Higher rank sheaves on threefolds and functional equations

We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf is not locally free. On a threefold it has dimension $\leq 1$. We consider the open subset of moduli space consisting of sheaves with empty or 0-dimensional singularity set. For fixed Chern classes $c_1,c_2$ and summing over $c_3$, we show that the generating function of topological Euler characteristics of these open subsets equals a power of the MacMahon function times a Laurent polynomial. This Laurent polynomial is invariant under $q \leftrightarrow q^{-1}$ (upon replacing $c_1 \leftrightarrow -c_1$). For some choices of $c_1,c_2$ these open subsets equal the entire moduli space. The proof involves wall-crossing from Quot schemes of a higher rank reflexive sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret this sublocus in terms of the singularities of the reflexive sheaf.


Introduction
Let X be a smooth projective threefold. We consider torsion free sheaves F of homological dimension ≤ 1 on X, i.e. torsion free sheaves which are locally free or can be resolved by a 2-term complex of locally free sheaves. They have the property that E xt >1 (F , O X ) = 0 and E xt 1 (F , O X ) has dimension ≤ 1. Important examples are reflexive sheaves. They are precisely the torsion free sheaves F of homological dimension ≤ 1 for which E xt 1 (F , O X ) is zero or 0-dimensional. We refer to [HL,Prop. 1.1.10] for details.
We first study Quot schemes Quot X (F , n) of length n quotients F Q. We start with some notation. The reciprocal of a polynomial P (q) of degree d is P * (q) := q d P (q −1 ).
Moreover P is called palindromic when P * (q) = P (q). We denote the MacMahon function by Furthermore e(·) denotes topological Euler characteristic. Theorem 1.1. For any rank r torsion free sheaf F of homological dimension ≤ 1 on a smooth projective threefold X, we have ∞ n=0 e Quot X (F , n) q n = M(q) re(X) ∞ n=0 e Quot X (E xt 1 (F , O X ), n) q n .
We expect that the generating function of Euler characteristics of Quot schemes of 0-dimensional quotients of E xt 1 (F , O X ) in Theorem 1.1 is always a rational function. We prove this in the toric case in a sequel [GK4]. In any case, when F = R is reflexive it is a polynomial and satisfies a nice duality.
be the (possibly empty) open subset of isomorphism classes [F ] for which F * * /F is zero or 0-dimensional. We prove openness in Lemma 2.2.
The open subset (1) has an alternative description as follows. The singularity set of a coherent sheaf F is defined as the locus where F is not locally free [OSS]. When F is torsion free on X, the singularity set has dimension ≤ 1. In Lemma 2.2 we show that M H X (r, c 1 , c 2 , c 3 ) • is also the locus of sheaves with empty or 0-dimensional singularity set. is a Laurent polynomial in q satisfying P r,c 1 ,c 2 (q −1 ) = P r,−c 1 ,c 2 (q).
For fixed c 1 ∈ H 2 (X, Z), let c 2 ∈ H 4 (X, Z) be chosen such that c 2 · H is the minimal value for which there exist rank r µ H -stable torsion free sheaves on X with Chern classes c 1 , c 2 or −c 1 , c 2 . Such a minimal value exists by the Bogomolov inequality [HL,Thm. 7 for all c 3 (e.g. by [GKY,Prop. 3.1]). In this case Theorem 1.3 gives a functional equation for the complete generating function. Explicit examples of the Laurent polynomials appearing in Theorem 1.3 for X a smooth projective toric threefold can be found in [GKY,]. E.g. for X = P 3 , r = 2, c 1 = H, c 2 = H 2 , where H denotes the hyperplane class, we have P 2,c 1 ,c 2 (q) = 4q −1 + 4q.

