On the Voevodsky motive of the moduli stack of vector bundles on a curve

We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localising tensor subcategory generated by the motive of the curve, using Bialynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.


Introduction
Let C be a smooth projective geometrically connected curve of genus g over a field k. We denote the moduli stack of vector bundles of rank n and degree d on C by Bun n,d ; this is a smooth algebraic stack of dimension n 2 (g − 1). The cohomology of Bun n,d has been studied using a wide array of different techniques, and together with Harder-Narasimhan stratifications, these results are used to study the cohomology of moduli spaces of semistable vector bundles. In this paper, we study the motive of Bun n,d in the sense of Voevodsky.
Let us start with a chronological survey of the various results on the cohomology of Bun n,d .
To describe the motives of the varieties Div n,d (lD 0 ), we use a G m -action and a Bia lynicki-Birula decomposition [7] as in [8]. The varieties Div n,d (lD 0 ) come with natural actions of GL n , and the morphisms Div n,d (lD 0 ) → Div n,d ((l + 1)D 0 ) are equivariant with respect to the action. If we restrict the action to a generic one-parameter subgroup G m ⊂ GL n , then the connected components of the fixed point locus can be identified with products of symmetric powers of C. We can then apply the Bia lynicki-Birula decomposition (cf. Theorem 3.2) and its motivic counterpart [10,12,27] (cf. Theorem 3.4) to deduce the following result. Theorem 1.2. Assume that R is a Q-algebra; then M (Bun n,d ) lies in the localising tensor triangulated category of DM(k, R) generated by M (C). Hence, M (Bun n,d ) is an abelian motive.
The assumption that R is a Q-algebra is used to show that the motive of a symmetric power of C is a direct factor of the motive of a power of C. For C = P 1 , as symmetric products of P 1 are projective spaces, we deduce that M (Bun n,d ) is a Tate motive for any coefficient ring R.
To prove this formula, one needs to understand the behaviour of the transition maps in the inductive system given in Theorem 1.1 with respect to the motivic Bia lynicki-Birula decompositions. We formulate a conjecture (cf. Conjecture 4.11) on the behaviour of these transitions maps with respect to these decompositions; this is equivalent to a conjecture concerning the intersection theory of the the smooth projective Quot schemes Div n,d (D) (cf. Conjecture 4. 12 and Remark 4.13). In Theorem 4.20, we prove that Conjecture 4.11 implies Conjecture 1.3; hence, it suffices to solve the conjecture on the intersection theory of these Quot schemes, which is ongoing work of the authors.
The assumption that C has a rational point is needed so that Abel-Jacobi maps from sufficiently large symmetric powers of C to Jac(C) are projective bundles (cf. Remark 4.15). In fact, these projective spaces then contribute to the motive of BG m (cf. Example 2.20).
Finally let us state some evidence to support Conjecture 1.3, as well as some consequences of this conjecture. First, using Poincaré duality for smooth stacks (cf. Proposition 2.34), we can deduce a formula for the compactly supported motive of Bun n,d (cf. Theorem 5.1), which is better suited to comparisons with the results concerning the topology of Bun n,d mentioned above. In §5.2, we explain how this conjectural formula for M c (Bun n,d ) is compatible with the Behrend-Dhillon formula [6] by using a category of completed motives inspired by work of Zargar [45] (cf. Lemma 5.4). In fact, in [6], it is also implicitly assumed that C has a rational point in order to use the same argument involving Abel-Jacobi maps. In §5.3, we deduce from Conjecture 4.11 formulae for the motive (and compactly supported motive) of the substack Bun L n,d of vector bundles with fixed determinant L and the stack Bun SLn of principal SL n -bundles over C.
The structure of this paper is as follows: in §2, we summarise the key properties of motives of schemes and, after proving some elementary results about homotopy (co)limits, we define motives of smooth exhaustive stacks and prove some analogous properties. In §3, we explain the geometric and motivic Bia lynicki-Birula decompositions. Then in §4, we prove Theorems 1.1 and 1.2, state Conjecture 1.3, and prove that this conjecture follows from a conjecture concerning the intersection theory of the smooth projective Quot schemes Div n,d (D) (cf. Theorem 4. 20). In §5, we deduce from Conjecture 1.3 a formula for the compactly supported motive of Bun, which we compare with previous formulae in §5.2 and we deduce formulae for the motives of Bun L n,d and Bun SLn from Conjecture 4.11 (cf. Theorems 5.6 and 5.7). Finally, in Appendix A, we explain and compare alternative approaches for definingétale motives of stacks.
1.1. Notation and conventions. Throughout we fix a field k and all schemes and stacks are assumed to be defined over k. By an algebraic stack, we mean a stack X for the fppf topology with an atlas given by a representable, smooth surjective morphism from a scheme that is locally of finite type. 1 For a closed substack Y of an algebraic stack X, we define the codimension of Y in X to be the codimension of Y × X U in U for an atlas U → X; this is independent of the choice of atlas.
For n ∈ N and a quasi-projective variety X, the symmetric group Σ n acts on X ×n and the quotient is representable by a quasi-projective variety X (n) , the n-th symmetric power of X.

Motives of schemes and stacks
Let k be a base field and R be a commutative ring of coefficients. If the characteristic p of k is positive, then we assume either that p is invertible in R or that k is perfect and admits the resolution of singularities by alterations. We let DM(k, R) := DM Nis (k, R) denote Voevodsky's category of (Nisnevich) motives over k with coefficients in R; this is a monoidal triangulated category. For a separated scheme X of finite type over k, we can associate both a motive M (X) ∈ DM(k, R), which is covariantly functorial in X and behaves like a homology theory, and a motive with compact supports M c (X) ∈ DM(k, R), which is covariantly functorial for proper morphisms and behaves like a Borel-Moore homology theory.
Without going into the details of the construction, we recall that objects in DM(k, R) can be represented by (symmetric) T -spectra in complexes of Nisnevich sheaves with transfers (i.e., with additional contravariant functoriality for finite correspondences) of R-modules on the category of smooth k-schemes, where Here R eff tr (X) denotes the sheaf of finite correspondences into X with R-coefficients for X ∈ Sm k . We write R tr (X) for the suspension spectrum Σ ∞ T R eff tr (X).
Remark 2.1. The main results of this paper hold in the category DM eff (k, R) of effective motives. We refrained from writing everything in terms of DM eff (k, R) for two reasons.
• Under our assumptions on k and R, the functor DM eff (k, R) → DM(k, R) is fully faithful [43] [38], so that results in DM eff (k, R) follow immediately from their stable counterparts. • The motive with compact support M c (X) of an Artin stack, however it is defined, is almost never effective (see Section §2.7).

