Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces

For any free oriented Borel-Moore homology theory $A$, we construct an associative product on the $A$-theory of the stack of Higgs torsion sheaves over a projective curve $C$. We show that the resulting algebra $A\mathbf{Ha}_C^0$ admits a natural shuffle presentation, and prove it is faithful when $A$ is replaced with usual Borel-Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose $A$-theory admits an $A\mathbf{Ha}_C^0$-action. These triples can be interpreted as certain sheaves on $\mathbb P(T^*C)$. In particular, we obtain an action of $A\mathbf{Ha}_C^0$ on the cohomology of Hilbert schemes of points on $T^*C$.


Introduction
Let C be a hereditary abelian category over finite field F q , such that all Hom-and Ext-spaces have finite dimension. We have two important examples of such categories: • for a finite quiver Q, the category of finite dimensional representations Rep Q = Rep Fq Q; • for a smooth projective curve C over F q , the category of coherent sheaves Coh C. Given a category C satisfying the conditions above, one can associate to it the Hall algebra H(C), as defined in [46]. Broadly speaking, its basis is given by isomorphism classes of objects in C, and the product is given by the sum of all non-isomorphic extensions. In the case C = Rep Q, where Q is a quiver of Dynkin type, a famous theorem by Ringel [44] describes the Hall algebra H(Rep Q) as the positive half of the quantum group U ν (g Q ), specialized at ν = q 1/2 . Moreover, one can upgrade H(C) to a (twisted, topological) bialgebra, such that the Drinfeld double D(H(Rep Q)) is isomorphic to the quantum group itself.
By contrast, the Hall algebra H(Coh C) seems to be far less understood. For instance, an explicit description of (the spherical part of) H(Coh C) by generators and relations is known only when C is rational [22] or elliptic [6]. Our principal motivation is to get a better understanding of this algebra. One way to do it is to study its representation theory. Unfortunately, since we do not possess an explicit combinatorial description of H(Coh C) in terms of generators and relations (see, however, [46,Section 4.11] for partial results), we have to construct its representations indirectly.
We use an approach close in spirit to the well-known construction of Nakajima [37], which realizes irreducible representations of the universal enveloping algebra U (g) of a simple Lie algebra g as homology groups of certain varieties. Let us summarize a variant of this construction, following the point of view from [52]. Namely, for a finite type quiver Q = (I, E) and a projective CQ-module P with top of graded dimension w ∈ Z I + , one considers the algebraic stack T * Rep ←P v Q, where Rep ←P v Q parametrizes pairs (V, ϕ) with V ∈ Rep Q, dimV = v, and ϕ ∈ Hom CQ (P, V ). The C-points of this stack can be identified with representations of a quiver Q ♥ , satisfying certain conditions [16,Section 5]. For every dimension vector v ∈ Z I + , one introduces a stability condition on these representations, such that subrepresentations of stable representations are stable, and the moduli stack of stable representations forms a smooth variety M(v, w). Inside these varieties, one has Lagrangian subvarieties L(v, w) := π −1 v (0), where π v : M(v, w) → Spec Γ(O M(v,w) ) is the affinization map. Finally, one considers a correspon- Denoting the projections on the first and second factor by Φ v and Ψ v correspondingly, we have the following operators in Borel-Moore homology: where ǫ i is the dimension vector of the simple representation at vertex i ∈ I. Then e i := v e i,v , f i := v f i,v give rise to an action of U (g Q ) on M w = v H(M(v, w)), and moreover its restriction to v H(L(v, w)) is the irreducible highest module of weight w.
In fact, this action can be extended to a much bigger algebra, so-called Yangian. This can be achieved by realizing it inside the cohomological Hall algebra [48,52], isomorphic to v H(T * Rep v Q) as a vector space (see [31] for another perspective on Yangians). The latter algebra then acts on M w by correspondences similar to the ones described above. The purpose of this paper is to begin investigation of analogous algebras and their representations in the context of curves.
In order to apply the same set of ideas to our situation, we have to introduce several modifications to our context. First, we have to consider T * Coh C instead of Coh C; note that the former stack is isomorphic to the stack of Higgs sheaves Higgs C. Secondly, we will study a homological version of Hall algebra. It will be modeled on the vector space A(Higgs C), where A is either Borel-Moore homology or an arbitrary free oriented Borel-Moore homology theory (see [30,Chapter 5] for the definition of the latter).
Optimistically, our program is as follows: (1) construct a (bi-)algebra structure AHa C on A(Higgs C); (2) define a suitable stability condition on T * Coh ←F C, where Coh ←F C is the stack of pairs (E, α) with E ∈ Coh C, α ∈ Hom(F, E); (3) construct an action of the Drinfeld double D(AHa C ) on the A-theory A(M) of the moduli of stable objects. In the present article, we treat a very particular case of the plan above. Namely, we restrict our attention to the category of torsion sheaves on C. Then, we have the following result: Theorem 0.1. There exists an associative product on d A(Higgs 0 d C), which makes it into an algebra AHa 0,C (Theorem 2.2).
The proof uses the techniques found in [49,52]. Because of our restrictions on the rank of sheaves, all stacks we consider can be explicitly realized as global quotients, and thus we can forget their stacky nature and work with equivariant A-theory of their atlases instead. In positive rank the stack Coh r,d is only locally a quotient stack, so that one has to check that separate constructions in each patch can be glued together. This was done in [45].
Note that we do not construct a coproduct on AHa 0 C . However, if we denote by AHa 0,T C the version of AHa 0 C equivariant with respect to the scaling action of G m on the cotangent fibers, one can define a certain algebra ASh C with explicit formulas for multiplication and construct a map ρ : AHa 0,T C → ASh C . Roughly speaking, ASh C looks like the space of formal series with coefficients in A(C), and the product is given by twisted symmetrization (see Definition 3.3). We expect ρ to be injective (Conjecture 4.12). This prediction is supported by the following theorem: Theorem 0.2. If A = H are the usual Borel-Moore homology groups, the map ρ : HHa 0,T 0,C → HSh C becomes injective after tensoring by Frac(A T (pt)) (Corollary 4.5).
If the conjecture is true, this map can be used to find relations in AHa 0,T C via direct computations, and also to transport a natural coproduct from ASh C .
Next, let us pick a locally free sheaf F as framing.
Theorem 0.3. Let C be a smooth projective curve, and d, n positive integers.
(1) The moduli of stable Higgs triples of degree d and frame F is represented by a smooth quasi-projective variety B(d, F) (Theorem 5.8); (2) Let F = n ⊗ O. Then for any n, the space AM n = d A(B(d, n ⊗ O)) is equipped with a structure of an AHa 0 C -module (Corollary 5.10). The second part of this theorem is proved by the same methods as Theorem 0.2. We strongly expect that the same result holds for any locally free F. As for the first part, it is done by realizing stable Higgs triples as sheaves on a compactification of T * C. Namely, we have the following theorem: Theorem 0.4. The variety B(d, F) is isomorphic to the moduli space of f -semisimple torsionfree sheaves on P C (ω ⊕ O), equipped with framing at infinity and satisfying certain numerical conditions. In particular, B(d, O) is isomorphic to the Hilbert scheme of points Hilb d T * C (Section 7).
This isomorphism can be understood as a relative version of classical derived equivalence between the category of sheaves on P 1 and of representations of the Kronecker quiver [3]. We refer the reader to Section 7 for definitions and precise statement.
Unfortunately, it is not entirely clear how to extend a AHa 0,T C -module structure on AM T n to a Yetter-Drinfeld module [43] with respect to some coproduct on AHa 0,T C . Still, the isomorphism B(d, O) ≃ Hilb d (T * C) suggests that AM T n should admit an action of the Drinfeld double of AHa 0,T C , similar to [38,Chapter 8]. In higher rank, we expect the moduli of stable Higgs triples to retain a close relation to the moduli of sheaves on P C (ω ⊕ O) framed at infinity. This is evidenced by the fact that similar objects appear in the works of Neguţ [40,39], where for any smooth projective surface S he defines an action of a certain W-algebra on the K-theory of moduli of stable sheaves on S. We expect that for S = T * C, these algebras get embedded into a suitable completion of KHa 0 C . In general, since Higgs sheaves on C can be thought of as coherent sheaves with proper support on T * C via BNR-correspondence [2], one can imagine a much more general picture: Guiding principle. Let S be a smooth projective surface together with a smooth divisor D ⊂ S. Denote by Coh(S, D) the stack of O S -modules with support disjoint from D, and by Coh 0 (S, D) its substack of O S -modules of finite length. Then AHa S = A(Coh(S, D)) should admit a Hall-like structure of an associative algebra, such that AHa 0 S = A(Coh 0 (S, D)) is a subalgebra containing A-theoretic W-algebra. Furthermore, the Drinfeld double D(AHa S ) should act on the A-theory of stable sheaves on S framed at D, for a certain stability condition.
For this principle to hold true, one will certainly need additional technical assumptions, such as transversality of the divisor defining stability condition with D. However, this discussion reaches far beyond the scope of this article.
Since the first draft of the present paper has appeared, some additional progress has been made in generalizing its results. As mentioned above, the definition of algebra AHa 0 C was extended to positive rank Higgs sheaves in [45]. In K-theory, the rank 0 algebra KHa 0 S was defined for any smooth surface S in [53]. In homology, the full algebra HHa S was defined in [24]. Moreover, it was shown HHa S acts on the Borel-Moore homology groups of rank 1 semi-stable sheaves on S.
Let us finish the introduction with a brief outline of the structure of the paper. In Section 1 we choose explicit presentations of Coh 0,d and Higgs 0,d as global quotient stacks, given by certain Quot-schemes. We also recollect basic facts about these schemes. In Section 2 we recall a construction introduced in works of Schiffmann and Vasserot, which permits us to define an associative product on d A(Higgs 0,d ). In Section 3 we introduce global shuffle algebras ASh g , prove that these algebras satisfy some quadratic relations, and obtain a shuffle presentation ρ of AHa 0,T C for a certain choice of g. The map ρ is obtained by localizing our product diagrams to the fixed point sets of a certain torus T. In passing, we also propose a geometric interpretation of the difference between two types of shuffle product, appearing in literature in similar context (Corollary 3.10). In Section 4, we prove that for A = H the shuffle presentation ρ is faithful. The proof uses the scaling torus action and weight filtration in a crucial way, so that it cannot be easily translated to other homology theories. However, we conjecture that ρ is faithful for general A. In Section 5 we introduce the moduli stack of Higgs triples, construct an action of AHa 0 C on d A(B(d, n)), and discuss how it can be related to the classical action of Heisenberg algebra on cohomology groups of Hilbert schemes of points on T * C [38]. In Section 6, we collect some technical facts about quiver sheaves for later use. In Section 7, we provide an alternative description of B(d, n) as a moduli of sheaves on a compactification of T * C. We also briefly describe the relation between our work and the W -algebras of Neguţ. Finally, in Appendix A we recall the notion of oriented Borel-Moore homology functor, following the monograph by Levine and Morel [30], and gather the statements necessary for our proofs. In particular, we adapt the localization theorem of Borel-Atiyah-Segal to this framework.
Acknowledgements: This paper constitutes a part of author's Ph.D. thesis, written under direction of Olivier Schiffmann. The author would like to thank him for his perpetual support and constant encouragement. I would also like to thank Quoc Ho, Sergei Mozgovoy, Andrei Neguţ, Francesco Sala and Gufang Zhao for their help and illuminating discussions, and the anonymous referee for their valuable suggestions.

