aa r X i v : . [ m a t h . AG ] A ug STACKY ABELIANIZATION OF ALGEBRAIC GROUPS
MASOUD KAMGARPOUR
Abstract.
We prove a conjecture of Drinfeld regarding restric-tion of central extensions of an algebraic group G to the commuta-tor subgroup. As an application, we construct the “true commuta-tor” of G . The quotient of G by the action of the true commutatoris the universal commutative group stack to which G maps.AMS subject classification: 20G15Keywords: central extensions, algebraic groups, commutator, abelian-ization, group stacks Contents
1. Introduction 12. Central extensions of algebraic groups 43. Relationship between central covers of G and [ G, G ] 94. True commutator and stacky abelianization 13Appendix A. Crossed modules and gr-stacks 15Appendix B. Computing fundamental group schemes 20References 22 Introduction
One of the goals of geometric character theory is constructing sheaveson an algebraic group G over a finite field F q whose “trace of Frobeniusfunctions” are the irreducible characters of G ( F q ) [Lus85], [Lus03],[BD06]. A one-dimensional character sheaf is easy to define: it isan irreducible local system K equipped with an isomorphism m ∗ K ∼ = K ⊠ K , where m : G × G → G is the group multiplication; see forinstance, [Gai03], § Author was supported by NSERC PGS D2 - 331456
Address : Department of Mathematics, University of Chicago, Chicago, IL 60637
E-mail : [email protected] Date : November 1, 2018. the following table central ext. 1 → A → ˜ G π −→ G → homomorphisms G ( F q ) → A G ( F q ) → Q × ℓ Here A is a finite abelian group. To go from left to right one takesthe trace of Frobenius acting on the stalks of the sheaf (in the case ofcentral extensions, the sheaf of local sections of π ). To go from top tobottom, one needs the extra data of a homomorphism A → Q × ℓ .Recall that every one-dimensional character of G ( F q ) is trivial onthe commutator subgroup [ G ( F q ) , G ( F q )]. There exist, however, exam-ples of one-dimensional character sheaves whose restriction to [ G, G ] isnontrivial. For instance, we have the central extension(1.1) 1 → µ n → SL n → PGL n → . Our goal is to construct a commutator for G which is suitable for doinggeometric character theory. To accomplish this, we need to study therelationship between central extensions of G and those of [ G, G ].Henceforth, let G denote a connected algebraic group over a perfectfield k . We will prove that there exists a pro-´etale group scheme Π et ( G )such that for every commutative ´etale group scheme A , Hom(Π et ( G ) , A )equals the set of isomorphism classes of central extensions of G by A . Example . Let G a denote the additive group over ¯ F q . ThenΠ et ( G a ) = Hom(¯ F q , C × ) . Restricting central extensions of G to [ G, G ] defines a morphism ofgroup schemes(1.2) Π et ( i ) : Π et ([ G, G ]) → Π et ( G ) . The following theorem was conjectured by V. Drinfeld.
Theorem 1.2.
The image of Π et ( i ) is finite. Let A et be the image of Π et ( i ) and let(1.3) 1 → A et → [ G, G ] true → [ G, G ] → et ([ G, G ]) ։ A et . We propose to think of [ G, G ] true as the “truecommutator” of G . True commutator is characterized by the followingproperties:P1) [ G, G ] true is a connected ´etale central extension of [ G, G ]. This is the geometric analogue of the following fact: For a perfect group Γ,Hom(H (Γ , Z ) , A ) = H (Γ , A ); see for instance, [Mil72]. TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 3
P2) The pullback of every ´etale central extension of G to [ G, G ] true is trivial.P3) The commutator map has a lift to [ G, G ] true ; that is to say, thereexists a morphism of algebraic varieties G × G → [ G, G ] true suchthat the following diagram commutes[ G, G ] true (cid:15) (cid:15) G × G c ' ' PPPP [ G, G ] , where c ( g, h ) = [ g, h ] := g − h − gh for g, h ∈ G . Remark . Over an algebraically closed field, the fact that P1)-P3)characterize the true commutator is equivalent to the exactness of thefollowing complex of profinite groups:(1.4) π ( G × G ) → Π et ([ G, G ]) → Π et ( G ) . Remark . If k is algebraically closed of characteristic zero and G isreductive, then the true commutator of G is the simply connected coverof the semisimple group [ G, G ], see Example 4.1. The fact that thecommutator map lifts to [
G, G ] true was observed by P. Deligne [Del79], § G by the true commutator. Note thatthe composition [ G, G ] true ։ [ G, G ] ֒ → G defines an action of [ G, G ] true on G by left translation. The quotient stack(1.5) G ab , st := [ G/ [ G, G ] true ]is called the stacky abelianization of G . Observe that stabilizers ofthe action of [ G, G ] true on G are isomorphic to A et . In particular,stacky abelianization is a Deligne-Mumford stack [DM69]. We willshow, moreover, that it is a strictly commutative Picard stack [Del73].Roughly speaking, this means that G ab , st is equipped with a commu-tative group structure. The following theorem states that G ab , st is theuniversal commutative group stack to which G maps. Theorem 1.5.
Let P be a strictly commutative Deligne-Mumford Pi-card stack, and let G → P be a 1-morphism of group stack. Then there For a more precise formulation, see Theorem 4.10.
MASOUD KAMGARPOUR exists a 1-morphism of Picard stacks G ab , st → P such that the followingdiagram commutes: G ab , st (cid:15) (cid:15) G ; ; vvvv $ $ IIII P . Organization of the text.
