RATIONAL POINTS OF QUIVER MODULI SPACES

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ANNALES DE

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LesAnnales de l’institut Fouriersont membres du Centre Mersenne pour l’édition scientifique ouverte

Victoria Hoskins & Florent Schaffhauser Rational points of quiver moduli spaces Tome 70, n^o3 (2020), p. 1259-1305.

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RATIONAL POINTS OF QUIVER MODULI SPACES

by Victoria HOSKINS & Florent SCHAFFHAUSER (*)

Abstract. —For a perfect base fieldk, we investigate arithmetic aspects of moduli spaces of quiver representations over k: we study actions of the absolute Galois group ofkon thek-valued points of moduli spaces of quiver representations overkand we provide a modular interpretation of the fixed-point set using quiver representations over division algebras, which we reinterpret using moduli spaces of twisted quiver representations (we show that those spaces provide differentk-forms of the initial moduli space of quiver representations). Finally, we obtain that stable k-representations of a quiver are definable over a certain central division algebra over their field of moduli.

Résumé. —Etant donné un corps parfaitket une clôture algébriquekdek, les espaces de modules dek-représentations semistables d’un carquoisQsont des k-variétés algébriques dont nous étudions ici les propriétés arithmétiques, en par- ticulier les points rationnels et leur interprétation modulaire. Outre les représen- tations à coefficients dansk, apparaissent naturellement certaines représentations rationnelles dites tordues, à coefficients dans une algèbre à division définie sur k et qui donnent lieu à différentes k-formes de la variété des modules initiale. En guise d’application, on montre qu’une k-représentation stable du carquois Qest définissable sur une algèbre à division centrale bien précise, elle-même définie sur le corps des modules de la représentation considérée.

1. Introduction

For a quiver Q and a field k, we consider moduli spaces of semistable k-representations of Q of fixed dimension d ∈ N^V, which were first constructed for an algebraically closed fieldkusing geometric invariant theory

Keywords:Algebraic moduli problems, Geometric Invariant Theory, Representations of quivers.

2020Mathematics Subject Classification:14D20, 14L24, 16G20.

(*) The authors thank the Institute of Mathematical Sciences of the National Uni- versity of Singapore, where part of this work was carried out, for their hospitality in 2016, and acknowledge the support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures And Representation varieties”

(the GEAR Network). The first author is supported by the Excellence Initiative of the DFG at the Freie Universität Berlin.

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(GIT) by King [10]. For an arbitrary fieldk, one can use Seshadri’s extension of Mumford’s GIT to construct these moduli spaces. More precisely, these moduli spaces are constructed as a GIT quotient of a reductive group GQ,d acting on an affine space Rep_Q,d with respect to a character χθ determined by a stability parameter θ ∈ Z^V. The stability parameter also determines a slope-type notion ofθ-(semi)stability fork-representations of Q, which involves testing an inequality for all proper non-zero subrepre- sentations. When working over a non-algebraically closed field, the notion of θ-stability is no longer preserved by base field extension, so one must instead considerθ-geometrically stable representations (that is, representations which areθ-stable after any base field extension), which correspond to the GIT stable points in Rep_Q,d with respect toχ_θ.

We let M^θ−ss_Q,d (resp. M^θ−gs_Q,d ) denote the moduli space of θ-semistable (resp.θ-geometrically stable)k-representations ofQof dimensiond; these are both quasi-projective varieties overkand are moduli spaces in the sense that they corepresent the corresponding moduli functors (cf. Section 2). For a non-algebraically closed fieldk, the rational points ofM^θ−gs_Q,d are not nec- essarily in bijection with the set of isomorphism classes ofθ-geometrically stable d-dimensional k-representations of Q. In this paper, we give a de- scription of the rational points of this moduli space for perfect fieldsk. More precisely, for a perfect field k, we study the action of the absolute Galois group Gal_k= Gal(k/k) onM^θ−ss_Q,d (k), whose fixed locus is the setM^θ−ss_Q,d (k) ofk-rational points. We restrict the action of Galk toM^θ−gs_Q,d ⊂ M^θ−ss_Q,d , so we can use the fact that the stabiliser of every GIT stable point in Rep_Q,d is a diagonal copy of Gm, denoted ∆, inGQ,d (cf. Corollary 2.14) to de- compose the fixed locus of Galk acting onM^θ−gs_Q,d (k) in terms of the group cohomology of Gal_k with values in ∆ or the (non-Abelian) group G_Q,d. More precisely, we obtain the following decomposition of the set of rational points, indexed by the Brauer group Br(k) ofk.

Theorem 1.1. — For a perfect fieldk, let

T :M^θ−gs_Q,d (k)−→H²(Galk;k^×)∼= Br(k)

be the type map introduced in Proposition 3.4. Then there is a decomposition

M^θ−gs_Q,d (k)' G

[cu]∈ImT

uRep^χ_Q,d^θ^−s(k)^Gal^k/uGQ,d(k)^Gal^k

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where uRep^χ_Q,d^θ^−s(k)^Gal^k/_uGQ,d(k)^Gal^k is the set of isomorphism classes of θ-geometrically stable d-dimensional representations of Q that are k- rational with respect to the twisted Gal_k-action Φ^u onRep_Q,d(k)defined in Proposition 3.10.