Conventions:
We denote by Sch/ the category of -schemes of finite type over ; pt stands for the terminal object Spec ∈ Sch/ . For any X ∈ Sch/ , the category of coherent O X -sheaves is denoted by Coh X. We will usually denote coherent sheaves by calligraphic letters, and implicitly identify locally free sheaves with corresponding vector bundles. For any E, F ∈ Coh X, we write Ext i (E, F) for Ext-functors, Hom := Ext 0 , and Ext i (E, F) for Ext-sheaves, Hom := Ext 0 . More generally, for any two complexes of sheaves E • , F • we denote by Àom(E • , F • ) the space of morphisms in the derived category D b (Coh X), and xt i (E • , F • ) := Àom(E • , F • [i]). Finally, we will liberally use the language of stacks; see [28] or [42] for background.

Coherent sheaves and Quot-schemes
Let be an algebraically closed base field of characteristic 0. Let C be a smooth projective curve defined over , and O its structure sheaf. Then one can define the following algebraic stacks (over Sch/ in étale topology): • Coh 0,d , the stack of torsion sheaves on C of degree d [28, Théorème 4.6.2.1]; • for any F ∈ Coh C, the stack Coh ←F 0,d of pairs (E ∈ Coh 0,d C, α ∈ Hom(F, E)) [15, Section 4.1]; • the cotangent stack Higgs 0,d := T * Coh 0,d . It is defined as the relative Spec of the symmetric algebra of the tangent sheaf; see [28,Chapitre 14,17] for the relevant definitions.
Remark 1.1. Note that the tangent sheaf of a stack is not the same as the tangent complex, but is rather its zeroth cohomology.
All of the stacks above can be realized as global quotient stacks. Below we will make an explicit choice of such presentation for computational purposes. Definition 1.2. Let Quot 0,d be the following functor: , Moreover, let us consider its open subfunctor Quot • 0,d ⊂ Quot 0,d , consisting of quotients (1) Quot 0,d and Quot • 0,d are representable by smooth schemes Quot 0,d and Quot • 0,d respectively, and Quot 0,d is a projective variety of dimension d 2 ; (1), see [29]. The claim (2) follows from the observation that any torsion sheaf of degree d is generated by its global sections, and every isomorphism of torsion sheaves is completely determined by its action on global sections. Finally, for (3) let us consider the natural map Coh ←V ⊗O 0,d → Coh 0,d . This is a vector bundle, which is trivialized in the atlas given by Quot • 0,d :  [20] for more details.
Recall that for any algebraic group G and any smooth G-variety X the cotangent bundle T * X is naturally equipped with a Hamiltonian G-action. Let µ : T * X → g * be the corresponding moment map, where g is the Lie algebra of G, and put T * G X := µ −1 (0). Note that the infinitesimal G-action provides a morphism g ⊗ O X µ * − → T X , where T X is the tangent sheaf of X. Lemma 1.5. Let X be a smooth variety equipped with an action of G. Then we have a natural isomorphism of stacks T * [X/G] ≃ [T * G X/G]. Proof. It follows from the definition of the moment map that the composition and we obtain the desired isomorphism after descending to [X/G].
We introduce the following notations for later use: We also fix isomorphisms Proof. In order to prove that Quot 0,d• is a closed subvariety of Quot • 0,d , let us recall the construction of Quot-schemes in [29,Chapter 4]. Namely, fix n ≫ 0 and an ample line bundle O(1) on C.
be the Grassmanian of subspaces of codimension d in d ⊗ H, and consider the following map: It is a closed embedding for n big enough. Now, for each quotient d ⊗ O ϕ − → E and for each i ∈ [1, k] we have the restricted short exact sequence is injective for all i, and thus h 0 (E i ) ≥ d i for all i; moreover, ϕ belongs to Quot 0,d• precisely when all the previous inequalities turn into equalities. Tensoring (1) by O(n) and taking global sections, we get an exact sequence for n big enough and all K i . Since E i is torsion sheaf, there exists an isomorphism E i (n) ≃ E i , and the exact sequence above implies that where the equality holds if and only if ϕ belongs to Quot 0,d• . Therefore The second set is closed in Grass( d ⊗ H), and thus Quot 0,d• is closed in Quot • 0,d as well. In order to prove that Quot 0,d• is smooth, consider the following diagram: , and q is the projection on the first coordinate. Note that we have the following map between short exact sequences for any point in Quot 0,d• : It follows that the map ϕ| d i / d i−1 : induces an isomorphism on global sections, and thus the image of p belongs to Quot • 0,d• × G d• . Moreover, the diagram above also implies that p is an affine fibration over Since Quot • 0,d• is smooth and q is a trivial G d• -torsor, this observation implies the smoothness of Quot 0,d• .
Remark 1.9. In the proof above we chose an n big enough so that all K's and K i 's cease to have higher cohomology groups and become generated by global sections after tensoring by O(n). It is possible because all our sheaves are parametrized by a finite union of Quot-schemes, and thus form bounded families (see Lemma 4.4.4 in [29]).
Notation. Throughout the paper, for any quotient d ⊗ O ϕ − → E in Quot 0,d we will denote Ker ϕ by K, and the inclusion K ֒→ d ⊗ O by ι. We will also decorate K, E, ϕ and ι with appropriate indices and markings.
Next, we recall the description of tangent spaces of Quot-schemes.
− → E be a point in Quot 0,d , and let K = Ker ϕ. Then the tangent space T ϕ Quot 0,d at ϕ is naturally isomorphic to Hom(K, E). Moreover, if ϕ ∈ Quot 0,d• we have Proof. The proof of the first claim can be found in [29,Chapter 8]. The second claim can be proved in a similar fashion, keeping track of the condition of admitting a sub-quotient throughout the proof of the first claim.
Because of this proposition, we will usually regard T * Quot • 0,d as a variety, whose -points are identified with pairs Additionally, let us define for later purposes the nilpotent part (T ϕ Quot 0,d• ) nilp of T ϕ Quot 0,d• :

Note that
with the second isomorphism being induced by H 0 (ϕ) : d ∼ − → H 0 (E). In these terms the moment map for the G d -action on T * Quot • 0,d can be written as follows: Since E is a torsion sheaf, we have Ext 1 ( d ⊗ O, E) = 0, and thus over each ϕ ∈ Quot • 0,d the restriction µ ϕ of the map µ to T * ϕ Quot • 0,d can be embedded in a long exact sequence: This implies that µ −1 ϕ (0) ≃ Ext 1 (E, E) * , and we get an identification on the level of -points 11. Let C = A 1 . Even though this curve is not projective, we can fix an isomorphism A 1 = P 1 \ {∞}, and define

Then the open subvariety
where the G d -action on the left gets identified with the adjoint action on the right. Thus In light of the example above, we will refer to T * G d Quot • 0,d as the commuting variety of C, and denote it by C d = C d (C). We will also write Example 1.12. Let us fix a geometric point x ∈ C( ), and consider the punctual Quot-scheme Such quotient is completely determined by its localization at x. More explicitly, since C is smooth, the completionÔ x is (non-canonically) isomorphic to t . The stalk of α at x is thus of the form d t where E is a t -module, and α 1 induces an isomorphism of -vector spaces d t 0 ≃ E. Such quotient is in its turn uniquely determined by a nilpotent operator T on d , the correspondence given by Thus we see that Quot • 0,d (x) is isomorphic to the nilpotent cone N d ⊂ g d together with the adjoint action of G d . Moreover, under this identification the cotangent space Hom(K, E) * in Quot • 0,d of a point α gets identified with g d , and the restriction of the moment map µ :

