Quotient Stacks

Jacob Gross Lincoln College

8 March 2018

Abstract These are notes from a talk on quotient stacks presented at the Reading Group on Algebraic Stacks; meeting weekly in the Quillen Room of the Andrew Wiles Building during the Hilary Term of 2018. We motivate the construction of quotient stacks via the moduli problem of semistable coherent sheaves and the moduli problem of representations of a finite quiver. These stacks are then constructed rigorously (following Laumon– Moret-Bailey). We finish by describing the process of ridigifcation. Read at your own risk. These notes have not been edited by persons other than their author.

Contents

1 Context 1

2 A Motivating Example 3

3 Conventions 5

4 Quotient Stacks 5

5 Rigidification 11

1 Context

“Moduli theory” is an umbrella term. It encompasses the topics of geometric in- variant theory, period maps, deformation theory, and representability problems. This reading group is really about representability problems and deformation theory. This talk is really about representability problems–more commonly called “moduli problems.” There are several kinds of moduli problems. op The simplest kind of moduli problem is a functor M : SchS → Set from the opposite category of schemes over a base S to the category of sets. The simplest kind of solution to this moduli problem is an S-scheme M such that ∼ M = HomSchS (−,X).

1 problem solution example op SchS → Set algebraic space closed subschemes op SchS → Gpd algebraic stack semistable sheaves op SchS → ∞-Gpd higher algebraic stack complexes of sheaves Remark 1.1. The entries of the rightmost column should really be read with certain qualifiers. “Closed subschemes” really means “flat families of closed sub- schemes.” Likewise, “semistable sheaves” really means “flat families of coherent semistable sheaves.” And “complexes of sheaves” means “perfect complexes of coherent sheaves in the derived category Db(Coh(X)).” A complex is called perfect if it is locally quasi-isomorphic to finite length complex of locally-free sheaves. Perfection is the correct notion of flatness for the derived category (see [8]).

Stacks become necessary when objects have automorphisms (similarly, higher stacks become necessary when objects have higher automorphisms). The notion of iso-triviality explains this. A non-trivial iso-trivial family is a family X → S for which all of its fi- bres are isomorphic but which is not the trivial family. Given a non-trivial automorphism, one can often construct a non-trivial iso-trivial family. As explained in [7], this ruins any possibility of representability by a scheme. To see this, suppose M is a fine moduli space with universal family U → M and suppose that X → B is a non-trivial iso-trivial family. Then there is a map B → M such that X U

B M is a pullback square. By iso-triviality, B → M must be a constant map. This contradicts the uniqueness of pullbacks. The author knows of no general categorical theorem that construct a non- trivial iso-trivial family from a given non-trivial automorphism. But the author also does not know of an example where this does not happen. Certainly, the reader is owed at least one example (taken from [2]).

Example 1.2. Let C be a hyperelliptic curve with hyperellipic involution τ. Let S be a variety with a fixed point-free involution. Then the quotient

(C × S)/(τ × i) is iso-trivial over the surface S/i–every fibre is isomorphic to C. Therefore Mg is not representable by a scheme.

A quotient stack is a particular example of an algebraic stack. Many of the stacks used in algebraic geometry and in representation theory are of this form. The points of a quotient stack [X/G] are meant to be orbits of an algebraic

2 group G acting on scheme X. When the action is free the orbit space is an algebraic space. Otherwise, it is a stack. And, indeed, the orbits have non- trivial automorphisms: they are the stabilisers of this non-free group action. In general, this stack [X/G] is an Artin stack. If X acts with finite stabilizers, then [X/G] is a Deligne-Mumford stack. Actually, these stacks are even defined for the action of an S-space G en groupes on an algebraic S-space X (or even an algebraic S-stack; see [9]). Al- though, the author does not know of a geometric interpretation of such general quotients.

2 A Motivating Example

Let (X,L) be a polarized scheme (which means that only that L is an ample line bundle). Let E be a coherent sheaf of X. The slope of E is

deg(E) µ(E) := rank(E) if E is torsion-free and +∞ if E has torsion. Here the degree of E is given by Z n−1 deg(E) := c1(E) ∧ c1(L) , X rank of E is the generic rank of E; torsion-free coherent sheaves are locally free outside a codimension 2 subset.

Definition 2.1. A coherent sheaf E is called (semi)-stable if for all sub- sheaves E0 ⊂ E we have µ(E0)(≤)µ(E).

