Group actions on algebraic stacks via butterflies
GROUP ACTIONS ON ALGEBRAIC STACKS VIA
BUTTERFLIES
BEHRANG NOOHI
Abstract. We introduce an explicit method for studying actions of a group
stack G on an algebraic stack X. As an example, we study in detail the case
where X = P(n0, · · · , nr) is a weighted projective stack over an arbitrary
base S. To this end, we give an explicit description of the group stack of
automorphisms of P(n0, · · · , nr), the weighted projective general linear 2-group
PGL(n0, · · · , nr). As an application, we use a result of Colliot-Thélène to show
that for every linear algebraic group G over an arbitrary base field k (assumed
to be reductive if char(k) > 0) such that Pic(G) = 0, every action of G on
P(n0, · · · , nr) lifts to a linear action of G on Ar+1.
1. Introduction
The aim of this work is to propose a concrete method for studying group actions
on algebraic stacks. Of course, in its full generality this problem could already
be very difficult in the case of schemes. The case of stacks has yet an additional
layer of difficulty due to the fact that stacks have two types of symmetries: 1-
symmetries (i.e., self-equivalences) and 2-symmetries (i.e., 2-morphisms between
self-equivalences).
Studying actions of a group stack G on a stack X can be divided into two sub-
problems. One, which is of geometric nature, is to understand the two types of sym-
metries alluded to above; these can be packaged in a group stack AutX. The other,
which is of homotopy theoretic nature, is to get a hold of morphisms G → AutX.
Here, a morphism G → AutX means a weak monoidal functor; two morphisms
f, g : G→ AutX that are related by a monoidal transformation ϕ : f ⇒ g should be
regarded as giving rise to the “same” action.
Therefore, to study actions of G on X one needs to understand the group stack
AutX, the morphisms G→ AutX, and also the transformations between such mor-
phisms. Our proposed method, uses techniques from 2-group theory to tackle these
problems. It consists of two steps:
1) finding suitable crossed module models for AutX and G;
2) using butterflies [No3, AlNo1] to give a geometric description
of morphisms G → AutX and monoidal transformations between
them.
Finding a ‘suitable’ crossed module model for AutX may not always be easy,
but we can go about it by choosing a suitable ‘symmetric enough’ atlas X → X.
This can be used to find an approximation of AutX (Proposition 6.2), and if we are
lucky (e.g., when X = P(n0, · · · , nr)) it gives us the whole AutX.
2 BEHRANG NOOHI
Once crossed module models for G and AutX are found, the butterfly method
reduces the action problem to standard problems about group homomorphisms and
group extensions, which can be tackled using techniques from group theory.
Organization of the paper
Sections §3–§5 are devoted to setting up the basic homotopy theory of 2-group
actions and using butterflies to formulate our strategy for studying actions. To
illustrate our method, in the subsequent sections we apply these ideas to study
group actions on weighted projective stacks. In §6 we define weighted projective
general linear 2-groups PGL(n0, n1, · · · , nr) and prove (see Theorem 6.3) that they
model AutP(n0, · · · , nr); we prove this over any base scheme S, generalizing the
case S = SpecC proved in [BeNo]:
Theorem 1.1. Let AutPS(n0, n1, · · · , nr) be the group stack of automorphisms of
the weighted projective stack PS(n0, n1, · · · , nr) relative to an arbitrary base scheme
S. Then, there is a natural equivalence of group stacks
PGLS(n0, n1, · · · , nr)→ AutPS(n0, n1, · · · , nr).
Here, PGLS(n0, n1, · · · , nr) stands for the group stack associated to the crossed
module PGLS(n0, · · · , nr) (see §6 for the defnition).
We analyze the structure of PGL(n0, n1, · · · , nr) in detail in §7. In Theorem 7.7
we make explicit the structure of PGL(n0, · · · , nr) = [Gm → G] by writing G as
a semidirect product of a reductive part (product of general linear groups) and a
unipotent part (successive semidirect product of linear affine groups).
In light of the two step approach discussed above, Theorems 6.3 and 7.7 enable
us to study actions of group schemes (or group stacks, for that matter) on weighted
projective stacks in an explicit manner. This is discussed in §9. We classify actions
of a group scheme G on PS(n0, n1, · · · , nr) in terms of certain central extensions
of G by the multiplicative group Gm. We also describe the stack structure of the
corresponding quotient (2-)stacks. As a consequence, we obtain the following (see
Theorem 9.1).
Theorem 1.2. Let k be a field and G a connected linear algebraic group over k,
assumed to be reductive if char(k) > 0. Let X = P(n0, n1, · · · , nr) be a wighted
projective stack over k. Suppose that Pic(G) = 0. Then, every action of G on X
lifts to a linear action of G on Ar+1.
In a forthcoming paper, we use the results of this paper (more specifically, The-
orem 6.3), together with the results of [No2], to give a complete classification, and
explicit construction, of twisted forms of weighted projective stacks; these are the
weighted analogues of Brauer-Severi varieties.
Contents
1. Introduction 1
2. Notation and terminology 3
3. Review of 2-groups and crossed modules 3
4. 2-groups over a site and group stacks 4
4.1. Presheaves of weak 2-groups over a site 5
4.2. Group stacks over C 5
4.3. Equivalences of group stacks 6
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 3
5. Actions of group stacks 7
5.1. Formulation in terms of crossed modules and butterflies 8
6. Weighted projective general linear 2-groups PGL(n0, n1, ..., nr) 9
6.1. Automorphism 2-group of a quotient stack 9
6.2. Weighted projective general linear 2-groups 12
7. Structure of PGL(n0, n1, ..., nr) 13
8. Some examples 18
9. 2-group actions on weighted projective stacks 20
9.1. Description of the quotient 2-stack 21
References 22
2. Notation and terminology
Our notation for 2-groups and crossed modules is that of [No1] and [No3], to
which the reader is referred to for more on 2-group theory relevant to this work.
In particular, we use mathfrak letters G, H for 2-groups or crossed modules. By a
weak 2-group we mean a strict monoidal category G with weak inverses (Definition
3.1). If the inverses are also strict, we call G a strict 2-group.
By a stack we mean a presheaf of groupoids (and not a category fibered in
groupoids) over a Grothendieck site that satisfies the decent condition. We use
mathcal letters X, Y,... for stacks.
Given a presheaf of groupoids X over a site, its stackification is denoted by Xa.
We use the same notation for the sheafification of a presheaf of sets (or groups).
Them-dimensional general linear group scheme over SpecR is denoted by GL(m,R).
When R = Z, this is abbreviated to GL(m). The corresponding projectivized gen-
eral linear group scheme is denoted by PGL(m); this notation does not conflict with
the notation PGL(n0, n1, · · · , nr) for a weighted projective general linear 2-group
(§6) because in the latter case we always assume r ≥ 1.
3. Review of 2-groups and crossed modules
A strict 2-group is a group object in the category of groupoids. Equivalently, a
strict 2-group is a strict monoidal groupoid G in which every object has a strict
inverse; that is, multiplication by an object induces an isomorphism from G onto
itself. A morphism of 2-groups is, by definition, a strict monoidal functor.
The weak 2-groups we will encounter in this paper are less weak than the ones
discussed in [No1, No3]. We hope that this change in terminology is not too con-
fusing for the reader.
Definition 3.1. A weak 2-group is a strict monoidal groupoid G in which multi-
plication by an object induces an equivalence of categories from G to itself. By a
morphism of weak 2-groups we mean a strict monoidal functor. By a weak mor-
phism we mean a weak monoidal functor. (We will not encounter weak morphisms
until later sections.)
The set of isomorphism classes of objects in a 2-group G is denoted by π0G; this
is a group. The automorphism group of the identity object 1 ∈ ObG is denoted by
π1G; this is an abelian group.
4 BEHRANG NOOHI
Weak 2-groups and strict monoidal functors between them form a category
W2Gp which contains the category 2Gp of strict 2-groups as a full subcategory.1
Morphisms in W2Gp induce group homomorphisms on π0 and π1. In other words,
we have functors π0, π1 : W2Gp → Gp; the functor π1 indeed lands in the full
subcategory of abelian groups. A morphism between weak 2-groups is called an
equivalence if the induced homomorphisms on π0 and π1 are isomorphisms. Note
that an equivalence may not have an inverse.