Next we give a modular interpretation of the decomposition above, by re- calling that Br(k) can be identified with the set of central division algebras overk. We first prove that for a division algebraD ∈ Br(k) to lie in the image of the type map, it is necessary that the index ind(D) :=p

dimk(D) divides the dimension vector d (cf. Proposition 3.12). As a corollary, we deduce that ifdis not divisible by any of the indices of non-trivial central division algebras overk, thenM^θ−gs_Q,d (k) is the set of isomorphism classes of d-dimensional k-representations of Q. We can interpret the above de- composition by using representations ofQover division algebras overk.

Theorem 1.2. — Let k be a perfect field. For a division algebra D ∈ ImT ⊂ Br(k), we have d = ind(D)d⁰_D for some dimension vector d⁰_D ∈ N^V and there is a modifying family uD and smooth affine k-varieties Rep_Q,d⁰

D,D (resp.GQ,d⁰_D,D) constructed by Galois descent such that Rep_Q,d⁰

D,D(k) =M

a∈A

Hom_Mod(D)(D^d⁰^D,t(a), D^d⁰^D,h(a))

=_u_DRep_Q,d(k)^Gal^k and

GQ,d⁰_D,D(k) = Y

v∈V

Aut_Mod(D)(D^d⁰^D,v) =u_DGQ,d(k)^Gal^k. Furthermore, we have a decomposition

M^θ−gs_Q,d (k)∼= G

D∈ImT

Rep^θ−gs_Q,d0

D,D(k)/G_Q,d⁰

D,D(k),

where the subset indexed by D is the set of isomorphism classes of d⁰_D- dimensionalθ-geometrically stableD-representations ofQ.

For example, if k = R ,→ k = C, then as Br(R) = {R,H}, there are two types of rational points inM^θ−gs_Q,d (R), namelyR-representations and H-representations of Q and the latter can only exists if d is divisible by 2 = ind(H) (cf. Example 3.23).

We can also interpret Br(k) as the set of isomorphism classes of Gm- gerbes over Speck, and show that the type map T can be defined for any field k using the fact that the moduli stack ofθ-geometrically stable

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d-dimensionalk-representations ofQis aGm-gerbe overM^θ−gs_Q,d (cf. Corol- lary 4.6). For any fieldk, we introduce a notion of twistedk-representations of a quiverQin Definition 4.7, analogous to the notion of twisted sheaves due to Căldăraru, de Jong and Lieblich [4, 9, 13], and we describe the moduli of twisted quiver representations. In particular, we show that twisted representations of Q are representations of Q over division algebras, by using Căldăraru’s description of twisted sheaves as modules over Azu- maya algebras; therefore, the decomposition in Theorem 1.2 can also be expressed in terms of twisted quiver representations (cf. Theorem 4.12).

Consequently, we construct moduli spaces of twistedθ-geometrically stable k-representations of Q and show, for Brauer classes in the image of the type mapT, that these moduli spaces give different forms of the moduli spaceM^θ−gs_Q,d .

Theorem 1.3. — For a fieldkwith separable closurek^s, letα:X−→

Speck be a Gm-gerbe over k and letD be the corresponding central division algebra over k. Then the stack of α-twisted θ-geometrically stable d⁰-dimensionalk-representations

αM^θ−gs_Q,d0,k∼= [Rep^θ−gs_Q,d0,D/GQ,d⁰,D] is aGm-gerbe over its coarse moduli space

M^θ−gs_Q,d0,D:= Rep^θ−gs_Q,d0,D/G_Q,d⁰_,D

(in the sense of stacks). The moduli space M^θ−gs_Q,d0,D is a coarse moduli space for:

(1) the moduli functor ofθ-geometrically stabled⁰-dimensionalD-representations ofQ, and

(2) the moduli functor ofα-twistedθ-geometrically stabled⁰-dimensionalk-representations ofQ.

If, moreover,D lies in the image of the type map T, then d = ind(D)d⁰ for some dimension vectord⁰ andM^θ−gs_Q,d0,D is ak-form of the moduli space M^θ−gs_Q,d,ks.

As an application of these ideas, we define a Brauer class which is the obstruction to the existence of a universal family onM^θ−gs_Q,d and show that this moduli space admits a twisted universal family of quiver representations (cf. Proposition 4.18).

The structure of this paper is as follows. In Section 2, we explain how to construct moduli spaces of representations of a quiver over an arbitrary field k following King [10], and we examine how (semi)stability behaves under base field extension. In Section 3, we study actions of Galk for a

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perfect field k and give a decomposition of the rational points of M^θ−gs_Q,d indexed by the Brauer group, using only elementary considerations from group cohomology. In Section 4, we see the benefit of rephrasing the work of Section 3 in the more sophisticated language of stacks and gerbes, which gives a quicker and more conceptual way to understand the arithmetic aspects of quiver representations over a field (Theorem 4.12). In particular, we interpret our decomposition result using twisted quiver representations and show that moduli spaces of twisted quiver representations give different forms of the moduli spaceM^θ−gs_Q,d . Finally, in Section 5, we apply Theorem 1.2 to briefly discuss the classical problem of fields of moduli vs.

fields of definition in the context of quiver representations.

Notation. — For a scheme S over a field k and a field extension L/k, we denote by S_L the base change of S to L. For a point s ∈ S, we let κ(s) denote the residue field ofs. A quiver Q= (V, A, h, t) is an oriented graph, consisting of a finite vertex setV, a finite arrow set A, a tail map t:A−→V and a head maph:A−→V.

Acknowledgements.We thank the referees of a previous version of this paper, for suggesting that we relate our results to twisted quiver representations and the referee of the current version for observations that have helped improve the paper. V.H. would like to thank Simon Pepin Lehalleur for several very fruitful discussions, which helped turned the former sug- gestion into what is now Section 4.

2. Quiver representations over a field

LetQ= (V, A, h, t) be a quiver and letkbe a field.

Definition 2.1. — A representation ofQ in the category of k-vector spaces(or k-representationofQ) is a tuple

W := ((Wv)_v∈V,(ϕa)_a∈A) where:

• Wv is a finite-dimensionalk-vector space for allv∈V;

• ϕ_a :W_t(a)−→W_h(a)is a k-linear map for alla∈A.

There are natural notions of morphisms of quiver representations and subrepresentations. The dimension vector of ak-representationW is the tupled= (dimkWv)v∈V; we then sayW isd-dimensional.

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2.1. Slope semistability

Following King’s construction of moduli spaces of quiver representations over an algebraically closed field [10], we introduce a stability parameter θ := (θv)_v∈V ∈ Z^V and the associated slope function µθ, defined for all non-zerok-representationsW ofQ, by

µθ(W) :=µ^k_θ(W) :=

P

v∈V θvdimkWv

P

v∈V dimkWv

∈Q.

Definition 2.2. — Ak-representationW ofQis:

(1) θ-semistable if µ_θ(W⁰) 6 µ_θ(W) for all k-subrepresentation 0 6=

W⁰⊂W.

(2) θ-stable ifµθ(W⁰)< µθ(W)for all k-subrepresentation 0 6=W⁰ ( W.

(3) θ-polystableif it is isomorphic to a direct sum ofθ-stable representations of equal slope.

The category ofθ-semistablek-representations ofQwith fixed slopeµ∈ Qis an Abelian, Noetherian and Artinian category, so it admits Jordan–

Hölder filtrations. The simple (resp. semisimple) objects in this category are precisely the stable (resp. polystable) representations of slopeµ(proofs of these facts are readily obtained by adapting the arguments of [23] to the quiver setting). The graded object associated to any Jordan–Hölder filtration of a semistable representation is by definition polystable and its isomorphism class as a graded object is independent of the choice of the filtration. Twoθ-semistable k-representations ofQare calledS-equivalent if their associated graded objects are isomorphic.

Definition 2.3. — LetWbe ak-representation ofk; then ak-subrepresentation U ⊂W is said to be strongly contradicting semistability (scss) with respect toθif its slope is maximal among the slopes of all subrepresentations of W and, for any W⁰ ⊂ W with this property, we have U ⊂ W⁰⇒U =W⁰.

For a proof of the existence and uniqueness of the scss subrepresentation, we refer to [17, Lemma 4.4]. The scss subrepresentation satisfies Hom(U, W/U) = 0. Using the existence and uniqueness of thescss, one can inductively construct a unique Harder–Narasimhan filtration; for example, see [17, Lemma 4.7].

We now turn to the study of how the notions of semistability and stability behave under a field extensionL/k. A k-representation W = ((W_v)_v∈V, (ϕa)a∈A) ofQdetermines anL-representationL⊗kW := ((L⊗kWv)v∈V,

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(IdL⊗ϕa)a∈A) (or simply L⊗W), where L⊗kW_v is equipped with its canonical structure ofL-vector space and IdL⊗ϕa is the extension of the k-linear mapϕ_a byL-linearity. Note that the dimension vector ofL⊗_kW as anL-representation is the same as the dimension vector ofW as ak- representation. It can then be proved following for instance the proof of the analogous statement for sheaves given in [12, Proposition 3] and [8, Theorem 1.3.7], that semistability of quiver representations is invariant under base field extension:

Proposition 2.4. — Let L/k be a field extension and let W be ak- representation. For a stability parameterθ∈Z^V, the following statements hold.

(1) IfL⊗k W is θ-semistable (resp. θ-stable) as an L-representation, thenW isθ-semistable (resp.θ-stable) as ak-representation.

(2) If W is θ-semistable as a k-representation, then L ⊗k W is θ- semistable as anL-representation.

Moreover, if (Wⁱ)16i6l is the Harder–Narasimhan filtration of W, then (L⊗kWⁱ)16i6l is the Harder–Narasimhan filtration ofL⊗kW.

Remark 2.5. — Part (2) of Proposition 2.4 is not true if we replace semistability by stability, as is evident if we setk =R and L =C: for a θ-stableR-representationW, its complexificationC⊗W is aθ-semistable C-representation by Proposition 2.4 and either, for allC-subrepresentations U ⊂C⊗W, one has µ^C_θ(U)< µ^C_θ(C⊗W), in which case C⊗W is actu- allyθ-stable as a C-representation; or there exists a C-subrepresentation U ⊂L⊗W such that µ^C_θ(U) =µ^C_θ(C⊗W). In the second case, let τ(U) be theC-subrepresentation ofC⊗W obtained by applying the non-trivial element of Aut(C/R) to U. Note that τ(U) 6= U, as otherwise it would contradict the θ-stability ofW as an R-representation (as in the proof of Part (2) of Proposition 2.4). It is then not difficult, adapting the arguments of [16, 20], to show thatU is a θ-stable C-representation and that C⊗W 'U⊕τ(U); thusC⊗W is onlyθ-polystable as aC-representation.