The product
Let us once and for all fix a free oriented Borel-Moore homology theory (OBM) A; for the definition and basic facts about this notion, see Appendix A. As explained there, we abuse the notation somewhat and consider the usual Borel-Moore homology H * as if it were a free OBM. We also equip the cotangent bundle T * C with an action of G m , given by dilations along the fibers; let us denote this torus by T .
We begin by recalling a general construction from [47]. Let G be an algebraic group with fixed Levi and parabolic subgroups H ⊂ P . Assume we are given smooth quasi-projective varieties X ′ , Y , V , equipped with actions of G, H, P respectively, and H-equivariant morphisms such that p is an affine fibration and q is a closed embedding. Set W = G × P V , X = G × P Y , where the P -action on Y is induced by the natural projection P → H, and consider the following maps of G-varieties: The map (f, g) : W → X × X ′ is a closed embedding, so from now on we will identify the smooth variety W with its image in X × X ′ . Let Z = T * W (X × X ′ ) be the conormal bundle. Projections on factors define two maps: Lemma 7.3(b)], and we have the following induced diagram: Now, Ψ and Ψ G are projective, Φ is an lci map, so that we get the following morphisms in A-groups:

By composing these two maps and using the induction isomorphism
Let us apply this general construction to a particular case of Quot-schemes of rank 0. Namely, , and let P = P d• be the parabolic group preserving the flag d 1 ⊂ . . . ⊂ d . We use Gothic letters g, p, h for corresponding Lie algebras, and p − for the parabolic algebra opposite to p. Next, put By Proposition 1.8 we have a closed embedding g : V ֒→ X ′ and an affine fibration f = gr : V ։ Y .
The following lemma will help us to identify all the terms in diagram (4).
Lemma 2.1. Let G, P, H, X ′ , X, V, Y be as above.
(1) There exist natural isomorphisms of G-varieties where (T ϕ Quot 0,d• ) nilp is defined as in (2). For each (ϕ, β) ∈ T * Quot 0,d• we have Φ((g, ϕ, β) mod P ) = (g, µ(ϕ, β), gr(ϕ, β)) mod P, (2) There are isomorphisms of G-varieties (1) The first isomorphism is obvious, so we start with T * X: Let us also note that the moment map µ : T * X → g * is given by By the same reasoning, Next, let us compute Z = T * W (X × X ′ ). We have W = G × P Quot 0,d• , and therefore But the conormal bundle T * W (X × X ′ ) can be expressed as the kernel of the following map of vector bundles: Therefore, we finally obtain Note that the desired formula for Φ follows from the first equality, and the formula for Ψ is evident. The claim (2) follows from the explicit descriptions of moment maps T * X ′ → g * , The general construction thus produces a map

and we get a bilinear map
We denote AHa 0 is an associative algebra. Proof. We begin by introducing some notations. Let Define the following varieties: . These varieties are Quotand Quot-bundles over certain partial flag varieties, so we may identify their -points as pairs (flag, quotient). Adopting "mod 3"-notation for indices, we have obvious Lemma 2.3. Using the notations above, Proof. First of all, we introduce a small abuse of notation. Namely, for any morphism of sheaves E f − → F and for any subsheaf With that in mind, we have Next, consider a commutative diagram (1). We have the following equalities on the level of -points: The natural map can be thus seen to be a bijection. The fiber product W 1 × X 2 W 3 is normal by [18, Proposition 6.14.1], W 2 is connected, therefore Zariski's main theorem implies that p is an isomorphism.
(2). To prove that our intersection is transversal, we need to show that for any By Proposition 1.10 we have the following isomorphisms: Let us fix a flag D • of dimension d • , and a quotient ϕ ∈ Quot 0,d such that (D • , ϕ) ∈ W 2 . Then the equalities above allow us to compute all the tangent spaces in question: where p ′ , p ⊂ g are parabolic subalgebras associated to flags D ′ • , D • respectively, and where ξ red denotes the image of ξ under the quotient map g/p → g/p ′ . The space above is isomorphic to T x W 2 by means of the map This proves the transversality.

Let us now put
The lemma above, combined with Theorem 2.7.26 in [9], tells us that the projection Therefore, we obtain the following diagrams with cartesian squares: , so that the associativity of multiplication in AHa 0 C follows. Note that all varieties in the definition of AHa 0 C admit a T = G m -action (by dilation along the fibers of cotangent and conormal bundles) and all maps we consider are T -equivariant. Therefore, the construction above also defines an associative product on Example 2.4. Let C = A 1 , and equip it with a natural action of weight 1 of another torus T ′ = G m . In this case KHa 0,T ×T ′ C and HHa 0,T ×T ′ C are precisely the K-theoretic and cohomological Hall algebras studied in [49] and [48] respectively.

Global shuffle algebra
In this section we focus our attention on the algebra AHa 0,T C . In order to study its product, we will utilize the localization theorem A.14. Let We will also denote by T d ⊂ H the maximal torus, which consists of operators diagonal with respect to the standard basis v 1 , . . . , v d of d . The Weyl group of G is then naturally isomorphic to S d , and the Weyl group of H is isomorphic Denote T = T d × T , and write t 1 , . . . , t d for the basis of character lattice of T d corresponding to the standard basis of d . In the same way, let t be the character of T of weight 1. One can think about characters of T as equivariant line bundles over a point. In this fashion, we identify We choose an integer N and an ample line bundle L such that for any ϕ the sheaf L N ⊗ K ϕ is generated by its global sections. Then K ϕ is uniquely determined by the subspace V : But we know that the only subspaces stable under the torus actions are direct sums of subspaces of weight spaces.
Recall that Quot 0,1 ≃ C (see Example 1.6), and let p ij : Quot 0,1 × Quot 0,1 × C → C × C denote the projection along the unnamed factor. Lemma 3.2. Let K, E ∈ Coh(Quot 0,1 × C) ≃ Coh(C × C) be the universal families of kernels and images of quotients O → E respectively. Then Proof. It is easy to see that E ≃ O ∆ , K ≃ O(−∆). Since K is locally free and E is a torsion sheaf over any point ϕ ∈ Quot 0,1 , the higher Ext-sheaves This proves the first equality. For the second one, we conclude by a similar computation: Notation. In order to keep notation concise, for any two sheaves A, B ∈ Coh(Quot 0,1 × C) we will write Hom(A, B) instead of p 12 * Hom(p * 13 A, p * 23 B) (here the pushforward p 12 * is underived). Let j H : C d• ֒→ T * Quot • 0,d• denote the closed embedding, and let i H : be the inclusion of the fixed point set. Recall that by localization theorem A.14 the map i * H becomes an isomorphism upon tensoring with the fraction field of A T (pt). Consider the following composition: where the isomorphism on the left is given by Proposition A.7, Weyl group acts on the right-hand side by Remark A.18, and Quot d 0,1 is identified with C d . In the same way, we can define a map The goal of this section is to construct a map Υ loc (between localized A-groups), such that the following diagram commutes: As in [49, Section 10], one expects Υ loc to be some incarnation of shuffle product. Let z be a formal variable, and let g ∈ (A * (C × C)((t))) ((e(z))) be an A * T (C × C) loc -valued formal Laurent series in e(z), where we interpret the latter as a formal symbol. We will also abuse the notations and write g as a function of z. For any positive d 1 , Let us also fix the following set of representatives of classes in Definition 3.3. The shuffle algebra associated to g is the vector space The formula (9) requires some explanation. First of all, the product between f and h is given by the map . Next, after replacing z i 's by t i 's and taking Euler classes, the function g d 1 ,d 2 becomes an honest cohomology class, which then operates on the product f · h. Finally, the natural action of σ ∈ Sh(d 1 , d 2 ) (see Remark A.18) simultaneously permutes t i 's and factors in the direct product C d 1 +d 2 .
It is easy to check that this product is associative. We will be mainly concerned with two specific choices of g: • the global shuffle algebra, denoted by ASh C , is the shuffle algebra associated to g C = e(tz −1 )e(zO(−∆))e(tzO(∆)) e(z) ; • the normalized global shuffle algebra, denoted by ASh norm C , is the shuffle algebra associated to e(zO(−∆))e(tzO(∆)) e(z)e(tz) .
By invoking the formal group law ⋆ associated to A, we can deduce that both functions are Laurent series in e(z) (see also discussion before Proposition 3.12).
Theorem 3.5. The collection of maps ρ d : Proof. Let us first introduce some notations. Define Also, let us denote the space Recall the notations of Section 2, specifically Lemma 2.1 (2). Our proof will proceed in two steps. First, consider the following diagram: Let us denote all vertical compositions by ρ, leaving out the subscripts. For any closed embedding of smooth varieties M ⊂ N , we denote by T M N the normal bundle of M .
Lemma 3.6. We have Proof. Everything in this diagram commutes, except for the lower left square. Moreover, by Proposition A.15 this square becomes commutative after multiplying by an appropriate Euler class. Note that since This (trivial) vector bundle has the same T -fixed points as its zero section T * Quot • 0,d• . Therefore, the Proposition A.15 tells us that the required Euler class is and we are done.
Next, consider another diagram: Proof. Once again, all squares in this diagram commute, except for the lower right one, which commutes up to multiplication by a certain Euler class (see Lemma A.12 (2) and Proposition A.15). Therefore, we have: It is left to compute the product of Euler classes in parentheses. We have the following chain of equalities: (8) provide us with explicit expressions for tangent spaces of various Quot-schemes: Hom(K j , E i ); Therefore by Lemma 3.2 and a straightforward computation shows that The statement of lemma follows.
Combining the results of two lemmas, we get: , which proves the theorem.
Remark 3.8. In order to recover the shuffle presentation in [49,Theorem 10.1], we can add an action of another torus as in Example 2.4. If we denote by q the T ′ -character of weight by −1, we get O(∆) = q −1 , and we obtain the desired presentation after further replacing t by qt. Unfortunately, we do not have a succinct explanation for this change of variables. Morally speaking, it occurs because in the natural compactification A 2 ⊂ P 2 the divisor at infinity is "diagonal", and for T * C ⊂ P(T * C) it is "horizontal".
Even though we have got an explicit formula, the morphism ρ depends on the embedding of C d into a smooth ambient variety T * Quot • 0,d . Unfortunately, the scheme C d is highly singular; for instance, the inclusion C d ֒→ C d is not known to be lci, so that we can not localize to T-fixed points directly. Still, we can do a little better. Letμ be the composition We introduce the following auxiliary variety, analogous to the one in [25]: .
Proof. First of all, the statement is true for C = A 1 [25]. For general C, it suffices to prove that For any ν = (ν 1 ≥ · · · ≥ ν k ) partition of d let Then S d C = ν⊢d S ν C, and this defines a stratification of C ∆ d : Consider the restriction of σ to these strata. For any point x ∈ S ν C, we have a G d -equivariant map The image of this map is an open subset where the vector subspaces defined by points in Grassmanians do not intersect. At each such point, the G d -action induces an isomorphism between the preimage of τ and i C n,•,∆ Since in particular this applies to C = A 1 , we have be the natural closed embeddings, and consider the composition Proof. Denote the closed embedding C ∆ d ֒→ T * Quot • 0,d by j ′ d . By Corollary A.16, we have the following identity: Note that the mapμ is T -equivariant. Since T contracts T * Quot • 0,d to a subvariety of C ∆ d and dim C ∆ d = dimμ, the argument similar to the one found in [14, Proposition 2.3.2] shows thatμ is flat. In particular, , by base change; see [13, B.7.4]. Therefore, we have Since both ρ and RN are morphisms of algebras, ̺ is as well.
Remark 3.11. For any function g, one can equip the algebra ASh g with a topological coproduct, analogous to the coproduct in [41,Section 4]. If the morphism ρ is injective (see Section 4 for discussion and partial results), it can be used to induce a coproduct on AHa 0 C . However, it is less clear how to construct such coproduct without using shuffle presentation.
Let us conclude this section by computing some relations in the algebra ASh g for an arbitrary rational function g(z). We write g(z) = h 1 (z)/h 2 (z), where h 1 , h 2 are polynomials. Given a line bundle L on C, define a bi-infinite series where z is a formal variable, and we consider e(z) to be a formal expression. Using the formal group law ⋆ associated to A (see Appendix A), we have the following equality for some f ∈ A * (pt) u, v : Therefore, by implicit function theorem for formal series [50, Exercise 5.59] e(z −1 ) can be interpreted as a formal series in e(z). In particular, g (w/z) is a formal series in e(w), e(z).
Proposition 3.12. Let L 1 , L 2 be two line bundles on C, and g(z) a rational function. Suppose that e(zw) is a polynomial in e(z) and e(w), and e(z −1 ) is a Laurent polynomial of e(z). Then the following equality holds: , the product between E L 1 and E L 2 is taken in ASh g , and we consider both sides as bi-infinite series in e(z), e(w) with coefficients in ASh g [2].
Proof. In order to unburden the notation, denote Z = e(z), W = e(w), T i = e(t i ), L i = e(L i ).
Let us also introduce bi-infinite series δ(z) = i∈Z z i . Note that for any Laurent polynomial f (z) the following identity, which we call change of variables, is satisfied: We have: Therefore, the equality (10) is equivalent to the following: However, using change of variables (11) for LHS we get: By the same reasoning RHS is also equal to zero. Therefore (12) is satisfied, and we may conclude.
In particular, if we set L 1 = t −1 1 and L 2 = t −1 2 , the equality (10) assumes a simpler form: where E(z) = i∈Z e(t −1 1 ) −i e(z) i . Remark 3.13. Note that the conditions of Proposition 3.12 are extremely restrictive. While they are satisfied for A = H, already for A = K the Euler class e(z −1 ) is not a Laurent polynomial of e(z). However, if we denoteẽ(z) = 1 − e(z), thenẽ(z −1 ) =ẽ(z) −1 , and thus the proof of relations (10) goes through if we replace E L (z) by In particular, we recover the identity (3.4) in [40]. This slight discrepancy is related to the fact that our K-theory, considered in the context of OBM homology theories, has a different set of equivariant generators from the usual K-theory, as defined for instance in [9,Chapter 5]. Nevertheless, the two are isomorphic up to a certain completion, see Remark A.4.

