78.12 Quasi-coherent sheaves on groupoids
Please compare with Groupoids, Section 39.14.
Definition 78.12.1. Let $B \to S$ as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. A quasi-coherent module on $(U, R, s, t, c)$ is a pair $(\mathcal{F}, \alpha )$, where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ U$-module, and $\alpha $ is a $\mathcal{O}_ R$-module map
\[ \alpha : t^*\mathcal{F} \longrightarrow s^*\mathcal{F} \]
such that
the diagram
\[ \xymatrix{ & \text{pr}_1^*t^*\mathcal{F} \ar[r]_-{\text{pr}_1^*\alpha } & \text{pr}_1^*s^*\mathcal{F} \ar@{=}[rd] & \\ \text{pr}_0^*s^*\mathcal{F} \ar@{=}[ru] & & & c^*s^*\mathcal{F} \\ & \text{pr}_0^*t^*\mathcal{F} \ar[lu]^{\text{pr}_0^*\alpha } \ar@{=}[r] & c^*t^*\mathcal{F} \ar[ru]_{c^*\alpha } } \]
is a commutative in the category of $\mathcal{O}_{R \times _{s, U, t} R}$-modules, and
the pullback
\[ e^*\alpha : \mathcal{F} \longrightarrow \mathcal{F} \]
is the identity map.
Compare with the commutative diagrams of Lemma 78.11.4.
The commutativity of the first diagram forces the operator $e^*\alpha $ to be idempotent. Hence the second condition can be reformulated as saying that $e^*\alpha $ is an isomorphism. In fact, the condition implies that $\alpha $ is an isomorphism.
Lemma 78.12.2. Let $B \to S$ as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $(\mathcal{F}, \alpha )$ is a quasi-coherent module on $(U, R, s, t, c)$ then $\alpha $ is an isomorphism.
Proof.
Pull back the commutative diagram of Definition 78.12.1 by the morphism $(i, 1) : R \to R \times _{s, U, t} R$. Then we see that $i^*\alpha \circ \alpha = s^*e^*\alpha $. Pulling back by the morphism $(1, i)$ we obtain the relation $\alpha \circ i^*\alpha = t^*e^*\alpha $. By the second assumption these morphisms are the identity. Hence $i^*\alpha $ is an inverse of $\alpha $.
$\square$
Lemma 78.12.3. Let $B \to S$ as in Section 78.3. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoid in algebraic spaces over $B$. Then pullback $f^*$ given by
\[ (\mathcal{F}, \alpha ) \mapsto (f^*\mathcal{F}, f^*\alpha ) \]
defines a functor from the category of quasi-coherent sheaves on $(U', R', s', t', c')$ to the category of quasi-coherent sheaves on $(U, R, s, t, c)$.
Proof.
Omitted.
$\square$
Lemma 78.12.4. Let $B \to S$ as in Section 78.3. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in algebraic spaces over $B$. Assume that
$f : U \to U'$ is quasi-compact and quasi-separated,
the square
\[ \xymatrix{ R \ar[d]_ t \ar[r]_ f & R' \ar[d]^{t'} \\ U \ar[r]^ f & U' } \]
is cartesian, and
$s'$ and $t'$ are flat.
Then pushforward $f_*$ given by
\[ (\mathcal{F}, \alpha ) \mapsto (f_*\mathcal{F}, f_*\alpha ) \]
defines a functor from the category of quasi-coherent sheaves on $(U, R, s, t, c)$ to the category of quasi-coherent sheaves on $(U', R', s', t', c')$ which is right adjoint to pullback as defined in Lemma 78.12.3.
Proof.
Since $U \to U'$ is quasi-compact and quasi-separated we see that $f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Morphisms of Spaces, Lemma 67.11.2). Moreover, since the squares
\[ \vcenter { \xymatrix{ R \ar[d]_ t \ar[r]_ f & R' \ar[d]^{t'} \\ U \ar[r]^ f & U' } } \quad \text{and}\quad \vcenter { \xymatrix{ R \ar[d]_ s \ar[r]_ f & R' \ar[d]^{s'} \\ U \ar[r]^ f & U' } } \]
are cartesian we find that $(t')^*f_*\mathcal{F} = f_*t^*\mathcal{F}$ and $(s')^*f_*\mathcal{F} = f_*s^*\mathcal{F}$ , see Cohomology of Spaces, Lemma 69.11.2. Thus it makes sense to think of $f_*\alpha $ as a map $(t')^*f_*\mathcal{F} \to (s')^*f_*\mathcal{F}$. A similar argument shows that $f_*\alpha $ satisfies the cocycle condition. The functor is adjoint to the pullback functor since pullback and pushforward on modules on ringed spaces are adjoint. Some details omitted.
$\square$
Lemma 78.12.5. Let $B \to S$ be as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits.
Proof.
Let $i \mapsto (\mathcal{F}_ i, \alpha _ i)$ be a diagram over the index category $\mathcal{I}$. We can form the colimit $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ which is a quasi-coherent sheaf on $U$, see Properties of Spaces, Lemma 66.29.7. Since colimits commute with pullback we see that $s^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits s^*\mathcal{F}_ i$ and similarly $t^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits t^*\mathcal{F}_ i$. Hence we can set $\alpha = \mathop{\mathrm{colim}}\nolimits \alpha _ i$. We omit the proof that $(\mathcal{F}, \alpha )$ is the colimit of the diagram in the category of quasi-coherent modules on $(U, R, s, t, c)$.
$\square$
Lemma 78.12.6. Let $B \to S$ as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. If $s$, $t$ are flat, then the category of quasi-coherent modules on $(U, R, s, t, c)$ is abelian.
Proof.
Let $\varphi : (\mathcal{F}, \alpha ) \to (\mathcal{G}, \beta )$ be a homomorphism of quasi-coherent modules on $(U, R, s, t, c)$. Since $s$ is flat we see that
\[ 0 \to s^*\mathop{\mathrm{Ker}}(\varphi ) \to s^*\mathcal{F} \to s^*\mathcal{G} \to s^*\mathop{\mathrm{Coker}}(\varphi ) \to 0 \]
is exact and similarly for pullback by $t$. Hence $\alpha $ and $\beta $ induce isomorphisms $\kappa : t^*\mathop{\mathrm{Ker}}(\varphi ) \to s^*\mathop{\mathrm{Ker}}(\varphi )$ and $\lambda : t^*\mathop{\mathrm{Coker}}(\varphi ) \to s^*\mathop{\mathrm{Coker}}(\varphi )$ which satisfy the cocycle condition. Then it is straightforward to verify that $(\mathop{\mathrm{Ker}}(\varphi ), \kappa )$ and $(\mathop{\mathrm{Coker}}(\varphi ), \lambda )$ are a kernel and cokernel in the category of quasi-coherent modules on $(U, R, s, t, c)$. Moreover, the condition $\mathop{\mathrm{Coim}}(\varphi ) = \mathop{\mathrm{Im}}(\varphi )$ follows because it holds over $U$.
$\square$
Comments (2)
Comment #5385 by Will Chen on
Comment #5619 by Johan on