The rank 1 case
Suppose C ⊂ X is a Cohen-Macaulay curve on a smooth projective threefold. Then its ideal sheaf I C has homological dimension 1. By dualizing the ideal sheaf sequence, we obtain an isomorphism In turn, dualizing any short exact sequence where Q is 0-dimensional, produces a Pandharipande-Thomas pair [PT1] 0 where F := E xt 2 (F , O X ), Q := E xt 3 (Q , O X ), and This uses [HL,Prop. 1.1.6,1.1.10]. These arguments show that the Quot schemes on the RHS in Theorem 1.1 are in bijective correspondence 1 to moduli spaces PT X (C, n) of Pandharipande-Thomas pairs (F, s) on 1 ↑ Here we only described a set theoretic bijection. This can be made into a morphism of schemes which is bijective on C-valued points much like in Theorem 3.4.
X with support curve C and χ(F) = χ(O C ) + n. Moreover, the Quot schemes on the LHS in Theorem 1.1 are in bijective correspondence with moduli spaces I X (C, n) of ideal sheaves I Z ⊂ I C such that I C /I Z is 0-dimensional of length n. So in this case, Theorem 1.1 reduces to the DT/PT correspondence on a fixed Cohen-Macaulay curve C proved by Stoppa-Thomas [ST] ∞ n=0 e I X (C, n) q n = M(q) e(X) ∞ n=0 e PT X (C, n) q n .
When X is Calabi-Yau, a weighted Euler characteristic version of this formula was proved by A. Ricolfi (for smooth C [Ric1,Ric2]) and G. Oberdieck (for any CM curve C [Obe]). Theorem 1.1 can be seen as a generalization of Stoppa-Thomas's result to arbitrary torsion free sheaves of homological dimension 1. Suppose X is Calabi-Yau, r = 1, and c 1 = 0. Then M H X (1, 0, c 2 , c 3 ) are Hilbert schemes of subschemes of dimension ≤ 1 of X and is a rational function invariant under q ↔ q −1 . This was proved by Y. Toda [Tod1]. A Behrend function version of this statement was proved by T. Bridgeland [Bri2]. This established the famous rationality and functional equation of Pandharipande-Thomas theory [PT1]. For general X, r, c 1 , c 2 we expect is a rational function and again we prove it in the toric case in [GK4]. Examples of this rational function were calculated in [GKY] and they are certainly not invariant under q → q −1 in general [GKY,Ex. 3.8].

The proof
Theorems 1.2, 1.3 are proved in Section 2. A special case of Theorem 1.1 was proved in [GK2], i.e. for F = R a rank 2 reflexive sheaf satisfying the following two additional conditions 2 • there exists a cosection R → O X such that the image is the ideal sheaf of a 1-dimensional scheme.
The proof of [GK2] uses the Serre correspondence and a rank 2 version of a Hall algebra calculation by Stoppa-Thomas [ST]. In this paper we first reduce to an affine version of Theorem 1.1 on Spec A (Section 2) with Let M be the A-module corresponding to F | Spec A . By a theorem of Bourbaki (see Theorem 3.1), there exists an A-module homomorphism M → A with free kernel. This cosection M → A allows us to prove an affine version of Theorem 1.1 in the rank 2 case following ideas of [GK2]. The higher rank case builds on the rank 2 case by a new inductive construction. The proof of the affine version of Theorem 1.1 occupies Sections 3 and 4.

Remark 1.4.
It is an interesting question to what extent the methods of this proof work without reduction to the affine case Spec A. Let X be a smooth projective threefold and F a torsion free sheaf of homological dimension 1 on X. Suppose there exists a cosection F → O X with locally free kernel. Then it appears that most methods of the proof of this paper (in particular Theorem 3.4) work globally on X as well. As mentioned above, when F = R is reflexive of rank 2, such a cosection exists (possibly after twisting by a line bundle). In general, the authors only have the required cosection in the affine case (Theorem 3.1), which leads to the current approach.
We expect that Theorem 1.1 is related to Toda's recent work on the higher rank DT/PT correspondence on Calabi-Yau threefolds [Tod2], which involves J. Lo's notion of higher rank Pandharipande-Thomas pairs [Lo1]. The latter were also related to Quot schemes of E xt 1 (R, O X ), where R is a µ H -stable reflexive sheaf, in [Lo1,Rem. 4.4]. Recently, after this paper appeared, the relationships between these works have been clarified and further investigated by Lo [Lo2] and S. Beentjes and Ricolfi [BR].
Theorem 1.1 has the nice property that it holds for any torsion free sheaf of homological dimension ≤ 1 (not necessarily stable or reflexive) on any smooth projective threefold (not necessarily Calabi-Yau).
Acknowledgements. We thank J. Lo, D. Maulik, R. Skjelnes, and R. P. Thomas for useful discussions. A.G. was partially supported by NSF grant DMS-1406788. M.K. was supported by Marie Skłodowska-Curie Project 656898.
Notation. All sheaves in this paper are coherent and all modules are finitely generated. The length of a 0-dimensional sheaf Q is denoted by (Q).
For any sheaf E with support of codimension c on a smooth variety X and any line bundle L on X, we define In this paper we often use [HL,Prop. 1.1.6,1.1.10]. Although these results are stated for smooth projective varieties, they also hold for smooth varieties and even for regular Noetherian C-schemes. 4 The topological Euler characteristic of a C-stack is by definition the naive Euler characteristic of the associated set of isomorphism classes of C-valued points as defined in [Joy1,Def. 4.8].

Consequences and reduction to affine case
In the first subsection, we prove that Theorem 1.1 implies Theorems 1.2 and 1.3. In the second subsection, we reduce Theorem 1.1 to the affine case.