This observation motivates the following definition.

Definition 2.6. — A k-representation W is θ-geometrically stable if L⊗kW isθ-stable as anL-representation for all extensions L/k.

Evidently, the notion of geometric stability is invariant under field extension. In fact, ifk =k, then being geometrically stable is the same as being stable: this can be proved directly, as in [8, Corollary 1.5.11], or as a consequence of Proposition 2.11 below. This implies that ak-representation

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W isθ-geometrically stable if and only ifk⊗kW isθ-stable (the proof is the same as in Part (2) - Case (iv) of Proposition 2.4).

2.2. Families of quiver representations

A family of k-representations of Qparametrised by a k-scheme B is a representation ofQin the category of vector bundles overB, denotedE= ((Ev)v∈V,(ϕa)a∈A)−→B. Ford= (dv)v∈V ∈N^V, we say a familyE −→B isd-dimensional if, for allv∈V, the rank ofEvisdv. For a morphism ofk- schemesf :B⁰−→B, there is a pullback familyf^∗E:= (f^∗Ev)v∈V overB⁰. Forb∈B with residue fieldκ(b), we letEb denote theκ(b)-representation obtained by pulling backE alongu_b: Specκ(b)−→B.

Definition 2.7. — A familyE −→Bofk-representations ofQis called:

(1) θ-semistable if, for allb ∈B, the κ(b)-representationE_b isθ-semistable.

(2) θ-geometrically stableif, for allb∈B, theκ(b)-representationE_b is θ-geometrically stable.

For a family E −→ B of k-representations of Q, the subset of points b ∈B for which E_b is θ-semistable (resp. θ-geometrically stable) is open;

one can prove this by adapting the argument in [8, Proposition 2.3.1]. By Proposition 2.4 and Definition 2.6, the pullback of a θ-semistable (resp.

θ-geometrically stable) family is semistable (resp. geometrically stable).

Therefore, we can introduce the following moduli functors:

(2.1) F_Q,d^θ−ss: (Sch_k)ôp−→Sets and F_Q,d^θ−gs: (Sch_k)ôp−→Sets, where (Schk)ôpdenotes the opposite category of the category ofk-schemes and, for B ∈ Sch_k, we have that F_Q,d^θ−ss(B) (resp. F_Q,d^θ−gs(B)) is the set of isomorphism classes of θ-semistable (resp. θ-geometrically stable) d- dimensional families overB ofk-representations ofQ.

We follow the convention in that a scheme Mis a coarse moduli space for the moduli functor F : (Schk)^op −→ Sets is F if it comes equipped with a universal natural transformationF−→Hom(·,M) inducing bijec- tionsM(Ω) ' F(Ω) for all algebraically closed fields Ω. When referring to the first condition only, it will sometimes be convenient to say thatM corepresentsthe functorF.

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2.3. The GIT construction of the moduli space

Fix a field k and dimension vector d = (dv)v∈V ∈ N^V; then every d- dimensionalk-representation ofQis isomorphic to a point of the following affine space overk

Rep_Q,d:= Y

a∈A

Matd_h(a)×dt(a). The reductive group G_Q,d := Q

v∈V GL_d_v over k acts algebraically on Rep_Q,d by conjugation: for g = (gv)v∈V ∈ GQ,d and M = (Ma)a∈A ∈ Rep_Q,d, we have

(2.2) g·M := (g_h(a)Mag_t(a)⁻¹)_a∈A.

There is a tautological familyF −→Rep_Q,dofd-dimensionalk-representations ofQ, where F_v is the trivial rankd_v vector bundle on Rep_Q,d.

Lemma 2.8. — The tautological family F −→ Rep_Q,d has the local universal property; that is, for every familyE= ((Ev)_v∈V,(ϕa)_a∈A)−→B of representations ofQover ak-schemeB, there is an open covering B=

∪_i∈IBi and morphismsfi:Bi−→Rep_Q,d such thatE|B_i∼=f_i^∗F.

Proof. — Take an open cover ofBon which all the (finitely many) vector bundlesEv are trivialisable, then the morphismsfi are determined by the

morphismsϕ_a.

We will construct a quotient of the GQ,d-action on Rep_Q,d via geometric invariant theory (GIT) using a linearisation of the action by a stability parameter θ = (θ_v)_v∈V ∈ Z^V. Let us set θ⁰ := (θ⁰_v)_v∈V where θ⁰_v :=θvP

α∈V dα−P

α∈V θαdα for all v ∈V; then one can easily check thatθ⁰-(semi)stability is equivalent to θ-(semi)stability. We define a char- acterχθ:GQ,d−→Gmby

(2.3) χ_θ((g_v)v∈V) := Y

v∈V

(detg_v)^−θ^v⁰.