The following lemma is straightforward.
Lemma 3.2. Let f : H→ G be a morphism of weak 2-groups. Then f , viewed as a
morphism of underlying groupoids, is fully faithful if and only if π0f : π0H→ π0G is
injective and π1f : π1H→ π1G is an isomorphism. It is an equivalence of groupoids
if and only if both π0f and π1f are isomorphisms.
A crossed module G = [∂ : G1 → G0] consists of a pair of groups G0 and G1,
a group homomorphism ∂ : G1 → G0, and a (right) action of G0 on G1, denoted
−a. This action lifts the conjugation action of G0 on the image of ∂ and descends
the conjugation action of G1 on itself. In other words, the following axioms are
satisfied:
• ∀β ∈ G1,∀a ∈ G0, ∂(βa) = a−1∂(β)a;
• ∀α, β ∈ G1, β∂(α) = α−1βα.
It is easy to see that the kernel of ∂ is a central (in particular abelian) subgroup
of G1; we denote this abelian group by π1G. The image of ∂ is always a normal
subgroup of G0; we denote the cokernel of ∂ by π0G. A morphism of crossed
modules is a pair of group homomorphisms which commute with the ∂ maps and
respect the actions. Such a morphism induces group homomorphisms on π0 and
Crossed modules and morphisms between them form a category, which we denote
by XMod. We have functors π0, π1 : XMod → Gp; the functor π1 indeed lands
in the full subcategory of abelian groups. A morphism in XMod is said to be an
equivalence if it induces isomorphisms on π0 and π1. Note that an equivalence may
not have an inverse.
There is a well-known natural equivalence of categories 2Gp ∼= XMod; see
[No1], §3.3. This equivalence respects the functors π0 and π1. This way, we can
think of a crossed module as a strict 2-group, and vice versa. For this reason, we
may sometimes abuse terminology and use the term (strict) 2-group for an object
which is actually a crossed module; we hope that this will not cause any confusion.
Note that W2Gp contains 2Gp as a full subcategory.
4. 2-groups over a site and group stacks
First a few words on terminology. For us a stack is presheaf of groupoids over a
Grothendieck site (and not a category fibered in groupoids) that satisfies the descent
condition. This may be somewhat unusual for algebraic geometers who are used to
categories fibered in groupoids, but it makes the exposition simpler. Of course, it
is standard that this point of view is equivalent to the one via categories fibered
in groupoids. Just to recall how this equivalence works, to any category fibered in
groupoids X one can associate a presheaf X of groupoids over C which is defined
1Both W2Gp and 2Gp are 2-categories but we will ignore the 2-morphisms for the time being
and only look at the underlying 1-category.
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 5
as follows. By definition, X is the presheaf that assigns to an object U ∈ C the
groupoids X(U) := Hom(U,X), where U stands for the presheaf of sets represented
by U and Hom is computed in the category of stacks over C. Conversely, to any
presheaf of groupoids one associates a category fibered in groupoids defined via the
Grothendieck construction. For more on this we refer the reader to [Ho], especially
§5.2.
4.1. Presheaves of weak 2-groups over a site. Let C be a Grothendieck site.
Let W2GpC be the category of presheaves of weak 2-groups over C; that is, the
category of contravariant functors from C to W2Gp. We define 2GpC and XModC
analogously. There is a natural equivalence of categories 2GpC ' XModC. In
particular, we can think of a presheaf of crossed modules as a presheaf of (strict)
2-groups. Note that W2GpC contains 2GpC as a full subcategory.
Let X be a presheaf of groupoids over C. To X we associate a presheaf of weak 2-
groups AutX ∈W2GpC which parametrizes auto-equivalences of X. By definition,
AutX is the functor that associates to an object U in C the weak 2-group of self-
equivalences of XU , where XU is the restriction of X to the comma category CU .
(The ‘comma category’, or the ‘over category’, CU is the category of objects in C
over U .) Notice that in the case where X is a stack, AutX, viewed as a presheaf of
groupoids, is also a stack. Indeed, AutX is almost a group object in the category of
stacks over C. To be more precise, AutX is a group stack in the sense of Definition
4.1 below.
Let G ∈ W2GpC be a presheaf of weak 2-groups on C. We define π
0 G to
be the presheaf U 7→ π0
, and π0G to be the sheaf associated to π
Similarly, π
1 G is defined to be the presheaf U 7→ π1
, and π1G to be the
sheaf associated to π
We define π0G and π1G for a presheaf of crossed modules G ∈ XModC in a
similar manner. The equivalence of categories between 2GpC and XModC respects
0 , π0, π
1 and π1. Lemma 3.2 remains valid in this setting if instead of π0 and
π1 we use π
0 and π
4.2. Group stacks over C. We recall the definition of a group stack from [Bre].
We modify Breen’s definition by assuming that our group stacks are strictly as-
sociative and have strict units. This is all we will need because the group stack
AutX of self-equivalences of a stack X (indeed, any presheaf of groupoids X) has
this property, and that is all we are concerned with in this paper.
Definition 4.1 ([Bre], page 19). Let C be a Grothendieck site. By a group stack
over C we mean a stack G that is a strict monoid object in the category of stacks
over C and for which weak inverses exist. By a morphism of group stacks we mean
a strict monoidal functor. That is, a morphism of stacks that strictly respects the
monoidal structures. By a weak morphism we mean a weak monoidal functor.
The condition on existence of weak inverses means that for every U ∈ ObC and
every object a in the groupoid G(U), multiplication by a induces an equivalence of
categories from G(U) to itself (or equivalently, an equivalence of stacks from XU to
itself). This condition is equivalent to saying that, for every U ∈ ObC, X(U) is a
weak 2-group. More concisely, it is equivalent to
(pr,mult)
−→ G× G
being an equivalence of stacks.
6 BEHRANG NOOHI
Remark 4.2. It is well known that a weak group stack can always be strictified
to a strict one. So, theoretically speaking, the strictness of monoidal structure in
Definition 4.1 is not restrictive. However, given fixed (strict) group stacks G and H,
strict morphisms H → G are not adequate. We will see in the subsequent sections
that when studying group actions on stacks we can not avoid weak morphisms. In
this section, however, we will only discuss strict morphisms.
Let grStC be the category of group stacks and strict morphisms between them
(Definition 4.1); this is naturally a full subcategory of W2GpC. There are natural
functors
W2GpC → grStC and XModC → grStC.
The former is the stackification functor that sends a presheaf of groupoids to its as-
sociated stack; note that since the stackification functor preserves products, we can
carry over the monoidal structure from a presheaf of groupoids to its stackification.
The latter functor is obtained from the former by precomposing with the natural
fully faithful functor XModC → W2GpC (see the beginning of §4.1). Given a
presheaf of crossed modules [∂ : G1 → G0], the associated group stack has as un-
derlying stack the quotient stack [G0/G1], where G1 acts on G0 by multiplication
on the right (via ∂).
Definition 4.3. Let X be a presheaf of groupoids over C. We define πpreX to be
the presheaf that sends an object U in C to the set of isomorphism classes in X(U).
We denote the sheaf associated to πpreX by πX. For a global section e of X, we
define AutX(e) to be sheaf associated to the presheaf that sends an object U in C
to the group of automorphisms, in the groupoid X(U), of the object eU ; note that
when X is a stack this presheaf is already a sheaf and no sheafification is needed.
Note that when G is the underlying presheaf of groupoids of a presheaf of weak
2-groups G ∈W2GpC, then π0G = πG and π1G = AutG(e), where e is the identity
section of G.
4.3. Equivalences of group stacks. There are two ways of defining the notion
of equivalence between group stacks. One way is to regard them as stacks and use
the usual notion of equivalence of stacks. The other is to regard them as presheaves
of weak 2-groups and use π0 and π1 (see §4.1). The next lemma shows that these
two definitions agree.