2.A. Consequences
Proof of Theorem 1.2. Any reflexive sheaf R on a smooth threefold X has homological dimension ≤ 1 ([Har2, Prop. 1.3] and the Auslander-Buchsbaum formula). Therefore we have a resolution where E 0 , E 1 are locally free. Dualizing this short exact sequence and breaking up the resulting long exact sequence gives 3 ↑ This differs slightly from the notation of [HL], where E D denotes E xt c (E, ω X ). 4 ↑ Indeed the only place where projectivity is used, is in the proof of the vanishing statement of Prop. 1.1.6(i), namely E xt i (F , O X ) = 0 for all i < c where c is the codimension of the support of F . However this vanishing also holds at the level of local rings using "local duality" [Hun,Thm. 4.4], thereby avoiding projectivity.
where C is the image of E * 0 → E * 1 . Dualizing (2) gives where we recall that E xt 1 (R, O X ) is zero or 0-dimensional because R is reflexive [HL,Prop. 1.1.10]. Therefore we conclude Assume E xt 1 (R, O X ) is non-zero, because otherwise R is locally free and there is nothing to prove. Recall the definition of P R (q) from the statement of Theorem 1.2 and let d := deg P R (q), which equals (E xt 1 (R, O X )). We claim that there is an isomorphism Indeed any short exact sequence Note that (Q D ) = (Q) and Q DD Q for any 0-dimensional sheaf Q on X. 5 The claim follows. The first statement of the theorem follows from the following computation where we first used (5) and then (4). Finally we note that for any line bundle L we have The second statement of the theorem follows from the fact that for any rank 2 reflexive sheaf R we have Before we prove that Theorem 1.1 implies Theorem 1.3, we need two lemmas.
(2) If R is µ H -stable and reflexive, then so is R * .
Proof. Part (2) is a standard result. For (1) we take a locally free resolution resolution which exists as we saw in the previous proof. Dualizing gives where By [GK1,Prop. 3.6] and the fact that F * * is rank r µ H -stable with Chern classes c 1 , c 2 , we deduce that c 3 (F * * ) is bounded above: where C := C(X, H, r, c 1 , c 2 ) is a constant only depending on X, H, r, c 1 , c 2 . We can choose this constant such that C − c 3 ≥ 0 is even and we set N := 1 2 (C − c 3 ). From (6), (7), and (8), we conclude Here for any two polynomials We claim that inclusion (9) is an equality. Indeed if Q is 1-dimensional, then where E xt 1 (F , O X ) has dimension ≤ 1 and E xt 2 (F , O X ) has dimension ≤ 0 by [HL,Prop. 1.1.10]. The second statement of the lemma easily follows from the fact that the singularity set of a reflexive sheaf on X has dimension ≤ 0.
We are now ready to prove Theorem 1.3. We recall some notation and basic facts about constructible functions (e.g. see [Joy1] and references therein). Let Γ be any abelian group. Suppose φ : X → Y is a morphism of C-schemes, locally of finite type, and f : X(C) → Γ is a constructible function on the C-valued points X(C) of X. Then we define integration against the Euler characteristic measure by The following map is a constructible function [Joy1,Prop. 3.8] We will use the following key fact due to R. MacPherson [Mac] (see also [Joy1,Prop. 3.8]) We use a slight generalization of this. Suppose φ : X → Y is a set theoretic map between C-schemes of finite type. We call φ a constructible morphism when X can be written as a finite disjoint union n i=1 X i of locally closed subschemes X i ⊂ X of finite type such that φ| X i : X i → Y comes from a morphism of schemes. Then it is easy to see that φ * f (defined as above) is a constructible function, for any constructible function f : X(C) → Γ , and (10) holds.
Proof of Theorem 1.3. Fix r, c 1 , c 2 . Denote by N H X (r, c 1 , c 2 , c 3 ) the moduli space of rank r µ H -stable reflexive sheaves on X with Chern classes c 1 , c 2 , c 3 . By Lemma 2.1, we can consider the following two maps (·) * * : Restricted to M H X (r, c 1 , c 2 , c 3 ) • and N H X (r, c 1 , c 2 , c 3 ), these are constructible morphisms. 7 The first map is surjective on closed points and the second map is bijective on closed points. At the level of closed points, the fibre of (·) * * over [R] is a union of Quot schemes By [GK1,Prop. 3.6], there exists a constant C := C(X, H, r, c 1 , c 2 ) such that for any rank r µ H -stable reflexive sheaf R on X with Chern classes c 1 , c 2 we have Likewise there exists a constant C := C(X, H, r, −c 1 , c 2 ), such that for any rank r µ H -stable reflexive sheaf R on X with Chern classes −c 1 , c 2 we have Using Lemma 2.1, the Chern classes c 3 (R) and c 3 (R ) are also bounded below: where the last sum is finite because E xt 1 (R, O X ) is zero or 0-dimensional and P R (q) was introduced previously in Theorem 1.2. Let φ := (·) * * be the double dual map, then by (10) Since the fibres of φ = (·) * * are unions of Quot schemes Quot X (R, n), we find 8 where in the second line we use f = M(q −2 ) re(X) g (Theorem 1.1) and the third line is the definition of P r,c 1 ,c 2 (q). Since the sums over c 3 on the right hand side of (11) are finite, P r,c 1 ,c 2 (q) is a Laurent polynomial. By Lemma 2.1 we have c 3 (R) + c 3 (R * ) = 2 (E xt 1 (R, O X )). Therefore Theorem 1.2 gives