We wish to construct a moduli space of L-semistable coherent sheaves on X. To do this we must first introduce the quot scheme. Consider the moduli op problem QuotE/X/S : SchS → Set defined as follows: Given a T → S in SchS a family of quotients of E parameterized by T is a pair (F, q) such that

a. a coherent sheaf F on XT = X ×S T such that the schematic support of F is proper over T and F is flat over T , and

b. a surjective morphism q : ET → F (where ET denotes the pull-back of E under the canonical projection XT → X) one identifies two such families (F, q) and (F, q0) with ker(q) = ker(q0). Write op hF, qi for such an equivalence class. Then one defines QuotE/X/S : SchS → Sets by T 7→ {all hF, q parameterized by T i}.

Theorem 2.2. [Grothendieck]. QuotE/X/S is representable by a quasi-projective scheme QuotE/X/S.

3 This quot scheme, breaks up as

QuotE/X/S = qα∈K(X)Quot(E, α), where Quot(E, α) ⊂ Quot(E/X/S) denotes the open substack of quotients ET → F where ch(F) = α. Now given a Chern character α, there exists m >> 0 such that

χ(E(m)) = H0(E(m)) for any L-semistable coherent sheaf E. This is called boundedness. Let V be a K- vector space of dimension m. There is a Quot scheme parameterizing quotients

V ⊗ OX (−m) → E.

Write H := V ⊗ OX (−m). There is an open subscheme

QuotL(H, α) ⊂ Quot(H, α) of quotients where E is L-semistable. We want to eliminate the ambiguity of choice of V . Indeed, GL(V ) acts canonically on QuotL(H,P ). So our space of semistable sheaves would be

QuotL(H,P )/GL(V ). But this is problematic because semistables sheaves have automorphisms.

Lemma 2.2. If k is algebraically closed, then the automorphism group of any coherent semistable sheaf on a k-scheme is K∗.

There are two possible ways to deal with this 1. Use geometric invariant theory. This is a procedure guaranteed to yield a scheme QuotL(H,P )//GL(V ) that is a categorical quotient. But its points corresponds to L-polystable coherent sheaves, rather than L-semistable coherent sheaves. A coherent sheaf is L-polystable if it is a direct sum of L-stable sheaves.

2. Use quotient stacks: There is a stack [QuotL(H, α)/GL(V )] that repre- sents the moduli functor Schop → Gpd of families of L-semistable coher- ent sheaves on X It is algebraic. Quiver varieties are pretty similar. Morally speaking, the moduli space of K-representations of the path algebra of a finite quiver Q = (Q0,Q1, t, h) of fixed dimension vector v ∈ ZQ0 is the quotient

Πe∈Q1 HomK(v(t(e)), v(h(e))/Πv∈Q0 GL(v(v)). To guarantee a scheme, one can take the GIT quotient. The GIT quotient parameterizes only semi-simple representations. The quotient stack represents the full moduli problems of v-dimensional representations of Q.

4 3 Conventions

We are now following [5] instead of [10]. Therefore I will take a moment to lay out notational conventions, as they differ somewhat between these two sources. Throughout, let S be a scheme.

Definition 3.1. An S-space is a sheaf of sets on the site (Aff/S) (with the e´tale topology).

Definition 3.2. An algebraic S-space X is an S-space such that

• the diagonal ∆X : X → X × X is schematique and quasi-compact • there is a S-scheme X0 a morphism X0 → X of S-spaces that is surjective and ´etale. Definition 3.3. An S-space in groups is G is S-space such that for each affine S-scheme X, G(X) is a group.

Definition 3.4. Let U ∈ ob(Aff/S), and x, y ∈ ob(XU ), defone IsomX (x, y): Aff/U → Set by

(V → U) 7→ HomXV (xv, yV ).

Lemma 3.5. Let X be a stack. The diagonal ∆X : X → X ×X is representable, if and only if for every S-scheme T , and any x, y ∈ X (T ), IsomT (x,y) is an algebraic space.

Proof. Observe that IsomT (x,y) fits into the 2-Cartesian diagram

IsomT (x,y) T

X X × X .

4 Quotient Stacks

Consider an algebraic group G acting on a scheme X. Note that if an S-space in groups is representable by a scheme G then G is automatically an algebraic S-group. If the action of G is free, then the orbit space X/G is not a scheme. It is, however, a stack–usually written [X/G]. This stack remembers non-trivial sta- bilizers. Recall that stacks over S form a (2, 1)-category StS with objects given by S-stacks, 1-morphisms give by functors, and 2-morphisms given by natural isomorphisms.