Any such character χ: GQ,d −→Gm defines a lifting of theGQ,d-action on Rep_Q,d to the trivial line bundle Rep_Q,d×A¹, where G_Q,d acts on A¹ via multiplication byχ. As the subgroup ∆⊂GQ,d, whose set ofR-points (forR ak-algebra) is

(2.4) ∆(R) :={(tId_v)_v∈V :t∈R^×} ∼=Gm(R),

acts trivially on Rep_Q,d, invariant sections only exist if χ^(R)(∆(R)) = {1_R×} for all R; this holds forχ_θ, as P

v∈V θ⁰_vd_v = 0. Let L_θ denote the line bundle Rep_Q,d×A¹ endowed with theGQ,d-action induced byχθ and

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byLⁿ_θ itsn-th tensor power forn>1 (endowed with the action ofχⁿ_θ). The invariant sections ofLⁿ_θ areχθ-semi-invariant functions; that is, morphisms f : Rep_Q,d−→A¹ satisfyingf(g·M) =χ_θ(g)ⁿf(M) for allg∈G_Q,d and allM ∈Rep_Q,d.

Definition 2.9. — A pointM ∈Rep_Q,d is called:

(1) χ_θ-semistableif there exists an integern >0and a G_Q,d-invariant sectionf ofLⁿ_θ such thatf(M)6= 0.

(2) χ_θ-stableif there exists an integern >0and aG_Q,d-invariant sectionf ofLⁿ_θ such thatf(M)6= 0, the action of GQ,d on(Rep_Q,d)f

is closed and dim_κ(M)(Stab(M)/∆_κ(M₎) = 0, where Stab(M) ⊂ GQ,d,κ(M)is the stabiliser group scheme ofM.

We denote the set ofχθ-(semi)stable points inRep_Q,d byRep^χ_Q,d^θ^−(s)s. Evidently, Rep^χ_Q,d^θ^−ssand Rep^χ_Q,d^θ^−sareG_Q,d-invariant open subsets. More- over, these subsets commute with base change (cf. [14, Proposition 1.14]

and [24, Lemma 2]). Mumford’s GIT (or, more precisely, Seshadri’s extension of GIT [24]) provides a categorical and good quotient of the GQ,d- action on Rep^χ_Q,d^θ^−ss

π: Rep^χ_Q,d^θ^−ss−→Rep_Q,d//_χ_θG_Q,d:= ProjM

n>0

H⁰(Rep_Q,d,Lⁿ_θ)^G^Q,d, which restricts to a geometric quotient

π|_Repχθ−s Q,d

: Rep^χ_Q,d^θ^−s−→Rep^χ_Q,d^θ^−s/GQ,d.

Given a geometric pointM : Spec Ω−→Rep_Q,d, let us denote by Λ(M) the set of 1-parameter subgroupsλ:G^m,Ω−→GQ,d,Ω such that the morphism Gm,Ω −→ Rep_Q,d,Ω, given by the λ-action on M, extends to A¹Ω. As Rep_Q,d is separated, if this morphism extends, its extension is unique.

If M₀ denotes the image of 0 ∈ A¹Ω, the weight of the induced action of Gm,Ω onLθ,Ω|M₀ is (χθ,Ω, λ)∈Z, where (·,·) denotes the natural pairing of characters and 1-parameter subgroups.

Proposition 2.10 (Hilbert–Mumford criterion [10]). — For a geometric pointM : Spec Ω→Rep_Q,d, we have

(1) M isχ_θ-semistable if and only if(χ_θ,Ω, λ)>0 for allλ∈Λ(M);

(2) M isχθ-stable if and only if (χθ,Ω, λ)> 0 for allλ ∈Λ(M), and (χ_θ,Ω, λ) = 0impliesImλ⊂Stab(M), where Stab(M)⊂G_Q,d,Ω is the stabiliser group scheme ofM.

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Proof. — If k is algebraically closed and Ω = k, this is [10, Proposi- tion 2.5]; then the above result follows as GIT (semi)stability commutes

with base change.

Before we relate slope (semi)stability and GIT (semi)stability for quiver representations, let Rep^θ−ss_Q,d (resp. Rep^θ−gs_Q,d ) be the open subset of points in Rep_Q,d over which the tautological family F is θ-semistable (resp. θ- geometrically stable).

Proposition 2.11. — For θ∈Z^V, we have the following equalities of k-schemes:

(1) Rep^θ−ss_Q,d = Rep^χ_Q,d^θ^−ss; (2) Rep^θ−gs_Q,d = Rep^χ_Q,d^θ^−s.

Proof. — Since all of thesek-subschemes of Rep_Q,dare open, it suffices to verify these equalities onk-points, for which one uses [10, Proposition 3.1]

(we note that we use the opposite inequality to King in our definition of slope (semi)stability, but this is rectified by the minus sign appearing

in (2.3) for the definition ofχθ).

Proposition 2.11 readily implies the result claimed at the end of Sec- tion 2.1, which we state here for future reference.

Corollary 2.12. — Ak-representationW isθ-geometrically stable if and only if k⊗kW is θ-stable. In particular, if k =k, then θ-geometric stability is equivalent toθ-stability.