Injectivity of shuffle presentation
Let ω = ω C be the canonical bundle of C. Applying Serre duality to (3), one sees that -points of Higgs 0,d are given by pairs (E, θ), where E ∈ Coh 0,d , and θ ∈ Hom(E, E ⊗ ω). We call θ the Higgs field.
We denote the stack of nilpotent Higgs torsion sheaves by Higgs nilp 0,d . It is a closed substack of Higgs 0,d , which has the following global quotient presentation:  14) x ∈ X : G.x ∩ Λ = ∅ = Λ, and assume that for any x ∈ X the intersection G m .x ∩ Λ is not empty. Then the pushforward along i induces an isomorphism of localized A-groups: Proof. Note that our assumptions imply X G×Gm ⊂ Λ. Furthermore, by Proposition A.7 we can assume that G is a torus. Take x ∈ X, and let (g, t) ∈ G × G m lie in the stabilizer of x. Suppose that t has infinite order. Then t −1 .x = g.x, and by consequence G.x ∩ Λ = (G × T ).x ∩ Λ is non-empty, so that x ∈ Λ. We conclude that for any x ∈ X \ Λ there exists a positive number . Since torus actions on finite type schemes always possess finitely many stabilizers, one can assume that N = N (x) does not depend on x. Let us consider the following character of G × G m : It is clear that for any x ∈ X \ Λ one has Stab G×Gm (x) ⊂ Ker χ. Therefore by Proposition A.13 one has an isomorphism which implies the desired result. Proof. Take X = C d , Λ = C •,n d . Any point in (p, β) ∈ X \ Λ is separated from Λ by the characteristic polynomial of β. Therefore condition (14) is verified. Moreover, the action of T contracts any Higgs field to zero, that is for any x ∈ X the intersection T.x ∩ Quot • 0,d ⊂ T.x ∩ Λ is not empty. We conclude by invoking Proposition 4.2.
From now on till the end of the section we suppose that = C, and A is the usual Borel-Moore homology H. Corollary 4.5. The morphism ρ : HHa 0,T C → HSh C of Theorem 3.5 becomes injective after tensoring by Frac(H T (pt)).
We will prove Theorem 4.4 in three steps: (1) shrink localizing set; (2) reduce the question to Coh 0,d ⊂ Higgs nilp 0,d ; The H * G (pt)-action on the latter space is given as follows. The natural free Q[z]-module structure on H * (C)[z] defines embeddings of algebras: The upper horizontal map defines us the desired action. Note that since lower horizontal ). In particular, this implies that the H * G×T (pt)-module H T * (Coh 0,d ) loc,I is torsionfree. Putting together the arguments above, we get the following result: Proposition 4.6. The group H T * (Higgs nilp 0,d ) loc,I is torsion-free as a H * G×T (pt)-module. Next, let us break down the stack of nilpotent Higgs sheaves into more manageable pieces. Recall the following stratification of Higgs nilp 0,d due to Laumon [27].  Proof. See the proof of Proposition 5.2 in [34].
As a consequence of this proposition, for any ν ⊢ d we have an isomorphism Before continuing with the rest of the proof, let us recall some basic properties of weight filtration from [10,11] and references therein. For any algebraic variety X, Deligne constructed the weight filtration W k on cohomology groups H i (X). This filtration is compatible with Künneth isomorphisms. Moreover, it is strictly compatible with natural maps, in the sense that an element in target group belongs to W k if and only if it is an image of an element in W k . We say that the weight filtration on . This is the case for any smooth projective variety X, as well as for classifying spaces BG. Weight filtration also exists for Borel-Moore homology and in equivariant setting; it can thus be extended to homology groups of quotient stacks. Proof. The weight filtration is strictly compatible with all maps in the long exact sequence. In particular, since H G i (U ) and H G i−1 (Z) are pure and have different weights, the connecting homomorphism vanishes. Furthermore, by strict compatibility we have the following short exact sequences for each j: Let us choose a total order ≺ on the set of partitions of d such that for any two partitions For each ν, this order gives rise to a long exact sequence in Borel-Moore homology: The homology groups H * (Coh 0,d ) comprise the S d -invariant part of H * (C × BG m ) ⊗d . Since the latter group has pure weight filtration, the same is true for the former as well, and by (16) for H T * (N il ν ) for any ν. A straightforward induction on ν using Lemma 4.9 shows that both H T * (N il ≺ν ) and H T * (N il ν ) are also pure. Additionally, the long exact sequence (17) splits into short exact sequences: where ∆ ∈ H 2 (C × C) is the class of diagonal. Using this explicit expression, we can rewrite the identity (10) as a set of relations. In particular, for For general L 1 and L 2 the relations become more complicated.

Conjecture 4.12.
For any oriented Borel-Moore theory A, the morphism ρ : AHa 0,T C → ASh C of Theorem 3.5 is injective.
We hope to prove Conjecture 4.12 in subsequent work by analyzing the action of AHa 0,T C on modules AM T n for varying n, defined in next section.

