Quotient Stacks. Jacob Gross. Lincoln College. 8 March 2018

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1 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 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 invariant 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. The simplest kind of moduli problem is a functor M : Sch op S 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 = Hom SchS (, ). 1

2 problem solution example Set algebraic space closed subschemes Gpd algebraic stack semistable sheaves -Gpd higher algebraic stack complexes of sheaves Sch op S Sch op S Sch op S Remark 1.1. The entries of the rightmost column should really be read with certain qualifiers. Closed subschemes really means flat families of closed subschemes. 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 D b (Coh()). 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 S for which all of its fibres 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 B is a non-trivial iso-trivial family. Then there is a map B M such that U B 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 nontrivial 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 M (C S)/(τ i) is iso-trivial over the surface S/i every fibre is isomorphic to C. Therefore M g 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 [/G] are meant to be orbits of an algebraic 2

3 group G acting on scheme. When the action is free the orbit space is an algebraic space. Otherwise, it is a stack. And, indeed, the orbits have nontrivial automorphisms: they are the stabilisers of this non-free group action. In general, this stack [/G] is an Artin stack. If acts with finite stabilizers, then [/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 (or even an algebraic S-stack; see [9]). Although, the author does not know of a geometric interpretation of such general quotients. 2 A Motivating Example Let (, L) be a polarized scheme (which means that only that L is an ample line bundle). Let E be a coherent sheaf of. The slope of E is µ(e) := deg(e) rank(e) if E is torsion-free and + if E has torsion. Here the degree of E is given by deg(e) := c 1 (E) c 1 (L) n 1, 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 subsheaves E E we have µ(e )( )µ(e). We wish to construct a moduli space of L-semistable coherent sheaves on. To do this we must first introduce the quot scheme. Consider the moduli problem Quot E//S : Sch op S Set defined as follows: Given a T S in Sch S a family of quotients of E parameterized by T is a pair (F, q) such that a. a coherent sheaf F on T = 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 : E T F (where E T denotes the pull-back of E under the canonical projection T ) one identifies two such families (F, q) and (F, q ) with ker(q) = ker(q ). Write F, q for such an equivalence class. Then one defines Quot E//S : Sch op S Sets by T {all F, q parameterized by T }. Theorem 2.2. [Grothendieck]. Quot E//S is representable by a quasi-projective scheme Quot E//S. 3

4 This quot scheme, breaks up as Quot E//S = α K() Quot(E, α), where Quot(E, α) Quot (E//S) denotes the open substack of quotients E T F where ch(f) = α. Now given a Chern character α, there exists m >> 0 such that χ(e(m)) = H 0 (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 O ( m) E. Write H := V O ( m). There is an open subscheme Quot L (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 Quot L (H, P ). So our space of semistable sheaves would be Quot L (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 Quot L (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 [Quot L (H, α)/gl(v )] that represents the moduli functor Sch op Gpd of families of L-semistable coherent sheaves on 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 = (Q 0, Q 1, t, h) of fixed dimension vector v Z Q0 is the quotient Π e Q1 Hom K (v(t(e)), v(h(e))/π v 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

5 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 is an S-space such that the diagonal : is schematique and quasi-compact there is a S-scheme a morphism of S-spaces that is surjective and étale. Definition 3.3. An S-space in groups is G is S-space such that for each affine S-scheme, G() is a group. Definition 3.4. Let U ob(aff/s), and x, y ob( U ), defone Isom (x, y) : Aff/U Set by (V U) Hom V (x v, y V ). Lemma 3.5. Let be a stack. The diagonal : is representable, if and only if for every S-scheme T, and any x, y (T ), Isom T (x,y) is an algebraic space. Proof. Observe that Isom T (x,y) fits into the 2-Cartesian diagram Isom T (x,y) T. 4 Quotient Stacks Consider an algebraic group G acting on a scheme. 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 /G is not a scheme. It is, however, a stack usually written [/G]. This stack remembers non-trivial stabilizers. Recall that stacks over S form a (2, 1)-category St S with objects given by S-stacks, 1-morphisms give by functors, and 2-morphisms given by natural isomorphisms. 5

6 Definition 4.1. Let be an S-stack. A point (or, really, an S-valued point) is a morphism : S. 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 M A of objects in A. It is locally of finite type. Geometric points p : Spec(k) M A of A are the same as objects E ob(a). The autmorphism group of p is isomorphic to Aut A (E). The automorphism groups of a quotient stack, say [/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 : C op Group on a functor F : C op 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 functor C op Grp into the category of groups whose composite with the forgetful functor Group Set equals h G. Group objects of C can action on objects of C. Definition 4.6. An action of a group objet G on an object is an action of the functor h G : C op Grp on h : C op Set. Proposition 4.7. Giving a left action of a group object G on an object is equivalent to assigning an arrow α : G, such that the following two diagrams commute (i). The identity of G acts like the identity on : pt e G id G (ii). The action is associative with respect to multiplication on G: α 6

7 G G G id G α m G id G α G Proof. This is easy to check in the category of sets Therefore, the result follows from Yoneda s lemma. Definition 4.8. Let and Y be C-objects, equipped with an action of G. Then, an arrow f : Y is called G-equivariant if for all objects U of C the induced function (U) Y (U) is G(U)-equivariant. Since we are almost there, I might as well rigorously define the notion of categorical 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 be an action of G on. Then a categorical quotient of an C-object is a C-morphism π : Y such that (i). π : Y is invariant so that the following diagram commutes α G pr π ρ Y π (ii). π : Y is universal with respect to this propoerty: for any morphism π : Z such that π ρ = π pr there is a unique C-morphism Y Z such that the diagram Y π π Z 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 étale topology as it were. 7

