The Stacks project

39.15 Colimits of quasi-coherent modules

In this section we prove some technical results saying that under suitable assumptions every quasi-coherent module on a groupoid is a filtered colimit of “small” quasi-coherent modules.

Lemma 39.15.1. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s, t$ are flat, quasi-compact, and quasi-separated. For any quasi-coherent module $\mathcal{G}$ on $U$, there exists a canonical isomorphism $\alpha : t^*s_*t^*\mathcal{G} \to s^*s_*t^*\mathcal{G}$ which turns $(s_*t^*\mathcal{G}, \alpha )$ into a quasi-coherent module on $(U, R, s, t, c)$. This construction defines a functor

\[ \mathit{QCoh}(\mathcal{O}_ U) \longrightarrow \mathit{QCoh}(U, R, s, t, c) \]

which is a right adjoint to the forgetful functor $(\mathcal{F}, \beta ) \mapsto \mathcal{F}$.

Proof. The pushforward of a quasi-coherent module along a quasi-compact and quasi-separated morphism is quasi-coherent, see Schemes, Lemma 26.24.1. Hence $s_*t^*\mathcal{G}$ is quasi-coherent. With notation as in Lemma 39.13.4 we have

\[ t^*s_*t^*\mathcal{G} = \text{pr}_{1, *}\text{pr}_0^*t^*\mathcal{G} = \text{pr}_{1, *}c^*t^*\mathcal{G} = s^*s_*t^*\mathcal{G} \]

The middle equality because $t \circ c = t \circ \text{pr}_0$ as morphisms $R \times _{s, U, t} R \to U$, and the first and the last equality because we know that base change and pushforward commute in these steps by Cohomology of Schemes, Lemma 30.5.2.

To verify the cocycle condition of Definition 39.14.1 for $\alpha $ and the adjointness property we describe the construction $\mathcal{G} \mapsto (s_*t^*\mathcal{G}, \alpha )$ in another way. Consider the groupoid scheme $(R, R \times _{t, U, t} R, \text{pr}_0, \text{pr}_1, \text{pr}_{02})$ associated to the equivalence relation $R \times _{t, U, t} R$ on $R$, see Lemma 39.13.3. There is a morphism

\[ f : (R, R \times _{t, U, t} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02}) \longrightarrow (U, R, s, t, c) \]

of groupoid schemes given by $s : R \to U$ and $R \times _{t, U, t} R \to R$ given by $(r_0, r_1) \mapsto r_0^{-1} \circ r_1$; we omit the verification of the commutativity of the required diagrams. Since $t, s : R \to U$ are quasi-compact, quasi-separated, and flat, and since we have a cartesian square

\[ \xymatrix{ R \times _{t, U, t} R \ar[d]_{\text{pr}_0} \ar[rr]_-{(r_0, r_1) \mapsto r_0^{-1} \circ r_1} & & R \ar[d]^ t \\ R \ar[rr]^ s & & U } \]

by Lemma 39.13.5 it follows that Lemma 39.14.4 applies to $f$. Thus pushforward and pullback of quasi-coherent modules along $f$ are adjoint functors. To finish the proof we will identify these functors with the functors described above. To do this, note that

\[ t^* : \mathit{QCoh}(\mathcal{O}_ U) \longrightarrow \mathit{QCoh}(R, R \times _{t, U, t} R, \text{pr}_1, \text{pr}_0, \text{pr}_{02}) \]

is an equivalence by the theory of descent of quasi-coherent sheaves as $\{ t : R \to U\} $ is an fpqc covering, see Descent, Proposition 35.5.2.

Pushforward along $f$ precomposed with the equivalence $t^*$ sends $\mathcal{G}$ to $(s_*t^*\mathcal{G}, \alpha )$; we omit the verification that the isomorphism $\alpha $ obtained in this fashion is the same as the one constructed above.

Pullback along $f$ postcomposed with the inverse of the equivalence $t^*$ sends $(\mathcal{F}, \beta )$ to the descent relative to $\{ t : R \to U\} $ of the module $s^*\mathcal{F}$ endowed with the descent datum $\gamma $ on $R \times _{t, U, t} R$ which is the pullback of $\beta $ by $(r_0, r_1) \mapsto r_0^{-1} \circ r_1$. Consider the isomorphism $\beta : t^*\mathcal{F} \to s^*\mathcal{F}$. The canonical descent datum (Descent, Definition 35.2.3) on $t^*\mathcal{F}$ relative to $\{ t : R \to U\} $ translates via $\beta $ into the map

\[ \text{pr}_0^*s^*\mathcal{F} \xrightarrow {\text{pr}_0^*\beta ^{-1}} \text{pr}_0^*t^*\mathcal{F} \xrightarrow {can} \text{pr}_1^*t^*\mathcal{F} \xrightarrow {\text{pr}_1^*\beta } \text{pr}_1^*s^*\mathcal{F} \]

Since $\beta $ satisfies the cocycle condition, this is equal to the pullback of $\beta $ by $(r_0, r_1) \mapsto r_0^{-1} \circ r_1$. To see this take the actual cocycle relation in Definition 39.14.1 and pull it back by the morphism $(\text{pr}_0, c \circ (i, 1)) : R \times _{t, U, t} R \to R \times _{s, U, t} R$ which also plays a role in the commutative diagram of Lemma 39.13.5. It follows that $(s^*\mathcal{F}, \gamma )$ is isomorphic to $(t^*\mathcal{F}, can)$. All in all, we conclude that pullback by $f$ postcomposed with the inverse of the equivalence $t^*$ is isomorphic to the forgetful functor $(\mathcal{F}, \beta ) \mapsto \mathcal{F}$. $\square$

Remark 39.15.2. In the situation of Lemma 39.15.1 denote

\[ F : \mathit{QCoh}(U, R, s, t, c) \to \mathit{QCoh}(\mathcal{O}_ U),\quad (\mathcal{F}, \beta ) \mapsto \mathcal{F} \]

the forgetful functor and denote

\[ G : \mathit{QCoh}(\mathcal{O}_ U) \to \mathit{QCoh}(U, R, s, t, c),\quad \mathcal{G} \mapsto (s_*t^*\mathcal{G}, \alpha ) \]

the right adjoint constructed in the lemma. Then the unit $\eta : \text{id} \to G \circ F$ of the adjunction evaluated on $(\mathcal{F}, \beta )$ is given by the map

\[ \mathcal{F} \to s_*s^*\mathcal{F} \xrightarrow {\beta ^{-1}} s_*t^*\mathcal{F} \]

We omit the verification.

Lemma 39.15.3. Let $f : Y \to X$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module, let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module, and let $\varphi : \mathcal{G} \to f^*\mathcal{F}$ be a module map. Assume

  1. $\varphi $ is injective,

  2. $f$ is quasi-compact, quasi-separated, flat, and surjective,

  3. $X$, $Y$ are locally Noetherian, and

  4. $\mathcal{G}$ is a coherent $\mathcal{O}_ Y$-module.

Then $\mathcal{F} \cap f_*\mathcal{G}$ defined as the pullback

\[ \xymatrix{ \mathcal{F} \ar[r] & f_*f^*\mathcal{F} \\ \mathcal{F} \cap f_*\mathcal{G} \ar[u] \ar[r] & f_*\mathcal{G} \ar[u] } \]

is a coherent $\mathcal{O}_ X$-module.

Proof. We will freely use the characterization of coherent modules of Cohomology of Schemes, Lemma 30.9.1 as well as the fact that coherent modules form a Serre subcategory of $\mathit{QCoh}(\mathcal{O}_ X)$, see Cohomology of Schemes, Lemma 30.9.3. If $f$ has a section $\sigma $, then we see that $\mathcal{F} \cap f_*\mathcal{G}$ is contained in the image of $\sigma ^*\mathcal{G} \to \sigma ^*f^*\mathcal{F} = \mathcal{F}$, hence coherent. In general, to show that $\mathcal{F} \cap f_*\mathcal{G}$ is coherent, it suffices the show that $f^*(\mathcal{F} \cap f_*\mathcal{G})$ is coherent (see Descent, Lemma 35.7.1). Since $f$ is flat this is equal to $f^*\mathcal{F} \cap f^*f_*\mathcal{G}$. Since $f$ is flat, quasi-compact, and quasi-separated we see $f^*f_*\mathcal{G} = p_*q^*\mathcal{G}$ where $p, q : Y \times _ X Y \to Y$ are the projections, see Cohomology of Schemes, Lemma 30.5.2. Since $p$ has a section we win. $\square$

Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid in schemes over $S$. Assume that $U$ is locally Noetherian. In the lemma below we say that a quasi-coherent sheaf $(\mathcal{F}, \alpha )$ on $(U, R, s, t, c)$ is coherent if $\mathcal{F}$ is a coherent $\mathcal{O}_ U$-module.

Lemma 39.15.4. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that

  1. $U$, $R$ are Noetherian,

  2. $s, t$ are flat, quasi-compact, and quasi-separated.

Then every quasi-coherent module $(\mathcal{F}, \beta )$ on $(U, R, s, t, c)$ is a filtered colimit of coherent modules.

Proof. We will use the characterization of Cohomology of Schemes, Lemma 30.9.1 of coherent modules on locally Noetherian scheme without further mention. We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i$ as the filtered colimit of coherent submodules $\mathcal{H}_ i \subset \mathcal{F}$, see Cohomology of Schemes, Lemma 30.10.4. Given a quasi-coherent sheaf $\mathcal{H}$ on $U$ we denote $(s_*t^*\mathcal{H}, \alpha )$ the quasi-coherent sheaf on $(U, R, s, t, c)$ of Lemma 39.15.1. Consider the adjunction map $(\mathcal{F}, \beta ) \to (s_*t^*\mathcal{F}, \alpha )$ in $\mathit{QCoh}(U, R, s, t, c)$, see Remark 39.15.2. Set

\[ (\mathcal{F}_ i, \beta _ i) = (\mathcal{F}, \beta ) \times _{(s_*t^*\mathcal{F}, \alpha )} (s_*t^*\mathcal{H}_ i, \alpha ) \]

in $\mathit{QCoh}(U, R, s, t, c)$. Since restriction to $U$ is an exact functor on $\mathit{QCoh}(U, R, s, t, c)$ by the proof of Lemma 39.14.6 we obtain a pullback diagram

\[ \xymatrix{ \mathcal{F} \ar[r] & s_*t^*\mathcal{F} \\ \mathcal{F}_ i \ar[r] \ar[u] & s_*t^*\mathcal{H}_ i \ar[u] } \]

in other words $\mathcal{F}_ i = \mathcal{F} \cap s_*t^*\mathcal{H}_ i$. By the description of the adjunction map in Remark 39.15.2 this diagram is isomorphic to the diagram

\[ \xymatrix{ \mathcal{F} \ar[r] & s_*s^*\mathcal{F} \\ \mathcal{F}_ i \ar[r] \ar[u] & s_*t^*\mathcal{H}_ i \ar[u] } \]

where the right vertical arrow is the result of appplying $s_*$ to the map

\[ t^*\mathcal{H}_ i \to t^*\mathcal{F} \xrightarrow {\beta } s^*\mathcal{F} \]

This arrow is injective as $t$ is a flat morphism. It follows that $\mathcal{F}_ i$ is coherent by Lemma 39.15.3. Finally, because $s$ is quasi-compact and quasi-separated we see that $s_*$ commutes with colimits (see Cohomology of Schemes, Lemma 30.6.1). Hence $s_*t^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits s_*t^*\mathcal{H}_ i$ and hence $(\mathcal{F}, \beta ) = \mathop{\mathrm{colim}}\nolimits (\mathcal{F}_ i, \beta _ i)$ as desired. $\square$

Here is a curious lemma that is useful when working with groupoids on fields. In fact, this is the standard argument to prove that any representation of an algebraic group is a colimit of finite dimensional representations.

Lemma 39.15.5. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that

  1. $U$, $R$ are affine,

  2. there exist $e_ i \in \mathcal{O}_ R(R)$ such that every element $g \in \mathcal{O}_ R(R)$ can be uniquely written as $\sum s^*(f_ i)e_ i$ for some $f_ i \in \mathcal{O}_ U(U)$.

Then every quasi-coherent module $(\mathcal{F}, \alpha )$ on $(U, R, s, t, c)$ is a filtered colimit of finite type quasi-coherent modules.

Proof. The assumption means that $\mathcal{O}_ R(R)$ is a free $\mathcal{O}_ U(U)$-module via $s$ with basis $e_ i$. Hence for any quasi-coherent $\mathcal{O}_ U$-module $\mathcal{G}$ we see that $s^*\mathcal{G}(R) = \bigoplus _ i \mathcal{G}(U)e_ i$. We will write $s(-)$ to indicate pullback of sections by $s$ and similarly for other morphisms. Let $(\mathcal{F}, \alpha )$ be a quasi-coherent module on $(U, R, s, t, c)$. Let $\sigma \in \mathcal{F}(U)$. By the above we can write

\[ \alpha (t(\sigma )) = \sum s(\sigma _ i) e_ i \]

for some unique $\sigma _ i \in \mathcal{F}(U)$ (all but finitely many are zero of course). We can also write

\[ c(e_ i) = \sum \text{pr}_1(f_{ij}) \text{pr}_0(e_ j) \]

as functions on $R \times _{s, U, t}R$. Then the commutativity of the diagram in Definition 39.14.1 means that

\[ \sum \text{pr}_1(\alpha (t(\sigma _ i))) \text{pr}_0(e_ i) = \sum \text{pr}_1(s(\sigma _ i)f_{ij}) \text{pr}_0(e_ j) \]

(calculation omitted). Picking off the coefficients of $\text{pr}_0(e_ l)$ we see that $\alpha (t(\sigma _ l)) = \sum s(\sigma _ i)f_{il}$. Hence the submodule $\mathcal{G} \subset \mathcal{F}$ generated by the elements $\sigma _ i$ defines a finite type quasi-coherent module preserved by $\alpha $. Hence it is a subobject of $\mathcal{F}$ in $\mathit{QCoh}(U, R, s, t, c)$. This submodule contains $\sigma $ (as one sees by pulling back the first relation by $e$). Hence we win. $\square$

We suggest the reader skip the rest of this section. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid in schemes over $S$. Let $\kappa $ be a cardinal. In the following we will say that a quasi-coherent sheaf $(\mathcal{F}, \alpha )$ on $(U, R, s, t, c)$ is $\kappa $-generated if $\mathcal{F}$ is a $\kappa $-generated $\mathcal{O}_ U$-module, see Properties, Definition 28.23.1.

Lemma 39.15.6. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $\kappa $ be a cardinal. There exists a set $T$ and a family $(\mathcal{F}_ t, \alpha _ t)_{t \in T}$ of $\kappa $-generated quasi-coherent modules on $(U, R, s, t, c)$ such that every $\kappa $-generated quasi-coherent module on $(U, R, s, t, c)$ is isomorphic to one of the $(\mathcal{F}_ t, \alpha _ t)$.

Proof. For each quasi-coherent module $\mathcal{F}$ on $U$ there is a (possibly empty) set of maps $\alpha : t^*\mathcal{F} \to s^*\mathcal{F}$ such that $(\mathcal{F}, \alpha )$ is a quasi-coherent modules on $(U, R, s, t, c)$. By Properties, Lemma 28.23.2 there exists a set of isomorphism classes of $\kappa $-generated quasi-coherent $\mathcal{O}_ U$-modules. $\square$

Lemma 39.15.7. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that $s, t$ are flat. There exists a cardinal $\kappa $ such that every quasi-coherent module $(\mathcal{F}, \alpha )$ on $(U, R, s, t, c)$ is the directed colimit of its $\kappa $-generated quasi-coherent submodules.

Proof. In the statement of the lemma and in this proof a submodule of a quasi-coherent module $(\mathcal{F}, \alpha )$ is a quasi-coherent submodule $\mathcal{G} \subset \mathcal{F}$ such that $\alpha (t^*\mathcal{G}) = s^*\mathcal{G}$ as subsheaves of $s^*\mathcal{F}$. This makes sense because since $s, t$ are flat the pullbacks $s^*$ and $t^*$ are exact, i.e., preserve subsheaves. The proof will be a repeat of the proof of Properties, Lemma 28.23.3. We urge the reader to read that proof first.

Choose an affine open covering $U = \bigcup _{i \in I} U_ i$. For each pair $i, j$ choose affine open coverings

\[ U_ i \cap U_ j = \bigcup \nolimits _{k \in I_{ij}} U_{ijk} \quad \text{and}\quad s^{-1}(U_ i) \cap t^{-1}(U_ j) = \bigcup \nolimits _{k \in J_{ij}} W_{ijk}. \]

Write $U_ i = \mathop{\mathrm{Spec}}(A_ i)$, $U_{ijk} = \mathop{\mathrm{Spec}}(A_{ijk})$, $W_{ijk} = \mathop{\mathrm{Spec}}(B_{ijk})$. Let $\kappa $ be any infinite cardinal $\geq $ than the cardinality of any of the sets $I$, $I_{ij}$, $J_{ij}$.