Moduli of stable Higgs triples
In this section we introduce an action of AHa 0 C on the A-theory of certain varieties, which can be regarded as generalization of the Hilbert schemes of points on T * C (see Section 7).
We start with the stack Coh ←F 0,d , where F ∈ Coh C is a fixed coherent sheaf on C. The following proposition seems to be well-known (compare to [20, Theorem 4.1.(i)] and the entirety of [17]), but we did not manage to find a precise reference.  Hom(E, I 1 ) Hom(E, I 2 ) · · · Hom(F, I 0 ) Hom(F, I 1 ) Taking cohomology of its total complex, we get But by Yoneda construction, pullback of the extension 0 → E → I 0 → Ker d 1 → 0 gives a bijection between self-extensions of E and morphisms E → Ker d 1 up to the ones factorizing through I 0 . Associating to every element ρ ∈ Ext 1 (E, E) the corresponding extension 0 → E → E ρ πρ − → E → 0, we get which is precisely the space of infinitesimal deformations of (E, α) as seen above in the diagram (19).

Definition 5.2.
A Higgs triple of rank r, degree d and frame F is the data (E, α, θ) of a coherent sheaf E ∈ Coh r,d C, a map α : F → E, and an element θ ∈ xt 1 (E, (F α − → E) ⊗ ω). Given two Higgs triples Thanks to Serre duality and Proposition 5.1, the -points of the stack T * Coh ←F 0,d are precisely Higgs triples of rank 0, degree d and frame F. More generally, its T -points for any scheme T are given by families of triples (E T , α T , θ T ), where E T is flat over T .

Definition 5.3. A Higgs triple is called stable if there is no subsheaf E ′ ⊂ E such that:
• Im α ⊂ E ′ , and • a(θ) ∈ Im(b), where a, b are the maps below, induced by inclusion E ′ ⊂ E: In other words, a triple is stable if the image of α generates E under θ. We denote by T * Coh ←F 0,d st ⊂ T * Coh ←F 0,d the substack of stable Higgs triples of rank 0. Recall that for an abelian category C of homological dimension 1 every complex in the bounded derived category D b (C ) is quasi-isomorphic to the direct sum of its shifted cohomology objects (see [ Remark 5.5. Note that the stability condition does not depend on θ e in this form. Proof. Replacing the complex F α − → E by the sum of its kernel and cokernel, the diagram (20) splits into two: Note that the map b e is an isomorphism. Therefore, the condition a(θ) ∈ Im(b) is equivalent to We say that a morphism of triples is a quotient, if the underlying map of sheaves is surjective.
The following lemma can be viewed as an avatar of Schur's lemma. Proof. Let T = (E, α, θ) be a stable Higgs triple, and suppose f ∈ End(E) induces an automorphism of T . We pose and thus θ h (E ′ ) ⊂ E ′ / Im α. This means that E ′ is a destabilizing subsheaf, which can only happen for E ′ = E. Thus f = id E .
From now on, we will only consider Higgs triples of rank 0.
Theorem 5.8. Let F be a locally free sheaf on C. Then the moduli stack of stable Higgs triples of rank 0, degree d and frame F is represented by a smooth quasi-projective variety B(d, F). In particular, B(d, O) ≃ Hilb d T * C.
We will prove this theorem in Section 7 by realizing B(d, F) as a moduli of torsion-free sheaves on a ruled surface. It is also possible to prove it directly by relating stability of Higgs triples to Mumford's GIT stability [36] on an atlas of T * Coh ←F 0,d , which was the approach used in a previous version of this paper.
Let us further assume that F ≃ n ⊗ O is a trivial sheaf of rank n. To simplify the notation, we will write 1 Br n 0,d : In the remainder of this section we will produce an action of AHa 0 C on the A-theory of moduli spaces B(d, n). In order to do this, we will use the general machinery from the beginning of Section 2. Let and F a vector space of dimension n. As before, we note G = G d , P = P d• . We put: We have a natural closed embeddingg :Ṽ ֒→X ′ and an affine fibrationf :Ṽ ։Ỹ . The formula (5) gives rise to a map in A-theory For instance, in the case k = 2 we get a map: Collecting these maps for all d 1 , d 2 , we get a map Proposition 5.9. The map m defines an AHa 0 C -module structure on AM n . Proof. The proof is mostly analogous to the proof of Theorem 2.2. Namely, using notations of that proof, let us consider the following varieties: . Again, we have inclusionsW i ֒→X i−1 ×X i+1 . Taking into account Hom-terms, the proof of Lemma 2.3 easily implies thatW 2 =W 1 ×X 2W 3 , and that the intersection and contemplating the diagram with cartesian square below: we may conclude as in the proof of Theorem 2.2. 1 Br stands for Bradlow, as in "Bradlow pairs" [51] Recall that we have open embeddings B(d, n) ⊂ T * Br n 0,d . If we denote the collection of these embeddings defines us a map of graded vector spaces which is surjective if A = H by Proposition A.10.
Corollary 5.10. There exists a AHa 0 C -module structure on AM n , such that the map (24) commutes with the action of AHa 0 C . Proof. Let us consider the following diagram: Recall that quotients of stable triples are stable by Lemma 5.6. Therefore we have an equalitỹ which shows that the mapΨ ′ is proper, and right square in the diagram above is cartesian. Hence, we have i * •Ψ * =Ψ ′ * • i * by Lemma A.12. This shows us that the diagram (25) defines a commutative square Moreover, if we replace all varieties in diagram (23) by open subvarieties of stable points as above, we can equally see that the upper right square remains cartesian. Therefore the map m ′ defines an AHa 0 C -module structure on AM n .
Since the whole construction is T -equivariant, we also obtain an action of AHa 0,T C on AM T n := d A T * (T * Br n 0,d ) and AM T n := d A T * (B(d, n)). We finish this section by comparing our results with the classical construction of Grojnowski and Nakajima. Recall [38,Chapter 8] that for any smooth surface S there exists an action of Heisenberg algebra on d H * (Hilb d S). More precisely, for any positive k and any homology class α ∈ H * (X) we possess an operator P α [k], given as follows: We now suppose that S = T * C. Let us compare this action with the HHa 0 C -action on HM 1 . In view of Theorem 5.8, HM 1 = H * (Hilb d T * C). Recall that Higgs 0,k ≃ Coh k (T * C), where the latter stack parametrizes coherent sheaves of length k on T * C. Therefore, the correspondence defining the HHa 0 C -module structure on HM 1 can be identified with the lower row in the following diagram with cartesian square: i is the natural closed embedding Coh ∆ k (T * C) ֒→ Coh k (T * C), and s : Coh ∆ k (T * C) → T * C sends each coherent sheaf to its support. One would like to prove an equality of the form so that the operators P α [k] are realized by action of certain elements in HHa 0 C , supported at diagonals Coh ∆ k (T * C). Unfortunately, the map s is too singular for a pullback to be well-defined. However, one can easily check that it is a locally trivial fibration with a fiber isomorphic to If the local system I k s on T * C, given by homology groups of fibers of s, were trivial, H * (Coh ∆ k (T * C)) would be isomorphic to the direct product H * (T * C) ⊗ H G k * (C n,n k ), and one would be able to define the pullback s ! by c → c ⊠ 1. After that, the identity (26) would follow once we proved that p ! = (Φ ∆ ) ! • (s ! × id). In light of these considerations, let us state the following conjecture: Conjecture 5.12. The local system I k s is trivial, and the action of P α [i] on HM 1 ≃ d H * (Hilb d T * C) is given by ι * (α ⊠ 1) ∈ HHa 0 C .
Note that Conjecture 5.12 is trivially satisfied for k = 1. Indeed, Coh ∆ 1 (T * C) ≃ Coh 1 (T * C) ≃ T * C × BG m , thus the diagram above takes the following form: Since the scheme Z is smooth by [8], pullbacks along all of the maps in triangle are well-defined, and therefore which gives us a realization of operators P α [1].

Quiver sheaves
In this section we recollect some properties of quiver sheaves, as introduced in [17]. Let X be a scheme over . Let Q = (I, E) be a finite quiver with head and tail maps h, t : E → I, and assume that Q has no cycles. For each edge a ∈ E, pick a locally free sheaf M a ∈ Coh X, and set M i = O X for all i ∈ I.
Observe that A 0 = i∈I M i is a sheaf of O X -algebras with coordinate-wise multiplication. We equip A 1 = a∈E M a with an A 0 -bimodule structure, where the map is the natural isomorphism if h(a) = i, t(a) = j, and zero otherwise.

By definition of A, it decomposes into the direct sum
so that we have an equality of left A-modules A = i∈I P i and of right A-modules A = i∈I I i . Note that the multiplication map (27) ensures we have maps of A-modules m (i) a : M a ⊗ I t(a) → I h(a) , m (p) a : P h(a) ⊗ M a → P t(a) . An element V ∈ A-mod can be equivalently defined as a collection (V i , ϕ a ) of coherent O Xmodules V i , i ∈ I, together with morphisms ϕ a : M a ⊗ V t(a) → V h(a) for all a ∈ E. Under this identification, we have a natural isomorphism of O X -modules I i ⊗ A V ≃ V i . Since the forgetful functor A-mod → A 0 -mod is faithful, we deduce that the functor I i ⊗ A − is exact.