5 Definition 4.1. Let X be an S-stack. A point ∗ (or, really, an S-valued point) is a morphism ∗ : S → X . The automorphism group of ∗ is the group of 2- morphisms ∗ −→∗∼ .

Example 4.2. Let k be a field and let A be a k-linear Abelian category. There is a moduli k-stack MA of objects in A. It is locally of finite type. Geometric points p : Spec(k) → MA of A are the same as objects E ∈ ob(A). The aut- morphism group of p is isomorphic to AutA(E).

The automorphism groups of a quotient stack, say [X/G], at a point x : S → G ought to be the stabilizers of the group action. To make this construction rigorous, one uses G-torsors.

Categorical Group Actions This subsection follows [10, Section 2.2] very closely.

Definition 4.3. A left action α of a functor G : Cop → Group on a functor F : Cop → Set is a natural transformation G × F → F , such that for any object U of C, the induced function G(U) × F (U) → F (U) is an action of the group G(U) on the set F (U).

Definition 4.4. A group object of C is an object G of C, together with s func- tor Cop → Grp into the category of groups whose composite with the forgetful functor Group → Set equals hG.

Group objects of C can action on objects of C.

Definition 4.6. An action of a group objet G on an object X is an action of op op the functor hG : C → Grp on hX : C → Set.

Proposition 4.7. Giving a left action of a group object G on an object X is equivalent to assigning an arrow α : G × X → X, such that the following two diagrams commute

(i). The identity of G acts like the identity on X:

pt × X eG×idX G × X

α X

(ii). The action is associative with respect to multiplication on G:

6 G × G × X mG×idG G × X

idG×α α G × X α X

Proof. This is easy to check in the category of sets Therefore, the result follows from Yoneda’s lemma. Definition 4.8. Let X and Y be C-objects, equipped with an action of G. Then, an arrow f : X → Y is called G-equivariant if for all objects U of C the induced function X(U) → Y (U) is G(U)-equivariant.

Since we are almost there, I might as well rigorously define the notion of cat- egorical quotient mentioned earlier in connexion with geometric invariant theory.

Definition 4.9. Let C be an category, let G be a group object in C, and let ρ : G × X → X be an action of G on X. Then a categorical quotient of an C-object X is a C-morphism π : X → Y such that (i). π : X → Y is invariant so that the following diagram commutes

ρ X × G X

prX π X π Y

(ii). π : X → Y is universal with respect to this propoerty: for any morphism 0 0 0 π : X → Z such that π ◦ ρ = π ◦ prX there is a unique C-morphism Y → Z such that the diagram

0 X π Z π Y

commutes. Again, GIT is a means of producing a categorical quotient in the category of schemes. Although it is not actually a categorical quotient of the original scheme, but rather of an open subset of it. The set of points of this GIT quotient scheme, in general, does not biject with the set of orbits e.g. it is not, in general, a geometric quotient.

Torsors G-Torsors are meant to be principal G-bundles, but it the ´etaletopology as it were.

7 Definition 4.10. Let U be an affine S-scheme and let G be a U-space in groups equipped with a G-action. Let P be a U-space. Then P is called a G-torsor if there exists a cover {Ui → U}i∈I such that ∼ P ×U Ui = G ×U Ui,

G-equivariantly, for each i ∈ I.

Example 4.11. Let G : (Aff/K)op → Set be the constant group K-space e.g. there exists a group G such that for any affine S-scheme X we have

G(X) =∼ G.

Take U := Spec(K). Then G is a Spec(K)-space. Then any G-torsor is a map G → Spec(K) e.g. a K-scheme structure on G.

The geometric points of the quotient stack [X/G] will be G-torsors over X. In particular, point of the quotient BG := [∗/G] of ∗ := Spec(k) by any algebraic k-group G will be structure maps G → ∗. Note that the automorphism group of G → ∗ is not Aut(G). We are consider- ing automorphism of G → ∗ as a G-torsor. Such automorphisms are morphisms φ : G → G such that φ(g) = g · φ(1G) so that φ is determined by φ(1G). And so the automorphism group of G → ∗ is G itself–as expected.

Quotient Stacks Finally, we give a rigorous construction of quotient stacks. This follows [5].

Definition 4.12. Let X be an S-space and let Y be an X -space (i.e. an S-space equipped with a morphism Y → X ) that has a G-action. Write [Y/G/X ] for the following S-groupoid: Given any object U ∈ ob(Aff/S) the fibre over U is the category with objects: triples (x, P, α), x ∈ X (U), P is a G ×X ,x U-torsor and α : P → Y ×X ,x U is a morphism of U-spaces which is G ×X ,x U-equivariant.

