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99.7 The functor of quotients

In this section we discuss some generalities regarding the functor Q_{\mathcal{F}/X/B} defined below. The notation \mathrm{Quot}_{\mathcal{F}/X/B} is reserved for a subfunctor of \text{Q}_{\mathcal{F}/X/B}. We urge the reader to skip this section on a first reading.

Situation 99.7.1. Let S be a scheme. Let f : X \to B be a morphism of algebraic spaces over S. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. For any scheme T over B we will denote X_ T the base change of X to T and \mathcal{F}_ T the pullback of \mathcal{F} via the projection morphism X_ T = X \times _ B T \to X. Given such a T we set

\text{Q}_{\mathcal{F}/X/B}(T) = \left\{ \begin{matrix} \text{quotients }\mathcal{F}_ T \to \mathcal{Q}\text{ where } \mathcal{Q}\text{ is a} \\ \text{quasi-coherent } \mathcal{O}_{X_ T}\text{-module flat over }T \end{matrix} \right\}

We identify quotients if they have the same kernel. Suppose that T' \to T is a morphism of schemes over B and \mathcal{F}_ T \to \mathcal{Q} is an element of \text{Q}_{\mathcal{F}/X/B}(T). Then the pullback \mathcal{Q}' = (X_{T'} \to X_ T)^*\mathcal{Q} is a quasi-coherent \mathcal{O}_{X_{T'}}-module flat over T' by Morphisms of Spaces, Lemma 67.31.3. Thus we obtain a functor

99.7.1.1
\begin{equation} \label{quot-equation-q} \text{Q}_{\mathcal{F}/X/B} : (\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets} \end{equation}

This is the functor of quotients of \mathcal{F}/X/B. We define a subfunctor

99.7.1.2
\begin{equation} \label{quot-equation-q-fp} \text{Q}^{fp}_{\mathcal{F}/X/B} : (\mathit{Sch}/B)^{opp} \longrightarrow \textit{Sets} \end{equation}

which assigns to T the subset of \text{Q}_{\mathcal{F}/X/B}(T) consisting of those quotients \mathcal{F}_ T \to \mathcal{Q} such that \mathcal{Q} is of finite presentation as an \mathcal{O}_{X_ T}-module. This is a subfunctor by Properties of Spaces, Section 66.30.

In Situation 99.7.1 we sometimes think of \text{Q}_{\mathcal{F}/X/B} as a functor (\mathit{Sch}/S)^{opp} \to \textit{Sets} endowed with a morphism \text{Q}_{\mathcal{F}/X/S} \to B. Namely, if T is a scheme over S, then an element of \text{Q}_{\mathcal{F}/X/B}(T) is a pair (h, \mathcal{Q}) where h a morphism h : T \to B and \mathcal{Q} is a T-flat quotient \mathcal{F}_ T \to \mathcal{Q} of finite presentation on X_ T = X \times _{B, h} T. In particular, when we say that \text{Q}_{\mathcal{F}/X/S} is an algebraic space, we mean that the corresponding functor (\mathit{Sch}/S)^{opp} \to \textit{Sets} is an algebraic space. Similar remarks apply to \text{Q}^{fp}_{\mathcal{F}/X/B}.

Remark 99.7.2. In Situation 99.7.1 let B' \to B be a morphism of algebraic spaces over S. Set X' = X \times _ B B' and denote \mathcal{F}' the pullback of \mathcal{F} to X'. Thus we have the functor Q_{\mathcal{F}'/X'/B'} on the category of schemes over B'. For a scheme T over B' it is clear that we have

Q_{\mathcal{F}'/X'/B'}(T) = Q_{\mathcal{F}/X/B}(T)

where on the right hand side we think of T as a scheme over B via the composition T \to B' \to B. Similar remarks apply to \text{Q}^{fp}_{\mathcal{F}/X/B}. These trivial remarks will occasionally be useful to change the base algebraic space.

Remark 99.7.3. Let S be a scheme, X an algebraic space over S, and \mathcal{F} a quasi-coherent \mathcal{O}_ X-module. Suppose that \{ f_ i : X_ i \to X\} _{i \in I} is an fpqc covering and for each i, j \in I we are given an fpqc covering \{ X_{ijk} \to X_ i \times _ X X_ j\} . In this situation we have a bijection

\left\{ \begin{matrix} \text{quotients }\mathcal{F} \to \mathcal{Q}\text{ where } \\ \mathcal{Q}\text{ is a quasi-coherent } \\ \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{families of quotients }f_ i^*\mathcal{F} \to \mathcal{Q}_ i \text{ where } \\ \mathcal{Q}_ i\text{ is quasi-coherent and } \mathcal{Q}_ i\text{ and }\mathcal{Q}_ j \\ \text{ restrict to the same quotient on }X_{ijk} \end{matrix} \right\}