We start § et . Wewill see that over an algebraically closed field, Π et is a quotient of theGrothendieck fundamental group. We conclude the section by provid-ing a summary of our calculations of fundamental group schemes. Thedetails of these calculations appear in Appendix B.In § G andthose of [ G, G ]. In particular, we provide examples of central extensionsof unipotent groups whose restriction to [
G, G ] is nontrivial. Moreover,we give a necessary and sufficient criterion for “lifting” central exten-sion of [
G, G ] to G . The proof of Theorem 1.2 is also given in thissection.In §
4, we study the true commutator and the stacky abelianization.We will show that the commutator map lifts to the true commutatorand use this to prove that the stacky abelianization is Picard. We thenprove the characterization of the true commutator and the universalproperty of the stacky abelianization (Theorem 1.5).Throughout the text we assume that the reader is familiar with groupstacks and their relationship to crossed modules. For the convenienceof the reader, we have gathered all the relevant results in Appendix A.The only part of this appendix which appears to be new is the stackyanalogue of the First Isomorphism Theorem (Lemma A.11).1.2.
Acknowledgments.
I wish to express my profound gratitude tomy thesis advisors Vladimir Drinfeld and Mitya Boyarchenko. I alsolike to thank Justin Noel, Larry Breen, Behrang Noohi, and TravisSchedler for many useful conversations.2.
Central extensions of algebraic groups
Constant morphisms.
Let ϕ : X → Y be a morphism of schemesover a noetherian base-scheme S . Definition 2.1. ϕ is constant if for every scheme T over S , the mapof sets ϕ T : Hom S ( T, X ) → Hom S ( T, Y ) is constant.
TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 5
Let S ′ be a scheme over S . Lemma 2.2. (i) If ϕ is constant, so is ϕ × S S ′ . (ii) Assume S ′ → S is flat and Y → S is finite. Then ϕ is constantif and only if ϕ × S S ′ is constant.Proof. Statement (i) follows from the fact that for every scheme T ′ over S ′ , Hom S ′ ( T ′ , X × S S ′ ) = Hom S ( T ′ , X ) . Under the conditions of (ii), for every scheme T over S , the basechange map Hom S ( T, Y ) → Hom S ′ ( T × S S ′ , Y × S S )is an isomorphism (EGA 1, ch. 0, § (cid:3) Lemma 2.3.
Assume X admits an S -section σ : S → X . Then thefollowing are equivalent: (1) ϕ is constant. (2) ϕ has a factorization X → S → Y .Proof. If ϕ is constant then ϕ = ( ϕ ◦ σ ) ◦ h , where h : X → S isthe structure map. Conversely, if ϕ has a factorization X → S → Y then ϕ T (Hom S ( T, X )) is the set consisting of one element; namely, thecomposition T → S → Y . (cid:3) Proposition 2.4.
Let k be a perfect field, A a finite scheme over k ,and X a connected reduced scheme over k which admits a k -section.Then every morphism ϕ : X → A of k -schemes is constant.Remark . As k is perfect, X is reduced if and only if it is geometri-cally reduced (EGA IV, part 2, prop. 4.6.1). As X admits a k -section,it is connected if and only if it is geometrically connected (loc. cit.,cor. 4.5.14). Proof of Proposition 2.4.
In view of Lemma 2.2 and Remark 2.5, wemay assume that k is algebraically closed. We may suppose, moreover,that X is affine.Let X = spec( R ) and A = spec( M ), where R and M are k -algebras.Let Φ : M → R be the k -algebra homomorphism corresponding to ϕ .Let N ⊆ R be the image of Φ. Since R has no nontrivial idempo-tents, N is a finite dimensional local k -algebra. As R is reduced, N isisomorphic to k . By Lemma 2.3, ϕ is constant. (cid:3) MASOUD KAMGARPOUR
Remark . (M. Boyarchenko) Let X be as in Proposition 2.4, S be a reduced scheme over k , and A be a finite scheme over S . Then one canshow that every S -morphism X × spec( k ) S → A is constant. On the otherhand, if S is not reduced, then this results fails. For example, let T bea k -algebra containing a nonzero element u satisfying u d = 0, for somepositive integer d . Let X = A k , S = spec( T ), and A = spec( T [ ǫ ] /ǫ d ).The map ǫ u.x extends to a T -algebra homomorphism T [ ǫ ] /ǫ d → T [ x ]. The corresponding morphism X × spec( k ) S → A of schemes over S is not constant.2.2. Fundamental group schemes of algebraic groups.
Let k bea perfect field. Let G denote the category of connected algebraic groupsover k . Lemma 2.7.
Fiber products exists in G .Proof. Given a diagram G → G ← G in G , its fiber product equals( G × G G ) : the reduced neutral connected component of the scheme-theoretic fibre product G × G G . (cid:3) Definition 2.8.
A morphism ˜ G → G in G is an isogeny if it is surjec-tive and its kernel is finite. (Of course, the kernel is computed in thecategory of all group schemes.) Definition 2.9. A group cover of G is a connected algebraic group ˜ G equipped with an isogeny ˜ G → G . The group ˜ G is a central cover ifthe kernel of ˜ G → G is central. Let
Cov ( G ) denote the category of coverings of G . (An arrow in Cov ( G ) is a morphism ˜ G → ˜ G ′ of group schemes over G .) It is clearthat Cov ( G ) is essentially small. Lemma 2.10. Cov ( G ) is anti-equivalent to a partially ordered directedset I = I ( G ) .Proof. Proposition 2.4 shows that there is at most one morphism be-tween two coverings of G . Thus, Cov ( G ) is a partially ordered set.The supremum of two element of Cov ( G ) is given by their fibre prod-uct (Lemma 2.7). (cid:3) For each i ∈ I ( G ), let G i → G be the corresponding object of Cov ( G ), and let A i := ker( G i → G ) . The kernels A i ’s form an in-verse system: there is a morphism A i → A j if and only if there is amorphism G i → G j in Cov ( G ). The set I( G ) has the following subsets:(2.1) I cent ( G ) ⊇ I et ( G ) ⊇ I disc ( G ) , It is easy to show that an ´etale group cover is central.
TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 7 corresponding to group covers of G with central, ´etale, and discretekernel. (A discrete group scheme is a finite ´etale scheme on which theGalois group acts trivially.)Every element of I dominated by an element of I cent belongs to I cent .Furthermore, the supremum of two elements of I cent belongs to I cent .These facts remain true if I cent is replaced by I et or I disc . Convention . Set Π cent ( G ) := lim ←− i ∈ I cent ( G ) A i , Π et ( G ) := lim ←− i ∈ I et ( G ) A i , Π disc ( G ) := lim ←− i ∈ I disc ( G ) A i . We refer to these profinite group schemes as the fundamental groupschemes of G . Remark . The group scheme Π et ( G ) is the maximal pro´etale quo-tient of Π cent ( G ). The group Π disc ( G ) is the maximal quotient of Π et ( G )on which the absolute Galois group of k acts trivially.The following table summarizes our computations of fundamentalgroup schemes of certain connected algebraic groups. See Appendix Bfor the details. Convention . For a semisimple group G , π ss1 ( G ) denotes the weightlattice modulo the root lattice [Bor69], § G , π ( G ) denotes the algebraic fundamental group of G [GR72]. Incharacteristic zero, π ss1 ( G ) = π ( G ). G defined over C Π et ( G ) = π ( G ) G semisimple Π et ( G ) = π ss1 ( G ) G commutative defined over F q Π disc ( G ) = G ( F q ) G = G a additive group over F q Π et ( G ) = Hom( k, Q / Z )2.3. Classifying central extensions.
Let A be a finite commutativegroup scheme over k . Let1 → A → ˜ G → G → G ) → G is a central cover. Let f ˜ G : Π cent ( G ) → A denote the compositionΠ cent ( G ) ։ ker(( ˜ G ) → G ) ֒ → A. MASOUD KAMGARPOUR
Proposition 2.14.
The map ˜ G f ˜ G defines a canonical isomorphism (2.2) H ( G, A ) ≃ −→ Hom(Π cent ( G ) , A ) . If A is ´etale ( resp. discrete ) , then we can replace Π cent ( G ) by Π et ( G )( resp. Π disc ( G )) .Proof. Let ϕ ∈ Hom(Π et ( G ) , A ). Let B := im( A ). The group B is afinite quotient of Π cent ( G ); thus, B ∼ = A i for some i ∈ I cent . Let G ϕ bethe pushforward of the central extension1 → A i → G i → G → A i ≃ −→ B ֒ → A . One checks that ϕ G ϕ is theinverse of ˜ G → f ˜ G . (cid:3) et is a quotient of π . In this section, we assume k is alge-braically closed. Let π ( G ) := π ( G, e ) denote the fundamental groupof G in the sense of [GR72]. For every abelian group A one has ahomomorphism(2.3) Hom(Π et ( G ) , A ) f A (cid:15) (cid:15) { central extensions of G by A } Hom( π ( G ) , A ) { A -torsors on G trivialized over e } . Since f A is functorial in A , it comes from a homomorphism f : π ( G ) → Π et ( G ). Lemma 2.15. (i) f A is injective. (ii) ϕ : π ( G ) → A belongs to the image of f A if and only if thediagram (2.4) π ( G × G ) m ∗ / / (cid:15) (cid:15) π ( G ) / / Aπ ( G ) × π ( G ) (cid:15) (cid:15) A × A sssssssssssssssssssssssssss commutes.Proof. A based A -torsor ˜ G → G is a central extension if and only if m ∗ ˜ G is isomorphic to ˜ G ⊠ ˜ G (as A -torsors on G × G ). This proves (i).To prove (ii), let ˜ G → G denote the based A -torsor correspondingto ϕ . One checks that the commutativity of the diagram is equivalent TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 9 to the existence of an ( A × A )-equivariant morphism of based schemes˜ m : ˜ G × ˜ G → ˜ G such that the diagram˜ G × ˜ G / / (cid:15) (cid:15) ˜ G (cid:15) (cid:15) G × G / / G commutes. Proposition 2.4 then implies that ˜ m satisfies the axiomsdefining a group. (cid:3) The following is a reformulation Lemma 2.15.
Corollary 2.16. Π et ( G ) equals the coequalizer of the following homo-morphisms: π ( G × G ) ab m ∗ −→ π ( G ) ab (2.5) π ( G × G ) ab / / π ( G ) ab × π ( G ) ab + / / π ( G ) ab . Remark . Corollary 2.16 remains valid if one replaces Π et ( G ) withΠ cent ( G ) and π ( G ) with the fundamental group scheme of G defined [Nor82].3. Relationship between central covers of G and [ G, G ]3.1.
Restricting central covers of G to [ G, G ] . Let k be a perfectfield, G a connected algebraic group over k , and ˜ G → G a central cover(Definition 2.9). It is clear that ( ˜ G × G [ G, G ]) is a central cover of[ G, G ]. Convention . We call ( ˜ G × G [ G, G ]) the restriction of ˜ G → G to[ G, G ]. Restricting central covers defines a homomorphismΠ cent ([ G, G ]) → Π cent ( G ) . In the introduction, we mentioned that if ˜ G → G is a nontrivialcentral cover of a semisimple non-simply-connected group G , then therestriction of ˜ G to [ G, G ] is nontrivial. We now give examples of coversof unipotent groups whose restrictions to [
G, G ] is nontrivial.
