nLab rigidification of a stack

Idea

Given a stack $\mathcal{S}$ over a site $\mathcal{C}$. One often wants to rigidify (kill off a flat subgroup of the inertia) in order to realize the stack as a gerbe over an algebraic space.

Alternative idea:

Given a moduli stack classifying some kind of structure, one sometimes wants to “remove the automorphisms” inside it such as to be left with just a moduli space. This is sometimes called “rigidification”. The archetypical example is the passage from the the groupoid of line bundles over a space to its decategorification given by the (set underlying the) Picard group. Doing this over all spaces means passing from the stack of line bundles to the Picard scheme. The general process of “rigidification” is supposed to be a mechanism that generalizes this process (ACV, 5.1.1).

Definition

We first give the simple general definition of rigidification

• General definition

Then we discuss specifically the case for algebraic stacks where one may add a bunch of technical assumptions

• Definition for algebraic stacks

General

For $\mathbf{H}$ an (∞,1)-topos and $X \in \mathbf{H}$ any object, write $\mathbf{Aut}(X) \in Grp(\mathbf{H})$ for its internal automorphism ∞-group. Consider a braided ∞-group $H \in BrGrp(\mathbf{H})$ and an ∞-group homomorphism

$$\iota \;\colon\; \mathbf{B}H \to \mathbf{Aut}(X)$$

of ∞-groups. This defines an ∞-action of $\mathbf{B}H$ on $X$, hence a fiber sequence in $\mathbf{H}$ of the form

$$\array{ X &\to& X//\mathbf{B}H \\ && \downarrow \\ && \mathbf{B}^2 H } \;\;\;\; inside \;\;\;\; \array{ X &\to& X//\mathbf{Aut}(X) \\ && \downarrow \\ && \mathbf{B} \mathbf{Aut}(X) } \,.$$

Definition

The ∞-quotient $X//\mathbf{B}H$ is what is sometimes called the “rigidification” of $X$, especially if $H$ is maximal such that there is a homomorphism $\mathbf{B}H \to \mathbf{Aut}(X)$.

For algebraic stacks

Let $X$ be a scheme. Let $\mathcal{S}\to X$ be an algebraic stack fibered in groupoids over $X$. Let $H$ be a finitely presented, separated, group scheme over $X$ such that for each $\xi\in\mathcal{S}(T)$ there is an embedding $H(T)\to Aut_T(\xi)$ compatible with pullback.

It follows that $H$ must be abelian (because $H(T)$ lies in the center of $Aut_T(\xi)$). The condition on $H$ is trivially satisfied whenever $\mathcal{S}$ is banded by $H$.

Define the $H$-rigidification of $\mathcal{S}$ to be $\mathcal{S}^H$. (ACV, def. 5.1.4).

Theorem (A-C-V, theorem 5.1.5): The space $\mathcal{S}^H$ exists such that there is a smooth surjective finitely presented morphism of stacks $\mathcal{S}\to \mathcal{S}^H$ satisfying the following:

1. For any $\xi\in \mathcal{S}(T)$ with image $\eta\in \mathcal{S}^H(T)$, we have $H(T)$ lies in the kernel of $Aut_T(\xi)\to Aut_T(\eta)$.
2. The map $\mathcal{S}\to \mathcal{S}^H$ is universal with respect to stack morphisms satisfying (1).
3. If $T$ is the spectrum of an algebraically closed field, then $Aut_T(\eta)=Aut_T(\xi)/H(T)$.
4. A moduli space for $\mathcal{S}$ is also a moduli space for $\mathcal{S}^H$.

and if $\mathcal{S}$ is a Deligne-Mumford stack, then $\mathcal{S}^H$ is also a Deligne-Mumford stack and $\mathcal{S}\to \mathcal{S}^H$ is etale.

Examples

We discuss some examples. First, to get rid of all distraction introduced by the dependence on objects of a site of definition, we consider the special case where the underlying site is the point, hence where stacks are just plain groupoids – geometrically discrete groupoids for emphasis.

• For a geometrically discrete groupoid

Then we discuss aspects of regidification for algebraic stacks

• For an algebraic stack.

For a geometrically discrete groupoid

If $\mathbf{H} =$ ∞Grpd and $X \in \infty Grpd$ is a 1-truncated object, hence just a groupoid, then $\mathbf{Aut}(X)$ is its automorphism 2-group. Its objects are naturally identified with those functors $\alpha \colon X \to X$ that are equivalences, and its morphisms with the natural isomorphisms $g \colon \alpha \to \beta$ between these. In particular if $\alpha = \beta = id$ is the identity automorphism, then such a $g$ is a function which to each object $\xi \in X$ assigns an automorphism $g_\xi \colon \xi \to \xi$ in $X$ such that for each morphism $\phi \colon \xi \to \eta$ in $X$ the naturality square

$$\array{ \xi &\stackrel{\phi}{\to}& \eta \\ \downarrow^{\mathrlap{g_\xi}} && \downarrow^{\mathrlap{g_\eta}} \\ \xi &\stackrel{\phi}{\to}& \eta } \,.$$

commutes.