Namely, let (f_ i^*\mathcal{F} \to \mathcal{Q}_ i)_{i \in I} be an element of the right hand side. Then since \{ X_{ijk} \to X_ i \times _ X X_ j\} is an fpqc covering we see that the pullbacks of \mathcal{Q}_ i and \mathcal{Q}_ j restrict to the same quotient of the pullback of \mathcal{F} to X_ i \times _ X X_ j (by fully faithfulness in Descent on Spaces, Proposition 74.4.1). Hence we obtain a descent datum for quasi-coherent modules with respect to \{ X_ i \to X\} _{i \in I}. By Descent on Spaces, Proposition 74.4.1 we find a map of quasi-coherent \mathcal{O}_ X-modules \mathcal{F} \to \mathcal{Q} whose restriction to X_ i recovers the given maps f_ i^*\mathcal{F} \to \mathcal{Q}_ i. Since the family of morphisms \{ X_ i \to X\} is jointly surjective and flat, for every point x \in |X| there exists an i and a point x_ i \in |X_ i| mapping to x. Note that the induced map on local rings \mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{X_ i, \overline{x_ i}} is faithfully flat, see Morphisms of Spaces, Section 67.30. Thus we see that \mathcal{F} \to \mathcal{Q} is surjective.

Lemma 99.7.4. In Situation 99.7.1. The functors \text{Q}_{\mathcal{F}/X/B} and \text{Q}^{fp}_{\mathcal{F}/X/B} satisfy the sheaf property for the fpqc topology.

Proof. Let \{ T_ i \to T\} _{i \in I} be an fpqc covering of schemes over S. Set X_ i = X_{T_ i} = X \times _ S T_ i and \mathcal{F}_ i = \mathcal{F}_{T_ i}. Note that \{ X_ i \to X_ T\} _{i \in I} is an fpqc covering of X_ T (Topologies on Spaces, Lemma 73.9.3) and that X_{T_ i \times _ T T_{i'}} = X_ i \times _{X_ T} X_{i'}. Suppose that \mathcal{F}_ i \to \mathcal{Q}_ i is a collection of elements of \text{Q}_{\mathcal{F}/X/B}(T_ i) such that \mathcal{Q}_ i and \mathcal{Q}_{i'} restrict to the same element of \text{Q}_{\mathcal{F}/X/B}(T_ i \times _ T T_{i'}). By Remark 99.7.3 we obtain a surjective map of quasi-coherent \mathcal{O}_{X_ T}-modules \mathcal{F}_ T \to \mathcal{Q} whose restriction to X_ i recovers the given quotients. By Morphisms of Spaces, Lemma 67.31.5 we see that \mathcal{Q} is flat over T. Finally, in the case of \text{Q}^{fp}_{\mathcal{F}/X/B}, i.e., if \mathcal{Q}_ i are of finite presentation, then Descent on Spaces, Lemma 74.6.2 guarantees that \mathcal{Q} is of finite presentation as an \mathcal{O}_{X_ T}-module. \square

Sanity check: \text{Q}_{\mathcal{F}/X/B}, \text{Q}^{fp}_{\mathcal{F}/X/B} play the same role among algebraic spaces over S.

Lemma 99.7.5. In Situation 99.7.1. Let T be an algebraic space over S. We have

\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})}(T, \text{Q}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_ T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and} \\ \mathcal{Q}\text{ is quasi-coherent and flat over }T \end{matrix} \right\}

where \mathcal{F}_ T denotes the pullback of \mathcal{F} to the algebraic space X \times _{B, h} T. Similarly, we have

\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})}(T, \text{Q}^{fp}_{\mathcal{F}/X/B}) = \left\{ \begin{matrix} (h, \mathcal{F}_ T \to \mathcal{Q}) \text{ where } h : T \to B \text{ and} \\ \mathcal{Q}\text{ is of finite presentation and flat over }T \end{matrix} \right\}

Proof. Choose a scheme U and a surjective étale morphism p : U \to T. Let R = U \times _ T U with projections t, s : R \to U.