This translates into
Strictly speaking, f is not a constructible function, because it is an infinite sum. However, it is constructible modulo q −A for arbitrary A > 0. Therefore the following equations should be read modulo q −A . Since A > 0 is arbitrary, the final equality, Finally in the rank 2 case, we have c 3 (R) = (E xt 1 (R, O X )). Therefore Theorem 1.2 implies which gives P 2,c 1 ,c 2 (q) = P 2,c 1 ,c 2 (q −1 ).

2.B. Reduction to affine case
In this section we first prove Theorem 1.1 in the rather well-known case where F is locally free. Subsequently we reduce Theorem 1.1 to the "affine case". When X is a smooth variety and Z ⊂ X is a closed subscheme, one can look at the formal neighbourhood of Z ⊂ X which we denote by X Z . On an open affine subset U = Spec A ⊂ X, where Z is given by the ideal I, we define the formal neighbourhood X Z by 9 Since the map X Z → X is flat, we can use X Z as part of an fpqc cover along which we can glue sheaves [Sta,Tag 023T,Tag 03NV].
In this paper we will be using Quot schemes of 0-dimensional quotients of sheaves on several types of schemes. Originally Grothendieck's construction implies (in particular) that the Quot functor of 0-dimensional quotients is representable by a scheme Quot X (F , n) when X is a projective C-scheme and F is a coherent sheaf. This was extended to X quasi-projective in [AK]. T. S. Gustavsen, D. Laksov, R. M. Skjelnes [GLS] showed representability of the Quot functor of 0-dimensional quotients for X = Proj S (any graded C-algebra S) or X = Spec A (any C-algebra A) and F quasi-coherent. We need this level of generality because we will encounter the case A = C [[x, y, z]], which is Noetherian but not of finite type. Proposition 2.3. Let X be a smooth quasi-projective threefold and let F be a rank r locally free sheaf on X. Then ∞ n=0 e Quot X (F , n) q n = M(q) re(X) .
Proof. Let P be any closed point in X and consider the punctual Quot scheme We study the function There exists a torus action T = C * r on Quot X (O ⊕r X , n) 0 defined by scaling the summands of O ⊕r X . The fixed locus of this action is 9 ↑ Note that we define formal schemes by Spec of a ring rather than Spf of a ring.
where Hilb n (X) 0 denotes the punctual Hilbert scheme of length n subschemes of X supported at P . Combining (12) and (13), we obtain where the second equality follows from a result of J. Cheah [Che]. 10 Next we consider the Hilbert-Chow type morphism where P i are the C-valued points of the support of Q and n i is the length of Q at P i . Consider the constructible function where the second equality follows from [BK,App. A.2] and the third equality follows from (14).
This proposition implies Theorem 1.1 when F is locally free. We now reduce Theorem 1.1 to the affine case.
Proposition 2.4. Let X be a smooth projective threefold and let F be a rank r torsion free sheaf on X of homological dimension ≤ 1. We assume the following: if Spec A is an affine scheme of one of the following types • Spec A a Zariski open subset of X, • Spec A = Spec O X,P , where O X,P is the completion of the stalk O X,P at a closed point P ∈ X, where e(Spec A) = 1 for A = O X,P . Then Theorem 1.1 is true for X and F .
Proof. Take X and F as in the proposition. Assume (15) is true for any Spec A as described in the proposition. We will show that the formula of Theorem 1.1 is true for X and F . Let S ⊂ X be the scheme-theoretic support of E xt 1 (F , O X ). Then S has dimension ≤ 1 by [HL,Prop. 1.1.10]. Let C be the union of the 1-dimensional connected components of S and let Z be the union of the 0-dimensional connected components of S. We note that F | X\S is locally free (by [OSS, Ch. II]) and F | X\C is reflexive by [HL,Prop. 1.1.10].
10 ↑ This can be seen by noting that e(Hilb n (X) 0 ) = e(Hilb n (Spec C[[x, y, z]]). Using the standard C * 3 -action on Spec C [[x, y, z]], this is the number of monomial ideals in x, y, z defining a 0-dimensional scheme of length n.
There exists a closed subset H ⊂ X such that C ∩H is 0-dimensional, H ∩Z = ∅, and the complement U of H is affine. This can be seen by embedding X in a projective space and intersecting with a general hyperplane. Write U = Spec A ⊂ X. Let P 1 , . . . , P be the closed points of C ∩ H. Define We take the following fpqc cover of X By fpqc descent and the fact that we are considering 0-dimensional quotients (so there are no gluing conditions on overlaps), we obtain a geometrically bijective constructible morphism 11 from Quot X (F , n) to The expressions We conclude that (16) is equal to 11 ↑ The definition of a constructible morphism was given before the proof of Thm. 1.3. Such a map is called geometrically bijective, when it is a bijection on C-valued points. If φ : X → Y is a geometrically bijective constructible morphism, then e(X) = e(Y ) by (10).
12 ↑ As formulated, Proposition 2.3 only applies to a smooth quasi-projective threefold. However the same proof works in the present setting. The essential point is that for any C-valued point P of X • H • , the formal neighborhood of P in X • H • is Spec C [[x, y, z]] by [EGA,(7.8.3)] and [Sta,TAG 07NU,TAG 0323]. The Hilbert-Chow morphism is constructed at the level of generality we need by D. Rydh [Ryd].

The proposition follows from
Here we used that X • H • and H • have homeomorphic subspaces of C-valued points and each Spec O X,P i has a single C-valued point (Lemma 2.5 below). Proof. The surjection A A/I induces a homeomorphism from Spec A/I onto its image in Spec A. This map sends a prime ideal q containing I to q. Furthermore this map sends a maximal ideal m containing I to the maximal ideal m and all maximal ideals of A arise in this way [Bou1,Ch. III.3.4,Prop. 8].

A → A A/I,
where the first map is injective by Krull's theorem [AM,Cor. 10.18]. The ideal q := p ∩ A is a C-valued point and therefore corresponds to a maximal ideal (A is finitely generated). We claim I ⊂ q and p = q, which implies p is maximal by the first part of the proof. By the first part of the proof, there exists a maximal ideal m containing I such that p ⊂ m (any prime ideal lies in a maximal ideal). Therefore Since q is maximal, we have q = m, so q contains I and p ⊂ q. By [AM,Prop. 1.17,10.13], we obtain the other inclusion p ⊃ q ⊗ A A = q.
This proves the claim. We conclude that the collection of C-valued points of A and the collection of closed points of A coincide.

Pandharipande-Thomas pairs and Quot schemes
In this section we assume A is a 3-dimensional Noetherian regular C-algebra and M is an A-module of homological dimension 1. Later we will be interested in the case A, M arise from X, F as in Proposition 2.4.

Γ (Spec A, ·) : QCoh(Spec A) → A-Mod
is an equivalence of categories and we denote the inverse by (·) as in [Har1,Sect. II.5]. We will mostly work in the latter category A-Mod.
We give a very brief outline of this section. Denote by T A the stack of 0-dimensional finitely generated A-modules. 13 We first construct a Cohen-Macaulay curve C ⊂ Spec A, with ideal I C ⊂ A, and an effective divisor S ⊂ Spec A related to M. It turns out that we can use C, S to construct injections for all Q ∈ T A . By varying Q over the stack T A , we obtain a closed locus Σ inside the moduli space of Pandharipande-Thomas pairs on C. The main result of this section is Theorem 3.4, which establishes a geometric bijection 14 from Quot A (Ext 1 (M, A) Quot A (Ext 1 (M, A), n) onto the locus Σ. In Section 4, the locus Σ naturally arises from a Hall algebra calculation involving The Hall algebra calculation of Section 4 together with the geometric bijection of Theorem 3.4 will lead to the proof of (15) and therefore Theorem 1.1.