Finally, we show the existence of coarse moduli spaces of θ-semistable (resp.θ-geometrically stable) k-representations of Qfor an arbitrary field k: For an algebraically closed field k, this result is proved in [10, Proposi- tion 5.2], and for generalk, it follows from the results of Seshadri in [24].

Theorem 2.13. — Thek-varietyM^θ−ss_Q,d := Rep_Q,d//χ_θGQ,dis a coarse moduli space for the functor F_Q,d^θ−ss and the natural map F_Q,d^θ−ss(k) −→

M^θ−ss_Q,d (k)is surjective. Moreover,

M^θ−gs_Q,d := Rep^χ_Q,d^θ^−s/G_Q,d

is an openk-subvariety of M^θ−ss_Q,d which is a coarse moduli space for the functorF_Q,d^θ−gsand the natural mapF_Q,d^θ−gs(k)−→ M^θ−gs_Q,d (k)is bijective.

We end this section with a result that is used repeatedly in Section 3.

Corollary 2.14. — ForM ∈Rep^θ−gs_Q,d , we have Stab(M) = ∆κ(M)⊂GQ,d,κ(M).

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Proof. — Stab(M)⊂GQ,d,κ(M)is isomorphic to Aut(FM), whereF −→

Rep_Q,dis the tautological family, andFM isθ-geometrically stable. The en- domorphism group of a stablek-representation ofQis a finite dimensional division algebra overk(cf. [8, Proposition 1.2.8]). Letκ(M) be an algebraic closure ofκ(M); then, asκ(M)⊗ FM isθ-stable andκ(M) is algebraically closed, End(κ(M)⊗ FM) =κ(M). Sinceκ(M)⊗End(FM)⊂End(κ(M)⊗ FM), it follows that End(FM) =κ(M) and thus Aut(FM)'∆_κ(M₎.

3. Rational points of the moduli space

Throughout this section, we assume k is a perfect field and we fix an algebraic closurekofk. For ak-schemeX, there is a left action of the Galois group Galk := Gal(k/k) on the set ofk-pointsX(k) as follows: forτ ∈Galk

andx: Speck−→X, we letτ·x:=x◦τ^∗, whereτ^∗ : Speck−→Speck is the morphism of k-schemes induced by the k-algebra homomorphism τ:k−→k. Askis perfect,X(k) =X(k)^Gal^k, where the right side denotes the fixed-point set the Galk-action onX(k). IfX_k = Speck×SpeckX, then X_k(k) =X(k) and Galk acts onX_k byk-scheme automorphisms and, ask is perfect, we can recoverX asX_k/Gal_k.

3.1. Rational points arising from rational representations

The moduli space M^θ−ss_Q,d constructed in Section 2 is ak-variety, so the Galois group Galk := Gal(k/k) acts on M^θ−ss_Q,d (k) as described above and the fixed points of this action are the k-rational points. Alternat- ively, we can describe this action using the presentation of M^θ−ss_Q,d as the GIT quotient Rep_Q,d//_χ_θG_Q,d. The Gal_k-action on Rep_Q,d(k) = Q

a∈AMatd_h(a)×dt(a)(k) and GQ,d(k) = Q

v∈V GLd_v(k) is given by applying ak-automorphism τ∈Galk = Aut(k/k) to the entries of the matrices (M_a)_a∈Aand (g_v)_v∈V. Both actions are by homeomorphisms in the Zariski topology and the second action is by group automorphisms and preserves the subgroup ∆(k) defined in (2.4). We denote these actions as follows (3.1) Φ : Galk×Rep_Q,d(k)−→Rep(k), τ,(Ma)_a∈A

7−→ τ(Ma)

a∈A

and

(3.2) Ψ : Galk×GQ,d(k)−→GQ,d(k), τ,(gv)_v∈V

7−→ τ(gv)

v∈V.

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They satisfy the following compatibility relation with the action ofGQ,d(k) on Rep_Q,d(k): for allg∈GQ,d(k), allM ∈Rep(k) and allτ∈Galk, one has (3.3) Φ_τ(g·M) = Ψ_τ(g)·Φ_τ(M)

i.e. theGQ,d(k)-action on Rep_Q,d(k) extends to an action of GQ,d(k)oGalk.

For convenience, we will often simply denote Φ_τ(M) by τ(M) and Ψ_τ(g) byτ(g).

Proposition 3.1. — TheGalk-action onRep_Q,d(k)preserves the open sub-variety Rep^χ_Q,d^θ^−(s)s(k). Moreover, if M1, M2 are two GIT-semistable points whose GQ,d(k)-orbits closures meet in Rep^χ_Q,d^θ^−ss(k), then, for all τ∈Galk, the same is true forτ(M1)andτ(M2).

Proof. — The first statement holds, as the Galk-action preserves the χ_θ-semi-invariant functions due to the compatibility relation (3.3), and moreover, forM ∈Rep_Q,d(k) andτ ∈Galk, we have

Stab_G

Q,d(k) τ(M)

=τ Stab_G

Q,d(k)(M) .

The second statement follows from (3.3) and the continuity of τ in the

Zariski topology of Rep^χ_Q,d^θ^−ss(k).

Proposition 3.1 combined with the compatibility relation (3.3) readily implies that Galk acts on the set of k-points of the k-varieties M^θ−ss_Q,d = Rep_Q,d//χ_θGQ,dandM^θ−gs_Q,d = Rep^θ−gs_Q,d /GQ,d. Explicitly, the Galk-action on the orbit space Rep^θ−gs_Q,d (k)/GQ,d(k) is given by

(3.4) (G_Q,d(k)·M)7−→(G_Q,d(k)·τ(M)).

Sincekis assumed to be a perfect field, this Galk-action on thek-varieties M^θ−ss_Q,d (k) and M^θ−gs_Q,d (k) suffices to recover the k-schemes M^θ−ss_Q,d and M^θ−gs_Q,d . In particular, the Galk-actions just described on M^θ−ss_Q,d (k) and M^θ−gs_Q,d (k) coincide with the ones described algebraically at the beginning of the present section.

Remark 3.2. — We can intrinsically define the Galois action on Rep_Q,d(k) by defining a Gal_k-action on arbitraryk-representations ofQas follows. If W = ((Wv)v∈V, (ϕa)a∈A) is ak-representation ofQ, then, for τ ∈ Galk, we defineW^τ to be the representation (W_v^τ, v∈V;φ^τ_a;a∈A) where:

• W_v^τis thek-vector space whose underlying Abelian group coincides with that ofW_v and whose external multiplication is given byλ·_τ w:=τ⁻¹(λ)wforλ∈kand w∈Wv.

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• The mapφ^τ_acoincides withφ_a, which isk-linear for the newk-vector space structures, asφ^τ_a(λ·τ w) = φa(τ⁻¹(λ)w) = τ⁻¹(λ)φa(w) = λ·_τφ^τ_a(w).

If ρ : W⁰ −→ W is a morphism of k-representations and τ ∈ Galk, we denote by ρ^τ : (W⁰)^τ −→ W^τ the induced homomorphism (which set- theoretically coincides with ρ). With these conventions, we have a right action, asW^τ¹^τ² = (W^τ¹)^τ². Moreover, if we fix a k-basis of eachWv, the matrix ofφ^τ_aisτ(M_a), whereM_ais the matrix ofφ_a, so we recover the Gal_k- action (3.1). We note that the constructionW 7−→W^τ is compatible with semistability andS-equivalence, thus showing in an intrinsic manner that Galk acts on the set of S-equivalence classes of semistable d-dimensional representations ofQ.

By definition of the moduli spacesM^θ−ss_Q,d andM^θ−gs_Q,d , we have natural maps

(3.5) F_Q,d^θ−ss(k)−→ M^θ−ss_Q,d (k) and F_Q,d^θ−gs(k)−→ M^θ−gs_Q,d (k), where F_Q,d^θ−ss and F_Q,d^θ−gs are the moduli functors defined at (2.1). As k is perfect, M^θ−ss_Q,d (k) = M^θ−ss_Q,d (k)^Gal^k and M^θ−gs_Q,d (k) = M^θ−gs_Q,d (k)^Gal^k. The goal of the present section is to use this basic fact in order to understand the natural maps (3.5). As a matter of fact, our techniques will only apply to F_Q,d^θ−gs(k) −→ M^θ−gs_Q,d (k), because M^θ−gs_Q,d (k) is the orbit space Rep^θ−gs_Q,d (k)/GQ,d(k) and all GIT-stable points in Rep_Q,d(k) have the Abelian group ∆(k)'k^× as their stabiliser for theGQ,d(k)-action.

Note first that, by definition of the functor F_Q,d^θ−gs, we have F_Q,d^θ−gs(k)'Rep^θ−gs_Q,d (k)/G_Q,d(k),

so the natural mapF_Q,d^θ−gs(k)−→ M^θ−gs_Q,d (k) may be viewed as the map fGal_k: Rep^θ−gs_Q,d (k)/GQ,d(k)−→ Rep^θ−gs_Q,d (k)/GQ,d(k)Gal_k

GQ,d(k)·M 7−→GQ,d(k)·(k⊗kM).

Proposition 3.3. — The natural mapF_Q,d^θ−gs(k)−→ M^θ−gs_Q,d (k)is injective.

Proof. — To prove this result, we identify this map withf_Gal_kand we will show that the non-empty fibres offGal_k are in bijection with the pointed set

ker H¹(Galk; ∆(k))−→H¹(Galk;GQ,d(k))

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where this map is induced by the inclusion ∆(k) ⊂ GQ,d(k). Then the result follows from this claim, asH¹(Galk; ∆(k)) ={1} by Hilbert’s 90th Theorem (for example, see [27, Theorem 4.3.1]. It remains to prove the above claim about the fibres of fGal_k. For this we consider M1, M2 in Rep^θ−gs_Q,d (k)^Gal^k such that GQ,d(k)·M₁ = GQ,d(k)·M₂. Then there ex- istsg ∈ GQ,d(k) such thatg·M2 =M1. Therefore, for all τ ∈ Galk, we have

g⁻¹·M1=M2=τ(M2) =τ(g⁻¹·M1) =τ(g⁻¹)·τ(M1), sogτ(g⁻¹)∈Stab_G

Q,d(k)(M1) = ∆(k). It is straight-forward to check that the map

βM₁,M₂ : Galk −→∆(k), τ7−→gτ(g⁻¹)

is a normalised ∆(k)-valued 1-cocycle whose cohomology class only depends on the GQ,d(k)^Gal^k-orbits of M₁ and M₂. Thus the cohomology class [βM₁,M₂] lies in the kernel of the pointed map H¹(Galk; ∆(k)) −→