Lemma 4.4. Let G and H be group stacks, and let f : H → G be a morphism of
group stacks. Then, the following are equivalent:
(i) f is an equivalence of stacks;
(ii) The induced maps π0f : π
0 H → π
0 G and π1f : π
1 H → π
1 G are
isomorphisms of presheaves of groups;
(iii) The induced maps π0f : π0H → π0G and π1f : π1H → π1G are isomor-
phisms of sheaves of groups.
Proof. The only non-trivial implication is (iii)⇒ (ii). In the proof we will use the
following standard fact from closed model category theory.
Theorem ([Hi], Theorem 3.2.13). Let M be a closed model cate-
gory, L a localizaing class of morphisms in M, and ML the localized
model category. Let X and Y be fibrant objects (i.e., L-local ob-
jects) in ML, and let f : Y→ X be a morphism in M that is a weak
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 7
equivalence in the localized model structure ML (that is, f is an
L-local weak equivalence). Then, f is a weak equivalence in M.
We will apply the above theorem with M being the model structure on the
category GpdC of presheaves of groupoids on C in which weak equivalences are
morphisms that induce isomorphisms (of presheaves of groups) on π
0 and π
and fibrations are objectwise. We take L to be the class of hypercovers. The
weak equivalences in the localized model structure will then be the ones inducing
isomorphism (of sheaves of groups) on π0 and π1. The main reference for this is
[Ho].
Let us now prove (iii)⇒ (ii). It is shown in [Ho] that G and H are L-local objects
(see §5.2 and §7.3 of [ibid.]). By hypothesis, f induces isomorphisms (of sheaves) on
π0 and π1, so it is a weak equivalence in the localized model structure. Therefore,
since G and H are L-local, f is already a weak equivalence in the non-localized model
structure. This exactly means that π0f : π
0 H→ π
0 G and π1f : π
1 H→ π
are isomorphisms of presheaves. �
Lemma 4.5. Let X be a presheaf of groupoids over C and ϕ : X→ Xa its stackifi-
cation. Then, we have the following (see Definition 4.3 for notation):
(i) The induced morphism πX→ π(Xa) is an isomorphism of sheaves of sets;
(ii) For every global section e of X, the natural map AutX(e)→ AutXa(e) is an
isomorphism of sheaves of groups.
Proof. This is a simple sheaf theory exercise. We include the proof of (i). Proof of
(ii) is similar.
First we prove that πϕ : πX → π(Xa) is injective. Let U ∈ ObC, and let x, y
be element in πX(U) such that πϕ(x) = πϕ(y). We have to show that x = y. By
passing to a cover of U , we may assume x and y lift to objects x̄ and ȳ in X(U). We
will show that there is an open cover of U over which x̄ and ȳ become isomorphic.
Since ϕ(x̄) and ϕ(ȳ) become equal in π(Xa), there is a cover {Ui} of U such that
there is an isomorphism αi : ϕ(x̄|Ui) ∼−→ ϕ(ȳ|Ui) in the groupoid Xa(Ui), for every
i. By replacing {Ui} with a finer cover, we may assume that αi come from X(Ui).
(More precisely, αi = ϕ(βi), where βi is a morphism in the groupoid X(Ui).) This
implies that, for every i, x̄|Ui and ȳ|Ui are isomorphic as objects of the groupoid
X(Ui). This is exactly what we wanted to prove.
Having proved the injectivity, to prove the surjectivity it is enough to show that
every object x in π(Xa)(U) is in the image of ϕ, possibly after replacing U by an
open cover. By choosing an appropriate cover, we may assume x lifts to Xa(U).
Since Xa is the stackification of X, we may assume, after refining our cover, that x
is in the image of X(U)→ Xa(U). The claim is now immediate. �
Lemma 4.6. Let G = [G1 → G0] be a presheaf of crossed modules, and let G =
[G0/G1] be the corresponding group stack. Then, we have natural isomorphisms of
sheaves of groups πiG ∼−→ πiG, i = 1, 2.
Proof. Apply Lemma 4.5. �
5. Actions of group stacks
In this section we present an interpretation of an action of a group stack on a
stack in terms of butterflies. We begin with the definition of an action.
8 BEHRANG NOOHI
Definition 5.1. Let X be a stack and G a group stack. By an action of G on X we
mean a weak morphism f : G→ AutX. We say two actions f and f ′ are equivalent
if there is a monoidal transformation ϕ : f → f ′.
In the case where G is a group (over the base site), it is easy to see that our
definition of action is equivalent to Definitions 1.3.(i) of [Ro]. A monoidal trans-
formation ϕ : f → f ′ between two such actions is the same as the structure of a
morphism of G-groupoids (in the sense of Definitions 1.3.(ii) of [Ro]) on the identity
map idX : X→ X, where the source and the target are endowed with the G-groupoid
structures coming from the actions f and f ′, respectively.
5.1. Formulation in terms of crossed modules and butterflies. Butterflies
were introduced in [No3] as a convenient way of encoding weak morphisms between
2-groups (rather, crossed modules representing the 2-groups). The theory was fur-
ther extended in [AlNo1] to the relative case (over a Grothendieck site). We will
use this theory to translate problems about 2-group actions on stacks to certain
group extension problems.
We begin by recalling the definition of a butterfly (see [No3], Definition 8.1 and
[AlNo1], §4.1.3).
Definition 5.2. Let G = [ϕ : G1 → G0] and H = [ψ : H1 → H0] be crossed
modules. By a butterfly from H to G we mean a commutative diagram of groups
H0 G0
in which both diagonal sequences are complexes, and the NE-SW sequence, that
is, G1 → E → H0, is short exact. We require that ρ and σ satisfy the following
compatibility with actions. For every x ∈ E, α ∈ G1, and β ∈ H1,
ι(αρ(x)) = x−1ι(α)x, κ(βσ(x)) = x−1κ(β)x.
A morphism between two butterflies (E, ρ, σ, ι, κ) and (E′, ρ′, σ′, ι′, κ′) is a mor-
phism f : E → E′ commuting with all four maps (it is easy to see that such an f is
necessarily an isomorphism). We define B(H,G) to be the groupoid of butterflies
from H to G.
This definition is justified by the following result (see [No3] and [AlNo1]).
Theorem 5.3. Let G = [G1 → G0] and H = [H1 → H0] be crossed modules
of sheaves of groups over the cite C. Let G = [G0/G1] and H = [H0/H1] be
the corresponding quotient group stacks. Then, there is a natural equivalence of
groupoids
B(H,G) ∼= Homweak(H,G).
Here, the right hand side stands for the groupoid whose objects are weak morphisms
of group stacks and whose morphisms are monoidal transformations.
The above result can be interpreted as follows. A butterfly as in the theorem
gives rise to a canonical zigzag in XModC
∼←− E→ G,
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 9
where E = [H1 ×G1
κ·ι−→ E]. After passing to the associated stacks, it gives rise to
a zigzag in grStC
∼←− E→ G,
which after inverting the left map (as a weak morphism), results in a weak morphism
H→ G. It follows from this description of a butterfly that π0 and π1 are functorial
with respect to butterflies. Furthermore, the equivalence of Theorem 5.3 respects
π0 and π1 (see Lemma 4.6).
When [G1 → G0] is a crossed module model for the group stack AutX of auto-
equivalences of a stack X, then it follows from Theorem 5.3 that an action of
H = [H0/H1] on X is the same thing as an isomorphism class of a butterfly as in
Definition 5.2. In other words, to give an action of H on X, we need to find an
extension E of H0 by G1, together with group homomorphisms κ : H1 → E and
ρ : E → G0 satisfying the conditions of Definition 5.2. This summarizes our strategy
for studying group actions on stacks. To show its usefulness, in the subsequent
sections we will apply this method to the case where X = PS(n0, n1, · · · , nr) is a
weighted projective stack over a base scheme S.
6. Weighted projective general linear 2-groups PGL(n0, n1, ..., nr)
In this section we introduce weighted projective general linear 2-group schemes
and prove that they model self-equivalences of weighted projective stacks (Theorem
6.3).
We begin by some general observations about automorphism 2-groups of quotient
stacks. From now on, we assume that C = SchS is the big site of schemes over
a base scheme S, endowed with a subcanonical topology (say, étale, Zariski, fppf,
fpqc, etc.).