Theorem 3.1 (Bourbaki). Let A be a Noetherian integrally closed ring and M a finitely generated rank r torsion free A-module. Then there exists an A-module homomorphism M → A with free kernel.
Therefore there exists an ideal I ⊂ A and a short exact sequence Note that I A because M is not locally free by assumption (M has homological dimension 1). Although our constructions will depend on the cosection M → A, our final result, equation (15), will not. 15 We start with a lemma. Proof. There exists an effective divisor S ⊂ Spec A and a closed subscheme C ⊂ Spec A of dimension ≤ 1 such that I I C (−S).
This follows by embedding I → I * * and observing that I * * /I has dimension ≤ 1 and that I * * ⊂ A is a line bundle.
Next we show that C is not empty or 0-dimensional. If it were, then according to [HL,Prop. 1.1.6] (see Notation in Section 1), we would get Ext 1 (I C (−S), A) Ext 2 ((A/I C )(−S), A) = 0. Therefore (17)  Ext 2 (I C (−S), A) Ext 2 (M, A) = 0, 14 ↑ A geometric bijection is a morphism of schemes, which is a bijection on C-valued points. 15 ↑ In [GK2] we assumed the existence of a global version of this cosection. For rank 2 reflexive sheaves, this global cosection is precisely the data featuring in Hartshorne's version of the Serre correspondence [Har2]. This led to the assumptions made in [GK2] and explained in the introduction.
where the last equality follows from the fact that M has homological dimension 1. Therefore
From now on, we will use the following version of (17) where S ⊂ X is the effective divisor and ∅ C ⊂ X is the Cohen-Macaulay curve of Lemma 3.2. We are interested in We can write this as where Hom(M, Q) onto is the set of surjective A-module homomorphisms, and ∼ is the equivalence relation induced by automorphisms of Q. In Section 4, we will perform a Hall algebra calculation, which relates this to The goals of this section are firstly to define this locus and secondly to give a nice geometric characterization of it (Theorem 3.4). We start with a lemma.
From the previous lemma, we obtain an inclusion for all Q ∈ T A . By the short exact sequence 0 → I C → A → A/I C → 0 and the fact that Q is 0-dimensional, we see that Ext 2 (Q, I C ) Ext 1 (Q, A/I C ).
The elements of Ext 1 (Q, A/I C ) correspond to short exact sequences where F is a 1-dimensional A-module. Such an extension is known as a Pandharipande-Thomas pair on Spec A whenever F is a pure A-module [PT1]. We denote the locus of PT pairs by Ext 1 (Q, A/I C ) pure . Put differently, a PT pair on Spec A consists of (F, s) where • F is a pure dimension 1 A-module, See [PT1] for details. Let PT A (C) be the moduli space of PT pairs (F, s) on Spec A where F has scheme theoretic support C. We can write this as where ∼ denotes the equivalence relation induced by automorphisms of Q. Using inclusion (19) and (20) This defines a closed subset Σ ⊂ PT A (C). The locus Σ depends on the choice of cosection (18). In the following theorem, we give a nice geometric characterization of Σ. Later, when taking Euler characteristics in Section 4.C, it leads to a proof of equation (15).

Theorem 3.4. Let A be a 3-dimensional regular Noetherian C-algebra and M a torsion free A-module of homological dimension 1. Fix a cosection (18). Then there exists a morphism of C-schemes
which induces a bijection between the C-valued points of Quot A (Ext 1 (M, A)) and the C-valued points of the locus Σ defined in (21).
The next two subsections are devoted to the proof of this theorem.

3.B. The rank 2 case
We first prove Theorem 3.4 for r = 2. This case is similar to the main result of [GK2], but with some minor modifications. We also show why the map of Theorem 3.4 is a morphism of schemes, which was not discussed in detail in [GK2].
Proof of Theorem 3.4 for r = 2. We first construct a set theoretic map at the level of C-valued points from Σ to Quot A (Ext 1 (M, A)) and show it is a bijection. In Step 4 we show that the inverse map is a morphism of schemes.
Step 1 using Notation of Section 1. By [Eis,Thm. 21.5], this is isomorphic to which is pure 1-dimensional by [HL,Sect. 1.1]. 16 We conclude that an element I C → I such that the triangle in the diagram commutes, where Ext 3 (Q, A(S)) =: Q D A(S) .
Step 2: In this step we show that an element in the image of the inclusion Step 3: In this step we construct a set theoretic bijective map at the level of C-valued points where Σ was defined in (21). For any 0-dimensional A-module Q and any pure dimension 1 A-module F we have [HL,Prop. 1.1.10] More generally a PT pair 0 → A/I C → F → Q → 0 dualizes to a short exact sequence where F D is pure and Q D is 0-dimensional. We summarize the results of Steps 1 and 2. Given a PT pair I • = {A → F} with cokernel Q, we can form the following diagram where ((A/I C ) D A(S) , s ξ ) was constructed from (18) in Step 1. Then I • lies in the image of the injection of Lemma 3.3 if and only if the indicated surjection exists. We have produced the map (23) at the level of sets.
Applying (·) D A(S) to the middle row of diagram (25) gives back the original PT pair I • = {A → F} by (24). From this fact, it is easy to construct the inverse of the map described above by starting from a surjection Ext 1 (M, A) Q and inducing the middle row of diagram (25).
Step 4: Finally we show that the inverse of the set theoretic bijection (23) constructed in the previous step is a morphism of schemes. Let U = Spec A and let C ⊂ U be the Cohen-Macaulay curve of Lemma 3.2. Since we work in the affine setting, we can use module notation as we have been doing so far, but for this step we prefer sheaf notation. Denote by F the torsion free sheaf corresponding to M and let B be any base C-scheme of finite type. Denote projection by π U : U × B → U . Suppose we are given a B-flat family of 0-dimensional quotients where K is the kernel of the composition We want to dualize the middle row of (26). Denote derived dual by (·) ∨ . Since C is Cohen-Macaulay we have [HL,Prop. 1.1.10 Therefore applying RH om(·, π * U O U (S)) to the middle row of (26) gives the following exact triangle (after a bit of rewriting) We claim that the induced map is a B-flat family of PT pairs. If so, then we have produced from a B-flat family of quotients a B-flat family of PT pairs and the inverse of the set theoretic map (23) is a morphism of schemes. We claim that the complexes in (27) are all concentrated in degree 0. Indeed for any closed point b ∈ B the pulled-back sheaves K b and Q b on the fibres U × {b} are pure sheaves of dimension 1 and 0. By [HL,Prop. 1.1.10] Using [Sch,Prop. 3.1] we obtain Furthermore are all concentrated in degree 0 for all closed points b ∈ B. Therefore the terms of (28) are all B-flat by cohomology and base change for Ext groups [Sch,Prop. 3.1]. Finally the exact sequence (28) pulls back to for all closed points b ∈ B. The second and third terms are pure of dimension 1 and 0 respectively, so we are done.

3.C. The higher rank case
In this section we prove Theorem 3.4 for the case r > 2. This requires a new inductive construction which builds on the rank r = 2 case.
Continuing inductively in this fashion, we obtain A-modules which fit in short exact sequences for all i = 1, . . . , r − 1 and j = 0, . . . , r − 1. From these short exact sequences we deduce at once that all modules M i are torsion free of homological dimension 1. We denote the corresponding extension classes by Dualizing (29) gives Here the map s ξ i sends 1 to ξ i . Fix an element Q ∈ T A . The original cosection (17)  Using (29) and Ext 2 (Q, A) = 0 [HL,Prop. 1.1.6], the inclusion of Lemma 3.3 factors as a sequence of inclusions Ext 2 (Q, M r (S)) → · · · → Ext 2 (Q, M 1 (S)) = Ext 2 (Q, I C ).
Step 1: Just like in Step 1 of Section 3.B, we ask when an element of Ext 1 (Q[−1], I C ) lies in the image of Ext 2 (Q, M r (S)) under inclusion (30). The same reasoning as in Step 1 such that the triangle commutes.
Step 2: We claim that an element I C → I in diagram (31) is a surjection. This is proved just as in Step 2 of Section 3.B.
Step 3: We have constructed a map Our goal in this step is prove that this is a geometric bijection. In fact we have constructed an entire sequence of maps Q∈T A Ext 2 (Q, M r (S)) pure / ∼

Hall algebra calculation
Let X be a smooth projective threefold and let F be a rank r torsion free sheaf on X of homological dimension ≤ 1. We assume Spec A is an affine scheme of one of the following types • Case 1: Spec A is a Zariski open subset of X, • Case 2: Spec A = Spec O X,P , where P ∈ X is a closed point.
We denote the A-module corresponding to F | Spec A by M Our goal is to prove equation (15), which finishes the proof of Theorem 1.1 by Proposition 2.4. We will achieve this by combining the geometric bijection of Theorem 3.4 with a higher rank variation on a Hall algebra computation by Stoppa-Thomas [ST], which we follow closely in this section. We assume M has homological dimension 1, because otherwise M is locally free in which case we already know (15) (Section 2.B). Throughout this section, X, F , A, M are fixed in this way.

4.A. Key lemma
Like in Section 3, we denote the stack of finitely generated 0-dimensional A-modules by T A . The following is the analog of [ST,Lem. 4.10]. Proof.
Step 1: Let ι : Spec A → X be the natural map and Q := ι * Q. We claim for all i. This will allow us to apply Serre duality on X in Step 2. We prove the first isomorphism; the second follows analogously. When ι : U = Spec A ⊂ X is a Zariski affine open, the isomorphism follows at once from the local-toglobal spectral sequence. Indeed which is only non-zero when p = 0 because U is affine. The spectral sequence collapses thereby giving the desired result.
In the case Spec A = Spec O X,P , we first observe that as complex vector spaces, where F P , Q P denote the stalks. This again follows by using the local-to-global spectral sequence on an affine open neighbourhood of the closed point P . Next let j : Spec O X,P → Spec O X,P be induced by formal completion. Then Q P = j * Q by definition. Note that j * is exact because j is affine and j * is exact by [AM,Prop. 10.14]. By adjunction Step 2: Since F has homological dimension ≤ 1 and Q is 0-dimensional, we have Ext ≥2 X (F , Q) = 0. By Serre duality on X we also have Ext ≤1 X (Q, F ) = 0 and the remaining Ext groups are Hom X (F , Q) Ext 3 X (Q, F ) * and Ext 1 X (F , Q) Ext 2 X (Q, F ) * . Furthermore The result follows from Step 1.