Let v : T \to \text{Q}_{\mathcal{F}/X/B} be a natural transformation. Then v(p) corresponds to a pair (h_ U, \mathcal{F}_ U \to \mathcal{Q}_ U) over U. As v is a transformation of functors we see that the pullbacks of (h_ U, \mathcal{F}_ U \to \mathcal{Q}_ U) by s and t agree. Since T = U/R (Spaces, Lemma 65.9.1), we obtain a morphism h : T \to B such that h_ U = h \circ p. By Descent on Spaces, Proposition 74.4.1 the quotient \mathcal{Q}_ U descends to a quotient \mathcal{F}_ T \to \mathcal{Q} over X_ T. Since U \to T is surjective and flat, it follows from Morphisms of Spaces, Lemma 67.31.5 that \mathcal{Q} is flat over T.

Conversely, let (h, \mathcal{F}_ T \to \mathcal{Q}) be a pair over T. Then we get a natural transformation v : T \to \text{Q}_{\mathcal{F}/X/B} by sending a morphism a : T' \to T where T' is a scheme to (h \circ a, \mathcal{F}_{T'} \to a^*\mathcal{Q}). We omit the verification that the construction of this and the previous paragraph are mutually inverse.

In the case of \text{Q}^{fp}_{\mathcal{F}/X/B} we add: given a morphism h : T \to B, a quasi-coherent sheaf on X_ T is of finite presentation as an \mathcal{O}_{X_ T}-module if and only if the pullback to X_ U is of finite presentation as an \mathcal{O}_{X_ U}-module. This follows from the fact that X_ U \to X_ T is surjective and étale and Descent on Spaces, Lemma 74.6.2. \square

Lemma 99.7.6. In Situation 99.7.1 let \{ X_ i \to X\} _{i \in I} be an fpqc covering and for each i, j \in I let \{ X_{ijk} \to X_ i \times _ X X_ j\} be an fpqc covering. Denote \mathcal{F}_ i, resp. \mathcal{F}_{ijk} the pullback of \mathcal{F} to X_ i, resp. X_{ijk}. For every scheme T over B the diagram

\xymatrix{ Q_{\mathcal{F}/X/B}(T) \ar[r] & \prod \nolimits _ i Q_{\mathcal{F}_ i/X_ i/B}(T) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod \nolimits _{i, j, k} Q_{\mathcal{F}_{ijk}/X_{ijk}/B}(T) }

presents the first arrow as the equalizer of the other two. The same is true for the functor \text{Q}^{fp}_{\mathcal{F}/X/B}.

Proof. Let \mathcal{F}_{i, T} \to \mathcal{Q}_ i be an element in the equalizer of \text{pr}_0^* and \text{pr}_1^*. By Remark 99.7.3 we obtain a surjection \mathcal{F}_ T \to \mathcal{Q} of quasi-coherent \mathcal{O}_{X_ T}-modules whose restriction to X_{i, T} recovers \mathcal{F}_ i \to \mathcal{Q}_ i. By Morphisms of Spaces, Lemma 67.31.5 we see that \mathcal{Q} is flat over T as desired. In the case of the functor \text{Q}^{fp}_{\mathcal{F}/X/B}, i.e., if \mathcal{Q}_ i is of finite presentation, then \mathcal{Q} is of finite presentation too by Descent on Spaces, Lemma 74.6.2. \square

Lemma 99.7.7. In Situation 99.7.1 assume also that (a) f is quasi-compact and quasi-separated and (b) \mathcal{F} is of finite presentation. Then the functor \text{Q}^{fp}_{\mathcal{F}/X/B} is limit preserving in the following sense: If T = \mathop{\mathrm{lim}}\nolimits T_ i is a directed limit of affine schemes over B, then \text{Q}^{fp}_{\mathcal{F}/X/B}(T) = \mathop{\mathrm{colim}}\nolimits \text{Q}^{fp}_{\mathcal{F}/X/B}(T_ i).

Proof. Let T = \mathop{\mathrm{lim}}\nolimits T_ i be as in the statement of the lemma. Choose i_0 \in I and replace I by \{ i \in I \mid i \geq i_0\} . We may set B = S = T_{i_0} and we may replace X by X_{T_0} and \mathcal{F} by the pullback to X_{T_0}. Then X_ T = \mathop{\mathrm{lim}}\nolimits X_{T_ i}, see Limits of Spaces, Lemma 70.4.1. Let \mathcal{F}_ T \to \mathcal{Q} be an element of \text{Q}^{fp}_{\mathcal{F}/X/B}(T). By Limits of Spaces, Lemma 70.7.2 there exists an i and a map \mathcal{F}_{T_ i} \to \mathcal{Q}_ i of \mathcal{O}_{X_{T_ i}}-modules of finite presentation whose pullback to X_ T is the given quotient map.

