A Luna étale slice theorem for algebraic stacks
Item Type Article
Authors Alper, Jarod; Hall, Jack; Rydh, David
Citation Alper, J., Hall, J., & Rydh, D. (2020). A Luna étale slice theorem
for algebraic stacks. Annals of Mathematics, 191(3), 675-738.
https://doi.org/10.4007/annals.2020.191.3.1
DOI 10.4007/annals.2020.191.3.1
Publisher ANNALS MATHEMATICS, FINE HALL
Journal ANNALS OF MATHEMATICS
Rights © 2020 Department of Mathematics, Princeton University.
Download date 09/08/2022 21:00:20
Item License http://rightsstatements.org/vocab/InC/1.0/
Version Final accepted manuscript
Link to Item http://hdl.handle.net/10150/641331
A LUNA
´
ETALE SLICE THEOREM FOR ALGEBRAIC STACKS
JAROD ALPER, JACK HALL, AND DAVID RYDH
Abstract. We prove that every algebraic stack, locally of finite type over an
algebraically closed field with affine stabilizers, is ´etale-locally a quotient stack
in a neighborhood of a point with linearly reductive stabilizer group. The
proof uses an equivariant version of Artin’s algebraization theorem proved in
the appendix. We provide numerous applications of the main theorems.
1. Introduction
Quotient stacks form a distinguished class of algebraic stacks which provide
intuition for the geometry of general algebraic stacks. Indeed, equivariant algebraic
geometry has a long history with a wealth of tools at its disposal. Thus, it has long
been desired—and more recently believed [Alp10, AK16]—that certain algebraic
stacks are locally quotient stacks. This is fulfilled by the main result of this article:
Theorem 1.1. Let X be a quasi-separated algebraic stack, locally of finite type over
an algebraically closed field k, with affine stabilizers. Let x ∈ X(k) be a point and
H ⊆ G
x
be a subgroup scheme of the stabilizer such that H is linearly reductive and
G
x
/H is smooth (resp. ´etale). Then there exists an affine scheme Spec A with an
action of H, a k-point w ∈ Spec A fixed by H, and a smooth (resp. ´etale) morphism
f :
[Spec A/H], w
→ (X, x)
such that BH
∼
=
f
−1
(BG
x
); in particular, f induces the given inclusion H → G
x
on stabilizer group schemes at w. In addition, if X has affine diagonal, then the
morphism f can be arranged to be affine.
This justifies the philosophy that quotient stacks of the form [Spec A/G], where
G is a linearly reductive group, are the building blocks of algebraic stacks near
points with linearly reductive stabilizers.
In the case of smooth algebraic stacks, we can provide a more refined descrip-
tion (Theorem 1.2) which resolves the algebro-geometric counterpart to the We-
instein conjectures [Wei00]—now known as Zung’s Theorem [Zun06, CF11, CS13,
PPT14]—on the linearization of proper Lie groupoids in differential geometry. Be-
fore we state the second theorem, we introduce the following notation: if X is
an algebraic stack over a field k and x ∈ X(k) is a closed point with stabilizer
group scheme G
x
, then we let N
x
denote the normal space to x viewed as a G
x
-
representation. If I ⊆ O
X
denotes the sheaf of ideals defining x, then N
x
= (I/I
2
)
∨
.
If G
x
is smooth, then N
x
is identified with the tangent space of X at x; see Section
3.1.
Date: Dec 19, 2019.
2010 Mathematics Subject Classification. Primary 14D23; Secondary 14B12, 14L24, 14L30.
During the preparation of this article, the first author was partially supported by the Australian
Research Council grant DE140101519, the National Science Foundation grant DMS-1801976 and
by a Humboldt Fellowship. The second author was partially supported by the Australian Research
Council grant DE150101799. The third author was partially supported by the Swedish Research
Council grants 2011-5599 and 2015-05554.
1
2 J. ALPER, J. HALL, AND D. RYDH
Theorem 1.2. Let X be a quasi-separated algebraic stack, locally of finite type over
an algebraically closed field k, with affine stabilizers. Let x ∈ |X| be a smooth and
closed point with linearly reductive stabilizer group G
x
. Then there exists an affine
and ´etale morphism (U, u) → (N
x
//G
x
, 0), where N
x
//G
x
denotes the GIT quotient,
and a cartesian diagram
[N
x
/G
x
], 0
[W/G
x
], w
f
//
oo
(X, x)
(N
x
//G
x
, 0) (U, u)
oo
such that W is affine and f is ´etale and induces an isomorphism of stabilizer groups
at w. In addition, if X has affine diagonal, then the morphism f can be arranged
to be affine.
In particular, this theorem implies that X and [N
x
/G
x
] have a common ´etale
neighborhood of the form [Spec A/G
x
].
The main techniques employed in the proof of Theorem 1.2 are
(1) deformation theory,
(2) coherent completeness,
(3) Tannaka duality, and
(4) Artin approximation.