Example . Let ˜ G be a connected noncommutative unipotent alge-braic group over a field of positive characteristic. Let A be an arbitraryfinite subgroup of the center Z ( ˜ G ) such that A ∩ [ ˜ G, ˜ G ] = { } . Let For example, A can be any nontrivial finite subgroup of C ( ˜ G ) := Z ( ˜ G ) ∩ [ ˜ G, ˜ G ].Note that dim( C ( ˜ G )) > G is a subgroup of C ( ˜ G ). In particular, C ( ˜ G ) has many finite subgroups. G := ˜ G/A . Then ˜ G π −→ G is a central cover of G whose restriction to[ G, G ] is nontrivial.
Convention . For a central cover ˜ G → G , let d ( ˜ G ) denote the degreeof the central cover ( ˜ G × G [ G, G ]) → [ G, G ]. Theorem 3.4.
There exists a constant C , depending only on G , suchthat d ( ˜ G ) < C for all central covers ˜ G → G . In view of Remark 2.12, the corresponding results for Π et and Π disc follow immediately. In particular, the image of the mapΠ et ([ G, G ]) → Π et ( G )is finite (Theorem 1.2).To prove Theorem 3.4, we need a lemma. For every positive integer n , let c n : G n → [ G, G ] denote the map(3.1) ( g , g , g , g , ..., g n − , g n ) [ g , g ][ g , g ] ... [ g n − , g n ] . Lemma 3.5.
There exists a positive integer n such that c n is surjec-tive. Proof.
See, for instance, [Mil06], cor. 11.13. (cid:3)
The following remark is a key observation used in the proof of The-orem 3.4.
Remark . Let ˜ G be a central cover of G and let H be its restrictionto [ G, G ]. The commutator map ˜ G × ˜ G → ˜ G descends to a morphismof algebraic varieties G × G → H . In this case, we say that the com-mutator map of G lifts to H . Proof of Theorem 3.4.
Choose n large enough so that c n is surjective.Remark 3.6 implies that c n has a lift to H ; that is to say, there existsa morphism of algebraic varieties G n → H such that the followingdiagram commutes:(3.2) H degree = d ( ˜ G ) (cid:15) (cid:15) G n : : c n GGGGGGGGG [ G, G ] In fact, one can take n = 2 dim( G ). TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 11 As c n is surjective, there exists a closed subvariety X ⊆ G n such thatthe generic fiber of the morphism c n | X : X → [ G, G ] is finite. Thenumber d ( ˜ G ) divides the degree of c n | X ; hence, it is bounded. (cid:3) Lifting central covers from [ G, G ] to G . We keep the notationand conventions of the previous section. Let H → [ G, G ] be a centralcover. The following result is a converse of Remark 3.6.
Proposition 3.7.
Suppose the commutator map of G lifts to H . Thenthere exists a central cover ˜ G → G whose restriction to [ G, G ] is iso-morphic to H . The proof of this proposition will take up the rest of this section.Let H be a central cover of [ G, G ] which has a lift of the commutatormap. Let δ : H → G denote the composition H ։ [ G, G ] ֒ → G . Lemma 3.8.
One can endow H δ −→ G with a structure of is a strictlystable crossed module (Definition A.7).Proof. Let {− , −} : G × G → H denote the lift of the commutator. Wemay assume that { , } = 1. Define a morphism of varieties G × H → H by ( g, h ) h g := h { δ ( h ) , g } . Using Proposition 2.4, it is easy to show that this morphism definesan action of G on H making H δ −→ G into a strictly stable crossedmodule. (cid:3) In the proof of Proposition 3.7, we employ a theorem about exten-sion of sheaves. To state this result, we need some notation. Let k fppf denote category of schemes over k equipped with the topology gener-ated by faithfully flat morphisms of finite type. In what follows, Extwill denote the extension of abelian sheaves in k fppf . Let Ab (resp. Fin k ) denote the category of finite abelian group (resp. the categoryof finite commutative group schemes over k ). Let J be a connectedcommutative algebraic group over k . Theorem 3.9.
Ext ( J, − ) : Fin k → Ab is an effaceable functor; thatis to say, given A ∈ Fin k and α ∈ Ext(
J, A ) , there exists a monomor-phism A ֒ → B in Fin k , such that the image of α under the map Ext ( J, A ) → Ext ( J, B ) is zero.Remark . This theorem was formulated and proved by V. Drin-feld (unpublished). It is closely related to the fact that Ext ( G, Q / Z )vanishes. For a proof of this vanishing result see [Boy07], lem. 3.2.2. Note that we do not use the fact that H is connected. Convention . All Picard stacks considered are assumed to be strictlycommutative, see Definition A.2.
Proof of Proposition 3.7.
