schemes. Set $\mathcal{A}' = g^*\mathcal{A}$. Let $F'$ be the functor associated to $(S', \mathcal{A}')$ above. Then there is a canonical isomorphism

$F' \cong h_{S'} \times _{h_ S} F$

of functors.

Proof. A pair $(f' : T \to S', \varphi ' : (f')^*\mathcal{A}' \to \mathcal{O}_ T)$ is the same as a pair $(f, \varphi : f^*\mathcal{A} \to \mathcal{O}_ T)$ together with a factorization of $f$ as $f = g \circ f'$. Namely with this notation we have $(f')^* \mathcal{A}' = (f')^*g^*\mathcal{A} = f^*\mathcal{A}$. Hence the lemma. $\square$

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

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

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. $\square$

Proof. We are going to use Schemes, Lemma 26.15.4.

First we check that $F$ satisfies the sheaf property for the Zariski topology. Namely, suppose that $T$ is a scheme, that $T = \bigcup _{i \in I} U_ i$ is an open covering, and that $(f_ i, \varphi _ i) \in F(U_ i)$ such that $(f_ i, \varphi _ i)|_{U_ i \cap U_ j} = (f_ j, \varphi _ j)|_{U_ i \cap U_ j}$. This implies that the morphisms $f_ i : U_ i \to S$ glue to a morphism of schemes $f : T \to S$ such that $f|_{I_ i} = f_ i$, see Schemes, Section 26.14. Thus $f_ i^*\mathcal{A} = f^*\mathcal{A}|_{U_ i}$ and by assumption the morphisms $\varphi _ i$ agree on $U_ i \cap U_ j$. Hence by Sheaves, Section 6.33 these glue to a morphism of $\mathcal{O}_ T$-algebras $f^*\mathcal{A} \to \mathcal{O}_ T$. This proves that $F$ satisfies the sheaf condition with respect to the Zariski topology.

Let $S = \bigcup _{i \in I} U_ i$ be an affine open covering. Let $F_ i \subset F$ be the subfunctor consisting of those pairs $(f : T \to S, \varphi )$ such that $f(T) \subset U_ i$.

We have to show each $F_ i$ is representable. This is the case because $F_ i$ is identified with the functor associated to $U_ i$ equipped with the quasi-coherent $\mathcal{O}_{U_ i}$-algebra $\mathcal{A}|_{U_ i}$, by Lemma 27.4.1. Thus the result follows from Lemma 27.4.2.

Next we show that $F_ i \subset F$ is representable by open immersions. Let $(f : T \to S, \varphi ) \in F(T)$. Consider $V_ i = f^{-1}(U_ i)$. It follows from the definition of $F_ i$ that given $a : T' \to T$ we gave $a^*(f, \varphi ) \in F_ i(T')$ if and only if $a(T') \subset V_ i$. This is what we were required to show.

Finally, we have to show that the collection $(F_ i)_{i \in I}$ covers $F$. Let $(f : T \to S, \varphi ) \in F(T)$. Consider $V_ i = f^{-1}(U_ i)$. Since $S = \bigcup _{i \in I} U_ i$ is an open covering of $S$ we see that $T = \bigcup _{i \in I} V_ i$ is an open covering of $T$. Moreover $(f, \varphi )|_{V_ i} \in F_ i(V_ i)$. This finishes the proof of the lemma. $\square$

Lemma 27.4.4. In Situation 27.3.1. The scheme $\pi : \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \to S$ constructed in Lemma 27.3.4 and the scheme representing the functor $F$ are canonically isomorphic as schemes over $S$.

Proof. Let $X \to S$ be the scheme representing the functor $F$. Consider the sheaf of $\mathcal{O}_ S$-algebras $\mathcal{R} = \pi _*\mathcal{O}_{\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A})}$. By construction of $\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A})$ we have isomorphisms $\mathcal{A}(U) \to \mathcal{R}(U)$ for every affine open $U \subset S$; this follows from Lemma 27.3.4 part (1). For $U \subset U' \subset S$ open these isomorphisms are compatible with the restriction mappings; this follows from Lemma 27.3.4 part (2). Hence by Sheaves, Lemma 6.30.13 these isomorphisms result from an isomorphism of $\mathcal{O}_ S$-algebras $\varphi : \mathcal{A} \to \mathcal{R}$. Hence this gives an element $(\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}), \varphi ) \in F(\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}))$. Since $X$ represents the functor $F$ we get a corresponding morphism of schemes $can : \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \to X$ over $S$.