Deformation theory produces an isomorphism between the nth infinitesimal neigh-
borhood N
[n]
of 0 in N = [N
x
/G
x
] and the nth infinitesimal neighborhood X
[n]
x
of x
in X. It is not at all obvious, however, that the system of morphisms {f
[n]
: N
[n]
→
X} algebraizes. We establish algebraization in two steps.
The first step is effectivization. To accomplish this, we introduce coherent com-
pleteness, a key concept of the article. Recall that if (A, m) is a complete local
ring, then Coh(A) = lim
←−
n
Coh(A/m
n+1
). Coherent completeness (Definition 2.1)
is a generalization of this, which is more refined than the formal GAGA results of
[EGA, III.5.1.4] and [GZB15] (see §4.4). What we prove in §2.1 is the following.
Theorem 1.3. Let G be a linearly reductive affine group scheme over a field k.
Let Spec A be a noetherian affine scheme with an action of G, and let x ∈ Spec A
be a closed point fixed by G. Suppose that A
G
is a complete local ring. Let X =
[Spec A/G] and let X
[n]
x
be the nth infinitesimal neighborhood of x. Then the natural
functor
(1.1) Coh(X) → lim
←−
n
Coh
X
[n]
x
is an equivalence of categories.
Tannaka duality for algebraic stacks with affine stabilizers was recently estab-
lished by the second two authors [HR19, Thm. 1.1] (also see Theorem 2.7). This
proves that morphisms between algebraic stacks Y → X are equivalent to sym-
metric monoidal functors Coh(X) → Coh(Y). Therefore, to prove Theorem 1.2,
we can combine Theorem 1.3 with Tannaka duality (Corollary 2.8) and the above
deformation-theoretic observations to show that the morphisms {f
[n]
: N
[n]
→ X}
effectivize to
b
f :
b
N → X, where
b
N = N ×
N
x
//G
x
Spec
b
O
N
x
//G
x
,0
. The morphism
b
f is
then algebraized using Artin approximation [Art69a].
The techniques employed in the proof of Theorem 1.1 are similar, but the
methods are more involved. Since we no longer assume that x ∈ X(k) is a non-
singular point, we cannot expect an ´etale or smooth morphism N
[n]
→ X
[n]
x
where
N = [N
x
/H]. Using Theorem 1.3 and Tannaka duality, however, we can produce a
A LUNA
´
ETALE SLICE THEOREM FOR ALGEBRAIC STACKS 3
closed substack
b
H of
b
N and a formally versal morphism
b
f :
b
H → X. To algebraize
b
f, we apply an equivariant version of Artin algebraization (Corollary A.19), which
we believe is of independent interest.
For tame stacks with finite inertia, Theorem 1.1 is one of the main results of
[AOV08]. The structure of algebraic stacks with infinite stabilizers has been poorly
understood until the present article. For algebraic stacks with infinite stabilizers
that are not—or are not known to be—quotient stacks, Theorems 1.1 and 1.2 were
only known when X = M
ss
g,n
is the moduli stack of semistable curves. This is
the central result of [AK16], where it is also shown that f can be arranged to be
representable. For certain quotient stacks, Theorems 1.1 and 1.2 can be obtained
using traditional methods in equivariant algebraic geometry, see §4.2 for details.
1.1. Some remarks on the hypotheses. We mention here several examples il-
lustrating the necessity of some of the hypotheses of Theorems 1.1 and 1.2.
Example 1.4. Some reductivity assumption of the stabilizer G
x
is necessary in
Theorem 1.1. For instance, consider the group scheme G = Spec k[x, y]
xy+1
→
A
1
= Spec k[x] (with multiplication defined by y 7→ xyy
0
+ y + y
0
), where the
generic fiber is G
m
but the fiber over the origin is G
a
. Let X = BG and x ∈ |X|
be the point corresponding to the origin. There does not exist an ´etale morphism
([W/G
a
], w) → (X, x), where W is an algebraic space over k with an action of G
a
.
Example 1.5. It is essential to require that the stabilizer groups are affine in a
neighborhood of x ∈ |X|. For instance, let X be a smooth curve and let E → X
be a group scheme whose generic fiber is a smooth elliptic curve but the fiber over
a point x ∈ X is isomorphic to G
m
. Let X = BE. There is no ´etale morphism
([W/G
m
], w) → (X, x), where W is an affine k-scheme with an action of G
m
.
Example 1.6. In the context of Theorem 1.1, it is not possible in general to find
a Zariski-local quotient presentation of the form [Spec A/G
x
]. Indeed, if C is the
projective nodal cubic curve with G
m
-action, then there is no Zariski-open G
m
-
invariant affine neighborhood of the node. If we view C (G
m
-equivariantly) as the
Z/2Z-quotient of the union of two P
1
’s glued along two nodes, then after removing
one of the nodes, we obtain a (non-finite) ´etale morphism [Spec(k[x, y]/xy)/G
m
] →
[C/G
m
] where x and y have weights 1 and −1. This is in fact the unique such
quotient presentation (see Remark 4.18).