Step 1: Let H be a central cover of [ G, G ]equipped with a lift of the commutator map. By Lemma 3.8, δ : H → G is a strictly stable crossed module. As explained in §§ A.3-A.4, thisimplies that the quotient stack P of G by the action of H is a Deligne-Mumford Picard stack. Set A := ker( δ ). Note that(3.3) π ( P ) = coker( δ ) = G ab , and π ( P ) = ker( δ ) = A. Step 2: It follows from [Del73], prop. 1.4.15, that the set of iso-morphism classes of Picard stacks with π = G ab and π = A equalsExt ( G ab , A ). Let α ∈ Ext ( G ab , A ) denote a representative for the iso-morphism class of P . By Theorem 3.9, there exists a monomorphism κ : A ֒ → B of finite commutative group schemes, such that the imageof α under the induced morphism Ext ( G ab , A ) → Ext ( G ab , B ) is zero.Let H ′ be the unique central extension of [ G, G ] for which the followingdiagram is commutative (3.4) 1 / / A / / κ (cid:15) (cid:15) H / / (cid:15) (cid:15) [ G, G ] / / (cid:15) (cid:15) / / B / / H ′ / / [ G, G ] / / . Step 3: The composition G × G {− , −} −→ H → H ′ endows H ′ with a liftof the commutator map. Applying Lemma 3.8, we conclude that H ′ → G is a strictly stable crossed module. Let P ′ denote the correspondingquotient stack. The class of P ′ in Ext ( G ab , B ) is trivial; therefore, P ′ ∼ = G/H × B − Tors . Here
G/H is is the discrete Picard stack defined by the commuta-tive algebraic group
G/H (Example A.3), and B − Tors denotes the(Deligne-Mumford Picard) stack of B -torsors on k fppf . The composition G → P ′ := [ G/H ′ ] ∼ −→ G × B − Tors → B − Tors , defines a 1-morphism of gr-stacks G → B − Tors .Step 4: Let Hom(
G, B − Tors ) (resp.
Cent ( G, B )) denote the Pi-card stack of 1-morphisms of gr-stacks G → B − Tors (resp. centralextensions of G by B ). According to [Gro72], § H ′ is the pushforward of H with respect to κ . TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 13 G → B − Tors defined in the previous step, gives rise to a central ex-tension 1 → B → G ′ → G → . This central extension is a lift of the central extension correspondingto the 1-morphism of gr-stacks[
G, G ] → A − Tors κ −→ B − Tors . It follows that the restriction of G ′ to [ G, G ] equals H ′ . Moreover,( H ′ ) ∼ = H , as required. (cid:3) True commutator and stacky abelianization
Stacky abelianization is Picard.
Let k be an algebraicallyclosed field and G be a connected algebraic group over k . Let A et be the image of Π et ([ G, G ]) → Π et ( G ). By Theorem 3.4, A et is finite.By Proposition 2.14, we obtain a central extension1 → A et → [ G, G ] true → [ G, G ] → . We call [
G, G ] true the true commutator of G . We refer to the quotientstack of G by the action of [ G, G ] true as the stacky abelianization of G and denote it by G ab , st . Example . Let G be a connected reductive group over an alge-braically closed field of characteristic zero. Let G ad := G/Z ( G ) be theassociated adjoint semisimple group. The group cover [ G, G ] → G ad defines an injection Π et ([ G, G ]) ֒ → Π et ( G ad ). It follows that the natu-ral morphism Π et ([ G, G ]) → Π et ( G ) is also injective. As we will see in § B.2, Π et ([ G, G ]) = π ss1 ([ G, G ]). Therefore [
G, G ] true identifies with thesimply connected cover of [ G, G ]. Lemma 4.2.
The true commutator is the restriction of a central coverof G (Conventions 3.1). To prove this lemma, we need an easy result from the theory ofprofinite groups whose proof we omit.
Lemma 4.3.
Let
A ֒ → C be an inclusion of a finite group into aprofinite abelian group. Then there exists an epimorphism C ։ B ,where B is a finite group, such that the composition A ֒ → C ։ B isan injection.Proof of Lemma 4.2. By Lemma 4.3, there exists a finite quotient B ofΠ et ( G ) such that the composition A et ֒ → Π et ( G ) ։ B is an injectivemorphism. Let ˜ G be the central cover of G corresponding to Π et ( G ) ։ B . The restriction of ˜ G to [ G, G ] is isomorphic to [
G, G ] true . (cid:3) Corollary 4.4.
The commutator map lifts to [ G, G ] true .Proof. This follows from Lemma 4.2 and Remark 3.6. (cid:3)
Corollary 4.5. G ab , st is a (strictly commutative) Picard stack.Proof. By Corollary 4.4 and Lemma 3.8, [
G, G ] true → G is a strictly sta-ble crossed module. Therefore, the corresponding quotient is a Picardstack. (cid:3) Characterization of the true commutator.
We keep the no-tation and conventions of the previous section. Let H be an ´etalecentral cover of [ G, G ]. Lemma 4.6.
The following are equivalent: (i)
The pullback of every central extension of G - by an ´etale groupscheme - to H is trivial. (ii) For every ´etale central cover ˜ G π −→ G , we have a morphism H → π − ([ G, G ]) of algebraic groups over [ G, G ] . Corollary 4.7.
Assume that the commutator map lifts to H . Then wehave a morphism [ G, G ] true → H of algebraic groups over [ G, G ] .Proof. By Proposition 3.7, H is a restriction of an ´etale central cover of G . By the previous lemma, we obtain a morphism [ G, G ] true → H . (cid:3) Proposition 4.8.
The true commutator is the unique, up to isomor-phism, ´etale central cover of [ G, G ] satisfying the following properties: P1)
The pullback of every central extension of G - by an ´etale groupscheme - to [ G, G ] true is trivial. P2)
The commutator map lifts to [ G, G ] true .Proof of Proposition 4.8. Let H be an ´etale central cover of [ G, G ].Suppose the true commutator lifts to H . Then by Corollary 4.7, wehave a morphism [ G, G ] true → H . Assume, furthermore, that the pull-back of every central extension of G - by an ´etale group scheme - to H istrivial. In view of the fact that [ G, G ] true is a restriction of an ´etale coverof G , Lemma 4.6 provides us with a morphism H → [ G, G ] true . (cid:3) Universal property of stacky abelianization.