6.1. Automorphism 2-group of a quotient stack. We define a crossed module
in S-schemes [∂ : G1 → G0] to be a pair of S-group schemes G0 and G1, an S-group
scheme homomorphism ∂ : G1 → G0, and a (right) action of G0 on G1 satisfying
the axioms of a crossed module. These are precisely the representable objects in
XModSchS ; in other words, a crossed module in schemes [∂ : G1 → G0] gives rise
to a presheaf of crossed modules
U 7→ [∂(U) : G1(U)→ G0(U)].
We often abuse terminology and call a crossed module in schemes over S simply
a strict 2-group scheme over S.
The following two propositions generalize Lemma 8.2 of [BeNo].
Proposition 6.1. Let S be a base scheme. Let A be an abelian affine group scheme
over S acting on a S-scheme X, and let X = [X/A] be the quotient stack. Let G
be those automorphisms of X which commute with the A action; this is a sheaf of
groups on SchS. We have the following:
(i) With the trivial action of G on A, the natural map ϕ : A → G becomes a
crossed modules in SchS-schemes.
(ii) Let G be the group stack associated to [ϕ : A→ G]. Then, there is a natural
morphism of group stacks G→ AutX. Furthermore, this morphism induces
an isomorphism of sheaves of groups π1G ∼−→ π1(AutX).
10 BEHRANG NOOHI
Proof. Part (i) is straightforward, because ϕ maps A to the center of G. Let G
denote the presheaf of 2-groups associated to [ϕ : A→ G]. To prove part (ii), it is
enough to construct a morphism of presheaves of 2-groups G → AutX and show
that it has the required properties. Stackification of this map gives us the desired
map (Lemma 4.6).
Let us construct the morphism G→ AutX. We give the effect of this morphism
on the sections over S. Since everything commutes with base change, the same
construction works for every U → S in the site SchS and gives rise to the desired
morphism. of presheaves.
To define G(S)→ AutX(S), recall the explicit description of the S-points of the
quotient stack [X/A]:
Ob[X/A](S) =
(T, α) | T an A-torsor over S
α : T → X an A-map
Mor[X/H](S)((T, α), (T
)) = {f : T → T ′ an A-torsor map s.t. α′ ◦ f = α}
Any element of g ∈ G(S) induces an automorphism of X relative to S (keep the
same torsor T and compose α with the action of g on X). Also, for any element
a ∈ A(S), there is a natural 2-isomorphism from the identity automorphism of X
to the automorphism induced by ϕ(a) ∈ G(S) (which is by definition the same as
the action of a). It is given by the multiplication action of a−1 on the torsor T
(remember that A is abelian) which makes the following triangle commute
Interpreted in the language of 2-groups, this gives a morphism of 2-groups
G(S)→ AutX(S).
To prove that G→ AutX induces an isomorphism on π1, we show that, for every
U → S in the site SchS , the morphism of 2-groups G(U) → AutX(U) induces
an isomorphism on π1. Again, we may assume that U = S. We know that the
group of 2-isomorphisms from the identity automorphism of X to itself is naturally
isomorphic to the group of global sections of the inertia stack of X. In the case
X = [X/A], this is naturally isomorphic to the group of elements of A(S) which act
trivially on X. Note that this group is naturally isomorphic to π1G(S). Therefore,
the map G(S)→ AutX(S) induces an isomorphism on π1. �
Proposition 6.2. Notation being as in Proposition 6.1, assume that X is a proper
Deligne-Mumford stack over S, and that X → S has geometrically connected and
reduced fibers. Also, assume that A fits in an extension
0→ A0 → A→ A/A0 → 0
where A/A0 is finite over S and A0 has geometrically connected fibers (this is auto-
matics, for example, in the case where A is smooth and the number of its geometric
connected components is a locally constant function on S). Then, G → AutX is
fully faithful (as a morphism of presheaves of groupoids). In particular, the induced
map π0G→ π0(AutX) of sheaves of groups is injective.
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 11
Proof. As in Proposition 6.1, let G denote the presheaf of 2-groups associated to
[ϕ : A → G]. We need to show that, for every U → S in the site SchS , G(U) →
AutX(U) is fully faithful; since AutX is a stack, it would then follow that the
stackified morphism G→ AutX is also fully faithful.
We may assume that U = S. By Proposition 6.1.(ii) and Lemma 3.2, it is enough
to prove that if the action of g ∈ G(S) on X is 2-isomorphic to the identity, then g
is of the form ϕ(a), for some a ∈ A(S). Let us fix such a 2-isomorphism. The effect
of this 2-isomorphism on the A-torsor on X corresponding the point X → [X/A],
viewed as an object in the groupoid [X/A](X) of X-points of [X/A], is given by
an A-torsor map F : A ×S X → A ×S X which makes the following A-equivariant
triangle commute:
A×S X
µ // X
A×S X
Here, A×S X is the trivial A-torsor on X and µ is the action of A on X.
Precomposing F with the canonical section X → A×S X (corresponding to the
identity element of A) and then projecting onto the first factor, we obtain a map
f : X → A relative to S. The proposition follows from the following.
Claim. The map f is constant, in the sense that it factors through an S-point
a : S → A of A. Furthermore, the effect of a on X (induced from the action of A
on X) is the same as the effect of g on X.
Let us prove the claim. It follows from the commutativity of the above diagram
that, for any point x in X, the effect of g on x is the same as the effect of f(x) on x.2
In other words, f(x)g−1 leaves x fixed. Applying this to ax instead of x, and using
the fact that a and f(x)g−1 commute, we find that f(ax)g−1 also leaves x fixed, for
every a ∈ A. This implies that, for any point x of X, and any a ∈ A, the element
r(a, x) := f(ax)f(x)−1 leaves x fixed. Therefore, the map ρ : A ×S X → A ×S X,
ρ(a, x) := (r(a, x), x) factors through the stabilizer group scheme τ : Σ→ X. Thus,
we have a commutative triangle
A×S X
Now, consider the short exact sequence
0→ A0 → A→ A/A0 → 0,
where A0 is a group scheme over S with geometrically connected fibers and A/A0 is
finite over S. Since τ : Σ→ X has discrete fibers (because X is Deligne-Mumford)
the restriction of ρ to A0 ×S X factors through the identity section. Hence, for
every a ∈ A0 and x ∈ X (over the same point in S), r(a, x) = f(ax)f(x)−1 is the
identity element of A. This implies that f : X → A is A0-equivariant (for the trivial
action of A0 on A). So, we obtain an induced map λ : [X/A0]→ A (relative to S).
Since [X/A0] is finite over [X/A], and [X/A] is proper over S, the structure map
2When we say a “point” of X we mean a scheme T over S and a morphism T → X relative to
12 BEHRANG NOOHI
π : [X/A0]→ S is proper. From our assumptions we have that π has geometrically
connected and reduced fibers. Base change then implies that π∗O[X/A0] = OS .
Since A is affine over S, it follows that λ is constant, i.e., factors through a section
a : S → A. Since f : X → A factors through λ, it also factors through a. By
construction, the effect of a on X is the same as the effect of g on X, which is what
we wanted to prove. �
6.2. Weighted projective general linear 2-groups. Since the construction of
the weighted projective stacks, and also of the weighted projective general linear
2-group schemes, commutes with base change, we can work over Z. We begin with
some notation. We denote the multiplicative group scheme over SpecZ by Gm,Z,
or simply Gm. The affine (r + 1)-space over a base scheme S is denoted by Ar+1S ;
when the base scheme is SpecR it is denoted by Ar+1R , and when the base scheme
is SpecZ simply by Ar+1. Since r will be fixed throughout this section, we will
usually denote Ar+1S − {0} by US . We will abbreviate USpecR and USpecZ to UR
and U, respectively. We fix a Grothendieck topology on SchS that is not coarser
than Zariski.