H¹(Gal_k;G_Q,d(k)). Hence, for

[M1] :=GQ,d(k)·M1∈Rep^θ−gs_Q,d (k)/GQ,d(k), there is a map

f_Gal⁻¹

k(f_Gal_k([M₁]))−→ker H¹(Gal_k; ∆(k))−→H¹(Gal_k,G_Q,d(k)) sending [M2] to βM₁,M₂. We claim this map is bijective. To prove surjec- tivity, suppose we have a 1-cocycleγ(τ) =gτ(g⁻¹)∈∆(k) that splits over GQ,d(k); thenτ(g⁻¹·M1) =g⁻¹·M1, since ∆(k) acts trivially onM1, so the cocycleβ defined using M₁ and M₂ :=g⁻¹·M₁ as above is equal to γ. To prove that the above map is injective, suppose that the ∆(k)-valued 1-cocycleβassociated toM₁andM₂:=g⁻¹·M₁splits over ∆(k) (i.e. that there existsa∈∆(k) such thatgτ(g⁻¹) =aτ(a⁻¹) for allτ∈Galk). Then, on the one hand,a⁻¹g∈GQ,d(k)^Gal^k, asτ(a⁻¹g) =a⁻¹g for allτ ∈Galk, and, on the other hand,

(a⁻¹g)⁻¹·M₁=g⁻¹·(a⁻¹·M₁) =g⁻¹·M₁=M₂, as ∆(k) acts trivially on Rep_Q,d(k). Therefore,

GQ,d(k)·M1=GQ,d(k)·M2. In order to study the image of the natural map

fGal_k:F_Q,d^θ−gs(k)−→ M^θ−gs_Q,d (k)

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we introduce a map T called the type map, from M^θ−gs_Q,d (k)^Gal^k to the Brauer group ofk, denoted by Br(k):

(3.6) T :M^θ−gs_Q,d (k)−→H²(Galk;k^×)∼= Br(k), which is defined as follows. Consider an orbit

(G_Q,d(k)·M)∈ M^θ−gs_Q,d (k) = (Rep^θ−gs_Q,d (k)/G_Q,d(k))^Gal^k,

of which a representative M has been chosen. As this orbit is preserved by the Galk-action, we have that, for all τ ∈ Galk, there is an element uτ ∈GQ,d(k) such that uτ·τ(M) =M. Note that for τ = 1Gal_k, we can simply takeu_τ = 1_G

Q,d(k), which we will. Since (τ₁τ₂)(M) =τ₁(τ₂(M)), it follows from the compatibility relation (3.3) that,

u⁻¹_τ₁_τ₂·M =τ1(u⁻¹_τ₂ ·M) =τ1(u⁻¹_τ₂ )·τ1(M) =τ(u⁻¹_τ₂ )u⁻¹_τ₁ ·M.

Therefore, for all (τ1, τ2) ∈ Galk×Galk, the element cu(τ1, τ2) :=

u_τ₁τ₁(u_τ₂)u⁻¹_τ

1τ₂ (which depends on the choice of the representativeM and the familyu:= (uτ)τ∈Gal_k satisfying, for allτ∈Galk,uτ·τ(M) =M) lies in the stabiliser ofM inGQ,d(k), which is ∆(k) sinceM is assumed to be χθ-stable.

Proposition 3.4. — The above map

c_u: Gal_k×Gal_k−→∆(k), (τ₁, τ₂)7−→u_τ₁τ₁(u_τ₂)u⁻¹_τ

1τ₂

is a normalised∆(k)-valued2-cocycle whose cohomology class only depends on theG_Q,d(k)-orbit ofM, thus this defines a map

T :M^θ−gs_Q,d (k)^Gal^k −→H²(Galk; ∆(k))'Br(k) that we shall call the type map.

Proof. — It is straightforward to check the cocycle relation c(τ1, τ2)c(τ1τ2, τ3) =τ1(c(τ2, τ3))c(τ1, τ2τ3)

for all τ₁, τ₂, τ₃ in Gal_k. If we choose a different family u⁰ := (u⁰_τ)_τ∈Gal_k such thatu⁰_τ·τ(M) =M for allτ∈Galk, then (u⁰_τ)⁻¹·M =uτ·M, thus a_τ :=u⁰_τu⁻¹_τ ∈∆(k) and it is straightforward to check, using that ∆(k) is a central subgroup ofGQ,d(k), that

u⁰_τ₁τ1(u⁰_τ₂) u⁰_τ₁_τ₂−1

= aτ₁τ1(aτ₂)a⁻¹_τ₁_τ₂

uτ₁τ1(uτ₂)u⁻¹_τ₁_τ₂ .

Therefore, the associated cocyclesc_uandc_u⁰ are cohomologous. If we now replaceM withM⁰=g·M forg∈GQ,d(k), then

τ(M⁰) =τ(g)·τ(M) =τ(g)u⁻¹_τ g⁻¹·M⁰