Convention . All stacks discussed below are Deligne-Mumford stacks.The 2-category of Deligne-Mumford gr-stacks is denoted by
DMgrst .All Picard stacks are assumed to be strictly commutative.Let Φ : G → G ab , st denote the canonical 1-morphism of gr-stacks. TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 15
Theorem 4.10.
Let P be a Picard stack over k . Then compositionwith Φ defines an equivalence of Picard groupoids Hom
DMgrst ( G ab , st , P ) ∼ −→ Hom
DMgrst ( G, P ) . Lemma 4.11.
Let C and C ′ be gr-stacks over k . Assume that π ( C ) isrepresentable by a connected algebraic group over k . Then, the groupoid Hom
DMgrst ( C , C ′ ) is discrete (i.e., the objects have no nontrivial auto-morphisms).Proof. Let F be a 1-morphism of stacks C → C ′ . By Remark A.6, a 2-morphism F = ⇒ F defines a morphism of schemes ǫ : π ( C ) → π ( C ′ ).As C ′ is a Deligne-Mumford stack, π ( C ′ ) is finite. By Proposition 2.4, ǫ is constant. (cid:3) Proof of Theorem 4.10.
In view of the above lemma, it is enough toshow that every 1-morphism of gr-stacks F : G → P has a canonicalfactorization(4.1) G → G ab , st Φ −→ P . By the First Isomorphism Theorem for gr-stacks (Lemma A.14) andthe remarks following it, there exists a strictly stable crossed mod-ule of algebraic groups δ : ˜ H → G such that F has a factorization G ։ [ G/ ˜ H ] ֒ → P . Let H := δ ( ˜ H ). Then H is a subgroup of [ G, G ]containing [
G, G ], and ˜ H is a central extension of H by an ´etale groupscheme. Without loss of generality, we may assume H = [ G, G ].By Remark 4.7, we have a morphism of algebraic groups [
G, G ] true → H , which in turn, defines a morphism of strictly stable crossed modules([ G, G ] true → G ) → ( H → G ). Hence, we obtain a 1-morphism ofPicard stacks G ab , st = [ G/ [ G, G ] true ] → [ G/ ˜ H ] ≃ −→ P , providing therequired factorization. (cid:3) Appendix A. Crossed modules and gr-stacks
A.1.
Gr-categories.
Convention
A.1 . All categories we consider are essentially small. Amonoidal category is denoted by ( M , ⊗ , ). In other words, we sup-press the associativity and unit constraints [Mac98]. Occasionally, wesuppress ⊗ and as well. A monoidal category is strict if the associa-tivity and unit constraints are trivial. With the usual abuse of notation, x ∈ M means x is an object of M . We denote the set of isomorphismclasses of objects of M with π ( M ). For every x ∈ M , π ( M , x )denotes the abelian group Aut M ( x ). We set π ( M ) := π ( M , ). Definition A.2. A gr-category is a monoidal groupoid all of whoseobjects have a weak inverse; that is to say, for every x ∈ M there exists y ∈ M such that x ⊗ y ∼ = y ⊗ x ∼ = [Sin75], § M is a strict gr-categoryif it is a strict monoidal groupoid such that for every x ∈ M there exists y ∈ M satisfying x ⊗ y = y ⊗ x = . A (strictly commutative) Picardgroupoid is a (strictly) symmetric gr-category [Del73], § Example
A.3 . Let G be a group. The discrete groupoid whose set ofobjects equals G is a gr-category which, by an abuse of notation, is alsodenoted by G . If G is commutative, the corresponding gr-category is astrictly commutative Picard groupoid. Remark
A.4 . Let M be a gr-category. For every x ∈ M , the map u u ⊗ id x defines an isomorphism π ( M , ) ≃ −→ π ( M , x ). Definition A.5. A of gr-categories (resp. Picard groupoids)is a monoidal functor (resp. symmetric monoidal functor) F : M →M ′ . F is a monomorphism (resp. epimorphism, resp. isomorphism)if it is fully faithful (resp. essentially surjective, resp. an equiva-lence). Gr-categories and Picard groupoids form a 2-category where2-morphism are monoidal natural transformations. Remark
A.6 . Let M and M ′ be gr-categories. Let F be a (not nec-essarily monoidal) functor M → M ′ , and let η be a natural trans-formation F = ⇒ F . For every m ∈ M , we obtain an element ofAut M ′ ( F ( m )) which, by the identification of Remark A.4, gives us anelement of π ( M ′ ). One checks that this element depends only on theisomorphism class of m in M . Therefore, we obtain a well-defined map η : π ( M ) → π ( M ′ ). Note that if η is trivial, then so is η .A.2. Crossed modules.Definition A.7.
A (right) crossed module G is the data consisting ofa group homomorphism δ : H → G and a right action of G on H ,denoted by h h g , such that for every g ∈ G and h , h ∈ H ,( h ) δ ( h ) = h − h h , δ ( h g ) = g − δ ( h ) g. G is strictly stable if there exists a map {− , −} : G × G → H such thatfor every g, g , g , g ∈ G and h, h , h ∈ H , we have:(1) δ { g , g } = [ g , g ](2) { δh , δh } = [ h , h ](3) { δh, g } = h − ( h g )(4) { g, δh } = ( h g ) − h (5) { g , g g } = { g , g }{ g , g } g (6) { g g , g } = { g , g } g { g , g } TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 17 (7) { g , g }{ g , g } = 1(8) { g, g } = 1.In view of axiom (1), {− , −} is called a lift of the commutator map to H . By an abuse of notation, we denote a crossed module (or a strictlystable crossed module) by H → G . Definition A.8.