Let n0, n1, · · · , nr be a sequence of positive integers, and consider the weight
(n0, n1, · · · , nr) action of Gm on U = Ar+1−{0}. (That is, for every scheme T , an el-
ement t ∈ Gm(T ) acts on UT by multiplication by (tn0 , tn1 , · · · , tnr ).) The quotient
stack of this action is called the weighted projective stack of weight (n0, n1, · · · , nr)
and is denoted by PZ(n0, n1, · · · , nr), or simply by P(n0, n1, · · · , nr). The weighted
projective general linear 2-group scheme PGL(n0, n1, · · · , nr) is defined to
be the 2-group scheme associated to the crossed module
[∂ : Gm → Gn0,n1,··· ,nr ],
where Gn0,n1,··· ,nr is the group scheme, over Z, of all Gm-equivariant (for the above
weighted action) automorphisms of U. More precisely, the T -points of Gn0,n1,··· ,nr
are automorphisms
f : UT → UT
that commute with the Gm-action. The homomorphism ∂ : Gm → Gn0,n1,··· ,nr is
the one induced from the Gm-action itself. We take the action of Gn0,n1,··· ,nr on
Gm to be trivial. The associated group stack is denoted by PGL(n0, n1, · · · , nr),
and is called the projective general linear group stack of weight (n0, n1, · · · , nr).
The following theorem says that a weighted projective general linear 2-group
scheme is a model for the group stack of self-equivalences of the corresponding
weighted projective stack. A special case of this theorem (namely, the case where
the base scheme is C) was proved in ([BeNo], Theorem 8.1). We briefly sketch how
the proof in [ibid.] can be modified to cover the general case.
Theorem 6.3. Let AutP(n0, n1, · · · , nr) be the group stack of automorphisms of
the weighted projective stack P(n0, n1, · · · , nr). Then, the natural map
PGL(n0, n1, · · · , nr)→ AutP(n0, n1, · · · , nr)
is an equivalence of group stacks. In particular, we have isomorphisms of sheaves
of groups
π0AutP(n0, n1, · · · , nr) ∼= π0PGL(n0, n1, · · · , nr) ∼= π0 PGL(n0, n1, · · · , nr),
π1AutP(n0, n1, · · · , nr) ∼= π1PGL(n0, n1, · · · , nr) ∼= π1 PGL(n0, n1, · · · , nr) ∼= µd,
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 13
where d = gcd(n0, n1, · · · , nr) and µd stands for the multiplicative group scheme
of dth roots of unity.
In order to prove our main result (Theorem 6.3) we need the following result
about line bundles on weighted projective stacks. For more details on this, the
reader is referred to [No4]. More general results about Picard stacks of algebraic
stacks can be found in [Bro].
Proposition 6.4. Let P = PS(n0, n1, · · · , nr), where S = SpecR is the spectrum
of a local ring. Then every line bundle on P is of the form O(d) for some d ∈ Z.
Proof. In the proof we use stack versions of Grothendieck’s base change and semi-
continuity results ([Ha], III. Theorem 12.11). We will assume that R is Noetherian.
In the case where R is a field, the assertion is easy to prove using the fact that
the Picard group of P is isomorphic to the Weil divisor class group. To prove the
general case, let x be the closed point of S = SpecR. Let L be a line bundle
on P. After twisting with on appropriate O(d), we may assume Lx ∼= O. We
will show that L is trivial. We have H1(Px,Lx) = H
1(Px,Ox) = 0. Hence, by
semicontinuity, H1(Py,Ly) = 0 for every point y of S. Base change implies that
R1f∗(L) = 0, and that R
0f∗(L) = f∗(L) is locally free (necessarily of rank 1).
Therefore, f∗(L) is free of rank 1 and, by base change, H
0(Py,Ly) is 1-dimensional
as a k(y)-vector space, for every y in S. In fact, this is true for every tensor
power L⊗n, n ∈ Z. So, Ly is trivial for every y in S. (Note that, when k is a
field, dimkH
0(Pk(n0, n1, · · · , nr),O(d)) is equal to the number of solutions of the
equation a1n0 + a2n1 + · · ·+ arnr = d in non-negative integers ai.)
Now let s be a generating section of f∗(L) ∼= R. It follows that f∗(s) is a
generating section of L. So L is trivial. �
Proof of Theorem 6.3. We apply Propositions 6.1 and 6.2 with S = SpecZ, X =
Ar+1 − {0}, and H = Gm. This implies that
PGL(n0, n1, · · · , nr)→ AutP(n0, n1, · · · , nr)
is a fully faithful morphism of stacks. That is, for every scheme U , the morphism
of groupoids
PGL(n0, n1, · · · , nr)(U)→ AutP(n0, n1, · · · , nr)(U)
is fully faithful. All that is left to show is that it is essentially surjective. Since
PGL(n0, n1, · · · , nr) and AutP(n0, n1, · · · , nr) are both stacks, it is enough to prove
this for U = SpecR, where R is a local ring. In this case, we know by Proposition
6.4 that PicP(n0, n1, · · · , nr) ∼= Z. We can now proceed exactly as in ([BeNo],
Theorem 8.1).
The isomorphisms stated at the end of the theorem follow from Lemma 4.4 and
Lemma 4.6. �
7. Structure of PGL(n0, n1, ..., nr)
In this section we give detailed information about the structure of the group
Gn0,n1,··· ,nr . We show that, as a group scheme over an arbitrary base, it splits as a
semi-direct product of a reductive group scheme and a unipotent group scheme. The
reductive part is a product of a copies of the general linear groups. The unipotent
part is a successive semi-direct product of vector groups; see Theorem 7.7.
14 BEHRANG NOOHI
Throughout this section, the action of Gm on U = Ar+1−{0} means the weight
(n0, n1, · · · , nr) action. To shorten the notation, we denote the group Gn0,n1,··· ,nr
by G. The rank m general linear group scheme over SpecR is denoted by GL(m,R).
When R = Z, this is abbreviated to GL(m). We always assume r ≥ 1. The
corresponding projectivized group scheme is denoted by PGL(m); this notation
does not conflict with the notation PGL(n0, n1, · · · , nr) for a weighted projective
general linear 2-group as in the latter case we have at least two variables.
We begin with a simple lemma.
Lemma 7.1. Let R be an arbitrary ring, and let f be a global section of the structure
sheaf of UR = Ar+1R − {0}, r ≥ 1. Then f extends uniquely to a global section of
Ar+1R .
Proof. Let Ui = SpecR[x0, · · · , xr, x−1i ] and consider the covering UR = ∪
i=1Ui.
We show that the restriction fi := f |Ui is a polynomial for every i. To see this,
observe that, except possibly for xi, all variables occur with positive powers in fi.
To show that xi also occurs with a positive power, pick some j 6= i and use the fact
that xi occurs with a positive power in fj |Ui∩Uj = fi|Ui∩Uj .
Therefore, for every i, fi actually lies in R[x0, · · · , xr, x−1i ]. Since fj |Ui = fi|Uj ,
it is obvious that all fi are actually the same and provide the desired extension of
f to UR. �
From now on, we will use a slightly different notation with indices. Namely, we
assume that the weights are m1 < m2 < · · · < mt, with each mi appearing exactly
ri ≥ 1 times in the weight sequence (so in the previous notation we would have
r + 1 = r1 + · · · + rt). We denote the corresponding projective general linear 2-
group by PGL(m1 : r1,m2 : r2, · · · ,mt : rt). We use the coordinates xij , 1 ≤ i ≤ t,
1 ≤ j ≤ ri, for Ar+1. We think of xij as a variable of degree mi. We will usually
abbreviate the sequence xi1, · · · , xiri to x
i. Similarly, a sequence F i1, · · · , F iri of
polynomials is abbreviated to Fi.
Let R be a ring. The following proposition tells us how a Gm,R-equivariant
automorphisms of UR looks like.
Proposition 7.2. Let F : UR → UR be a Gm-equivariant map. Then F is of the
form (Fi)1≤i≤t, where for every i, each component F
j ∈ R[x
j ; 1 ≤ i ≤ t, 1 ≤ j ≤ ri]
of Fi is a weighted homogeneous polynomial of weight mi.
Proof. The fact that components of F are polynomial follows from Lemma 7.1. The
statement about homogeneity of F ij is a simple exercise in polynomial algebra and
is left to the reader. �
In the above proposition, each F ij can be written in the form F
j = L
j + P
where Lij is linear in the variables x
1, · · · , xiri , and P
j is a homogeneous polynomial
of degree mi in variables x
b with a < i. Let LF := (L
i)1≤i≤t be the linear part of
F . It is again a Gm-equivariant endomorphism of U.