Proposition 6.2. We have an exact sequence of left
where all tensor products are considered over O X , p is the concatenation Ae i ⊗ e i A → A, and q is given by q(x, n, y) = m (n, y). Proof. The statement is local in X. When X = Spec R is affine, this is the standard resolution of the twisted path algebra as a bimodule over itself [7, (1.2)].
Let us now consider the derived category D b (A-mod). Corollary 6.3. For any V ∈ A-mod, we have a short exact sequence More generally, for any V • ∈ D b (A-mod) we have an exact triangle Apply the functor − ⊗ A V • to the exact sequence from Proposition 6.2.
Let (V i , ϕ a ), (W, ψ a ) ∈ A-mod, and consider the following complex of sheaves: where δ is given by Let us consider a closely related category A-mod D . Its objects are given by collections We have a functor F : where the maps ϕ a are induced by multiplication maps (27).
Lemma 6.5. The functor F is full and essentially surjective.
Proof. Given an object (V • i , ϕ a ) ∈ A-mod D , let V • be a mapping cone of the map By Corollary 6.3, we have F (V • ) = (V • i , ϕ a ), so that F is essentially surjective. Moreover, if we consider any morphism (V • i , ϕ a ) → (W • i , ψ a ) in A-mod D , the existence of a compatible morphism V • → W • follows from the axioms of a triangulated category. Thus F is full, and we may conclude.
Given a category C , let us denote by C the groupoid obtained from C by forgetting all non-invertible morphisms.
Corollary 6.6. The functor F induces an equivalence of groupoids F ′ : Proof. Consider the forgetful functor . It preserves isomorphisms and factors through F . Therefore, F ′ is faithful.
7. Torsion-free sheaves on P(T * C) In this section we prove Theorem 5.8 by realizing the moduli of Higgs triples as a certain moduli of sheaves on a surface.
Let X be a scheme over , not necessarily smooth. Pick a line bundle L over X, and consider the projectivization S = P X (L ⊕ O X ) of its total space Tot L. Denote the complement of Tot L in S by D; let also i : D ֒→ S be the natural embedding, and π : S → X the natural projection. Note that by definition of S and D we have Rπ * O(D) = π * O(D) = O X ⊕ L ∨ , and π induces an isomorphism D ≃ X.
Let T = O S (D) ⊕ O S , and consider the sheaf of O X -algebras π * Hom(T , T ). We can write it as a matrix algebra over X; the opposite algebra, which we denote by A, is then obtained by transposition: Note that A can be seen as a twisted path algebra of the following quiver: For any coherent sheaf E ∈ Coh S, the Hom-sheaf Hom(T , E) = π * (T ∨ ⊗ E) is naturally a left A-module, given by the quadruple (π * E(−D), π * E, ϕ 0 , ϕ 1 ), where (ϕ 0 , ϕ 1 ) is the natural composition we have a pair of adjoint functors As a left module over itself, A can be decomposed as a direct sum P 1 ⊕ P 2 , where are the left A-modules defined in Section 6. The following proposition should be known to experts (for example, see remark at the end of [3]), but we include the proof for completeness. Proof. The proof is based on Beȋlinson's lemma [3]. For any E ∈ Coh S, there exists n > 0 such that E(nD) has no higher cohomology, and the counit map π * π * E(nD) → E(nD) is surjective. By the seesaw principle [35,Corollary 5.6], the kernel of this map has the form π * (N )(−D), where N ∈ Coh X. Thus E admits a resolution of the form where N 1 , N 2 ∈ Coh X. Taking into account short exact sequences we see that as a triangulated category, D b (Coh S) is generated by Coh X and T = O S (D) ⊕ O S . Similarly, D b (A-mod) is generated by Coh X and A = P 1 ⊕ P 2 as a triangulated category by Corollary 6.3.
We have Rπ * Hom(T , O(D)) = P 1 , Rπ * Hom(T , O) = P 2 . Using Theorem 6.4, it is easy to check the following isomorphisms: Applying Beȋlinson's lemma, we conclude that the functor RHom X (T , −) is an equivalence of triangulated categories. Moreover, since the functor − ⊗ L A T is its left adjoint, it provides the inverse equivalence.
Let us apply this proposition to X = T × C, L = O T ⊠ ω C . Combining it with Corollary 6.6, we obtain an equivalence of groupoids (29) Θ : where p : T × C → C is the projection. Moreover, this equivalence commutes with base change in T whenever the latter preserves bounded derived categories, e.g. for flat maps T ′ → T .
Remark 7.2. It would be desirable to express this as an equivalence of presheaves in groupoids.
The problem is that groupoids on both sides of (29) are not functorial in T . Namely, boundedness of complexes is not preserved under pullbacks along general maps T ′ → T . Nevertheless, in the sequel we are only concerned with certain subgroupoids on both sides, see Proposition 7.7. Their objects will satisfy flatness condition over T , and therefor will be preserved under arbitrary base change, forming presheaves. We will thus abuse the notation for convenience, and say that the two sides of (29) form presheaves D b (Coh S) and A-mod D respectively.
From now on, let X = C, so that S = P C (ω ⊕ O) compactifies the cotangent bundle T * C. Let us recall some properties of sheaves on S; we will closely follow the exposition in [33, Section 2]. The Neron-Severi group of S is given by NS(S) = H 2 (S, Z) = ZD ⊕ Zf , where f is the class of a fiber of π : S → C. Thus, for any coherent sheaf E on S we will write the first Chern class c 1 (E) as a linear combination c 1,D (E)D + c 1,f (E)f . The product in NS(S) is determined by the following equalities: where the last one follows from the fact that O S (D)| D ≃ ω −1 . Moreover, the canonical divisor of S is K S = −2D. We will write elements of H even (S, Z) as triples (a, b, c) ∈ Z ⊕ NS(S) ⊕ Z; the same applies to H even (C, Z). In this fashion, Todd classes of S and C are respectively given by td S = (1, D, 1 − g), td C = (1, 1 − g), and the pushforward along π in cohomology is given by Given a sheaf E ∈ Coh S, the Chern character of its derived pushforward Rπ * E can be computed using Grothendieck-Riemann-Roch theorem. Namely, let a = c 1,D (E), b = c 1,f (E), r = rk E, and recall that ch(E) = r, c 1 (E), c 1 (E) 2 − 2c 2 (E) 2 = (r, aD + bf, a 2 (1 − g) + ab − c 2 (E)).
We have: The result of this computation can be rewritten as follows: For any nef divisor H on S, we can define a notion of H-semistability for sheaves on S. One example of nef divisor is given by f . Instead of giving general definitions, we will use the following characterization of f -semistable sheaves: . A torsion-free sheaf E on S is f -semistable if and only if its generic fiber over C is isomorphic to O P 1 (l) ⊕m for some l ∈ Z, m ∈ N.
Lemma 7.4. For a torsion-free f -semistable sheaf E, the following numerical conditions are equivalent: If these conditions are fulfilled, we further have H 0 (E) = H 2 (E) = 0, and H 1 (E) = H 0 (R 1 π * E).
Proof. Let us first prove the equivalence. 1 ⇒ 2: follows from the first formula in (30); 2 ⇒ 3: rank is a generic invariant, therefore we have Since m is a positive number, this implies that l = −1.
3 ⇒ 1: since RΓ(O P 1 (−1)) = 0, both π * E and R 1 π * E have rank 0. Furthermore, let T ⊂ π * E be a torsion subsheaf. By adjunction, we exhibit a map π * T → E from a torsion sheaf to a torsion-free sheaf. It is a zero map if and only if T = 0; thus π * E is locally free. We conclude that π * E = 0.
In order to prove the second statement, recall that we have Leray spectral sequence Since R j π * E = 0 for j = 1, it degenerates to the equality H i+1 (E) = H i (R 1 π * E). Finally, R 1 π * E is a torsion sheaf, so that H i (E) is non-zero only for i = 1.
We will also need the following computation: Proof. Let us denote K = Ker ε. Since ε is surjective and becomes an isomorphism after applying π * , we have Rπ * K = 0. This means that at each point c ∈ C the fiber K c is isomorphic to a direct sum of several copies of O P 1 (−1) [35,Corollary 5.4]. In particular, K(D) c is trivial at each point c, and thus the natural map π * π * (K(D)) → K(D) is an isomorphism. Consider the following short exact sequence: Note that all these sheaves have globally generated fibers over C. Therefore, after applying π * we obtain π * (K(D)) ≃ Ker (π * O(D) ⊗ π * O(D) → π * O(2D)) . However, since π * O(D) = O ⊕ ω ∨ , we have Therefore K ≃ π * π * (K(D)) ⊗ O(−D) ≃ π * ω ∨ (−D), and we may conclude.
Remark 7.6. For later purposes, let us fix an isomorphism π * (K(D)) ≃ ω ∨ , so that the inclusion Let us return to the equivalence (29). Fix d > 0, and a locally free sheaf F ∈ Coh C of rank n. Consider subfunctors Coh F d S ⊂ D b (Coh S), A-mod F d ⊂ A-mod D , defined as follows: Proposition 7.7. For any d > 0, the equivalence (29) induces a natural transformation → Rπ * E is obtained by applying Rπ * to the first map in the short exact sequence Pick a point t ∈ T . We have π * E(−D) t = 0 by Lemma 7.4, so that Rπ * E(−D) t = R 1 π * E(−D) t [−1]. Moreover, the formulas (30) applied to E(−D) t show that rk(Rπ * E(−D) t ) = 0 and c 1 (Rπ * E(−D) t ) = −d. Therefore, R 1 π * E(−D) t is a torsion sheaf of degree d.
Finally, let us prove flatness. Let f : T × S → T , p : T × C → T be the natural projections. Using the second part of Lemma 7.4, a proof analogous to [29,Corollary 4.2.12] shows that R 1 f * E(−D) is a locally free sheaf. Let L be an ample line bundle on C, and k ∈ N. Since R 1 π * E(−D) t is a torsion sheaf for any t ∈ T , it is isomorphic to L k ⊗ R 1 π * E(−D) in the neighborhood of t. In particular, the fact that p * (R 1 π * E(−D)) ≃ R 1 f * E(−D) is locally free implies that p * (L k ⊗ R 1 π * E(−D)) is locally free for any k. We conclude that R 1 π * E(−D) is flat over T by [21,Proposition 2.1.2].

Consider rigidified functors Coh
where we fix the additional data of an isomorphism Ψ : E| D ∼ − → F. We will refer to elements of (Coh ←F d S)( ) as F-framed sheaves. Lemma 7.8. Any F-framed sheaf is locally free in a neighborhood of D.
Proof. Recall that for any torsion-free sheaf E, its double dual E ∨∨ is a vector bundle. Let (E, Ψ) be an F-framed sheaf, and consider the quotient U = E ∨∨ /E. It is a sheaf with zerodimensional support. If the intersection D ∩ supp U is non-empty, E| D is a proper subsheaf of E ∨∨ | D . However, and rk E| D = rk E ∨∨ | D , so that E| D = E ∨∨ | D . Therefore the support of U is disjoint from D, and we have an isomorphism E ≃ E ∨∨ in a neighborhood of D.
Let us recall a closely related notion of stable pairs. We specialize the definition in [5] to the case when polarization of S given by the divisor H = D + N f , and N > 2g − 2. Recall that for any locally free sheaf E on C, its slope is defined as µ(E) = deg E/ rk E. Definition 7.9. Let E be a torsion-free sheaf on S satisfying ch E = (n, deg F · f, −d), and Ψ : E| D ∼ − → F an isomorphism. Fix N > 2g − 2, and δ > 0. A pair (E, Ψ) is said to be (N, δ)-stable, if for any subsheaf E ′ ⊂ E with 0 < rk E ′ < n the following inequality holds: It is known that the moduli of (N, δ)-stable pairs is represented by a quasi-projective variety, see [5,Theorem 2.3].
Proposition 7.10. There exist N, δ big enough, such that every F-framed sheaf (E, Ψ) is (N, δ)stable. In particular, the functor Coh ←F d S is represented by a quasi-projective variety B(d, F).
Proof. The (N, δ)-stability condition is vacuous for sheaves of rank 1. Therefore, we will assume that n ≥ 2. The existence of Harder-Narasimhan filtration [29,Chapter 5] implies that for a locally free sheaf F on C, there exists a constant µ max (F), such that µ(F ′ ) < µ max (F) for all F ′ ⊂ F. From now on, we will assume that δ > (µ max (F) − µ(F))n 2 , and N > 2g − 2 + δ.
. Consider the saturation E ′ of E ′ inside E. It has the same rank as E ′ , and c 1,D (E ′ ) ≤ 0. Moreover, since E is a vector bundle in the neighborhood of D by Lemma 7.8, E ′ is its subbundle in the same neighborhood. As a consequence, we have E ′ | D ⊂ E| D , and c 1 (E ′ )D = deg(E ′ | D ). Putting this together, we obtain (31) which is the desired estimate. Now, suppose E ′ ⊂ E(−D). In this case k 1 > 0, and E ′ is not contained in E(−(k 1 + 1)D). Let k be the maximal positive integer such that E ′ ⊂ E(−kD); we have k ≤ k 1 . In particular, E ′ (kD) is naturally a subsheaf of E, which is not contained in E(−D). Moreover, for a generic Therefore, the inequality (31) holds for E ′ (kD) by previous considerations. We have We can thus conclude that for our choice of N, δ every F-framed sheaf is (N, δ)-stable.
Remark 7.11. The divisor D ⊂ S is not nef when g(C) > 1, so that [5, Let us establish relation between A-mod ←F d and the stack of Higgs triples. Lemma 7.12. Let E, F ∈ Coh C, ϕ ∈ Hom(F, E), and C 1 , C 2 two cones of ϕ. Then there exists the unique map f ∈ Àom(C 1 , C 2 ) making the following diagram commute: where i, j are the natural maps.
Proof. The existence of map f is assured by axioms of triangulated category. Let f 1 , f 2 be two such maps, and consider their difference g = f 1 − f 2 : C 1 → C 2 . By definition, we have g • i = 0 and j • g = 0. Therefore, g lies in the image of composition Since both E and F lie in the heart of D b (Coh C), we have Àom(F [1], E) = 0. Thus g = 0, and the unicity of f follows.
Thanks to the lemma above, we can define a natural transformation τ : is the unique isomorphism given by Lemma 7.12.
Proposition 7.13. The functor τ is a natural equivalence.
Proof. Let us consider a natural transformation υ : T * Coh ←F 0,d → A-mod ←F d , defined on T -points by the following formula: Here ι is the natural map E[−1] → (F → E), and ∆ is obtained from the mapping cone of α: It is clear that τ • υ is the identity functor. On the other hand, the composition υ • τ sends The map f induces a natural equivalence υ •τ ≃ Id A-mod ←F d , so that τ and υ are mutually inverse equivalences.
Theorem 7.14. The composition ← − Θ = τ • Θ ← factors through the stack of stable Higgs triples, and induces an equivalence Proof. Since the functor ← − Θ is fully faithful, it is enough to compute its image on T -points of Coh ←F d S for every T . Let (E, α, θ) be a Higgs triple, and consider the morphism where ε : π * π * O(D) → O(D) is the counit map from Lemma 7.5, and we use the identification π * O(D) ≃ O ⊕ ω ∨ . After restricting to D, we get The map ξ| D can be naturally completed to a distinguished triangle Thus, we obtain an identification Ψ : up to remembering the framing. Combining the inverses of each functor in the composition provided by Proposition 7.13, Lemma 6.5 and Proposition 7.1, we obtain the following left inverse of Θ ← on the set of isomorphism classes of T -points: where κ : π * E(D) ⊕ π * α[1] → π * E(D) ⊕ π * α[1] multiplies first summand by 1, and the second one by −1. In what follows, we are only interested in the set-theoretic image of G T . As such, we will not concern ourselves with functoriality, and will liberally make use of (non-unique) cones of various maps.
Let us consider the map G between -points. Recall (see Section 5) that as a complex, α is quasi-isomorphic to K ⊕ J[−1], where K = Ker α, J = Coker α. Thus, we can express ξ ′ as a sum: Let M ∈ Ext 1 (π * (E ⊗ ω ∨ ), π * K(D)) be the extension given by ξ ′ e . Then the two-step complex given by ξ ′ is quasi-isomorphic to M → π * J(D), with arrow defined as the composition of ξ ′ h with the projection M ։ π * (E ⊗ ω ∨ ). Consequently, the cone of ξ ′ has length 1 if and only if ξ ′ h is surjective. Proof. By abuse of notation, we will write ξ = ξ ′ h ⊗ω throughout the proof, and study surjectivity of ξ. Thanks to Lemma 7.15, ξ is adjoint to where θ = θ h as in Section 5, and ι = ι h : E ։ J is the natural projection. Let c ∈ C, and s = π −1 (c) = P(ω c ⊕ O c ). If we choose an identification ω c ≃ O c , the stalk ξ s at the point s is given by a linear combination aθ c + bι c for some a, b ∈ . In particular, ξ is surjective if and only if it is surjective at each point s ∈ S, that is for every c ∈ C and [a : b] ∈ P 1 the map aθ c + bι c is surjective.
Suppose that ξ is not surjective. Then there exists a point s ∈ π −1 (c), c ∈ C where surjectivity fails. This means that J ′ c := Im(aθ c + bι c ) is a proper subsheaf of J c for some a = 0, b. Denote Then Im α ⊂ E ′ and θ(E ′ ) ⊂ J ′ , which precludes the triple (E, α, θ) from being stable. Conversely, suppose that (E, α, θ) is destabilized by a subsheaf E ′ ⊂ E. Denote J ′ = ι(E ′ ). Let us choose a point c ∈ C, such that E ′ c ⊂ E c is a proper subsheaf. By assumption ξ(E ′ c ) ⊂ J ′ c and E/E ′ ≃ J/J ′ . Suppose ξ is surjective. Then for any s ∈ π −1 (c) the stalk ξ s induces an automorphism of E c /E ′ c . In particular, the map a Id +bθ c is an automorphism of E c /E ′ c for each [a : b] ∈ P 1 . However, E is a torsion sheaf, therefore E c /E ′ c is finite-dimensional as a -module. Because of this, θ must possess an eigenvalue λ, so that θ − λ Id cannot be invertible. Thus ξ is not surjective. Lemma 7.16 shows that any family of Higgs triples which contains a non-stable one is mapped outside of Coh ←F d S by G T . This proves that the essential image of We now need to show that every flat T -family (E T , α T , θ T ) of stable Higgs triples lies in the image of ← − Θ , or equivalently its image G T (E T , α T , θ T ) lies in Coh ←F d S(T ). By Lemma 7.16, it is a coherent sheaf E T on S × T , equipped with an isomorphism Ψ : Moreover, by construction E T is a subsheaf of M T , with latter being obtained as an extension of a torsion-free sheaf π * K T (D) by π * E T . As a consequence, the torsion Tor E T is contained in the support of π * E T . However, since F is locally free, the existence of Ψ implies that the support of Tor E T must be disjoint from D × T . Since the support of every subsheaf of π * E intersects D × T , we conclude that E is torsion-free.
Pick a point t ∈ T . Outside of the support of π * E, the complex ξ[−1] is quasi-isomorphic to π * F. By Lemma 7.3, it implies that E t is f -stable.
Let us compute the Chern character of E t : It remains to show that E T is T -flat. For this, we will express E T in a different fashion. In what follows, we will drop the subscript T , implicitly assuming that all objects live in families over T .
Denote by ev : H 0 (E) ⊗ O ։ E the natural evaluation map. Let K be the kernel of (α, ev) : The octahedral axiom applied to the composition F where the first map is defined by the triangle above, and the second map comes from the quasi-isomorphism E ≃ ξ ′ ⊗ (−D). One more application of the octahedral axiom gives rise to the following diagram: Here, E is defined as a cone of the composition above. Note that since both E and H 0 (E) ⊗ O S are sheaves (as opposed to complexes of sheaves), E is also a sheaf.
Recall that if N is a T -flat sheaf, and is a short exact sequence, then M 1 is T -flat if and only if M 2 is. As a consequence of this, π * K is T -flat as the kernel of π * (α, ev); the middle row of diagram (32) shows E is T -flat; and finally, the middle column of (32) shows that E is T -flat as well.