We still have to check that, after possibly increasing i, the map \mathcal{F}_{T_ i} \to \mathcal{Q}_ i is surjective and \mathcal{Q}_ i is flat over T_ i. To do this, choose an affine scheme U and a surjective étale morphism U \to X (see Properties of Spaces, Lemma 66.6.3). We may check surjectivity and flatness over T_ i after pulling back to the étale cover U_{T_ i} \to X_{T_ i} (by definition). This reduces us to the case where X = \mathop{\mathrm{Spec}}(B_0) is an affine scheme of finite presentation over B = S = T_0 = \mathop{\mathrm{Spec}}(A_0). Writing T_ i = \mathop{\mathrm{Spec}}(A_ i), then T = \mathop{\mathrm{Spec}}(A) with A = \mathop{\mathrm{colim}}\nolimits A_ i we have reached the following algebra problem. Let M_ i \to N_ i be a map of finitely presented B_0 \otimes _{A_0} A_ i-modules such that M_ i \otimes _{A_ i} A \to N_ i \otimes _{A_ i} A is surjective and N_ i \otimes _{A_ i} A is flat over A. Show that for some i' \geq i M_ i \otimes _{A_ i} A_{i'} \to N_ i \otimes _{A_ i} A_{i'} is surjective and N_ i \otimes _{A_ i} A_{i'} is flat over A. The first follows from Algebra, Lemma 10.127.5 and the second from Algebra, Lemma 10.168.1. \square

Lemma 99.7.8. In Situation 99.7.1. Let

\xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Y \ar[r] & Y' }

be a pushout in the category of schemes over B where Z \to Z' is a thickening and Z \to Y is affine, see More on Morphisms, Lemma 37.14.3. Then the natural map

Q_{\mathcal{F}/X/B}(Y') \longrightarrow Q_{\mathcal{F}/X/B}(Y) \times _{Q_{\mathcal{F}/X/B}(Z)} Q_{\mathcal{F}/X/B}(Z')

is bijective. If X \to B is locally of finite presentation, then the same thing is true for Q^{fp}_{\mathcal{F}/X/B}.

Proof. Let us construct an inverse map. Namely, suppose we have \mathcal{F}_ Y \to \mathcal{A}, \mathcal{F}_{Z'} \to \mathcal{B}', and an isomorphism \mathcal{A}|_{X_ Z} \to \mathcal{B}'|_{X_ Z} compatible with the given surjections. Then we apply Pushouts of Spaces, Lemma 81.6.6 to get a quasi-coherent module \mathcal{A}' on X_{Y'} flat over Y'. Since this sheaf is constructed as a fibre product (see proof of cited lemma) there is a canonical map \mathcal{F}_{Y'} \to \mathcal{A}'. That this map is surjective can be seen because it factors as

\begin{matrix} \mathcal{F}_{Y'} \\ \downarrow \\ (X_ Y \to X_{Y'})_*\mathcal{F}_ Y \times _{(X_ Z \to X_{Y'})_*\mathcal{F}_ Z} (X_{Z'} \to X_{Y'})_*\mathcal{F}_{Z'} \\ \downarrow \\ \mathcal{A}' = (X_ Y \to X_{Y'})_*\mathcal{A} \times _{(X_ Z \to X_{Y'})_*\mathcal{A}|_{X_ Z}} (X_{Z'} \to X_{Y'})_*\mathcal{B}' \end{matrix}

and the first arrow is surjective by More on Algebra, Lemma 15.6.5 and the second by More on Algebra, Lemma 15.6.6.

In the case of Q^{fp}_{\mathcal{F}/X/B} all we have to show is that the construction above produces a finitely presented module. This is explained in More on Algebra, Remark 15.7.8 in the commutative algebra setting. The current case of modules over algebraic spaces follows from this by étale localization. \square

Remark 99.7.9 (Obstructions for quotients). In Situation 99.7.1 assume that \mathcal{F} is flat over B. Let T \subset T' be an first order thickening of schemes over B with ideal sheaf \mathcal{J}. Then X_ T \subset X_{T'} is a first order thickening of algebraic spaces whose ideal sheaf \mathcal{I} is a quotient of f_ T^*\mathcal{J}. We will think of sheaves on X_{T'}, resp. T' as sheaves on X_ T, resp. T using the fundamental equivalence described in More on Morphisms of Spaces, Section 76.9. Let

0 \to \mathcal{K} \to \mathcal{F}_ T \to \mathcal{Q} \to 0

define an element x of Q_{\mathcal{F}/X/B}(T). Since \mathcal{F}_{T'} is flat over T' we have a short exact sequence