This S-groupoid [Y/G/X ] is called the “quotient stack”; in scare quotes because we have not yet shown it is a stack.

Example 4.13. If Y = X , then write B(G/X ) is called the classifier of G/X . For every U ∈ ob(Aff/S), B(G/X ) is simply the category of G ×S U-torsors.

It remains to show that these quotient stacks are indeed algebraic.

Definition 4.14. An S-space in groupoids X is two S-spaces X0 and X1, and S-space maps s : X1 → X0, t : X1 → X), identity  : X0 → X1, and multiplication m : X1S,X0,tX1 → X1 such that

1. s ◦  = t ◦  = IdX0 , s ◦ i = t, t ◦ i = s, s ◦ m = s ◦ pr2, and t ◦ m = t ◦ pr1

8 2. (associativity) The compositions

m×id m X1 = X1 × s, X0X0 = X ×X0,t X1 X1 ×s,X0,b X1 X1

and

id×m m X1 = X1 × s, X0X0 = X ×X0,t X1 X1 ×s,X0,b X1 X1

are equals 3. (neutral element) Both compositions

×id m X1 = X1 ×s,X0 X0 = X0 ×X0,t X1 X1 ×s,X0,t X1 X1

and

Id× m X1 = X1 ×s,X0 X0 = X0 ×X0,t X1 X1 ×s,X0,t X1 X1

are both equal to IdX1 . 4. (inverse) The diagrams

i×Id X1 X1 ×s,X0,t X1

t m  X0 X1

and

i×Id X1 X1 ×s,X0,t X1

t  m X0 X1

commute. To such an S-space in groupoids, on associates the following S-groupoid [X·]: for every U ∈ ob(Aff/S), the category fiber [X·]U has X0(U) as its set of objects, X1(U) as its set of arrows, s as its source map, and t as its target map. ∗ For every morphism φ : V → U is Aff/S, the functor φ :[X·]U → [X·]V est de- fined by restriction. Note there is a canonical 1-morphism p : X0 → [X·] (which reduces to the function Id: X0(U) → ob([X·]U ) and a canonical 2-isomorphism ∼ p ◦ s −→ p ◦ t between the arrow of X1 and to those of [X·].

9 Remark 4.15. The S-groupoid associated to an S-space in groupoids is not, in general, an S-stack—one has only an S-prestack. And so one writes [X·] for 0 the stack associated to the prestack [X·] . One obtains a canonical morphism p : X0 → [X·] by post-composition with the canonical S-prestack morphism 0 [X·] → [X·].

Example 4.16. Take X· to be the following S-space in groupoids: X0 = Y,X1 = Y ×X G, s = µ where µ is the right action of G on Y, t = prY , , i and m are induced from the neutral element, the inverse, and the composition law (respectively) of G. Then [X·] = [Y/G/X ]. Fact 4.17. The S-groupoid [Y/G/X ] is an S-stack.

Proposition 4.18. Let X0 → X1 be an S-space in groupoids such that

1. X1 and X0 are algebraic S-spaces, and

2. p0, p1 are smooth (resp. ´etale)

3. The map (p1, p2): X1 → X0 ×S X0 is separated and quasi-compact Then, (p1, p2) is finite type and

X := [X1 → X0]

is algebraic (resp. Deligne-Mumford) and the canonical map X0 → X is an atlas.

Proof. The fact that (p1, p2) is finite type follows from the first assertion of (4.2). To see that X is algebraic, it suffices to show that the diagonal ∆X : X → X ×S X representable. So let V ∈ ob(Aff/S) and let x, y be objects of X (V ). It suffices to show that the V -space IsomX (x, y) is representable. This is certainly the case if x and y se releve a x, y ∈ X0(V ) as IsomX (x, y) is simply the fibre product

X ×(p1,p2),X0×S X0,(x,y) V. In the general case, following from the definition of quotient stack, there exists 0 a covering family with an element V → V such that x|V 0 and y|V 0 se releve 0 X0. And so IsomX (x, y) ×V V is representable and the result follows from (1.6.4).

10 5 Rigidification

Suppose there is a fixed flat group scheme H lying inside all automorphism groups. The idea of rigidification to remove all H from the automorphism groups of M to obtain a rigidified stack M\ H. op To be precise, let H : Sch → Set be a scheme in groups such that each HT is flat over T . Suppose for each x ∈ M(T ) there exists an injective morphism ix : HT → AutT (x) for which its formation respects base change. Note that HT acts on the right and on the left of MT by the formula

h1 · u· = iy(h1) ◦ u ◦ ix(u1).