8 Definition 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 {U i U} i I such that G-equivariantly, for each i I. P U U i = G U U i, Example 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 we have G() = 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 [/G] will be G-torsors over. 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 considering automorphism of G as a G-torsor. Such automorphisms are morphisms φ : G G such that φ(g) = g φ(1 G ) so that φ is determined by φ(1 G ). 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 Let be an S-space and let Y be an -space (i.e. an S-space equipped with a morphism Y ) that has a G-action. Write [Y/G/ ] 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 (U), P is a G,x U-torsor and α : P Y,x U is a morphism of U-spaces which is G,x U-equivariant. This S-groupoid [Y/G/ ] is called the quotient stack ; in scare quotes because we have not yet shown it is a stack. Example If Y =, then write B(G/ ) is called the classifier of G/. For every U ob(aff/s), B(G/ ) is simply the category of G S U-torsors. It remains to show that these quotient stacks are indeed algebraic. Definition An S-space in groupoids is two S-spaces 0 and 1, and S-space maps s : 1 0, t : 1 ), identity ɛ : 0 1, and multiplication m : 1S,0,t 1 1 such that 1. s ɛ = t ɛ = Id 0, s i = t, t i = s, s m = s pr 2, and t m = t pr 1 8

9 2. (associativity) The compositions and m id m 1 = 1 s, 0 0 = 0,t 1 1 s,0,b 1 1 id m m 1 = 1 s, 0 0 = 0,t 1 1 s,0,b 1 1 are equals 3. (neutral element) Both compositions and ɛ id m 1 = 1 s,0 0 = 0 0,t 1 1 s,0,t 1 1 Id ɛ m 1 = 1 s,0 0 = 0 0,t 1 1 s,0,t 1 1 are both equal to Id (inverse) The diagrams and commute. i Id 1 1 s,0,t 1 t ɛ 0 1 i Id m 1 1 s,0,t 1 t m 0 1 To such an S-space in groupoids, on associates the following S-groupoid [ ]: for every U ob(aff/s), the category fiber [ ] U has 0 (U) as its set of objects, 1 (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 : [ ] U [ ] V est defined by restriction. Note there is a canonical 1-morphism p : 0 [ ] (which reduces to the function Id: 0 (U) ob([ ] U ) and a canonical 2-isomorphism p s p t between the arrow of 1 and to those of [ ]. ɛ 9

10 Remark 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 [ ] for the stack associated to the prestack [ ]. One obtains a canonical morphism p : 0 [ ] by post-composition with the canonical S-prestack morphism [ ] [ ]. Example Take to be the following S-space in groupoids: 0 = Y, 1 = Y G, s = µ where µ is the right action of G on Y, t = pr Y, ɛ, i and m are induced from the neutral element, the inverse, and the composition law (respectively) of G. Then [ ] = [Y/G/ ]. Fact The S-groupoid [Y/G/ ] is an S-stack. Proposition Let be an S-space in groupoids such that and 0 are algebraic S-spaces, and 2. p 0, p 1 are smooth (resp. étale) 3. The map (p 1, p 2 ) : 1 0 S 0 is separated and quasi-compact Then, (p 1, p 2 ) is finite type and := [ 1 0 ] is algebraic (resp. Deligne-Mumford) and the canonical map 0 is an atlas. Proof. The fact that (p 1, p 2 ) is finite type follows from the first assertion of (4.2). To see that is algebraic, it suffices to show that the diagonal : S representable. So let V ob(aff/s) and let x, y be objects of (V ). It suffices to show that the V -space Isom (x, y) is representable. This is certainly the case if x and y se releve a x, y 0 (V ) as Isom (x, y) is simply the fibre product (p1,p 2), 0 S 0,(x,y) V. In the general case, following from the definition of quotient stack, there exists a covering family with an element V V such that x V and y V se releve 0. And so Isom (x, y) V V is representable and the result follows from (1.6.4). 10

11 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. To be precise, let H : Sch op Set be a scheme in groups such that each H T is flat over T. Suppose for each x M(T ) there exists an injective morphism i x : H T Aut T (x) for which its formation respects base change. Note that H T acts on the right and on the left of M T by the formula h 1 u = i y (h 1 ) u i x (u 1 ). Assumption 5.1. Suppose that for any T /T, any h H T, u Hom(x T, y T ) one has that u 1 hu H T. This property of H is called being normal in the Hom M (x, y). Theorem 5.2. [Abramovich-Corti-Vistoli]. Assume that H is normal in the sheaves Hom M (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 Aut T (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 : B be morphism in algebraic S-spaces. Define the category P /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 U = U b,b There natural map P /B Sch/S is an algebraic stack, called the Picard stack of π. By abuse of notation, we simply write P /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 ; 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 G m. Example 5.4. This example comes from differential geometry. We construct a moduli C -stack of connections on a principle G-bundle P over a smooth manifold. 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

12 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 R n 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 modelled on affine C -rings. Note that, in a sort of contrast to the theory of algebraic 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 M inst (Yang-Mills, G 2, 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. coskun/571.lec8.pdf [3] Joyce, D. Algebraic Geometry over C -Rings ariv: [4] Huybrechts, D. and Lehn, M. The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Braunschweig, Vieweg, [5] Kleiman, S.L. The Picard Scheme ariv: [6] Laumon, G. and Moret-Bailley, L. Champs Algébriques. Spring-Verlag, [7] Math Stack Exchange. [8] Pandharipande, R. and Thomas, R.P. Curve counting via stable pairs in the derived category, ariv:

13 [9] Romangy, M. Group actions on stacks and applications,2005, Michigan. Math J. 1: [10] Vistoli, A. Notes on Grothendieck topologies, fibered categories and descent theory ariv:math/