2.1.
Properties of motives of schemes. The category DM(k, R) was originally constructed in [42] and its deeper properties were established under the hypothesis that k is perfect and satisfies resolution of singularities (with no assumption on R). They were extended to the case where k is perfect by Kelly in [28], using Gabber's refinement of de Jong's results on alterations. Finally, the extension of scalars of DM from a field to its perfect closure was shown to be an equivalence in [14,Proposition 8.1.(d)].
The motive M (Spec k) := R tr (Spec k) of the point is the unit for the monoidal structure, and there are Tate motives R(n) ∈ DM(k, R) for all n ∈ Z. For any motive M and n ∈ Z, we write M (n) := M ⊗ R(n), and we write M {n} := M (n)[2n].
Let us list the main properties of motives that will be used in this paper.
• (Künneth formula): for schemes X and Y , we have • (A 1 -homotopy invariance): by construction of DM(k, R), for any Zariski-locally trivial affine bundle Y → X with fibre A r , the induced morphisms are isomorphisms. • (Projective bundle formula): for a vector bundle E → X of rank r + 1, there are isomorphisms • (Gysin triangles): for a closed immersion i : Z → X of codimension c between smooth k-schemes, there is a functorial distinguished triangle • (Localisation triangles): for a separated scheme X of finite type and Z any closed subscheme, there is a functorial distinguished triangle • (Motives with and without compact supports): for a separated finite type scheme X, there is a morphism M c (X) → M (X), which is an isomorphism if X is proper. • (Internal homs and duals): the category DM(k, R) has internal homomorphisms, which can be used to define the dual of any motive M ∈ DM(k, R) as M ∨ := Hom(M, R(0)).
• (Poincaré duality): for a smooth scheme X of pure dimension d, there is an isomorphism • (Algebraic cycles): for a smooth scheme X (say of pure dimension d for simplicity), a separated scheme Y of finite type and i ∈ N, there is an isomorphism where CH i denotes the Chow groups of cycles of dimension i. • (Compact generators): The triangulated category DM(k, R) is compactly generated, with a compact family of generators given by motives of the form M (X)(−n) for X smooth and n ∈ N. We denote by DM gm (k, R) the triangulated subcategory consisting of compact objects. For any separated finite type scheme X, both M (X) and M c (X) are compact. We recall that an object M ∈ DM(k, R) is compact if and only if for all families (N α ) α∈A ∈ DM(k, R) A indexed by a set A, the natural map α∈A Hom(M, N α ) → Hom(M, α∈A N α ) is an isomorphism.
Remark 2.2. If we instead work with the category ofétale motives DMé t (k, R), this would lead to weaker results (as this category does not capture the information about integral and torsion Chow groups), but defining motives of stacks is technically simpler as we explain in Appendix A.
2.2. Homotopy (co)limits. As DM(k, R) is a compactly generated triangulated category, it admits arbitrary direct sums and arbitrary direct products [33,Proposition 8.4.6]; hence, one can define arbitrary homotopy colimits and homotopy limits for N-indexed systems in DM(k, R) using only the triangulated structure together with direct sums and products as follows.
Definition 2.3. The homotopy colimit of an inductive system F * : where we write σ for any of the maps Note that, by construction, for any given choice of such a cone, there is a compatible system of maps, i.e. an element of lim i Hom(F i , hocolim F * ) (resp. lim j Hom(holim G * , G j )).
Remark 2.4. Using [33, Lemma 1.7.1], it is easy to extend this definition to homotopy colimits indexed by filtered partially ordered sets I such that there exists a cofinal embedding N → I.
The reader unfamiliar with this definition should compare it with the definition of the limit and its derived functor R 1 lim for N-indexed diagrams of abelian groups in [44, Definition 3.5.1]. Since homotopy (co)limits are defined by the choice of a cone, they are only unique up to the a non-unique isomorphism. However, in the case where the N-indexed system actually comes from the underlying model category (that is, it can be realised as a system of T -spectra of complexes of Nisnevich sheaves with transfers and morphisms between them), then homotopy colimits can be realised in a simple, canonical way as follows. Proof. We have a short exact sequence where the injectivity of the first map follows from the fact that the abelian category of Tspectra of complexes of Nisnevich sheaves with transfers is a Grothendieck abelian category, so in particular has exact filtered colimits. This exact sequence provides a distinguished triangle in the derived category, and so also in DM(k, R), which exhibits the colimit as a cone of the map id − σ, and thus as a homotopy colimit.
Nevertheless homotopy colimits are functorial in the following relatively weak sense. Lemma 2.6. For two N-indexed inductive (resp. projective) systems F * , F * (resp. G * , G * ) in DM(k, R), fix a choice of homotopy (co)limits F, F (resp. G, G); that is, a specific choice of cones of the morphisms in Definition 2.3. Then, modulo these choices, there is a uniquely determined short exact sequence Hom(F n , F ) → 0 (resp. 0 → R 1 lim n Hom(G, G n [−1]) → Hom(G, G) → lim n Hom(G, G n ) → 0). In particular, for a morphism f * : F * → F * of inductive systems, we can choose a morphism f : F → F such that for any n ∈ N the diagram commutes, and f is uniquely determined up to the R 1 lim term appearing in the above exact sequence (and there is a similar statement for homotopy limits). By abuse of notation, we sometimes denote such a morphism by M (f * ).
Proof. This follows directly from the definition of the functor R 1 lim.
The compatibility between the weak functoriality and the triangulated structure is as follows.
. For any choice homotopy colimits of those systems and compatible morphisms between them as in Lemma 2.6, the triangle The analogous statement holds for homotopy limits.
Proof. This follows directly from the fact that direct sums (resp. direct products) of distinguished triangles are distinguished and the nine lemma.
Let us state some results about simple homotopy (co)limits.
Proof. This follows from the definition of the homotopy colimit, as the tensor product commutes with direct sums in a tensor triangulated category.
Lemma 2.9. Let I be a set and let F i * : N → DM(k, R) be an inductive system for all i ∈ I. Then there is an isomorphism hocolim n i∈I Proof. This follows from the definition of the homotopy colimit, as a direct sum of distinguished triangles is distinguished.

2.3.
Vanishing results for homotopy (co)limits. In order to compute homotopy (co)limits, we will frequently rely on various vanishing results which we collect together in this section.
where DM(k, R) m denotes the smallest localising subcategory of DM(k, R) containing M c (X)(n) for all separated schemes X of finite type over k and all integers n with dim(X) + n ≤ m.
Note that analoguous filtrations appear in the literature dealing with classes of stacks in the Grothendieck ring of varieties (for example, see [6]). Totaro proves the following result. Moreover, for a projective system G * : N op → DM(k, R) and a sequence of integers a n → −∞ such that G n ∈ DM(k, R) an for all n, it follows that holim n G n ≃ 0. Corollary 2.12. Let (M n ) n∈N ∈ DM(k, R) N be a family of motives and (a n ) n∈N be a sequence of integers such that a n → −∞ and M n ∈ DM(k, R) an for all n. Then the natural morphism n∈N M n → n∈N M n is an isomorphism.
Proof. The cone of this morphism lies in DM(k, R) an for all n ∈ N, thus is zero by Proposition 2.11 (for further details, see the proof of [40,Lemma 8.7]).
We will also need a vanishing result for homotopy colimits. However, the dual result to Propositon 2.11 does not hold in DM(k, R): if k has infinite transcendance dimension, there is an inductive system ( . . . R(n)[n] → R(n + 1)[n + 1] . . . ) whose homotopy colimit is non-zero (cf. [4, Lemma 2.4]). Hence, the intersection ∩ n≥0 DM eff (k, R)(n) is non-zero. Fortunately, with some control over the Tate twists and shifts, we can prove the following vanishing result.
Proposition 2.13. Let U * ⊂ X * be an inductive system of open immersions of smooth finite type k-schemes; that is, we have inductive systems U * , X * : N → Sm k and a morphism U * → X * such that U n ֒→ X n is an open immersion for all n ∈ N. Let c n be the codimension of the complement of X n − U n in X n . If c n → ∞, then the morphism is an A 1 -weak equivalence.
Proof. Let C n be the level-wise mapping cone (as a T -spectrum of complexes of sheaves with transfers) of the morphism R tr (U n ) → R tr (X n ). There are induced maps C n → C n+1 and we get a distinguished triangle in DM(k, R). Thus it suffices to show that colim n C n is A 1 -equivalent to 0.
Let W n = X n − U n , and consider a finite locally closed stratification (W i n ) mn i=1 with each W i n smooth over k (which exists because W n is geometrically reduced and of finite type over k). For each 1 ≤ j ≤ m n , we have a Gysin distinguished triangle } By inductively applying the octahedral axiom to these distinguished triangles, we conclude that C n is a successive extension of the motives M (W i n ){codim X (W i n )}. Since the category DM(k, R) is compactly generated by motives of the form M (X)(a), with X ∈ Sm k and a ∈ Z, it suffices to show for all X ∈ Sm k and a, b ∈ Z that As c n → ∞, we have c n − b + a − dim(X) > 0 for n >> 0. Thus, for n >> 0 and c ≥ c n , is strictly positive. Then we deduce that (1) holds by using Lemma 2.14 below.
Lemma 2.14. Let X (resp. Z) be a variety of dimension at most d (resp. e). For k, l ∈ Z with l > k + d + e, we have Proof. By our standing assumption on k and R and [14, Proposition 8.1.(d)], we can assume that k is a perfect field. Let us first prove the claim when Z is proper.
where R(k) cdh is the cdh-sheaffification of the Suslin-Voevodsky motivic complex [28]. The cohomological dimension of (Z × k X) cdh is at most d + e by [37,Theorem 5.13], and the complex R(k) cdh is 0 in cohomological degrees greater than k, which implies the result.
We now turn to the general case. LetZ be any compactification of Z (not necessarily smooth). From the localisation triangle for the closed pair (Z − Z,Z), we obtain a long exact sequence As bothZ andZ − Z are proper with dimensions bounded by e and e − 1 respectively, we deduce that the outer terms vanish by the proper case, and so we obtain the desired result.

2.4.
Motives of stacks. The definition of motives of stacks in general is complicated by the fact that the category DM(k, R) does not satisfy descent for theétale topology (as this is already not the case for Chow groups); hence naive approaches to defining motives of even Deligne-Mumford stacks in terms of an atlas do not work. In Appendix A, we explain and compare alternative approaches for definingétale motives of stacks, provide references to the literature, and define motives with compact supports.
To define the motive of a quotient stack X = [X/G] independently of the presentation of X as a quotient stack, we need an appropriate notion of "algebraic approximation of the Borel construction X × G EG". We use a variant of the definition of compactly supported motives of quotient stacks given by [40] and extend this to more general stacks. More precisely, we will define the motive of certain smooth stacks over k, possibly not of finite type, which are exhaustive in the following sense.
. . ⊂ be a filtration of an algebraic stack X by increasing open substacks X i ⊂ X which are quasi-compact and cover X; we will simply refer to this as a filtration. Then an exhaustive sequence of vector bundles on X with respect to this filtration is a pair (V • , W • ) given by a sequence of vector bundles V m over X m together with injective maps of vector bundles f m : ). An exhaustive stack is a stack which admits such an exhaustive sequence with respect to some filtration.
For this paper, we have in mind two important examples of exhaustive stacks: i) quotient stacks (cf. Lemma 2.16), and ii) the stack of vector bundles on a curve (cf. Proposition 4.2). Lemma 2.16. Let X = [X/G] be a quotient stack of a quasi-projective scheme X by an affine algebraic group G such that X admits a G-equivariant ample line bundle; then X is exhaustive.
Proof. Let X = [X/G] be a quotient stack; then there is an exhaustive sequences of vector bundles over X with respect to the constant filtration built from a faithful G-representation G → GL(V ) such that G acts freely on an open subset U ⊂ V . More precisely, we let W := V − U ⊂ V and for m ≥ 1 consider the G-action on V m , which is free on the complement of W m . First, we construct an exhaustive sequence (B • , C • ) of vector bundles over BG (with respect to the constant filtration) by taking B m = V m with the natural transition maps and C m = W m . We then form the exhaustive sequence on [X/G]; this works as the open complement (X × (B m − C m ))/G is a quasi-projective scheme by [32, Proposition 7.1] due to the existence of a G-equivariant ample line bundle on X.
Definition 2.17. Let X be a smooth exhaustive stack; then for an exhaustive sequence (V • , W • ) of vector bundles with respect to a filtration X = ∪ m X m , we define the motive of X as Before proving this definition is independent of the choice of filtration and exhaustive sequence, we first note that if X = X is a scheme, then this definition coincides with the usual definition of the motive of X. Indeed X is an exhaustive stack, as we can take the constant filtration and let V m = U m = X.
Lemma 2.19. The motive of a smooth exhaustive stack X does not depend (up to a canonical isomorphism) on the choice of the filtration or the exhaustive sequence of vector bundles.
Proof. We first fix a filtration of X = ∪ m X m and prove the resulting object does not depend on the exhaustive sequence. Let (V • , W • ) and (V ′ • , W ′ • ) be two exhaustive sequence of vector bundles on X. As in the proof of [40,Theorem 8 , which are injective vector bundle homomorphisms. Let us prove that this sequence satisfies properties (ii) and (iii) in Definition 2.15 (in fact, it satisfies property (i), but this is not needed below). For (ii), we note that U ′′ . . as in Definition 2.17 and to complete the proof, it suffices by symmetry to show that (thus a smooth scheme) and that the codimension of the complement satisfies In particular, these codimensions tend to infinity with m. Moreover, U m × Xm V ′ m → U m is a vector bundle. Hence we have the following two morphisms of inductive systems of T -spectra A filtered colimit of A 1 -weak equivalences is an A 1 -weak equivalence, so that the bottom morphism of systems induces an A 1 -weak equivalence on colimits. It is thus enough to show that the top morphism of systems also induces an A 1 -weak equivalence; by (3), this follows from Proposition 2.13. Now let us prove that the definition is independent of the filtration. Let (X n ) n≥0 and (X ′ n ) n≥0 be two such filtrations of X. Note that by quasi-compactness of the stacks X n and the fact that X = ∪ n X ′ n , for n ∈ N, there exists n ′ ∈ N such that X n ⊂ X ′ n ′ . Hence, we can find strictly increasing sequences ( and so on. Since colimits are stable under passing to a cofinal sequence, we obtain the result. R{j}. Proof. Part (i) holds, as filtered colimits in DM(k, R) are homotopy colimits and N is compact. For (ii), by Lemma 2.6, it suffices to show that R 1 lim Hom(M (U n ) [1], N ) = 0. We actually show that the map Hom(M (U n+1 ) [1], N ) → Hom(M (U n ) [1], N ) is an isomorphism for n large enough, which implies that the corresponding R 1 lim term vanishes.
Since By definition, the transition maps in the system Hom(M (V n ) [1], N ) are induced by the maps Both V n and i * n V n+1 are vector bundles over the same stack X n , and V n → i * n V n+1 is a map of vector bundles, which tends to infinity with n by assumption. To complete the proof, we use the same argument as in the paragraph above to deduce that Hom(M (V n+1 ) [1], N ) ≃ Hom(M (V n ) [1], N ) for n large enough.
This shows, in particular, that the following definition is not unreasonable (since it can be computed via any given exhaustive sequence).
Definition 2.22. Let X be a smooth exhaustive stack, and p, q ∈ Z. The motivic cohomology of X is defined as It is not immediately clear that the definition of the motive of an exhaustive stack is functorial, but it is relatively simple to prove a weak form of functoriality for certain representable morphisms. To have more systematic forms of functoriality it is better to work with other definitions of motives of stacks as explained in Appendix A.
Lemma 2.23. Let g : X → Y be a flat finite type representable morphism of smooth algebraic stacks such that Y is exhaustive, with an exhaustive sequence of vector bundles Proof. Since g is flat, we have that codim g * Vn (g −1 (W n )) ≥ codim Vn (W n ) which tends to infinity with n. By the fact that g is representable and of finite type, is a separated finite type k-scheme. We leave the verification of Property (iii) in Definition 2.15 to the reader. The morphism M (g) is then defined by taking colimits in the morphism of systems of T -spectra R tr (g −1 (U n )) → R tr (U n ).
Remark 2.24. The flatness condition was imposed only in order to prove the codimension condition on (g * V • , g −1 (W • )). There are many other cases in which the definition of M (f ) above works; for instance, for any finite type representable morphism between quotient stacks.
One can also prove Künneth isomorphisms and A 1 -homotopy invariance.
Proposition 2.25. Let X and Y be smooth exhaustive smooth stacks over k.
). Then we leave the reader to easily verify that on X × k Y there is an exhaustive sequence compatible with the filtration where we have used the commutation of tensor products of T -spectra with colimits and the fact that the diagonal N → N×N is cofinal. The proof of part (ii) follows by a similar argument using the pullback of any exhaustive sequence from X to E and homotopy invariance for schemes.
, and one can check that this morphism is independent of the presentation.

Chern classes of vector bundles on stacks.
For a vector bundle E over a smooth k-scheme X, one has motivic incarnations of Chern classes given by a morphism Let us extend this notion to vector bundles on smooth exhaustive stacks.
Definition 2.27. Let E → X be a vector bundle on a smooth exhaustive stack and let (V • , W • ) be an exhaustive system of vector bundles on X with respect to a filtration X = n X n ; then the pullback E n of E to the smooth scheme U n : Proof. We omit the details as the proof is very similar to that of Lemma 2.19.
We will need some basic functoriality results for these Chern classes.
Lemma 2.29. Let g : X → Y be a flat finite type representable morphism of smooth algebraic stacks such that Y is exhaustive (then X is exhaustive by Lemma 2.23). For a vector bundle E → Y and j ∈ N, we have a commutative diagram Proof. This follows from the proof of Lemma 2.23 and the functoriality of Chern classes for vector bundles on smooth schemes.

2.6.
Motives of G m -torsor. In this section, we prove a result for the motive of a G m -torsor, which will be used to compute the motive of the stack of principal SL n -bundles. We recall that a G m -torsor over a stack X is a morphism X → BG m , or equivalently a cartesian square Proof. Let us first verify the statement for schemes. Let X be a scheme and Y → X be a G m -torsor. If we let L = Y × Gm A 1 denote the associated line bundle; then Y = L − X. By . If X is smooth, then so is L and we can consider the Gysin triangle associated to the closed immersion X ֒→ L as the zero section: Then the map M (X) → M (X){1} is given by the first Chern class of L by [16, Example 1.25]. Now suppose we are working with smooth exhaustive stacks. We note that the morphism M (∆) is defined in Remark 2.26. Let (V • , W • ) be an exhaustive sequence on X with respect to a filtration X = ∪ n X n and let U n := V n − W n as usual. Then we let L n denote the pullback of the line bundle L → X to U n and let Y n ⊂ L n denote the complement to the zero section. Since U n are schemes, we have for all n, distinguished triangles Since the fibre product U n ∼ = L n × L n+1 U n+1 is transverse, the Gysin morphisms for (U n , L n ) and (U n+1 , L n+1 ) are compatible, and so by taking the homotopy colimit of the distinguished triangles ( R{n}. Using the notation of the proof above, the line bundle L n on U n = P n−1 is the tautological line bundle O P n (−1), whose first Chern class c 1 (L n ) : M (P n−1 ) ≃ ⊕ n−1 j=0 R{j} → R{1} is just the projection onto this direct factor. It follows from this that the morphism R{n} is the natural projection.

2.7.
Compactly supported motives and Poincaré duality. One can define the compactly supported motive of an exhaustive stack as follows.
where the first map is the flat pullback for the open immersion U m+1 × X m+1 X m → U m and the second map is the localisation map for the immersion f m : Remark 2.33. One can prove that the definition of the compactly supported motive of an exhaustive stack is independent of the choice of filtration and exhaustive sequence similarly to Lemma 2.19 (see also [40,Theorem 8.5]). Furthermore, unlike in the motive case, we do not have to assume that X is smooth (as the argument uses Proposition 2.11 instead of Lemma 2.14).
We can show one part of the statement of Poincaré duality for exhaustive smooth stacks follows from Poincaré duality for schemes. Proof. This follows directly from the definitions of M (X) and M c (X) and Poincaré duality for schemes, as the dual of a homotopy colimit is a homotopy limit.
Note that, because the dual of an infinite product is not in general an infinite sum, it is not clear in general that the duality works the other way.

Motivic Bia lynicki-Birula decompositions
3.1. Geometric Bia lynicki-Birula decompositions. Let X be a smooth projective k-variety equipped with a G m -action. By a result of Bia lynicki-Birula [7], there exists a decomposition of X, indexed by the connected components of the fixed locus X Gm , with very good geometric properties. In fact, this decomposition exists in the following slightly more general context.
• X Gm is proper (and thus projective), and • for every point x ∈ X (not necessarily closed), the action map f x : G m → X given by t → t · x extends to a mapf x : A 1 → X. Since X is separated, the extension is unique and we write lim t→0 t · x for the limit pointf x (0) ∈ X.
In particular, any G m -action on a smooth projective variety is semi-projective. Note that the limit point lim t→0 t · x is necessarily a fixed point of the G m -action if it exists. Theorem 3.2 (Bia lynicki-Birula). Let X be a smooth quasi-projective variety over k with a semi-projective G m -action. Then the following statements hold.
(i) The fixed locus X Gm is smooth and projective. Write {X i } i∈I for its set of connected components and d i for the dimension of X i . (ii) For i ∈ I, write X + i for the attracting set of X i , i.e., the set of all points x ∈ X such that lim t→0 t · x ∈ X i . Then X + i is a locally closed subset of X and X = i∈I X + i . (iii) For every i ∈ I, the map of sets X + i → X i given by x → lim t→0 t·x underlies a morphism of schemes p + i : X + i → X i , which is a Zariski locally trivial fibration in affine spaces. For each i ∈ I, we have (v) Let n := |I|; then there is a bijection ϕ : {1, . . . , n} → I and a filtration of X by closed subschemes is a single attracting set (and thus, in particular, is smooth).
Proof. Points (i) -(iv) are all established in [7, Theorem 4.1] under the assumption that k is algebraically closed (the hypothesis that X is smooth and quasi-projective is used to ensure the existence of an open covering by G m -invariant affine subsets). The hypothesis that k is algebraically closed was then removed by Hesselink in [23].
As the proof of (v) is scattered through [21, §1], we recapitulate their argument. Let L be a very ample line bundle on the quasi-projective variety X. By [36, Theorem 1.6] applied to the smooth (hence normal) variety X, there exists an integer n ≥ 1 such that L ⊗n admits a G m -linearisation. In particular, this provides a projective space P with a linear G m -action and a G m -equivariant immersion ι : X → P. Let {P j } j∈J be the connected components of P Gm with corresponding attracting sets P + j for each j ∈ J; then by equivariance of ι, there is a (not necessarily injective) map τ : I → J such that ι(X + i ) ⊂ P + τ (i) for all i ∈ I. As each X i is connected, the group G m acts on L |X i via a character ω i ∈ Hom(G m , G m ) ≃ Z. For the partial order on I given by i < i ′ ⇔ ω i > ω i ′ , we claim that for i = i ′ ∈ I Indeed, we can similarly define a partial order on J such that i < i ′ if and only if τ (i) < τ (i ′ ) by equivariance of ι; then one can easily deduce that (6) holds for P from the linearity of the G m -action on P. We now deduce (6) for X from the corresponding ambient property for P; the only non-trivial case to consider is when i = i ′ have the same image j under τ , so that X + i and X + i ′ are both contained in P + j . In this case, if x ∈ X + i ∩ X + i ′ , then by passing to an algebraic closure of k if necessary, we can assume that there is a connected curve S ⊂ X with x ∈ S and S − x ⊂ X + i ; then S ⊂ P + j and as the action on X is semi-projective, p + j (S) ⊂ X Gm and this connects X i and X i ′ , contradicting i = i ′ .
Finally to prove the filterability of X, we choose any total ordering of I extending the above partial order and we view this ordering as a bijection ϕ : {1, . . . , n} → I. Then for 0 ≤ k ≤ n, (6) with Z n = ∅ and Z 0 = X. Remark 3.3. Let X be smooth projective with a fixed G m -action (t, x) → t · x. The opposite G m -action (t, x) → t −1 ·x has the same fixed point locus as the original action, but the associated Bia lynicki-Birula decomposition is different. We write X − i for the associated strata, c − i (resp. r − i ) for their codimension (resp. their rank as affine bundles), etc. By Theorem 3.2 (iv), we see that c − i = r + i and r − i = c + i , and that the strata X + i and X − i intersect transversally along X i . 3.2. Motivic consequences. Let X be a smooth quasi-projective variety with a semi-projective G m -action. The geometry exhibited in the previous sections implies a decomposition of the motive of X. There are in fact two natural such decompositions, one for the motive M (X) and one for the motive with compact support M c (X). These motivic decompositions have been studied in [10,12,27]; we explain and expand upon their results in this section. Recall that for two smooth k-schemes X and Y with X of dimension d and an integer i ∈ N, there is an isomorphism with CH i the Chow groups of cycles of dimension i; when this does not lead to confusion, we use the same notation for a cycle and the corresponding map of motives.
Theorem 3.4. Let X be a smooth quasi-projective variety with a semi-projective G m -action. With the notation of Theorem 3.2, for each i ∈ I, we let γ + i be the class of the algebraic cycle given by the closure Γ p + i of the graph of p + i : X + i → X i in X × X i , and we let (γ + i ) t be the class of the transposition of this graph closure. Then we have the following motivic decompositions (where we use without comment that M (X i ) ≃ M c (X i ) as X i is projective).
(i) (Bia lynicki-Birula decomposition for the motive): There is an isomorphism and is a twist of the Poincaré duality isomorphism for the motive of the smooth projective variety X i if i = j (noting the equality d − c + i − r + i = d i ). Proof. We first prove (i). By Theorem 3.2, there is a filtration ∅ = Z n ⊂ Z n−1 ⊂ . . . ⊂ Z 0 = X by closed subvarieties such that, for all 1 ≤ k ≤ n, we have that Z k−1 − Z k = X + k is an attracting cell. Let U k := X − Z k , which is an open subset and so in particular is smooth. For 1 ≤ i ≤ k ≤ n, let us write γ + i,k for the closure of Γ p i in U k × X i (this makes sense since X + i ⊂ U i ⊂ U k ) so that γ + i = γ + i,n . We will prove, by induction on 1 ≤ k ≤ n, that the map is an isomorphism. For k = 1, the statement holds trivially as U 0 = ∅ and so M (U 1 ) = M (X + 1 ) ≃ M (X 1 ) via p + 1 . Assume that the statement is true for k − 1. We have a closed immersion i k : X + k → U k between smooth schemes with codimension c + k and open complement U k−1 ; hence, there is a Gysin triangle where the left vertical map is an isomorphism by induction. This shows that the triangle splits. As p + k : X + k → X k is a Zariski locally trivial fibration of affine spaces, M (p + k ) is an isomorphism. It remains to show that the composition M ( Let us write γ • k,k for the graph of p + k considered as a subscheme of X + k × X k . Let us recall the functoriality of the Poincaré duality isomorphism with respect to algebraic cycles. Let Y 1 , Y 2 be smooth projective varieties of dimensions d 1 and d 2 , and γ ∈ CH c (Y 1 × Y 2 ), which induces morphisms γ : From this commutativity, it suffices to show that γ • k,k • pr * 1 Gy(i k ) : M (U k × X k ) → R(c + k + d k ) coincides with the map γ + k,k : M (U k × X k )→R(c + k + d k ). Let us denote by a k : γ + k,k → X + × k X k and b k : γ + k,k → U k × X k the closed immersions, so that b k = (i k × X k ) • a k . Consider the diagram in DM(k, R). The left triangle commutes because of the general behaviour of Gysin maps with respect to composition [16, Theorem 1.34]. The outer square and the bottom quadrilateral commute because of the compatibility of Gysin maps with fundamental classes of cycles of smooth subvarieties in motivic cohomology [15,Lemma 3.3]. This implies that the top triangle commutes. Since pr 1 is a smooth morphism, we have pr * 1 Gy(i k ) = Gy(i k × k X k ) by [16, Proposition 1.19 (1)] and the commutation of the top triangle is exactly the equality we want. This concludes the proof of (i).
Statements (ii) and (iii) are deduced from (i) by applying Poincaré duality and using the functoriality of the Poincaré duality isomorphism with respect to algebraic cycles recalled above in the proof of (i).
In the smooth projective case, one has M c (X) ≃ M (X) and one would like compare this decomposition with the decomposition obtained for the opposite G m -action. Question 3.5. Let X be a smooth projective variety with a G m -action. Then, via the isomorphism M c (X) ≃ M (X), do the motivic Bia lynicki-Birula decompositions of M (X) in Theorem 3.4 (i) and of M c (X) in Theorem 3.4 (ii) for the opposite G m -action coincide? In other words, for every (i, j) ∈ I 2 , is the composition zero if i = j and the identity if i = j (noting the equality r − i = c + i )?

