The Stacks project

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 $(\mathcal{F}, \beta ) \mapsto \mathcal{F}$. $\square$

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