nLab
rigidification of a stack

Contents

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

Given a stack 𝒮\mathcal{S} over a site 𝒞\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

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

General

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

ι:BHAut(X) \iota \;\colon\; \mathbf{B}H \to \mathbf{Aut}(X)

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

X X//BH B 2HinsideX X//Aut(X) BAut(X). \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//BHX//\mathbf{B}H is what is sometimes called the “rigidification” of XX, especially if HH is maximal such that there is a homomorphism BHAut(X)\mathbf{B}H \to \mathbf{Aut}(X).

For algebraic stacks

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

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

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

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

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

and if 𝒮\mathcal{S} is a Deligne-Mumford stack, then 𝒮 H\mathcal{S}^H is also a Deligne-Mumford stack and 𝒮𝒮 H\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 groupoidsgeometrically discrete groupoids for emphasis.

Then we discuss aspects of regidification for algebraic stacks

For a geometrically discrete groupoid

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

ξ ϕ η g ξ g η ξ ϕ η. \array{ \xi &\stackrel{\phi}{\to}& \eta \\ \downarrow^{\mathrlap{g_\xi}} && \downarrow^{\mathrlap{g_\eta}} \\ \xi &\stackrel{\phi}{\to}& \eta } \,.

commutes.

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

ι:BHAut(X) \iota \;\colon\; \mathbf{B}H \to \mathbf{Aut}(X)

has to send the essentially unique point of BH\mathbf{B}H to the identity functor id Xid_X and is hence equivalently a function that sends each element gGg \in G to a natural isomorphism g:id Xid Xg \colon id_X \to id_X, hence a function g ()g_{(-)} that sends each object ξX\xi \in X to a morphism g ξ:ξξg_\xi \colon \xi \to \xi in XX, such that the above diagram commutes. Moreover, this being a 2-group homomorphism means that for g 1,g 2Hg_1, g_2 \in H two elements, they are sent to the composite (g 2) ξ(g 1) ξ(g_2)_\xi\circ (g_1)_\xi in XX.

In other words, we have a functor

ρ:X×BHX, \rho \colon X \times \mathbf{B}H \to X \,,

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

X×BH×BH ρ×id BH X×BH id X× BH ρ X×BG ρ X. \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 BH\mathbf{B}H on the groupoid XX 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

(X 1× X 0X 1 X 1 X 0)×(H×H H *)(X 1× X 0X 1 X 1 X 0). \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 (ξ,*)ξ(\xi,\ast) \mapsto \xi, in degree 1 it is (with the symbols as above) the map (ϕ,g)g ηϕ=ϕg ξ(\phi,g) \mapsto g_\eta \circ \phi = \phi \circ g_\xi and so on.

Finally, the ∞-quotient X//BHX//\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× BHEBH(X×EBH)/BH 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

EBH(BH) Δ 1× BG* \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 BH\mathbf{B}H.

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

For an algebraic stack

The standard example is the 𝔾 m\mathbb{G}_m-rigidification of the Picard stack. Suppose X/kX/k is an irreducible variety over a field. One can say that the failure of the Picard stack, 𝒫𝒾𝒸 X\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 𝒫𝒾𝒸 X\mathcal{Pic}_X and representable. Moreover 𝒫𝒾𝒸 XPic X\mathcal{Pic}_X\to Pic_X is a 𝔾 m\mathbb{G}_m-gerbe, so 𝔾 m\mathbb{G}_m satisfies the conditions to rigidify.

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

References

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

Last revised on March 6, 2013 at 18:59:20. See the history of this page for a list of all contributions to it.