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Quotient Stacks 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 ! 1-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 +1 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! fall hF; q parameterized by T ig: 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 = qα2K(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 2 ZQ0 is the quotient Πe2Q1 HomK(v(t(e)); v(h(e))=Πv2Q0 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 2 ob(Aff=S), and x; y 2 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 2 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.
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