Proposition 7.3. Let F be as in the Proposition 7.2. The assignment F 7→ LF
respects composition of endomorphisms. In particular, if F is an automorphism,
then so is LF .
Proof. This follows from direct calculation, or, alternatively, by using the fact that
LF is simply the derivative of F at the origin. �
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 15
Corollary 7.4. There is a natural split homomorphism
φ : G→ GL(r1)×GL(r2)× · · · ×GL(rt).
Next we give some information about the structure of the kernel U of φ. It
consists of endomorphisms F = (F ij )i,j , where F
j has the form
F ij = x
j + P
Here, P ij is a homogeneous polynomial of degree mi in variables x
b with a < i.
Indeed, it is easily seen that, for an arbitrary choice of the polynomials P ij , the
resulting endomorphism F is automatically invertible. So, to give such an F ∈ U
is equivalent to giving an arbitrary collection of polynomials {P ij}1≤i≤t,1≤j≤ri such
that each P ij is a homogeneous polynomial of degree mi in variables x
b with a < i.
So, from now on we switch the notation and denote such an element of U by (P ij )i,j .
Proposition 7.5. For each 1 ≤ a ≤ t, let Ua ⊆ U be the set of those endomor-
phisms F = (P ij )i,j for which P
j = 0 whenever i 6= a. Let Ka denote the set of
monomials of degree ma in variables x
j, i < a, and let ka be the cardinality of Ka.
(In other words, ka is the number of solutions of the equation
zi,j = ma
in non-negative integers zi,j.) Then we have the following:
(i) Ua is a subgroup of U and is canonically isomorphic to the vector group
scheme Ara ⊗ AKa ∼= Araka . (Note: U1 is trivial.)
(ii) If a < b, then Ua normalizes Ub.
(iii) The groups Ua, 1 ≤ i ≤ t, generate U and we have Ua ∩ Ub = {1} if a 6= b.
Proof of (i). The action of (P ij )i,j ∈ Ua on A
r+1 is given by
(x1, · · · ,xa, · · · ,xt) 7−→ (x1, · · · ,xa + Pa, · · · ,xt).
So, if AKa stands for the vector group scheme on the basis Ka, there is a canonical
isomorphism
Ua ∼=
AKa ∼= Ara ⊗ AKa .
Proof of (ii). Let G = (Qij)i,j be an element in Ua and F = (P
j )i,j an element in
Ub. By (i), the inverse of G is G
−1 = (−Qij)i,j . Let us analyze the effect of the
composite G ◦ F ◦G−1 on Ar+1:
(x1, · · · ,xa, · · · ,xb, · · · ,xt) G
7−→ (x1, · · · ,xa −Qa, · · · ,xb, · · · ,xt)
F7−→ (x1, · · · ,xa −Qa, · · · ,xb + Rb, · · · ,xt)
G7−→ (x1, · · · ,xa, · · · ,xb + Rb, · · · ,xt).
Here the polynomial Rbk, 1 ≤ k ≤ rb, is obtained from P
k by substituting the
variables xaj with the polynomial x
Proof of (iii). Easy. �
16 BEHRANG NOOHI
Part (ii) implies that each Ua acts by conjugation on each of Ua+1, Ua+2,· · · ,
3 To fix the notation, in what follows we let the conjugate of an automorphism
f by an automorphism g to be g ◦ f ◦ g−1.
Notation. Let {Ua}ta=1 be a family of subgroups of a group U which satisfies the
following properties: 1) Each Ua normalizes every Ub with a < b; 2) No two distinct
Ua intersect; 3) The Ua generate U . In this case, we say that U is a successive
semi-direct product of the Ua, and use the notation U ∼= Ut o · · ·o U2 o U1.
The following is an immediate corollary of Proposition 7.5.
Corollary 7.6. There is a natural decomposition of U as a semi-direct product
U ∼= Ut o · · ·o U2 o U1,
where Ua ∼= Araka is the group introduced in Proposition 7.5. (Note that U1 is
trivial.)
In the next theorem we use the notation Am for two things. One that has already
appeared is the affine group scheme of dimension m. When there is a group scheme
G involved, we also use the notation Am for the trivial representation of G on Am.
Theorem 7.7. There is a natural decomposition of G as a semi-direct product
G ∼= Ut o · · ·o U2 o U1 o
GL(r1)× · · · ×GL(rt)
where Ua ∼= Araka and ka is as in Proposition 7.5. (Note that U1 is trivial.)
Furthermore, for every 1 ≤ a ≤ t, the action of GL(ra) leaves each Ub invariant.
We also have the following:
(i) When a > b the induced action of GL(ra) on Ub is trivial.
(ii) When a = b the induced action of GL(ra) on Ua is naturally isomorphic
to the representation ρ ⊗ AKa , where ρ is the standard representation of
GL(ra) and Ka is as in Proposition 7.5. (Recall that Ua is canonically
isomorphic to Ara ⊗ AKa .)
(iii) When a < b the action of GL(ra) on Ub is naturally isomorphic to the
representation ⊕
0≤l≤bmb
Arbdl ⊗ ρ̂⊗l.
Here ρ̂ stands for the inverse transpose of ρ, and dl is the number of mono-
mials of degree mb in variables x
j, i < b, i 6= a; so dl also depends on a
and b. (In other words, dl is the number of solutions of the equation
i 6=a
zi,j = mb − lma
in non-negative integers zi,j.)
Proof. Let g ∈ GL(ra) and F ∈ Ub. As in the proof of Proposition 7.5.i, we analyze
the effect of the composite g ◦ F ◦ g−1 on Ar+1. The element g ∈ GL(ra) acts
on Ar+1 as follows: it leaves every component xji invariant if i 6= a and on the
coordinates xa1 , · · · , xara it acts linearly (like the action of an ra × ra matrix on a
column vector).
3All group actions in this section are assumed to be on the left.
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 17
Proof of (i). The effect of g ∈ GL(ra) only involves the variables xa1 , · · · , xara and
does not see any other variable, whereas the effect of F ∈ Ub only involves the
variables xij , i ≤ b. Since b < a, these two are independent of each other. That is,
F and g commute.
Proof of (ii). Assume F = (P ij )i,j ; so P
j = 0 if i 6= a. The effect of g ◦ F ◦ g
−1 can
be described as follows:
(x1, · · · ,xa, · · · ,xt) g
7−→ (x1, · · · ,ya, · · · ,xt)
F7−→ (x1, · · · ,ya + Pa, · · · ,xt)
g7−→ (x1, · · · ,xa + Qa, · · · ,xt).
Here, yaj is the linear combination of x
1 , · · · , xara , the coefficients being the en-
tries of the jth row of the matrix g−1. Similarly, Qaj is the linear combination of
P a1 , · · · , P ara , coefficients being the entries of the j
th row of the matrix g.
Proof of (iii). Assume F = (P ij )i,j ; so P
j = 0 if i 6= b. Let y
a be as in (ii). The
effect of g ◦ F ◦ g−1 can be described as follows:
(x1, · · · ,xa, · · · ,xb, · · · ,xt) g
7−→ (x1, · · · ,ya, · · · ,xb, · · · ,xt)
F7−→ (x1, · · · ,ya, · · · ,xb + Rb, · · · ,xt)
g7−→ (x1, · · · ,xa, · · · ,xb + Rb, · · · ,xt).
Here the polynomials Rbk, 1 ≤ k ≤ rb, are obtained from P
k by substituting the
variable xaj with y
Let λ be the representation of GL(ra) on the space V of homogenous polynomials
of degree mb which acts as follows: it takes a polynomial P ∈ V and substitutes
the variables xaj , 1 ≤ j ≤ ra, with y
j . From the description above, we see that the
representation of GL(ra) on Ub is a direct sum of rb copies of λ. We will show that
0≤l≤bmb
Adl ⊗ ρ̂⊗l.
