The Stacks project

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
\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

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

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. $\square$

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 $f|_{U_ i} = f_ i$, 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 $(\pi , \varphi ) \in F(\underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}))$. 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 $F$ (, 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 (14)

Comment #3417 by Anon on

In the last equation of Lemma 27.4.2 (contructing the inverse map) it's easy to see that gives a map on global sections, but how is the map defined?

Also, I think there's a typo in the second last equation of Lemma 27.4.2. The map should take to .

Comment #3430 by Herman Rohrbach on

Typo in equation ( should be .

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 and a -module , how does one define the canonical map ? 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 . A more highbrow method is to use the adjunction mapping and then use that .

@#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 -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 instead of .

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 (not given as a scheme over ) the set of pairs where makes into a scheme over .

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

Comment #4511 by Théo de Oliveira Santos on

@#4508 Thanks for the clarification!

Comment #7185 by Jinyong An on

@#4508 I have a question. In the Lemma 27.4.4, is a scheme representing the functor . Then what is the -morphism structure on ? I guess that since we have , the -morphism structure is the first element in theh corresponding to . Is it true?

Comment #8435 by on

I think it's also interesing to think about the presheaf on that the object represents. Let be a category, and let be an object of . Let be a presheaf on . Consider the induced functor that sends to . Then is representable if and only if is representable (not difficult to show). In our situation, , , the functor sends to , and is

That is, in Görtz & Wedhorn words (see (11.2) of the 2nd ed.), this construction can be regarded as a globalized version of the natural isomorphim where is an -algebra.

Comment #8438 by on

I think it would be nice to add the following two remarks somewhere in this section. I will be adopting the POV explained in #8435 of representability of instead of .

(i) So far, we have an explicit description of the map Namely, it is the one induced by the universal element . However, we don't have yet an explicit description of the inverse of \eqref{map}. Here's one: Given an -linear map , the induced map is the unique morphism over such that for all open affine , the restricted map over equals , where is the map in 27.3.4 and the first arrow comes from Schemes, 26.6.4. Proof: First we note that is a sheaf on (basically because is a sheaf). Then we note that for any open , we have an induced map Namely, it is the one obtained by the universal element of , where is the inclusion and we are using the notation given in . From here, it is not difficult to verify that the diagram commutes. In other words, we have a morphism of sheaves on . Thus, we can characterize the inverse of \eqref{map} locally on and we finish by the proof in

(ii) An explanation of why is a functor: If we have a morphism of -algebras (we've used 27.4.6, (3)) then we obtain a canonical map over . By remark (i), it is the unique morphism of schemes over such that for every open affine , we have that identifies with via the isomorphisms of Lemma 27.3.4. It follows that is a contravariant functor from the category of quasi-coherent -algebras to the category of schemes over . On the other hand, equation \eqref{map} specializes to so the functor is fully faithful.

Comment #8439 by on

Final remarks on why I am writing all these comments in this section instead of just introducing a pull request on GitHub: In my opinion, it would be better to change this whole section to the study of the representability of instead of , and maybe mentioning the existence of just at the end plus the categorical remark in #8435. I do think so because (a) is easier to remember and more motivated than , see #8435 (I also think it's what #3430 had on his mind), and (b) proofs with are less wordy than those with . However, I was unsure whether such a big edit would be accepted or if only a proper subset of all the edits would pass. In consequence, I deemed it better to point it all out here.

Comment #8448 by on

I propose to add the following remark to the webpage: Let be a scheme and let be a morphism of quasi-coherent -algebras. As it was explained in #8438 (ii), we obtain a morphism over . This gives a transformation of functors that induce a transformation of functors , where is our old friend and is the functor taking a scheme over to . We claim that at the transformation is given by precomposition by . We begin with a morphism of -algebras that gives a morphism over . In turn, we get a composite . By #8438 (i), this gives a morphism of -algebras that on sections over an open affine reads as , i.e., it equals the composite , which is what we wanted to show.

Comment #9060 by on

8435, #8438, #8439, #8448. You are right that we can define the functor on the slice category (of course) and perhaps we should have done so. You are also right that it is easy to switch between the 2 versions, so I don't see an urgency in making changes. One of the problems with the material in this chapter is the inumerable identifications that are constantly used between different functors / sheaves and it is a bit hard to succintly tell the reader how to think about it. In hindsight, it is best to have as little as possible of this type of material.

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