Let G := ( H → G ) and G ′ := ( H ′ → G ′ ) be crossedmodules. A morphism of crossed modules is a pair of homomorphisms a : H → H ′ and b : G → G ′ such that the following diagram commutes H a (cid:15) (cid:15) / / G b (cid:15) (cid:15) H ′ / / G ′ , and for every h ∈ H and g ∈ G , a ( h g ) = a ( h ) b ( g ) .Suppose that G and G ′ are strictly stable crossed modules, and let {− , −} and {− , −} ′ be their respective lift of the commutator map.Then ( a, b ) is a morphism of strictly stable crossed modules if the fol-lowing diagram commutes: G × G a (cid:15) (cid:15) {− , −} / / H b (cid:15) (cid:15) G ′ × G ′ {− , −} ′ / / H ′ . A.3.
Relationship between gr-categories and crossed modules.
Crossed modules to gr-categories: The passage from crossed modulesto gr-categories uses the notion of quotient groupoid.
Definition A.9.
Let Γ be a group acting on a set X . The quotientgroupoid [ X/ Γ] is the groupoid whose objects are the elements of X .An arrow x → x ′ in [ X/ Γ] is an element γ ∈ Γ such that γ.x = x ′ .Let H → G be a crossed module. Then H acts on G by righttranslation. One can show that [ G/H ] is a gr-category. Furthermore,a morphism of crossed modules defines a strict M be a gr-category. Let M ′ be a strict model of M ; that is to say, M ′ is a strict gr-category whichis equivalent to M . Let G be the group of objects of M ′ and let H bethe group of arrows whose source is . There is a natural morphism H → G given by taking an arrow to its target. Furthermore, G actson H by conjugation. One can show that H → G is a crossed moduleand [ G/H ] is equivalent to M ; see, for instance, [Noo05].Our goal is to prove a first isomorphism theorem for gr-categories.Recall that the first isomorphism from group theory states that ev-ery homomorphism f : Γ → M has a canonical factorization Γ ։ Γ / ( f − (1)) ֒ → M . Convention
A.10 . Let F : C → D be a functor between groupoids. Let d ∈ D . Let F − ( d ) denote the following groupoid:objects = objects of C which map to d morphisms = morphisms of C which map to id d .Let G be a group and let M = ( M , ⊗ , ) be a gr-category. Let F : G → M be a monoidal functor (Example A.3). Let H := F − ( ). Lemma A.11. (First Isomorphism Theorem) (1) H → G is a crossed module. (2) F has a factorization G ։ [ G/H ] ֒ → M .Proof. By [Noo06], thm. 7.10, F has a factorization G F ′ −→ M ′ e −→ M where M ′ is a strict gr-category, F ′ is a strict monoidal functor, and e is an isomorphism of gr-category. Let H ′ := F ′− ( d ). Then, e definesan isomorphism of gr-categories H ′ ≃ −→ H . Therefore, we may assumethat M and F are strict.By definition, the objects of H are pairs ( x, ϕ ) where x ∈ G and ϕ is an isomorphism → F ( x ) in M . Define a multiplication on H by( x, ϕ ) . ( y, ψ ) := ( xy, ϕ ⊗ ψ ) . Note that ϕ ⊗ ψ is an isomorphism → F ( x ) ⊗ F ( y ) = F ( xy ); hence,this product is well-defined. As M is strictly associative, this productis associative. It is easy to check that the above multiplication makes H into a group, where the unit is (1 , id U ). Furthermore, the map H → G , defined by ( x, ϕ ) x , is a group morphism.Next, define an action of G on H by( x, ϕ ) g := ( g − xg, id F ( g ) − ⊗ ϕ ⊗ id F ( g ) ) . One checks that this map is indeed an action, making H → G into acrossed module. Finally, define a functor K : [ G/H ] → M as follows: K ( g ) := F ( g ) for g ∈ GK ( g h −→ gδ ( h )) := id g ⊗ ( h −→ δ ( h )). K is a monomorphism of gr-categories, giving rise to the required fac-torization. (cid:3) TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 19
Remark
A.12 . In the previous lemma, if M were a strictly commuta-tive Picard groupoid, then H → G would be a strictly stable crossedmodule.A.4. Gr-stacks and Picard stacks.
Let k be a field. Let k et denotethe category of sheaves on the category of schemes over k , equippedwith the ´etale topology. The 2-category of stacks (in groupoids) on k et is defined in [Gir71]. Roughly speaking, a stack S is a sheaf ofgroupoids: for every scheme U over k , the “sections” of S above U form a groupoid, which is denoted by S U . The gluing conditionsfor these groupoids is best expressed using fibred categories and de-scent [GR72], [Gir71]. In this text, we will only be concerned withDeligne-Mumford stacks [DM69]. Example
A.13 . Let G be an algebraic group acting on an algebraic va-riety X . Suppose the stabilizer of every point of X is an ´etale subgroupof G . Then the quotient stack [ X/G ] is a Deligne-Mumford stack.Gr-stacks (resp. Picard stack) are studied in [Bre92], § G be a connected algebraic group over k . (Note that G can alsobe considered as a Deligne-Mumford gr-stack on k .) Let P a Deligne-Mumford strictly commutative Picard stack and let F : G → M be a1-morphism of gr-stacks. Let H = F − ( ). Lemma A.14. (i) H is an algebraic group; that is to say, it is areduced scheme of finite type over k . (ii) H → G is a strictly stable crossed module. (iii) F has a factorization G ։ [ G/H ] ֒ → P .Proof. For every scheme U over spec( k ), and every object x ∈ H U , wehave Aut H U ( x ) = { id } . By [LMB06], cor. 8.1.1 (iii), H is representable(by an algebraic space). As H has a group structure, we conclude thatit is represented by a group scheme over k . Moreover, the kernel of thecanonical morphism H → G equals π ( P ) and is, therefore, an ´etale For every scheme U over k , [ X/G ] U equals the quotient groupoid[Hom( U, X ) / Hom(
U, G )], see Definition A.9. group scheme over k . It follows that H is an algebraic group over k ;i.e., it is reduced of finite-type.Statements (ii) and (iii) follow from Lemma A.11 and Remark A.12. (cid:3) Appendix B. Computing fundamental group schemes
B.1.