To obtain the above decomposition, simply note that a polynomial in V can be
uniquely written in the form ∑
0≤l≤bmb
SlTl,
where Tl is a homogenous polynomial of degree lma in variables x
1 , · · · , xara , and
Sl is a homogenous polynomial of degree mb− lma in the rest of the variables. The
action of GL(ra) leaves Sl intact and acts on Tl by the l
th power of the inverse
transpose of the standard representation. �
The actions of various pieces in the above semi-direct product decomposition,
though explicit, are tedious to write down, except for small values of t. We give
some examples in §8.
Let us denote Ut o · · ·o U2 o U1 by U and define the crossed module
PGL(n0, n1, · · · , nr)red := [∂ : Gm → GL(r1)× · · · ×GL(rt)],
where the kth factor of ∂(λ) is the size rk scalar matrix λ
mk . Theorem 7.7 then
implies that
PGL(n0, n1, · · · , nr) ∼= U o PGL(n0, n1, · · · , nr)red.
18 BEHRANG NOOHI
We think of U as the unipotent radical and PGL(n0, n1, · · · , nr)red as the reductive
part of PGL(n0, n1, · · · , nr).
Remark 7.8. It is perhaps useful to put the above result in the general context of
algebraic group theory. Recall that every algebraic group G over a field fits in a
short exact sequence
1→ U → G→ Gred → 1,
where U is the unipotent radical of G and Gred is reductive. The sequence is not
split in general. In our case, the group scheme Gn0,n1,··· ,nr admits such a short
sequence over an arbitrary base and, furthermore, the sequence is split.
The general theory of unipotent groups tells us that any unipotent group over
a perfect field admits a filtration whose graded pieces are vector groups. This
filtration splits, but only in the category of schemes (i.e., the splitting maps may
not be group homomorphisms). In our case, however, the group scheme U admits
such a filtration over an arbitrary base. Furthermore, the filtration is split group
theoretically.
8. Some examples
In this section we look at some explicit examples of weighted projective general
linear 2-groups.
Example 8.1. Weight sequence m < n,m - n. In this case we have t = 2, and
r1 = r2 = 1 and k1 = 0. So G ∼= Gm ×Gm.
Example 8.2. Weight sequence m < n,m | n. In this case we have t = 2, r1 = r2 =
1, and k1 = 1. So we have
G ∼= Ao (Gm ×Gm).
The action of an element (λ1, λ2) ∈ Gm ×Gm on an element a ∈ A is given by
(λ1, λ2) · a = λ2λ
More explicitly, an element in G is map of the form
(x, y) 7→ (λ1x, λ2y + ax
Note the similarity with the group of 2× 2 lower-triangular matrices.
Example 8.3. Weight sequence n = m. We obviously have G ∼= GL(2).
Example 8.4. Weight sequence 1, 2, 3. First we determine U . A typical element in
U is of the form
(x, y, z) 7→ (x, y + ax2, z + bx3 + cxy).
We have U2 = A and U3 = A2. The action of an element a ∈ U2 on an element
(b, c) ∈ U3 is given by (b− ac, c). That is, a acts on U3 = A2 by the matrix(
So, U ∼= A⊕2 oA. Finally, we have
G ∼= U o (Gm)3 = A⊕2 oAo (Gm)3,
where the action of an element (λ1, λ2, λ3) ∈ (Gm)3 on an element (a, b, c) ∈ U is
given by (λ−21 λ2a, λ
1 λ3b, λ
2 λ3c).
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 19
Example 8.5. Weight sequence 1, 2, 4. An element in U has the general form
(x, y, z) 7→ (x, y + ax2, z + bx4 + cx2y + dy2).
We have U2 = A and U3 = A3. The action of an element a ∈ U2 on an element
(b, c, d) ∈ U3 is given by the matrix
 1 −a a20 1 −2a
0 0 1
So, U ∼= A⊕3 oA.
Finally, we have
G ∼= U o (Gm)3 = A⊕3 oAo (Gm)3,
where the action of an element (λ1, λ2, λ3) ∈ (Gm)3 on an element (a, b, c, d) ∈ U
is given by
(λ−21 λ2a, λ
1 λ3b, λ
2 λ3c, λ
2 λ3d).
Next we look at PGL(m1 : r1,m2 : r2, · · · ,mt : rt). Recall that, as a crossed
module, this is given by [∂ : Gm → G], where ∂ is the obvious map coming from
the action of Gm on Ar+1, and the action of G on Gm is the trivial one.
Observe that the map ∂ factors though the component GL(r1)× · · · ×GL(rt) of
G. So, let us define L to be the cokernel of the following map:
r1︷ ︸︸ ︷
m1 , . . . , λ
,·····,
rt︷ ︸︸ ︷
mt , . . . , λ
) // GL(r1)× · · · ×GL(rt).
From Theorem 7.7 we immediately obtain the following.
Proposition 8.6. Let L be the group defined in the previous paragraph, and let
k = gcd(m1, · · · ,mt). We have natural isomorphisms of group schemes
π0 PGL(m1 : r1,m2 : r2, · · · ,mt : rt) ∼= Ut o · · ·o U2 o U1 o L,
π1 PGL(m1 : r1,m2 : r2, · · · ,mt : rt) ∼= µk.
Our final result is that, if all weights are distinct (that is, ri = 1), then the
corresponding projective general linear 2-group is split.
Proposition 8.7. Let {m1, · · · ,mt} be distinct positive integers, and consider the
projective general linear 2-group PGL(m1,m2, · · · ,mt). Then, the projection map
G → π0 PGL(m1,m2, · · · ,mt) splits. In particular, PGL(m1,m2, · · · ,mt) is split.
That is, it is completely classified by its homotopy group schemes:
π0 PGL(m1, · · · ,mt) ∼= Ut o · · ·o U2 o U1 o (Gm)t−1,
π1 PGL(m1, · · · ,mt) ∼= µk.
Proof. By Theorem 7.7 and Proposition 8.6 we know that G ∼= Ut o · · · o U2 o
U1 o (Gm)t and π0 PGL(m1,m2, · · · ,mt) ∼= Ut o · · ·oU2 oU1 oL, where L is the
cokernel of the map
α : Gm
(λm1 ,··· ,λmt ) // (Gm)t.
So it is enough to show that the image of µ is a direct factor. Note that if we
divide all the mi by their greatest common divisor, the image of α does not change.
20 BEHRANG NOOHI
So, we may assume gcd(m1, · · · ,mt) = 1. Let M be a t × t integer matrix whose
determinant is 1 and whose first column is (m1, · · · ,mt). The matrix M gives rise
to an isomorphism µ : (Gm)t → (Gm)t whose restriction to the subgroup Gm ×
{1}t−1 ∼= Gm is naturally identified with α. The subgroup µ({1} × (Gm)t−1) ⊂
(Gm)t is the desired complement of the image of α. �
Corollary 8.8. Let m, n be distinct positive integers, and let k = gcd(m,n). Then
PGL(m,n) is a split 2-group. That is, it is classified by its homotopy groups:
π0 PGL(m,n) ∼=
Gm, if m < n,m - n
AoGm, if m < n,m | n
π1 PGL(m,n) ∼= µk.
(In the case m | n, the action of Gm on A in the cross product A o Gm is simply
the multiplication action.)
Proof. Everything is clear, except perhaps a clarification is in order regarding the
parenthesized statement. Observe that the Gm appearing in the cross product
AoGm is indeed the cokernel of the map
α : Gm
(λm,λn) // (Gm)2,
which is naturally identified with the subgroup {1} ×Gm ⊂ (Gm)2. Therefore, by
the formula of Example 8.2, the action of an element λ ∈ Gm on an element a ∈ A
is given by λa. �
Finally, for the sake of completeness, we include the following.
Proposition 8.9. The 2-group PGL(k, k, · · · , k), k appearing t times, is given by
the following crossed module:
(λk,··· ,λk)// GL(t)].
We have π0 PGL(k, · · · , k) ∼= PGL(t) and π1 PGL(k, · · · , k) ∼= µk. In particular,
PGL(1, 1, · · · , 1), 1 appearing t times, is equivalent to the group scheme PGL(t).