0 \to f_ T^*\mathcal{J} \otimes _{\mathcal{O}_{X_ T}} \mathcal{F}_ T \xrightarrow {i} \mathcal{F}_{T'} \xrightarrow {\pi } \mathcal{F}_ T \to 0

and we have f_ T^*\mathcal{J} \otimes _{\mathcal{O}_{X_ T}} \mathcal{F}_ T = \mathcal{I} \otimes _{\mathcal{O}_{X_ T}} \mathcal{F}_ T, see Deformation Theory, Lemma 91.11.2. Let us use the abbreviation f_ T^*\mathcal{J} \otimes _{\mathcal{O}_{X_ T}} \mathcal{G} = \mathcal{G} \otimes _{\mathcal{O}_ T} \mathcal{J} for an \mathcal{O}_{X_ T}-module \mathcal{G}. Since \mathcal{Q} is flat over T, we obtain a short exact sequence

0 \to \mathcal{K} \otimes _{\mathcal{O}_ T} \mathcal{J} \to \mathcal{F}_ T \otimes _{\mathcal{O}_ T} \mathcal{J} \to \mathcal{Q} \otimes _{\mathcal{O}_ T} \mathcal{J} \to \to 0

Combining the above we obtain an canonical extension

0 \to \mathcal{Q} \otimes _{\mathcal{O}_ T} \mathcal{J} \to \pi ^{-1}(\mathcal{K})/i(\mathcal{K} \otimes _{\mathcal{O}_ T} \mathcal{J}) \to \mathcal{K} \to 0

of \mathcal{O}_{X_ T}-modules. This defines a canonical class

o_ x(T') \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_{X_ T}}(\mathcal{K}, \mathcal{Q} \otimes _{\mathcal{O}_ T} \mathcal{J})

If o_ x(T') is zero, then we obtain a splitting of the short exact sequence defining it, in other words, we obtain a \mathcal{O}_{X_{T'}}-submodule \mathcal{K}' \subset \pi ^{-1}(\mathcal{K}) sitting in a short exact sequence 0 \to \mathcal{K} \otimes _{\mathcal{O}_ T} \mathcal{J} \to \mathcal{K}' \to \mathcal{K} \to 0. Then it follows from the lemma reference above that \mathcal{Q}' = \mathcal{F}_{T'}/\mathcal{K}' is a lift of x to an element of Q_{\mathcal{F}/X/B}(T'). Conversely, the reader sees that the existence of a lift implies that o_ x(T') is zero. Moreover, if x \in Q_{\mathcal{F}/X/B}^{fp}(T), then automatically x' \in Q_{\mathcal{F}/X/B}^{fp}(T') by Deformation Theory, Lemma 91.11.3. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi).

Remark 99.7.10 (Deformations of quotients). In Situation 99.7.1 assume that \mathcal{F} is flat over B. We continue the discussion of Remark 99.7.9. Assume o_ x(T') = 0. Then we claim that the set of lifts x' \in Q_{\mathcal{F}/X/B}(T') is a principal homogeneous space under the group

\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X_ T}}(\mathcal{K}, \mathcal{Q} \otimes _{\mathcal{O}_ T} \mathcal{J})

Namely, given any \mathcal{F}_{T'} \to \mathcal{Q}' flat over T' lifting the quotient \mathcal{Q} we obtain a commutative diagram with exact rows and columns

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & \mathcal{K} \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{F}_ T \otimes \mathcal{J} \ar[r] \ar[d] & \mathcal{Q} \otimes \mathcal{J} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}' \ar[r] \ar[d] & \mathcal{F}_{T'} \ar[r] \ar[d] & \mathcal{Q}' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[d] \ar[r] & \mathcal{F}_ T \ar[d] \ar[r] & \mathcal{Q} \ar[d] \ar[r] & 0 \\ & 0 & 0 & 0 }

(to see this use the observations made in the previous remark). Given a map \varphi : \mathcal{K} \to \mathcal{Q} \otimes \mathcal{J} we can consider the subsheaf \mathcal{K}'_\varphi \subset \mathcal{F}_{T'} consisting of those local sections s whose image in \mathcal{F}_ T is a local section k of \mathcal{K} and whose image in \mathcal{Q}' is the local section \varphi (k) of \mathcal{Q} \otimes \mathcal{J}. Then set \mathcal{Q}'_\varphi = \mathcal{F}_{T'}/\mathcal{K}'_\varphi . Conversely, any second lift of x corresponds to one of the qotients constructed in this manner. If we ever need this remark we will turn this remark into a lemma, precisely formulate the result and give a detailed proof (in fact, all of the above works in the setting of arbitrary ringed topoi).


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