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. $\square$

Comments (7)

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!

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01LQ. Beware of the difference between the letter 'O' and the digit '0'.