Let $U \subset S$ be any affine open. Let $F_ U \subset F$ be the subfunctor of $F$ corresponding to pairs $(f, \varphi )$ over schemes $T$ with $f(T) \subset U$. Clearly the base change $X_ U$ represents $F_ U$. Moreover, $F_ U$ is represented by $\mathop{\mathrm{Spec}}(\mathcal{A}(U)) = \pi ^{-1}(U)$ according to Lemma 27.4.2. In other words $X_ U \cong \pi ^{-1}(U)$. We omit the verification that this identification is brought about by the base change of the morphism $can$ to $U$. $\square$

Definition 27.4.5. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras. The relative spectrum of $\mathcal{A}$ over $S$, or simply the spectrum of $\mathcal{A}$ over $S$ is the scheme constructed in Lemma 27.3.4 which represents the functor $F$ (27.4.0.1), see Lemma 27.4.4. We denote it $\pi : \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \to S$. The “universal family” is a morphism of $\mathcal{O}_ S$-algebras

$\mathcal{A} \longrightarrow \pi _*\mathcal{O}_{\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A})}$

The following lemma says among other things that forming the relative spectrum commutes with base change.

Lemma 27.4.6. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras. Let $\pi : \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \to S$ be the relative spectrum of $\mathcal{A}$ over $S$.

1. For every affine open $U \subset S$ the inverse image $\pi ^{-1}(U)$ is affine.

2. For every morphism $g : S' \to S$ we have $S' \times _ S \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) = \underline{\mathop{\mathrm{Spec}}}_{S'}(g^*\mathcal{A})$.

3. The universal map

$\mathcal{A} \longrightarrow \pi _*\mathcal{O}_{\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A})}$

is an isomorphism of $\mathcal{O}_ S$-algebras.

Proof. Part (1) comes from the description of the relative spectrum by glueing, see Lemma 27.3.4. Part (2) follows immediately from Lemma 27.4.1. Part (3) follows because it is local on $S$ and it is clear in case $S$ is affine by Lemma 27.4.2 for example. $\square$

Lemma 27.4.7. Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. By Schemes, Lemma 26.24.1 the sheaf $f_*\mathcal{O}_ X$ is a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras. There is a canonical morphism

$can : X \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ S(f_*\mathcal{O}_ X)$

of schemes over $S$. For any affine open $U \subset S$ the restriction $can|_{f^{-1}(U)}$ is identified with the canonical morphism

$f^{-1}(U) \longrightarrow \mathop{\mathrm{Spec}}(\Gamma (f^{-1}(U), \mathcal{O}_ X))$

coming from Schemes, Lemma 26.6.4.

Proof. The morphism comes, via the definition of $\underline{\mathop{\mathrm{Spec}}}$ as the scheme representing the functor $F$, from the canonical map $\varphi : f^*f_*\mathcal{O}_ X \to \mathcal{O}_ X$ (which by adjointness of push and pull corresponds to $\text{id} : f_*\mathcal{O}_ X \to f_*\mathcal{O}_ X$). The statement on the restriction to $f^{-1}(U)$ follows from the description of the relative spectrum over affines, see Lemma 27.4.2. $\square$

Comment #3417 by Anon on

In the last equation of Lemma 27.4.2 (contructing the inverse map) it's easy to see that $\varphi : f^{\*} \mathcal{A} \to \mathcal{O}_{T}$ gives a map on global sections, but how is the map $\Gamma(S, \mathcal{A}) \overset{f^{\*}}{\longrightarrow} \Gamma(A, f^{\*}\mathcal{A})$ defined?

Also, I think there's a typo in the second last equation of Lemma 27.4.2. The map should take $a$ to $(a^{\*} f_{univ}, a^{\*}\varphi_{univ})$.

Comment #3430 by Herman Rohrbach on

Typo in equation (26.4.0.1): $Sch^{opp}$ should be $(Sch/S)^{opp}$.

Comment #3479 by on

@#3417 Thanks for the typo. I am not sure what you first question was, but it might be related to the following general question: given a morphism $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ and a $\mathcal{O}_Y$-module $\mathcal{G}$, how does one define the canonical map $H^0(Y, \mathcal{G}) \to H^0(X, f^*\mathcal{G})$? A good answer is to go back to the definition of the pullback of a module in Section 6.24 and define it using the construction of the pullback $f^*\mathcal{G}$. A more highbrow method is to use the adjunction mapping $\mathcal{G} \to f_*f^*\mathcal{G}$ and then use that $H^0(Y, f_*f^*\mathcal{G}) = H^0(X, f^*\mathcal{G})$.

@#3430 Thanks for the typo. The change is here.

Comment #4425 by Théo de Oliveira Santos on

Very minor typo: "This is also the same as giving a $\mathcal{O}_S$-algebra map [...]"

Comment #4426 by Théo de Oliveira Santos on

Also, was the typo pointed out in Comment #3430 reversed back? (01LR) is currently displaying $Sch^{opp}$ instead of $(Sch/S)^{opp}$.

Comment #4508 by on

@#4425: Yes, I've changed this here.

@#4426: It was changed back because it was/is actually correct this way. Namely, the functor assigns to a scheme $T$ (not given as a scheme over $S$) the set of pairs $(F, \varphi)$ where $f$ makes $T$ into a scheme over $S$.

You were also added to the contributors in a different commit.

Comment #4511 by Théo de Oliveira Santos on

@#4508 Thanks for the clarification!

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