Characteristic zero.
Let G be a connected algebraic over a fieldof characteristic zero. The K¨unneth formula implies that the two ho-momorphisms in (2.5) are equal. It follows that π ( G ) = Π et ( G ).B.2. Semisimple groups.
Let G be a connected semisimple groupover an algebraically closed field. Let π ss1 ( G ) denote the weight latticemodulo the root lattice. We claim that Π et ( G ) = π ss1 ( G ).It is enough to show that a connected simply connected semisimplealgebraic group G does not admit a nontrivial central extension byfinite groups. Suppose 1 → A → ˜ G p −→ G → G where A is a finite abelian group, and ˜ G is connected. It isclear that ˜ G is semisimple. Let ˜ T be a connected maximal torus of˜ G , and let T := p ( ˜ T ) denote the corresponding maximal torus in G .Since the center of ˜ G is a subgroup of ˜ T , we have a central extension1 → A → ˜ T → T →
1. This, in turn, defines an inclusion X ( T ) ֒ → X ( ˜ T ) ֒ → Λ( T ) = Λ( ˜ T ) , where X (resp. Λ) denotes the root (resp. weight) lattice. As G issimply connected, X ( T ) = Λ( T ); thus, X ( T ) = X ( ˜ T ), implying that A is trivial.B.3. Finite fields.
Let F q denote a finite field with q elements. Let G be a connected algebraic group over F q . Let Fr denote the Frobeniusautomorphism x x q . Let A be a finite abelian group (considered asa discrete group scheme over F q ). A central extension1 → A → ˜ G π −→ G → f π : G ( F q ) → A as follows. Given g ∈ G ( F q ), pick g ′ ∈ π − ( g ). Note that Fr( g ′ ) = ag ′ for some a ∈ A . Lemma B.1. ( [Boy07] , § H ( G, A ) → Hom( G ( F q ) , A ) given by the above construction is an injection. If G is commutative,then it is an isomorphism. The choice of the maximal torus does not matter. This is an instance of Grothendieck’s sheaf-function correspondence. The sheafin question, is the sheaf of local sections of π . TACKY ABELIANIZATION OF ALGEBRAIC GROUPS 21
Corollary B.2.
Let G be a connected commutative algebraic groupover F q . Then Π disc ( G ) = G ( F q ) . B.4.
Additive group.
Let k be an algebraically closed field of positivecharacteristic. Let G denote the additive group over k . Our aim is toprove Π et ( G ) = Hom( k, Q / Z ) Remark
B.3 (Pontryagin Duality) . The category of profinite abeliangroups is anti-equivalent to that of discrete torsion abelian groups: toa profinite abelian group Π, one associates Γ := Hom continuous (Π , Q / Z ).The inverse functor is Γ → Hom(Γ , Q / Z ).By Pontryagin duality, it is enough to show that H ( G, Q / Z ) = k . As G is connected and commutative, the natural injection Ext( G, Q / Z ) ֒ → H ( G, Q / Z ) is an isomorphism. Thus, it is enough to show thatExt( G, Q / Z ) = k . On the other hand, G is killed by multiplication by p ; hence, the long exact sequence corresponding to0 → p Z / Z → Q / Z p −→ Q / Z → G, Q / Z ) = Ext( G, F p ). Thus, we are reduced to prov-ing Ext( G, F p ) = k .Observe thatExt( G, F p ) ⊆ H ( G, F p ) = H ( A k , F p ) = k [ x ] /A ( k [ x ]) , where A ( u ) = u p − u . Furthermore, every element of H ( G, F p ) can bewritten as h = X p ∤ i c i x i , where x i is the image of x i in the quotient k [ x ] /A ( k [ x ]). By Lemma2.15, h ∈ Ext( G, F p ) if and only if the polynomials X p ∤ i c i ( x i + y i ) , and X p ∤ i c i ( x + y ) i represent the same element of H ( A , F p ) = k [ x, y ] /A ( k [ x, y ]). Thishappens if and only if the polynomial t ( x, y ) = X p ∤ i c i [( x + y ) i − x i − y i ]is of the form u p − u . This is possible if and only if c i = 0 for all i = 1(otherwise, the degree of t is not divisible by p ). Therefore, Ext( G, F p )identifies with { cx | c ∈ k } ⊆ k [ x ] /A ( k [ x ]) = H ( A , F p ) . References [BD06] D. Boyarchenko and V. Drinfeld. A motivated introduction to charactersheaves and the orbit method for unipotent groups in positive character-istic. arXiv: math.RT/0609769 , 09/27/2006.[Bor69] A. Borel.
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