0 Assumption 5.1. Suppose that for any T /T , any h ∈ HT 0 , u ∈ Hom(xT 0 , yT 0 ) one has that −1 u hu ∈ HT 0 .

This property of H is called being normal in the HomM(x, y).

Theorem 5.2. [Abramovich-Corti-Vistoli]. Assume that H is normal in the sheaves HomM(x, y). Then, there exists an algebraic stack M\H and a smooth surjective morphism M → MH such that

(i). via f, elements of H ⊂ AutT (x) map to the identity, and f is universal for this property, (ii). if M is Deligne-Mumford, then so is M\ H (iii). if M is separated (resp. proper) then M\ H is separated (resp. proper) (iv). M\ H admits a coarse moduli space if and only if M admits one

Example 5.3. Let π : X → B be morphism in algebraic S-spaces. Define the category PicX/B as follows: • An objects is a triple (U, b, L) where U is an object of Sch/S, b : U → B is a morphism over S, and L is an invertible sheaef on XU = U ×b,B X

There natural map PicX/B → Sch/S is an algebraic stack, called the Picard stack of π. By abuse of notation, we simply write PicX/B for the Picard stack of π. The Picard scheme is a quasi-projective scheme that represents the moduli problem of invertible sheaves on a fixed scheme X; the existance of this scheme structure on the Picard group is a theorem of Grothendieck. The morphism from the Picard stack to the Picard scheme is a rigidification by Gm.

Example 5.4. This example comes from differential geometry. We construct a moduli C∞-stack of connections on a principle G-bundle P → X over a smooth manifold X. The correct notion of stack in differential geometry is that of a C∞-stack. We shall not delve into all the details here, but the interested reader can consult [3]. Essentially a C∞-ring is an algebraic object which generalizes

11 the structure of the ring of smooth functions C∞(M) on a smooth manifold M– literally it is finite-product preserving Set-valued functor F : Euc → Set on the category of Euclidean spaces Rn and smooth maps between them. Another way to think of C∞-rings the generalization of commutative rings such that all smooth operations make sense, rather than just polynomial ones. There is a notion of the spectrum of such a C∞-ring–these things are affine C∞-schemes. C∞-schemes are locally C∞-ringed spaces that are locally mod- elled on affine C∞-rings. Note that, in a sort of contrast to the theory of alge- braic schemes, all manifolds are affine as C∞-schemes.C∞-stacks and quotient C-stacks are defined similarly to algebraic stacks. Recall that a connection on a principal G-bundle P → M over a smooth manifold M is a g-valued one-form on P ; where g denotes the Lie algebra of G. Connections form an affine space A. The ring C∞(A) of C∞-functions on A gives a C∞-scheme Spec(C∞(A)). The gauge group G = Map(M,G) acts on A. The quotient C∞-stack

[Spec(C∞(A))/G] is the moduli stack of connection modulo gauge; it probably has closed substacks of instantons Minst (Yang-Mills, G2, etc.). Note that the the center of the gauge group Z(G) is contained in every stabilizer. Conbections with gauge group larger than Z(G) are called irreducible. To obtain a moduli space where only reducible connections have automorphisms one needs the ridigification [Spec(C∞(A))/G] \ Z(G).

References

[1] Cao, Y., Joyce, D., and Upmeier, M. Orientation data on moduli stack. In preparation. [2] Coskun, I. http://homepages.math.uic.edu/ coskun/571.lec8.pdf [3] Joyce, D. Algebraic Geometry over C∞-Rings. 2009. arXiv:1001.0023 [4] Huybrechts, D. and Lehn, M. The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Braunschweig, Vieweg, 1997. [5] Kleiman, S.L. The Picard Scheme. 2014. arXiv:1402.0409 [6] Laumon, G. and Moret-Bailley, L. Champs Alg´ebriques. Spring-Verlag, 2000.

[7] Math Stack Exchange. https://math.stackexchange.com/questions/609131/killing- the-automorphisms-to-make-a-functor-representable. [8] Pandharipande, R. and Thomas, R.P. Curve counting via stable pairs in the derived category, arXiv:0707.2348.

12 [9] Romangy, M. Group actions on stacks and applications,2005, Michigan. Math J. 1:53 209-236. [10] Vistoli, A. Notes on Grothendieck topologies, fibered categories and descent theory. 2004. arXiv:math/0412512

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