Proof of Theorem 5.8. Representability follows from Theorem 7.14 together with Proposition 7.10. By definition of f -stability, E| π −1 (c) ≃ O n P 1 for a generic point c ∈ C. Since O P 1 (−1) has no global sections, this implies that the image of ϕ must be a torsion sheaf. However, E(−D) is a torsion-free sheaf, so we may conclude.
For the second claim, let E 1 be a locally free sheaf of rank 1 on S, such that c 1 (E 1 ) = 0. By the seesaw principle, E 1 ≃ π * π * E 1 . In particular, if E 1 | D ≃ O C , then E 1 ≃ O S . By consequence, we have E ∨∨ ≃ O S for any (E, Ψ) ∈ B(d, 1), and fixing Ψ makes this isomorphism canonical. Therefore, the map In view of Theorem 5.8, it is instructive to compare our results with recent works of Neguţ [40,39]. For any smooth projective surface S and an ample divisor H, he considers the moduli space M of H-stable sheaves on S with varying second Chern class, and for every n ∈ Z defines an operator e n : K(M) → K(M × S) by Hecke correspondences. These operators generate a subalgebra A inside k>0 Hom(K(M) → K(M × S k )), which can be then projected to a shuffle algebra V sm . The content of Conjecture 3.20 in [40] is that this projection is supposed to be an isomorphism. This conjecture is proved under rather restrictive assumptions; for instance, it is required that K(S × S) ≃ K(S) ⊗ K(S).
Let us now take S = T * C together with a scaling action of T ≃ G m , and replace usual K-groups with their T -equivariant counterpart. In this case, the algebra V sm can be identified with the subalgebra of KSh norm C , generated by K(BG m ) ⊂ KSh norm C [1] ≃ K T (C × BG m ). If we further replace K-groups by Borel-Moore homology, then by Corollary 4.5 homological version of V sm is realized as a subalgebra of HHa 0,T C . Therefore, one can regard results of Section 5 as a "homological non-compact" version of Neguţ's conjecture for S = T * C, c 1,D = 0, and stability condition given by f . Another modest gain of our approach is that while A is given by operators on K-groups K(M), the definition of AHa 0,T C is independent from its natural representations, which allows to study this algebra without invoking torsion-free sheaves on T * C.
In general, one expects that the moduli of framed sheaves on P C (ω ⊕ O) with non-trivial first Chern class can be recovered from the moduli of stable Higgs triples of positive rank. Nevertheless, as stability condition for triples varies, Lemma 7.16 seems to suggest that the objects on S which correspond to stable Higgs sheaves do not have to lie in the usual heart of D b (Coh S). These questions will be investigated in future work [32].
Appendix A. Oriented Borel-Moore homology theories In this appendix we recall the notion of equivariant oriented Borel-Moore functor and recollect some of its properties. For a more detailed exposition, we refer the reader to the monograph [30] for a treatment of non-equivariant version, and to [19] for the equivariant case.
Definition A.1. An oriented Borel-Moore homology theory A on Sch/ (or OBM for short) is the data of: (1) for every object in X ∈ Sch/ , a graded abelian group A * (X); (2) for every projective morphism f : X → Y , a homomorphism f * : A * (X) → A * (Y ); (3) for every locally complete intersection (lci for short) morphism g : X → Y of relative dimension d, a homomorphism f * : A * (Y ) → A * +d (X); (4) an element ½ ∈ A 0 (pt), and for any X, Y ∈ Sch/ a bilinear pairing which is associative, commutative and has ½ as unit; satisfying the following conditions: (BM0) A * (X 1 ⊔ X 2 ) = A * (X 1 ) ⊕ A * (X 2 ); (BM1) Id * X = Id A * (X) , (f • g) * = g * • f * ; In [30], Levine and Morel define and study algebraic cobordism theory Σ * associated to the universal formal group law (L, F L ) on the Lazard ring L. Let us call an OBM A free if the natural map Σ * ⊗ L * A * (pt) → A * is an isomorphism. For this class of OBMs many properties will follow immediately after establishing them for Σ * .
Example A.2. Chow group functor CH * and the Grothendieck group of coherent sheaves K 0 are free OBMs.
Note that usual Borel-Moore homology is not an OBM, because of the presence of odddimensional part. Moreover, even the even-dimensional part fails to be a free OBM, which prevents us from translating results found in [30] in a straightforward way. Still, all of the results we need can be proved in a similar way for the usual Borel-Moore homology. We will thus abuse the language somewhat and allude to it as to a free OBM in the propositions below, giving separate proofs where needed; in the case of omitted proof, we will give a separate reference.
For any reductive group G, free OBM A, and a G-variety X Heller and Malagón-López [19] define equivariant homology groups A G * (X). Roughly speaking, the group G has a classifying space represented by a projective system {EG N } N ∈N of G-varieties, and we set For example, if G = GL d , the varieties EG N are just the Grassmanians Grass d (d, N ). Most of the constructions mentioned above for ordinary OBMs can be extended to the equivariant ones. Remark A.4. One can observe that in the case of algebraic K-theory we get K T (pt) = Z[1 − t −1 1 , . . . , 1 − t −1 d ], which is different from the usual ring of Laurent polynomials Z t ±1 1 , . . . , t ±1 d . However, the two become isomorphic after passing to completion. One can prove that this happens for any T -scheme X, using the argument in [1, Lemma 3.1] (the author would like to thank Gufang Zhao for pointing out this article).
From now on until the end of appendix, let us fix a free OBM A. Moreover, since we are not concerned with questions of integrality, we also assume that A * (pt) contains Q, so that all A-groups are Q-vector spaces. We will often omit homological grading from notations, and write A = A * , A G = A G * , A G = A * G . Remark A.5. To the best of author's knowledge, the notion of oriented Borel-Moore homology theory is not yet fully developed for arbitrary algebraic stacks. However, since all the stacks of interest in our paper are quotient stacks, we usually slightly abuse the notation and write for any quotient stack [X/G] (see also [19,Proposition 27]). where H acts on X × G diagonally.
Proposition A.7 ([19, Theorem 33]). Let G be a reductive simply connected algebraic group, T ⊂ G a maximal torus with normalizer N , W = N/T the Weyl group, and X a G-variety. Then W acts on A T (X), and we have a natural isomorphism Proposition A.8. Let Z be a non-reduced G-scheme, and denote by Z red its reduction. Then the pushforward map along the natural embedding is an isomorphism.
Proof. Follows from the definition of algebraic cobordism theory. For an explicit mention of this fact, see the proof of Proposition 3.4.1 in [30].
The following proposition holds only for universal OBMs, usual Borel-Moore homology not included.  Given G-equivariant regular embeddings j : P ֒→ N , i : N ֒→ M , Whitney product formula applied to the short exact sequence 0 → T P N → T P M → j * T N M → 0 tells us that (33) e(T P M ) = e(T P N ) · j * e(T N M ).
For our purposes, one of the most important pieces of data coming from an OBM is the Gysin pullback. Let us state several compatibility results about it.
Lemma A.12. The following properties of Gysin pullback are verified: (1) Gysin pullback commutes with composition, that is for any diagram with cartesian squares , provided that f 1 and f 2 are locally complete intersections; (2) let F : Y → X, G : X ′ → X, ι : Z → X be morphisms of schemes such that F is lci, G and ι are proper, and F and G are transversal. Consider the following diagram, where all squares are cartesian: Then we have an equality Proof. See [30, Theorem 6.6.6(3)] for (1) and [52, Lemma 1.14] for (2).
Proposition A.13. Let i : Y ֒→ X be a closed embedding of T -varieties, and {χ 1 , . . . , χ k } ⊂ T ∨ a finite set of characters. Suppose that X T is not empty, X T ⊂ Y , and for any point x ∈ X \ Y its stabilizer under the action of T is contained in k i=1 Ker(χ i ). Then the pushforward along i induces an isomorphism In the interest of brevity, we will abuse the notation and write χ i instead of c 1 (χ i ). Let us start with surjectivity. By Proposition A.10 we have an exact sequence Thus it suffices to prove that A T * (X \ Y )[χ −1 1 , . . . , χ −1 k ] = 0. By Lemma 2 in [12], there exists an open subvariety U ⊂ X \Y and a subgroup T 1 ⊂ T such that U ≃Ũ ×T /T 1 as T -variety, whereŨ is equipped with a trivial action of T . In particular, A T * (U ) ≃ A * (Ũ ) ⊗ A * (pt) A T * (T /T 1 ). Because of our hypotheses, one has T 1 ⊂ Ker(χ i ) for some i, and thus χ i A T * (U ) = 0. We conclude by Noetherian induction. Namely, let Z be the complement of U in X \ Y . We have the following exact sequence: . By induction pA T (Z) = 0, where p is a monomial in χ 1 , . . . χ k . Therefore A T (X) is annihilated by χ i p, and thus A T (X \ Y )[χ −1 1 , . . . , χ −1 k ] = 0. It is left to prove injectivity. If A = H, we may already conclude by invoking long exact sequence in homology. Otherwise, we follow an approach found in [4, 2.3, Corollary 2]. First, let us denote Ker(χ i ) .