The motive of the stack of vector bundles
Throughout this section, we let C be a smooth projective geometrically connected curve of genus g over a field k. We fix n ∈ N and d ∈ Z and let Bun n,d denote the stack of vector bundles over C of rank n and degree d; this is a smooth stack of dimension n 2 (g − 1). In this section, we give a formula for the motive of Bun n,d in DM(k, R), by adapting a method of Bifet, Ghione and Letizia [8] to study the cohomology of Bun n,d using matrix divisors. This argument was also used by Behrend and Dhillon [6] to give a formula for the virtual motivic class of Bun n,d in (a completion of) the Grothendieck ring of varieties (see §5.2).
We can define the motive of Bun n,d , as it is exhaustive: to explain this, we filter Bun n,d using the maximal slope of all vector subbundles. rk(E) . We define The maximal slope µ max (E) is equal to the slope of the first vector bundle appearing in the Harder-Narasimhan filtration of E. For µ ∈ Q, we let Bun ≤µ n,d denote the substack of Bun n,d consisting of vector bundles E with µ max (E) ≤ µ; this substack is open by upper semicontinuity of the Harder-Narasimhan type [34]. Any sequence (µ l ) l∈N of increasing rational numbers tending to infinity defines a filtration of (Bun ≤µ l n,d ) l∈N of Bun n,d . The stacks Bun ≤µ l n,d are all quasi-compact, as they are quotient stacks. Indeed, all vector bundles over C of rank n and degree d with maximal slope less than or equal to µ l form a bounded family (cf. [ compatible with the forgetful morphisms to Bun n,d , and so we can construct an ind-variety Div n,d := (Div n,d (D)) D of matrix divisors of rank n and degree d on C.
We will use the forgetful map Div n,d → Bun n,d to study the motive of Bun n,d in terms of that of Div n,d and, in particular, to define an exhaustive sequence of vector bundles on Bun n,d . where Div ≤µ l n,d (lD) is the open subvariety of Div n,d (lD) consisting of rank n degree d matrix Proof. For all vector bundles E with µ max (E) ≤ µ l , as µ max (E) < deg(lD) − 2g + 2, we have Indeed if this vector space was non-zero then by Serre duality, there would exist a non-zero homomorphism O C (lD) ⊕n ⊗ ω −1 C → E, but one can check that this is not possible by using the Harder-Narashiman filtration and standard results about homomorphisms between semistable bundles of prescribed slopes (see [26,Proposition 1.2.7] and also [8, §8.1]). Hence, there is a vector bundle V l := R 0 p 1 * (E ∨ univ ⊗ p * 2 O C (lD) ⊕n ) over Bun ≤µ l n,d , whose fibre over E is the space   and Φ(Hom ni (E, F (D))) ⊂ Hom <n (E, O ⊕n D ). Hence, (9) dim Hom ni (E, F (D)) ≤ dim Hom <n (E, O ⊕n D ) + dim Hom(E, F ). For a smooth irreducible variety X and a closed subset Y , recall that we have codim X (Y ) = dim(X) − dim(Y ). Therefore where the first inequality follows from (9), the second inequality follows from (7) and the final inequality follows from (8). This concludes the proof. We then obtain the left isomorphism by applying Proposition 2.13 to the inductive system of open immersions (Div ≤µ l n,d (lD) ֒→ Div n,d (lD)) l∈N . To apply this corollary, we need to check that the closed complements Div n,d (lD) − Div ≤µ l n,d (lD) have codimensions tending to infinity with l. Let E be a vector bundle with µ max (E) > µ l ; then µ max (E) ≥ µ l + 1 n 2 since slopes are rational numbers with denominator at most n. Thus µ max (E) + 2g − 1 ≥ µ l + 1 n 2 + 2g − 1 = deg(lD), and so by [8, Proposition 5.2 (4)], we have codim Div n,d (lD) (Div n,d (lD) − Div ≤µ l n,d (lD)) ≥ l deg D − c for a constant c independent of l, which completes the proof.
We recall that a motive is pure if it lies in the heart of Bondarko's Chow weight structure on DM(k, R) defined in [9]. In particular, the motive of any smooth projective variety is pure and, as M (Bun n,d ) is described as a homotopy colimit of motives of smooth projective varieties, we deduce the following result. This corollary sits well with the fact that the cohomology of Bun n,d , and more generally the cohomology of moduli stacks of principal bundles on curves, is known to be pure in various contexts; for instance, if k = C, the Hodge structure is pure by [39,Proposition 4.4], and over a finite field, the ℓ-adic cohomology is pure by [22,Corollary 3.3.2].

4.2.
The Bia lynicki-Birula decomposition for matrix divisors. We recall that the Quot scheme Div n,d (D) is a smooth projective variety of dimension n 2 deg D − nd. The group GL n acts on Div n,d (D) by automorphisms of O C (D) ⊕n . If we fix a generic 1-parameter subgroup G m ⊂ GL n of the diagonal maximal torus T = G n m , then the fixed points of this G m -action agree with the fixed points for the T -action. These actions and their fixed points were studied by Strømme [35]; the fixed points are matrix divisors of the form By specifying the degree m i of each F i we index the connected components of this torus fixed locus; more precisely, the components indexed by a partition m = (m 1 , . . . , m n ) of n deg D − d is the following product of symmetric powers of C C (m) := C (m 1 ) × · · · × C (mn) .
Strømme also studied the associated Bia lynicki-Birula decomposition (for C = P 1 ) and this was later used by Bifet, Ghione and Letizia [8] (for C of arbitrary genus) to study the cohomology of moduli spaces of vector bundles. Using the same ideas, del Baño showed that the Chow motive of Div n,d (D) (with Q-coefficients) is the (n deg D − d)-th symmetric power of the motive of C × P n−1 ; see [17,Theorem 4.2]. In order to consider such a Bia lynicki-Birula decomposition, let us fix G m ֒→ GL n of the form t → diag(t w 1 , . . . , t wn ) with decreasing integral weights w 1 > · · · > w n . The action of t ∈ G m on Div n,d (D) is given by precomposition with the corresponding automorphism of O C (D) ⊕n . The Bia lynicki-Birula decomposition for this G m -action gives a stratification of where Div n,d (D) + m is a smooth locally closed subvariety of Div n,d (D) consisting of points whose limit as t → 0 under the G m -action lies in C (m) . In fact, we can give a more precise modular description of the strata. First, let us introduce some notation.
For a subsheaf E ⊂ M with torsion quotient T , let E ≤i := E ∩ M ≤i and T ≤i := M ≤i /E ≤i , and let E i := E ≤i /E ≤i−1 and T i := T ≤i /T ≤i−1 . We note that E i is invertible and T i is torsion.
For the universal sequence 0 → E → π * C M → T → 0 on Div n,d (D) × C, we define E ≤i , E i , T ≤i , T i as above; however, in general, E ≤i and T ≤i are no longer flat over Div n,d (D). For any family F over S of rank n subsheaves of M, we define F ≤i and F i as above.
In particular, the stratum Div n,d (D) + m has codimension c + m := n i=1 (i − 1)m i . Proof. The description of the strata is given in [8, §3] and the normal spaces to the strata at fixed points are described in [6, §6], from which one can compute the ranks r + m and the codimensions, as c + m + r + m + n i=1 m i = dim Div n,d (D). To prove (iii), one can argue as in [8,Proposition 5.2 (3)] by identifying the strata with locally closed subschemes of certain flag Quot schemes (encoding the natural flag given by the G m -action). The tangent sheaf to Div n,d (D) is T Div n,d (D) ∼ = (π Div n,d (D) ) * Hom(E, T ) and the tangent sheaf of the flag Quot scheme is given by replacing Hom with the filtration preserving homomorphisms Hom − . After applying (π Div n,d (D) ) * to the short exact sequence 0 → Hom − (E, T ) → Hom(E, T ) → Hom + (E, T ) → 0 we also obtain a short exact sequence, as Ext 1 (E, T ) = 0 for a torsion sheaf T → C. Then, as in [8], one deduces N + m ∼ = (π Div n,d (D) ) * (Hom + (E, T )| Div n,d (D) + m ×C ), from which (iii) follows. We can now state the motivic Bia lynicki-Birula decomposition in this special case.

4.3.
A description of the motive of the stack of bundles. From the above results, we deduce the following description of M (Bun n,d ).
Theorem 4.9. Assume that R is a Q-algebra. Then the motive M (Bun n,d ) of the stack of rank n degree d vector bundles on C is contained in the smallest localising tensor triangulated category of DM(k, R) containing the motive of the curve C.
Proof. Since R is a Q-algebra, for any motive M ∈ DM(k, R) and any i ∈ N, there exists a symmetric power Sym i M which is a direct factor of M ⊗i and such that, for any X smooth quasi-projective variety, we have M (X (i) ) ≃ Sym i M (X) [41,Proposition 2.4]. Therefore the result follows from Theorem 4.4 and Corollary 4.8.
Since the motive of a curve is an abelian motive (that is, it lies in the localising subcategory generated by motives of abelian varieties), we immediately deduce the following result.

4.4.
A conjecture on the transition maps. In order to obtain a formula for this motive, we need to understand the functoriality of these motivic BB decompositions for the closed immersions i D,D ′ : Div n,d (D) → Div n,d (D ′ ) for divisors D ′ ≥ D ≥ 0. More precisely, the map i D,D ′ and the decompositions of Corollary 4.8 induce a commutative diagram (11) M (Div n,d (D)) where a D ′ −D : C (m 1 ) → C (m ′ 1 ) corresponds to adding n(D ′ − D). Based on some small computations, we make the following conjectural description for the morphisms k m,m ′ . One can also formulate this conjecture on the level of Chow groups, as there is also a BB decomposition for Chow groups. This leads to the following conjecture on the intersection theory of the Quot schemes Div n,d (D).   Conjecture 4.19). In fact, as we explain in §5.2, our conjectural formula for M (Bun n,d ) (or strictly speaking the formula for the compactly supported motive M (Bun n,d ) that follows by Poincaré duality) resembles many other classical formulas for invariants of Bun n,d , such as Harder's stacky point count of Bun n,d over a finite field and the Behrend-Dhillon formula for the class of Bun n,d in a dimensional completion of the Grothendieck ring of varieties. 4.5. The motive of the stack of line bundles. Throughout this section, we assume that C(k) = ∅ and we prove a formula for the motive of the stack of line bundles (cf. Corollary 4.16) by proving a result about the motive of an inductive system of symmetric powers of curves (cf. Lemma 4.14); the proof of this lemma is probably well-known, at least in cohomology, but we nevertheless include a proof as a generalisation of this argument is used in Theorem 4.20.
Lemma 4.14. Suppose that C(k) = ∅. For d ∈ Z and an effective divisor D 0 on C of degree d 0 > 0, consider the inductive system (C (ld 0 −d) ) l≥|d| with a D 0 : Proof. For notational simplicity, we prove the statement for d = 0; the proof is the same in general. We consider the Abel-Jacobi maps AJ l : C (ld 0 ) → Jac(C) defined using lD 0 which are compatible with the morphisms a D 0 : C (ld 0 ) → C ((l+1)d 0 ) . For ld 0 > 2g − 2 as C(k) = ∅, the Abel-Jacobi map is a P ld 0 −g -bundle: we have C (ld 0 ) ∼ = P(p * P l ), where P l is a Poincaré bundle on Jac(C) × C of degree ld 0 and p : Jac(C) × C → Jac(C) is the projection. In fact, we can assume that P l+1 = P l ⊗ q * (O C (D 0 )) for the projection q : Jac(C) × C → C. Then P l is a subbundle of P l+1 , and the induced map between the projectivisations is a D 0 .
By Example 2.20, we have M (BG m ) ≃ hocolim r M (P r ). As the inductive system (M (P ld 0 −g )) l is a cofinal subsystem of this system, we conclude the result using [33, Lemma 1.7.1].
Remark 4.15. If C(k) = ∅, then the Abel-Jacobi map from a sufficiently high symmetric power of C is not a projective bundle in general, but rather a Brauer-Severi bundle [30].
From this result we obtain a formula for the motive of the stack of line bundles. Remark 4.17. Alternatively, we can deduce this result as Bun 1,d → Pic d (C) ∼ = Jac(C) is a trivial G m -gerbe (or equivalently, Pic d (C) is a fine moduli space, which is true as by assumption C(k) = ∅ and thus the Poincaré bundle gives a universal family).

4.6.
A conjectural formula for the motive. Let us explain how to deduce a formula for the motive of Bun n,d from Theorem 4.4 and Conjecture 4.11. Throughout this section, we continue to assume that C(k) = ∅.
Definition 4. 18. Let X be a quasi-projective k-variety and N ∈ DM(k, R). The motivic zeta function of X at N is We can now state our main conjecture; for evidence supporting this conjecture, see §5.2.  i only depends on m ♭ , and we will also use the notation c m ♭ . For m ♭ ∈ N n−1 , we define an inductive system P m ♭ , * : N → DM(k, R) as follows: where the map P m ♭ ,l → P m ♭ ,l+1 is zero if m 1 (l + 1) < 0 and the morphism For each m ♭ and l, we have a generalised Abel-Jacobi map (14) AJ m ♭ ,l : C (m(l)) → Pic nld 0 −d (C) × C (m 2 ) × · · · × C (mn) ∼ = Jac(C) × C (m ♭ ) sending (F 1 , . . . , F n ) to (O C ( n i=1 F i ), F 2 , . . . , F n ). In fact, if m 1 (l) > 2g − 2, this morphism is a P m 1 (l)−g -bundle: we have that C (m(l)) ∼ = P(p * F) where p : Jac(C)× C (m ♭ ) × C → Jac(C)× C (m ♭ ) is the projection and F is the tensor product of the pullback of the degree nld 0 − d Poincaré bundle P → Jac(C) × C with the pullbacks of the duals of the universal line bundles L i → C (m i ) × C for 2 ≤ i ≤ n. In this case, by the projective bundle formula, we have M (C (m(l)) ) ≃ M (Jac(C)) ⊗ M (C (m ♭ ) ) ⊗ M (P m 1 (l)−g ) Then by Lemma 2.8 and Lemma 4.14, we have Hence, using the commutation of sums and tensor products, we have which completes the proof of the theorem.

Consequences and comparisons with previous results
As above, we let C be a smooth projective geometrically connected curve of genus g over a field k. We fix n ∈ N and d ∈ Z and let Bun n,d denote the stack of vector bundles over C of rank n and degree d. However homotopy limits rarely commute to inclusions of localising subcategories of compactly generated triangulated categories, and it is not difficult to deduce much from this formula. Let us first calculate the duals of the motives of the Jacobian of C and of the classifying space BG m , and of the motivic zeta function. As Jac(C) is smooth and projective of dimension g, we have by Poincaré duality M (Jac(C)) ∨ ≃ M c (Jac(C)){−g} ≃ M (Jac(C)){−g}.
As BG m is a smooth quotient stack of dimension −1, we have by Proposition 2.34 that As the dual of an infinite sum of motives is the infinite product of the dual motives, we have as the symmetric power C (j) of the curve C is a smooth projective variety of dimension j.
By Corollary 2.12, we have that for l ≥ 2, the natural morphism in DM(k, R) As Bun n,d is a smooth stack of dimension n 2 (g − 1), Poincaré duality gives which gives the above formula.

5.2.
Comparison with previous results. In this section, we compare our conjectural formula for the (compactly supported) motive of Bun n,d (cf. Theorem 5.1) with other results concerning topological invariants of Bun n,d . One of the first formulae to appear in the literature, was a computation of the stacky point count of Bun n,d over a finite field F q , which is defined as Theorem 5.2 (Harder, Siegel). Over a finite field F q , we have where ζ C denotes the classical Zeta function of C.
We note that our conjectural formula for M c (Bun n,d ) is a direct translation of this formula, when one replaces a variety (or stack) by its (stacky) point count, q k by R{k}, and the classical Zeta function by the motivic Zeta function.
Behrend and Dhillon [6] give a conjectural description for the class of the stack of principal G-bundles over C in a dimensional completion K 0 (Var k ) of the Grothendieck ring of varieties when G is a semisimple group and, moreover, they prove their formula for G = SL n by following the geometric arguments in [8]. By a minor modification of their computation, one obtains the following formula for the class of Bun n,d . It is possible to compare their formula with our conjectural formula for M c (Bun n,d ) if one passes to the Grothendieck ring of a certain dimensional completion of DM(k, R). We sketch this comparison here. First note that it does not make sense to work with the Grothendieck ring of DM(k, R), as this category is cocomplete and so by the Eilenberg swindle, K 0 (DM(k, R)) ≃ 0. Instead, we will follow the lines of Zargar [45, §3] and work with a completion of DM(k, R); however, note that Zargar works with effective motives and completes with respect to the slice filtration and we will instead complete with respect to the dimensional filtration.
For this completion process, one would like to take limits of projective systems of triangulated categories, but the category of triangulated categories is unsuitable for this task; hence, in the rest of this section, we let DM(k, R) denote the symmetric monoidal stable ∞-category underlying Voevodsky's category. As in Definition 2.10, given m ∈ Z, we write DM gm (k, R) r (resp. DM(k, R) r ) for the full sub-∞-category (resp. presentable sub-∞-category) of DM(k, R) generated by motives of the form M c (X)(r) with X being a separated finite type k-scheme with dim(X) + r ≤ n. The localisations DM (gm) (k, R)/DM (gm) (k, R) r are symmetric monoidal ∞-categories (presentable in the non-geometric case), as this filtration is symmetric monoidal. One can then define  R)). Note that, as in [45], it is not clear how small K 0 (DM ∧ gm (k, R)) is, and in particular if the natural map K 0 (DM gm (k, R)) → K 0 (DM ∧ gm (k, R)) is injective, which limits the interest of such a comparison. To go further, one could try to introduce an analogue of the ring M(k, R) in [45]; however, we do not pursue this here.

5.3.
Vector bundles with fixed determinant and SL n -bundles. For a line bundle L on C of degree d, we let Bun L n,d denote the stack of rank n degree d vector bundles over C with determinant isomorphic to L, and we let Bun ≃L n,d be the stack of pairs (E, φ) with E a rank n degree d vector bundle and φ : det(E) ≃ L. We have the following diagram of algebraic stacks with cartesian squares where the morphism det is smooth and surjective. In fact, the bottom left square is cartesian by the See-saw Theorem. We see that Bun L n,d is thus a closed smooth substack of Bun n,d of codimension g, and that Bun ≃L n,d → Bun L n,d is a G m -torsor. Moreover, when d = 0 and L = O C , the stack Bun ≃O C n,0 is actually isomorphic to the stack Bun SL n of principal SL n -bundles; indeed, the vector bundle associated to an SL n -bundle via the standard representation has its determinant bundle canonically trivialised. For more details on this picture, in the more general case of principal bundles, see [5, §1-2] We will use these facts to compute the motive of Bun SL n assuming Conjecture 4.11. First, we claim that the smooth stacks Bun L n,d and Bun ≃L n,d are exhaustive; the claim for the later follows from the claim for the former by Proposition 2.23, as the G m -torsor Bun ≃L n,d → Bun L n,d is representable and flat of finite type. To prove that Bun L n,d is exhaustive, we use matrix divisors with fixed determinant L. For every effective divisor D on C, let Div L n,d (D) be the subvariety of Div n,d (D) parametrising matrix divisors with determinant L; this is a smooth closed subscheme of codimension g (see [18, §6]). Then one can prove that Bun L n,d is exhaustive analogously to Proposition 4.2 by using matrix divisors with fixed determinant L; for the relevant codimension estimates, see [6, §6]. Hence, it suffices to verify that the pullbacks of the normal bundles of Div L n,d (D) ֒→ Div n,d (D) and Div n,d (D) + m ֒→ Div n,d (D) to Div L n,d (D) + m are isomorphic; this follows from the description of the normal bundle of the former given in [6, §6] and of the latter in Lemma 4.7.
Since j D is transverse to the BB strata, Conjecture 4.11 for the transition maps in the motivic BB decompositions of Div implies the analogous statement for Div L , as we obtain the analogous morphisms by intersecting with the classes of Div Theorem 5.6. Assume that Conjecture 4.11 holds and C(k) = ∅. In DM(k, R), we have and Proof. First one restricts Div n,d → Bun n,d to Div L n,d → Bun L n,d and, analogously to Theorem 4.4, one proves that for any non-zero effective divisor D 0 on C, there is an isomorphism in DM(k, R); for the relevant codimension estimates, one can use [6, §6] together with Lemma 4.3. We then consider the Bia lynicki-Birula decompositions for the smooth closed G m -invariant subvarieties Div L n,d (lD 0 ) ⊂ Div n,d (lD 0 ), whose fixed loci is the disjoint union of C (i) For an atlas A → X, we let A [n] := A × X · · · × X A be the n + 1-fold self-fibre product of A in X or equivalently the nthČech simplicial scheme associated to this atlas 2 . We define theétale motive of X with respect to this atlas as where R eff (A [•] ) denotes the complex ofétale sheaves associated to the simplicial sheaf of R-modules A [•] . (ii) Let f : X → Y be a representable morphism of stacks; then for any atlas A Y → Y, the map A X := A Y × Y X is an atlas for X and thus there is a morphism Mé t atlas (f ) : Mé t atlas,A X (X) → Mé t atlas,A Y (Y). (iii) We define the nerve-theoretic motive of X by taking the functor X : Sm k → Gpds valued in the category of groupoids and composing with the nerve N : Gpds → sSets to the category of simplicial sets and then finally composing with the singular complex sing : sSets → C * (abgp) which takes values in the category of complexes of abelian groups; this gives a complex ofétale sheaves of abelian groups aé t sing •N • X on Sm k , which we tensor with our coefficient ring R and put Mé t nerve (X) := Σ ∞ T (aé t (sing •N • X) ⊗ R). Remark A.2.
(i) By definition, the motive of an algebraic stack X with respect to an atlas U → X and the nerve-theoretic motive of X are both effective motives, as both definitions take place in DA eff,ét (k, R). (ii) If X is a scheme X, then clearly we have M (X) ≃ M atlas,X (X) and in fact this also agrees with the nerve-theoretic definition by Lemma A.3 below.
Lemma A.3. In DAé t (k, R) for an algebraic stack X and for any atlas A → X, we have Mé t atlas,A (X) ≃ Mé t nerve (X). In particular, the isomorphism class of M atlas,A (X) is independent of the atlas.
Proof. This follows as DAé t (−, R) satisfies cohomological descent with respect to h-hypercoverings [13,Theorem 14.3.4. (2)], thus in particular with respect toČech hypercoverings induced by smooth surjective maps. For a more detailed argument, see [11] which treats the case of a Deligne-Mumford stack X in DMé t (k, Q) (replacingétale descent by smooth descent).
For an exhaustive stack X, one can define the motive of X in DAé t (k, R) (or in fact, for any choice of topology) exactly as in Definition 2.17 by using an exhaustive sequence of vector bundles with respect to a filtration of X. In this appendix, to distinguish the motive in Definition 2.17 (resp. itsétale, without transfers analogue) from other definitions, we denote it by M mono (X) (resp. Mé t mono (X)) and say X is mono-exhaustive (rather than just exhaustive). In fact, dual to Definition 2.15 of an (injective) exhaustive sequence is the following notion of a surjective exhaustive sequence as in [40, §8].
Definition A.4. For a smooth stack X, a surjective exhaustive sequence of vector bundles on X with respect to a filtration X 0 i 0 → X 1 i 1 → . . . ⊂ X is a pair (V • , W • ) given by a sequence of vector bundles V m over X m together with surjective maps of vector bundles f m : V m+1 × X m+1 X m → V m and closed substacks W m ⊂ V m such that (i) the codimension of W m in V m tend towards infinity, (ii) the complement U m := V m − W m is a separated finite type k-scheme, and (iii) we have W m+1 × X m+1 X m ⊂ f −1 m (W m ).
In the category of T -spectra of complexes ofétale sheaves, we have For m ∈ N, let V m := V m × Xm A m and define W m and U m analogously; as A → X is representable, these are (smooth, finite type) k-schemes. The maps V m → V m+1 lift to maps V m → V m+1 . We have colim which is a filtered colimit of A 1 -equivalences, thus an A 1 -equivalence. We have V m × X A is an A 1 -equivalence since it is a filtered colimit of A 1 -equivalences. Putting everything together, we get an isomorphism in DAé t (k, R) Mé t atlas,A (X) ≃ colim m R(U m ) =: Mé t mono (X) which concludes the proof.