4.B. The relevant stacks
We write T X for the stack of 0-dimensional sheaves on X. Similar to [ST], we consider the following T X -stacks • Hom X (F , ·) is the stack whose fibre over Q ∈ T X is Hom X (F , Q), • Hom X (F , ·) onto is the stack whose fibre over Q ∈ T X is Hom X (F , Q) onto , • Ext 2 X (·, F ) is the stack whose fibre over Q ∈ T X is Ext 2 X (Q, F ), • C r (·) is the stack whose fibre over Q ∈ T X is C r (Q) .
Denote by ι : Spec A → X the natural map. Recall that we introduced two cases at the beginning of this section. In Case 1, ι * induces an isomorphism of T A onto the open substack of T X of sheaves supported on Spec A ⊂ X. In Case 2, ι * induces an isomorphism of T A onto the closed substack of T X of sheaves supported at the point P ∈ X.
Recall that in the proof of Lemma 4.1 we showed that for all Q ∈ T A , i = 0, 1, 2, 3, and Q := ι * Q. Pull-back along gives rise to the following T A -stacks: • Hom A (M, ·) is the stack whose fibre over Q ∈ T A is Hom A (M, Q), • Hom A (M, ·) onto is the stack whose fibre over Q ∈ T A is Hom A (M, Q) onto , • Ext 2 A (·, M) is the stack whose fibre over Q ∈ T A is Ext 2 A (Q, M). In Case 1, these are open substacks of Hom X (F , ·), Hom X (F , ·) onto , Ext 2 X (·, F ). In Case 2 these are closed substacks of Hom X (F , ·), Hom X (F , ·) onto , Ext 2 X (·, F ). As such, we will view them as T X -stacks. Next we fix a cosection M → I C (−S) as in (18) (which always exist by Theorem 3.1 and Lemma 3.2!). By Lemma 3.3 we get an induced injection for all Q ∈ T A . In equation (21) we defined Ext 2 A (Q, M(S)) pure as the intersection of Ext 2 A (Q, M(S)) with the locus of PT pairs Ext 1 A (Q, A/I C ) pure . This gives a substack Ext 2 A (Q, M(S)) pure ⊂ Ext 2 A (Q, M(S)). By 1 T X = 1 0 + 1 T • X is invertible with inverse 1 T X = 1 0 − 1 T • X + 1 T • X * 1 T • X − · · · We denote by P z (·) : H(T X ) −→ Q(z) [[q]] the virtual Poincaré polynomial. Here z is the variable of the virtual Poincaré polynomial and q keeps track of an additional grading as follows. Any element [U → T X ] ∈ H(T X ) is locally of finite type and can have infinitely many components. Let T X,n ⊂ T X be the substack of 0-dimensional sheaves of length n and define P z (U ) := ∞ n=0 P z (U × T X T X,n ) q n .
Then P z (·) is a Lie algebra homomorphism to the abelian Lie algebra Q(z) [[q]] by [ST,Thm. 4.32]. (15) Proof of equation (15). The inclusion ι * : T A → T X is a T X -stack, which we denote by 1 T A . Just like 1 T X , it is an invertible element of (H(T X ), * ). By the inclusion-exclusion principle, Hom A (M, Q) onto can be written as

4.C. Proof of equation
where < denotes strict inclusion. This leads to Bridgeland's generalization of Reineke's formula in our setting (see [ST,Bri2] for details) Hom A (M, ·) = Hom A (M, ·) onto * 1 T A , where we view all stacks as T X -stacks. We also use the following identity 17 from [ST] Ext 2 A (·, M) = 1 T A * Ext 2 A (·, M) pure .
By (36)  where the second line follows from Ext 2 A (·, M) pure Ext 2 A (·, M(S)) pure and Theorem 3.4. Since we have C r (·) Hom A (A ⊕r , ·), the analog of (35) with M replaced by A ⊕r gives lim z→1 P z (V )(q) = lim z→1 P z Hom A (A ⊕r , ·) onto (q) 17 ↑ The proof of this identity goes as follows. First we observe that 1 T A * Ext 1 A (·, A/I C ) pure = Ext 1 A (·, A/I C ) as in [ST]. This comes from a geometric bijection from LHS to RHS. An object of LHS consists of a short exact sequence 0 → Q 1 → Q → Q 2 → 0 in T A together with a PT pair 0 → A/I C → F → Q 2 → 0. To these data, we assign the induced short exact sequence given by 0 → A/I C → F × Q 2 Q → Q → 0. This geometric bijection restricts to a geometric bijection 1 T A * Ext 2 A (·, M) pure → Ext 2 A (·, M). This follows by noting that 0 → A/I C → F → Q 2 → 0 satisfies the property described in Step 1 of Sections 3.B and 3.C if and only if 0 → A/I C → F × Q 2 Q → Q → 0 satisfies this property.