The following two examples illustrate that in Theorem 1.1 it is not always pos-
sible to obtain a quotient presentation f : [Spec A/G
x
] → X, such that f is repre-
sentable or separated without additional hypotheses; see also Question 1.10.
Example 1.7. Consider the non-separated affine line as a group scheme G → A
1
whose generic fiber is trivial but the fiber over the origin is Z/2Z. Then BG admits
an ´etale neighborhood f : [A
1
/(Z/2Z)] → BG which induces an isomorphism of
stabilizer groups at 0, but f is not representable in a neighborhood.
Example 1.8. Let Log (resp. Log
al
) be the algebraic stack of log structures (resp.
aligned log structures) over Spec k introduced in [Ols03] (resp. [ACFW13]). Let
r ≥ 2 be an integer and let N
r
be the free log structure on Spec k. There is an ´etale
neighborhood [Spec k[N
r
]/(G
r
m
oS
r
)] → Log of N
r
which is not representable. Note
that Log does not have separated diagonal. Similarly, there is an ´etale neighbor-
hood [Spec k[N
r
]/G
r
m
] → Log
al
of N
r
(with the standard alignment) which is repre-
sentable but not separated. Because [Spec k[N
r
]/G
r
m
] → Log
al
is inertia-preserving,
Log
al
has affine inertia and hence separated diagonal; however, the diagonal is not
affine. In both cases, this is the unique such quotient presentation (see Remark
4.18).
4 J. ALPER, J. HALL, AND D. RYDH
1.2. Generalizations. Using similar arguments, one can in fact establish a gen-
eralization of Theorem 1.1 to the relative and mixed characteristic setting. This
requires developing some background material on deformations of linearly reduc-
tive group schemes, a more general version of Theorem 1.3 and a generalization of
the formal functions theorem for good moduli spaces. To make this article more
accessible, we have decided to postpone the relative statement until the follow-up
article [AHR19].
If G
x
is not reductive, it is possible that one could find an ´etale neighborhood
([Spec A/ GL
n
], w) → (X, x). However, this is not known even if X = B
k[]
G
where
G
is a deformation of a non-reductive algebraic group [Con10].
In characteristic p, the linearly reductive hypothesis in Theorems 1.1 and 1.2
is quite restrictive. Indeed, a smooth affine group scheme G over an algebraically
closed field k of characteristic p is linearly reductive if and only if G
0
is a torus and
|G/G
0
| is coprime to p [Nag62]. We ask however:
Question 1.9. Does a variant of Theorem 1.1 remain true if “linearly reductive”
is replaced with “reductive”?
We remark that if X is a Deligne–Mumford stack, then the conclusion of Theorem
1.1 holds. We also ask:
Question 1.10. If X has separated (resp. quasi-affine) diagonal, then can the mor-
phism f in Theorems 1.1 and 1.2 be chosen to be representable (resp. quasi-affine)?
If X does not have separated diagonal, then the morphism f cannot necessarily
be chosen to be representable; see Examples 1.7 and 1.8. We answer Question 1.10
affirmatively when X has affine diagonal (Proposition 3.2) or is a quotient stack
(Corollary 3.3), or when H is diagonalizable (Proposition 3.4).
1.3. Applications. Theorems 1.1 and 1.2 yield a number of applications to old
and new problems.
Immediate consequences. Let X be a quasi-separated algebraic stack, locally of finite
type over an algebraically closed field k with affine stabilizers, and let x ∈ X(k) be
a point with linearly reductive stabilizer G
x
.
(1) There is an ´etale neighborhood of x with a closed embedding into a smooth
algebraic stack.
(2) There is an ´etale-local description of the cotangent complex L
X/k
of X in
terms of the cotangent complex L
W/k
of W = [Spec A/G
x
]. If x ∈ |X| is a
smooth point (so that W can be taken to be smooth) and G
x
is smooth,
then L
W/k
admits an explicit description. If x is not smooth but G
x
is
smooth, then the [−1, 1]-truncation of L
W/k
can be described explicitly by
appealing to (1).
(3) For any representation V of G
x
, there exists a vector bundle over an ´etale
neighborhood of x extending V .
(4) If G
x
is smooth, then there are G
x
-actions on the formal miniversal de-
formation space
d
Def(x) of x and its versal object, and the G
x
-invariants
of
d
Def(x) is the completion of a finite type k-algebra. This observation is
explicitly spelled out in Remark 4.17.
(5) Any specialization y x of k-points is realized by a morphism [A
1
/G
m
] →
X. This follows by applying the Hilbert–Mumford criterion to an ´etale
quotient presentation constructed by Theorem 1.1.
Local applications. The following consequences of Theorems 1.1 and 1.2 to the local
geometry of algebraic stacks will be detailed in Section 4: