The Stacks project

77.12 Quasi-coherent sheaves on groupoids

Please compare with Groupoids, Section 39.14.

Definition 77.12.1. Let $B \to S$ as in Section 77.3. 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

  1. 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

  2. the pullback

    \[ e^*\alpha : \mathcal{F} \longrightarrow \mathcal{F} \]

    is the identity map.

Compare with the commutative diagrams of Lemma 77.11.4.

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 77.12.2. Let $B \to S$ as in Section 77.3. 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 77.12.1 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 77.12.3. Let $B \to S$ as in Section 77.3. 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 77.12.4. Let $B \to S$ as in Section 77.3. Consider a morphism $f : (U, R, s, t, c) \to (U', R', s', t', c')$ of groupoids in algebraic spaces over $B$. Assume that

  1. $f : U \to U'$ is quasi-compact and quasi-separated,

  2. the square

    \[ \xymatrix{ R \ar[d]_ t \ar[r]_ f & R' \ar[d]^{t'} \\ U \ar[r]^ f & U' } \]

    is cartesian, and

  3. $s'$ 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 77.12.3.

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 66.11.2). 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 68.11.2. 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 77.12.5. Let $B \to S$ be as in Section 77.3. 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 65.29.7. 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 77.12.6. Let $B \to S$ as in Section 77.3. 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

This is pretty pedantic, but it might be good to spell out what a morphism of quasicoherent sheaves on groupoids is. (I assume it's just a morphism of sheaves on such that the obvious square involving the 's commute?)

Similarly with the analogous section for groupoid schemes.

Comment #5619 by on

Yes. I am going to leave this for now. We should also reformulate the definition and just require to be an isomorphism. Then this will force to be the identity (this statement is the replacement for Lemma 77.12.2). Similar stuff happens in for example Section 35.34.


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 0440. Beware of the difference between the letter 'O' and the digit '0'.