The Stacks project

77.13 Colimits of quasi-coherent modules

This section is the analogue of Groupoids, Section 39.15.

Lemma 77.13.1. Let $B \to S$ as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. 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 Morphisms of Spaces, Lemma 66.11.2. Hence $s_*t^*\mathcal{G}$ is quasi-coherent. With notation as in Lemma 77.11.4 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 Spaces, Lemma 68.11.2.

To verify the cocycle condition of Definition 77.12.1 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 77.11.3. 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 77.11.5 it follows that Lemma 77.12.4 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 on Spaces, Proposition 73.4.1.

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 on Spaces, Definition 73.3.3) 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 77.12.1 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 77.11.5. 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 77.13.2. In the situation of Lemma 77.13.1 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 77.13.3. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. 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 Spaces, Lemma 68.12.2 as well as the fact that coherent modules form a Serre subcategory of $\mathit{QCoh}(\mathcal{O}_ X)$, see Cohomology of Spaces, Lemma 68.12.4. 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 on Spaces, Lemma 73.6.1). 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 Spaces, Lemma 68.11.2. Since $p$ has a section we win. $\square$

Let $B \to S$ be as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. 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 77.13.4. Let $B \to S$ be as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Assume that

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

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

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

Proof. We will use the characterization of Cohomology of Spaces, Lemma 68.12.2 of coherent modules on locally Noetherian algebraic spaces 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 Spaces, Lemma 68.15.1. 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 77.13.1. Consider the adjunction map $(\mathcal{F}, \beta ) \to (s_*t^*\mathcal{F}, \alpha )$ in $\mathit{QCoh}(U, R, s, t, c)$, see Remark 77.13.2. 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 77.12.6 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 77.13.2 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 77.13.3. 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. $\square$

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