Let $(\mathcal{F}, \alpha )$ be a quasi-coherent module on $(U, R, s, t, c)$. Set $M_ i = \mathcal{F}(U_ i)$, $M_{ijk} = \mathcal{F}(U_{ijk})$. Note that

\[ M_ i \otimes _{A_ i} A_{ijk} = M_{ijk} = M_ j \otimes _{A_ j} A_{ijk} \]

and that $\alpha $ gives isomorphisms

\[ \alpha |_{W_{ijk}} : M_ i \otimes _{A_ i, t} B_{ijk} \longrightarrow M_ j \otimes _{A_ j, s} B_{ijk} \]

see Schemes, Lemma 26.7.3. Using the axiom of choice we choose a map

\[ (i, j, k, m) \mapsto S(i, j, k, m) \]

which associates to every $i, j \in I$, $k \in I_{ij}$ or $k \in J_{ij}$ and $m \in M_ i$ a finite subset $S(i, j, k, m) \subset M_ j$ such that we have

\[ m \otimes 1 = \sum \nolimits _{m' \in S(i, j, k, m)} m' \otimes a_{m'} \quad \text{or}\quad \alpha (m \otimes 1) = \sum \nolimits _{m' \in S(i, j, k, m)} m' \otimes b_{m'} \]

in $M_{ijk}$ for some $a_{m'} \in A_{ijk}$ or $b_{m'} \in B_{ijk}$. Moreover, let's agree that $S(i, i, k, m) = \{ m\} $ for all $i, j = i, k, m$ when $k \in I_{ij}$. Fix such a collection $S(i, j, k, m)$

Given a family $\mathcal{S} = (S_ i)_{i \in I}$ of subsets $S_ i \subset M_ i$ of cardinality at most $\kappa $ we set $\mathcal{S}' = (S'_ i)$ where

\[ S'_ j = \bigcup \nolimits _{(i, j, k, m)\text{ such that }m \in S_ i} S(i, j, k, m) \]

Note that $S_ i \subset S'_ i$. Note that $S'_ i$ has cardinality at most $\kappa $ because it is a union over a set of cardinality at most $\kappa $ of finite sets. Set $\mathcal{S}^{(0)} = \mathcal{S}$, $\mathcal{S}^{(1)} = \mathcal{S}'$ and by induction $\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'$. Then set $\mathcal{S}^{(\infty )} = \bigcup _{n \geq 0} \mathcal{S}^{(n)}$. Writing $\mathcal{S}^{(\infty )} = (S^{(\infty )}_ i)$ we see that for any element $m \in S^{(\infty )}_ i$ the image of $m$ in $M_{ijk}$ can be written as a finite sum $\sum m' \otimes a_{m'}$ with $m' \in S_ j^{(\infty )}$. In this way we see that setting

\[ N_ i = A_ i\text{-submodule of }M_ i\text{ generated by }S^{(\infty )}_ i \]

we have

\[ N_ i \otimes _{A_ i} A_{ijk} = N_ j \otimes _{A_ j} A_{ijk} \quad \text{and}\quad \alpha (N_ i \otimes _{A_ i, t} B_{ijk}) = N_ j \otimes _{A_ j, s} B_{ijk} \]

as submodules of $M_{ijk}$ or $M_ j \otimes _{A_ j, s} B_{ijk}$. Thus there exists a quasi-coherent submodule $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{G}(U_ i) = N_ i$ such that $\alpha (t^*\mathcal{G}) = s^*\mathcal{G}$ as submodules of $s^*\mathcal{F}$. In other words, $(\mathcal{G}, \alpha |_{t^*\mathcal{G}})$ is a submodule of $(\mathcal{F}, \alpha )$. Moreover, by construction $\mathcal{G}$ is $\kappa $-generated.

Let $\{ (\mathcal{G}_ t, \alpha _ t)\} _{t \in T}$ be the set of $\kappa $-generated quasi-coherent submodules of $(\mathcal{F}, \alpha )$. If $t, t' \in T$ then $\mathcal{G}_ t + \mathcal{G}_{t'}$ is also a $\kappa $-generated quasi-coherent submodule as it is the image of the map $\mathcal{G}_ t \oplus \mathcal{G}_{t'} \to \mathcal{F}$. Hence the system (ordered by inclusion) is directed. The arguments above show that every section of $\mathcal{F}$ over $U_ i$ is in one of the $\mathcal{G}_ t$ (because we can start with $\mathcal{S}$ such that the given section is an element of $S_ i$). Hence $\mathop{\mathrm{colim}}\nolimits _ t \mathcal{G}_ t \to \mathcal{F}$ is both injective and surjective as desired. $\square$

Comments (0)

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 07TS. Beware of the difference between the letter 'O' and the digit '0'.