9. 2-group actions on weighted projective stacks
In this section we combine the method developed in §3–§5 with the results about
the structure of AutP(n0, n1, · · · , nr) to study 2-group actions on a weighted pro-
jective stack P(n0, n1, · · · , nr). The goal is to illustrate how one can classify 2-group
actions using butterflies and how to describe the corresponding quotient 2-stacks.
Below, all group scheme are assumed to be flat and of finite presentation over a
fixed base S.
Let H be a group stack and [ψ : H1 → H0] a presentation for it as a crossed
module in schemes. By Theorem 5.3, to give an action of H on P(n0, n1, · · · , nr)
is equivalent to giving a butterfly diagram
}} ρ &&
H0 Gn0,n1,··· ,nr
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 21
In other words, to give an action of H on X, we need to find a central extension
1→ Gm → E → H0 → 1
of H0 by Gm, together with
• a lift κ : H1 → E of ψ to E such that κ(βσ(x)) = x−1κ(β)x, for every
β ∈ H1 and x ∈ E;
• an extension of the weighted action of Gm to a linear action of E on Ar+1
which is trivial on the image of κ.
The following result is more or less immediate from the above description of a
group action.
Theorem 9.1. Let k be a field and H a connected linear algebraic group over k,
assumed to be reductive if char k > 0. Let X = P(n0, n1, · · · , nr) be a wighted
projective stack over k. Suppose that Pic(H) = 0. Then, every action of H on X
lifts to an action of H on Ar+1 via a homomorphism H → Gn0,n1,··· ,nr .
Proof. An action of H on X lifts to Ar+1 if and only if the corresponding butterfly
is equivalent to a strict one. By ([AlNo1], Proposition 4.5.3) this is equivalent to
the central extension
1→ Gm → E → H → 1
being split. By ([C-T], Corollary 5.7) such central extensions are classified by
Pic(H), which in our case is assumed to be trivial. Any choice of a splitting
amounts to a lift of the action of H to Ar+1. �
This result is essentially saying that the obstruction to lifting the H-action from
P(n0, n1, · · · , nr) to Ar+1 lies in Pic(H).
9.1. Description of the quotient 2-stack. Given an action of a group stack H on
the weighted projective stack P(n0, n1, · · · , nr), we can use the associated butterfly
to get information about the quotient 2-stack [P(n0, n1, · · · , nr)/H]. First, notice
that [κ : H1 → E] is also a crossed module in schemes. Denote the associated group
stack by H′. It acts on P(n0, n1, · · · , nr) via ρ. We have
[P(n0, n1, · · · , nr)/H] ∼= [(Ar+1 − {0})/H′].
Note that the right hand side is the quotient stack of the action of a group stack on
an honest scheme, namely, Ar+1 − {0}. It is easy to describe its 2-stack structure
by looking at cohomologies of the NW-SE sequence
κ−→ E ρ−→ Gn0,n1,··· ,nr
of the butterfly. Set
[P(n0, n1, · · · , nr)/H]1 := [(Ar+1 − {0})/ cokerκ],
[P(n0, n1, · · · , nr)/H]0 := [(Ar+1 − {0})/ im ρ].
(Here, cokerκ and im ρ are the sheaf theoretic cokernel and image of the corre-
sponding maps which we assume are representable.) Then, [P(n0, n1, · · · , nr)/H]1
is the best approximation of [P(n0, n1, · · · , nr)/H] by a 1-stack, in the sense that it
is obtained by killing off the 2-automorphisms of the 2-stack [P(n0, n1, · · · , nr)/H].
More precisely, there is a natural map
[P(n0, n1, · · · , nr)/H]→ [P(n0, n1, · · · , nr)/H]1
22 BEHRANG NOOHI
making the former a 2-gerbe over the latter for the 2-group [kerκ→ 1]. Similarly,
[P(n0, n1, · · · , nr)/H]0 is an orbifold (i.e., a Deligne-Mumford stack which is gener-
ically a scheme) obtained by quotienting out the generic 1-automorphisms. More
precisely, there is a natural map
[P(n0, n1, · · · , nr)/H]1 → [P(n0, n1, · · · , nr)/H]0
making the former a gerbe over the latter for the group ker ρ/ imκ (namely, the
middle cohomology of the NW-SE sequence).
It follows that the quotient 2-stack [P(n0, n1, · · · , nr)/H] is equivalent to a 1-
stack if and only if κ is injective; it is an orbifold if and only if the NW-SE sequence
is left exact.
Example 9.2. Suppose that H is an honest group scheme and denote it by H.
Then, to give an action of H on P(n0, n1, · · · , nr) is equivalent to giving a central
extension
1→ Gm → E → H → 1
of H by Gm, together with a linear action of E on Ar+1 extending the weighted
action of Gm. We have
[P(n0, n1, · · · , nr)/H] ∼= [(Ar+1 − {0})/E],
which is an honest 1-stack.
Example 9.3. Suppose that H is the group stack associated to [A → 1], where A
is an abelian group scheme. We rename H to A[1]. Then, to give an action of
A[1] on P(n0, n1, · · · , nr) is equivalent to giving a character κ : A → µd ⊂ Gm,
where d is the greatest common divisor of (n0, n1, · · · , nr). The quotient 2-stack
[P(n0, n1, · · · , nr)/A[1]] is a 1-stack if and only if κ is injective. Assume this to
be the case and identify A with the corresponding subgroup of µd. Then, roughly
speaking, the quotient stack [P(n0, n1, · · · , nr)/A[1]] is obtained by killing the A
in µd ⊆ Ix at every inertia group Ix of P(n0, n1, · · · , nr). (Note that the generic
inertia group of P(n0, n1, · · · , nr) is µd.) For example, if the base is an algebraically
closed field of characteristic prime to d, then
[P(n0, n1, · · · , nr)/A[1]] ∼= P(
, · · · ,
where a is the order of A.
References
[AlNo1] E. Aldrovandi, B. Noohi, Butterflies I: morphisms of 2-group stacks, Adv. Math. 221
(2009), no. 3, 687–773.
[BeNo] K. Behrend, B. Noohi, Uniformization of Deligne-Mumford analytic curves, J. reine
angew. Math. 599 (2006), 111–153.
[Bre] L. Breen, On the classification of 2-gerbes and 2-stacks, Astérisque No. 225, 1994.
[Bro] S. Brochard, Foncteur de Picard d’un champ algébrique, Math. Ann. 343 (2009), no. 3,
541–602.
[C-T] J-L. Colliot-Thélène, Résolutions flasques des groupes linéaires connexes, J. reine angew.
Math. 618 (2008), 77–133.
[Ha] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag,
New York-Heidelberg, 1977.
[Hi] P. S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Mono-
graphs, 99, American Mathematical Society, Providence, RI, 2003.
[Ho] Sh. Hollander, A homotopy theory for stacks, math.AT/0110247.
GROUP ACTIONS ON ALGEBRAIC STACKS VIA BUTTERFLIES 23
[No1] B. Noohi, Notes on 2-groupoids, 2-groups and crossed modules, Homotopy, Homology, and
Applications, 9 (2007), no. 1, 75–106.
[No2] , Group cohomology with coefficients in a crossed module, J. Inst. Math. Jussieu, 10
(2011), no. 2, 359–404.
[No3] , On weak maps between 2-groups, preprint, arXiv:math/0506313v3 [math.CT].
[No4] , Picard stack of a weighted projective stack, preprint available at
http://www.maths.qmul.ac.uk/∼noohi/research.html.
[Ro] M. Romagny, Group actions on stacks and applications, Michigan Math. J. 53, Issue 1
(2005), 209–236.
	1. Introduction
	2. Notation and terminology
	3. Review of 2-groups and crossed modules
	4. 2-groups over a site and group stacks
	4.1. Presheaves of weak 2-groups over a site
	4.2. Group stacks over C
	4.3. Equivalences of group stacks
	5. Actions of group stacks
	5.1. Formulation in terms of crossed modules and butterflies
	6. Weighted projective general linear 2-groups PGL(n0,n1,...,nr)
	6.1. Automorphism 2-group of a quotient stack
	6.2. Weighted projective general linear 2-groups
	7. Structure of PGL(n0,n1,...,nr)
	8. Some examples
	9. 2-group actions on weighted projective stacks
	9.1. Description of the quotient 2-stack
	References