Now for $H$ an abelian group there is the delooping groupoid $\mathbf{B}H$ which has a single object and $H$ as the group of morphisms from that object to itself. Both $\mathbf{Aut}(X)$ and $\mathbf{B}H$ are 2-groups in this case. A homomorphism of 2-groups

$$\iota \;\colon\; \mathbf{B}H \to \mathbf{Aut}(X)$$

has to send the essentially unique point of $\mathbf{B}H$ to the identity functor $id_X$ and is hence equivalently a function that sends each element $g \in G$ to a natural isomorphism $g \colon id_X \to id_X$, hence a function $g_{(-)}$ that sends each object $\xi \in X$ to a morphism $g_\xi \colon \xi \to \xi$ in $X$, such that the above diagram commutes. Moreover, this being a 2-group homomorphism means that for $g_1, g_2 \in H$ two elements, they are sent to the composite $(g_2)_\xi\circ (g_1)_\xi$ in $X$.

In other words, we have a functor

$$\rho \colon X \times \mathbf{B}H \to X \,,$$

which takes a pair of objects $(\xi,\ast)$ to $\xi$, takes a pair of morphisms of the form $(id_\xi, \ast \stackrel{g}{\to} \ast)$ to $(\xi \stackrel{g_\xi}{\to} \xi)$ and takes a pair of morphisms of the form $(\xi \stackrel{\phi}{\to} \eta, id_\ast)$ to $(\xi \stackrel{\phi}{\to} \eta)$; and which satisfies the action property,

$$\array{ X \times \mathbf{B}H \times \mathbf{B}H &\stackrel{\rho \times id_{\mathbf{B}H}}{\to}& X \times \mathbf{B}H \\ {}^{\mathllap{id_X \times \cdot_{\mathbf{B}H}}}\downarrow &\swArrow& \downarrow^{\rho} \\ X \times \mathbf{B}G &\stackrel{\rho}{\to}& X } \,.$$

In fact, with the groupoids explicitly presented the way we have discussed them, the natural transformation filling this diagram is the identity and hence we have exhibited the ∞-action of the 2-group $\mathbf{B}H$ on the groupoid $X$ by an ordinary action. More precisely, under passing to nerves of groupoids we have exhibited it as the action of a simplicial group on a Kan complex, which is just a simplicial diagram of ordinary actions of ordinary groups on plain sets. Since these are 2-coskeletal simplicial sets (being the nerves of just 1-groupoids), it is sufficient to consider them just in degrees 0,1,2. So then we have the following simplicial diagram of ordinary groups acting on ordinary sets

$$\left( \array{ X_1 \times_{X_0} X_1 \\ \downarrow \downarrow \downarrow \\ X_1 \\ \downarrow \downarrow \\ X_0 } \right) \times \left( \array{ H \times H \\ \downarrow \downarrow \downarrow \\ H \\ \downarrow \downarrow \\ \ast } \right) \to \left( \array{ X_1 \times_{X_0} X_1 \\ \downarrow \downarrow \downarrow \\ X_1 \\ \downarrow \downarrow \\ X_0 } \right) \,.$$

In degree 0 this is the identity map $(\xi,\ast) \mapsto \xi$, in degree 1 it is (with the symbols as above) the map $(\phi,g) \mapsto g_\eta \circ \phi = \phi \circ g_\xi$ and so on.

Finally, the ∞-quotient $X//\mathbf{B}H$ of an ∞-action of an ∞-group presented as an ordinary action of a simplicial group on a Kan complex this way is presented by the Borel construction, namely the ordinary quotient of simplicial sets

$$X \times_{\mathbf{B}H} \mathbf{E}\mathbf{B}H \coloneqq (X \times \mathbf{E}\mathbf{B}H)/\mathbf{B}H$$

(where now all symbols stand for the corresponding simplicial sets as described above). Here

$$\mathbf{E} \mathbf{B}H \coloneqq (\mathbf{B}H)^{\Delta^1} \times_{\mathbf{B}G} *$$

is a model for the total space of the universal principal 2-bundle over $\mathbf{B}H$.

So the Kan complex $X \times_{\mathbf{B}H} \mathbf{E}\mathbf{B}H$ presents the “rigidification” of $X$ with respect to the chosen $\iota \colon \mathbf{B}H \to \mathbf{Aut}(X)$.

For an algebraic stack

The standard example is the $\mathbb{G}_m$-rigidification of the Picard stack. Suppose $X/k$ is an irreducible variety over a field. One can say that the failure of the Picard stack, $\mathcal{Pic}_X$ to be representable comes from the fact that objects in fiber categories have automorphisms by the multiplicative group, so we would like to kill this group.

As pointed out in Picard scheme, the relative Picard scheme is the sheafification of $\mathcal{Pic}_X$ and representable. Moreover $\mathcal{Pic}_X\to Pic_X$ is a $\mathbb{G}_m$-gerbe, so $\mathbb{G}_m$ satisfies the conditions to rigidify.

By the universal property, the rigidification is exactly $Pic_X$, so in this case we see that the sheafification and the rigidification by the inertia are the same.

References

• Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted Bundles and Admissible Covers (arXiv:0106211)

• Matthieu Romagny, Group Actions on Stacks and Applications, Michigan Math. J. Volume 53, Issue 1 (2005), 209-236 (Project Euclid).

category: algebraic geometry