Moduli of PT-semistable objects II

We generalise the techniques of semistable reduction for flat families of sheaves to the setting of the derived category $D^b(X)$ of coherent sheaves on a smooth projective three-fold $X$. Then we construct the moduli of PT-semistable objects in $D^b(X)$ as an Artin stack of finite type that is universally closed. In the absence of strictly semistable objects, we construct the moduli as a proper algebraic space of finite type.


Introduction
In this paper, we continue our study of PT-semistable objects and the construction of their moduli spaces, using the results established in [Lo]. In [Lo], boundedness of the moduli of PT-semistable objects was established. We also showed that the stack of objects in the heart used for PT-stability is universally closed, and proved a series of technical lemmas that will now be applied.
Using semistable reduction in the derived category, we now show that PT-semistability is an open property for a flat family of complexes. This enables us to construct Artin stacks of PT-semistable objects that are of finite type and universally closed. When there are no strictly semistable objects, we construct proper algebraic spaces of finite type parametrising PT-stable objects. Overall, we not only have moduli stacks of objects of any Chern classes in the derived category on three-folds, but also offer a perspective of higher-rank analogues of stable pairs studied in [PT] (see also [Bay,Proposition 6.1.1]).
In precise terms, our main theorem is: Theorem 1.1. Let (X, H) be a polarised smooth projective three-fold over k.
(1) The PT-semistable objects on X of any fixed Chern character form a universally closed Artin stack of finite type. The stack is a substack of Lieblich's Artin stack of universally gluable complexes -see [Lie].
(2) When there are no strictly semistable objects, the PT-semistable objects on X of any fixed Chern character form an algebraic space of finite type that satisfies the valuative criterion for properness for an arbitrary discrete valuation ring. The algebraic space is a subfunctor of Inaba's algebraic space of simple complexessee [Ina].
Although many of the arguments here are written down only for a particular polynomial stability, namely PT-stability on three-folds, the same proofs apply to Gieseker stability for sheaves if one replaces our t-structure by the standard t-structure. The techniques in this paper should also work for a wider class of stability conditions, and on higher-dimensional varieties.
1.1. Related Work. Semistable reduction for sheaves is originally due to Langton [Lan], while polynomial stabilities were first defined by Bayer [Bay]. In the case of smooth projective three-folds X, no examples of Bridgeland stability conditions on D b (X) have been constructed for an arbitrary X. As approximations of Bridgeland stability conditions on three-folds, Bayer [Bay] and Toda [Tod1] independently came up with the notions of polynomial stability and limit stability, respectively. In Bayer's paper, he introduced a class of polynomial stability conditions on normal projective varieties, which includes Toda's limit stability (in fact, Toda's stability acts as a wall in the wall-crossing in [Bay]) as well as Gieseker stability for sheaves. One of the main results in Bayer [Bay,Proposition 6.1.1] states that, for objects in the heart A p = Coh ≤1 (X), Coh ≥2 (X)[1] with ch = (−1, 0, β, n) and trivial determinant, the stable objects with respect to a particular polynomial stability function are precisely the stable pairs described in Pandharipande and Thomas's work [PT]; for this reason, this particular stability function is called the PT-stability function in [Bay]. The moduli space of such stable pairs has been constructed by Le Potier [Pot] using geometric invariant theory (GIT) and shown to be projective. In general, however, it is not clear how to apply GIT to objects in the derived category, because very different-looking complexes can be isomorphic in the derived category.
In Toda's paper [Tod1], he showed that the moduli space of limit-stable objects in A p of ch = (−1, 0, β, n) and trivial determinant on a Calabi-Yau three-fold is a separated algebraic space of finite type [Tod1,Theorem 3.20].
There were earlier examples of moduli spaces of objects in the derived category that satisfy the valuative criterion for properness. For example, in Abramovich and Polishchuk's work [AP], they showed that the valuative criteria for separatedness and properness for Bridgeland-stable objects hold, under the assumption that the heart of t-structure in the stability condition is Noetherian, which A p is not. (The techniques in [AP] were generalised in [Pol].) On the other hand, Arcara-Bertram-Lieblich constructed projective moduli spaces of rank-zero Bridgeland-stable derived objects on surfaces with trivial canonical bundle [ABL]. We note that the idea of elementary modifications for objects in the derived category has already appeared in [ABL].
In the case of K3 and abelian surfaces X, Toda showed in [Tod2] that for stabilities σ lying in a particular connected component of the space Stab(X) of Bridgeland stability conditions, the moduli of semistable objects with respect to σ of any given numerical type and phrase is an Artin stack of finite-type.
1.2. Notation. The notations in this paper are the same as those in [Lo]. For convenience, we provide a summary of the notations we use. More details can be found in [Lo].
Throughout this paper, k will be an algebraically closed field of characteristic 0. And R will denote a discrete valuation ring (DVR), not necessarily complete, with uniformiser π and field of fractions K. Unless specified, X will always denote a smooth projective three-fold over k.
We will write X R := X ⊗ k R, and X K := X ⊗ R K. For any integer m ≥ 1, let X m := X ⊗ k R/π m , and let ι m : X m ֒→ X R denote the closed immersion. We will often write ι for ι 1 , and X k for the central fibre of X R . For integers 1 ≤ m ′ < m, let ι m,m ′ : X m ′ ֒→ X m denote the closed immersion. We also write for the open immersion. Note that the pushforward functor ι * : Coh(X k ) → Coh(X R ) is exact, while the pullback ι * : Coh(X R ) → Coh(X) is right-exact. Similarly for the pushforward ι m,m ′ * . On the other hand, j * : Coh(X R ) → Coh(X K ) is exact.
For a Noetherian scheme Y, we will always write Kom(Y) for the category of chain complexes of coherent sheaves on Y, D b (Y) for the bounded derived category of coherent sheaves, and D(Y) for the unbounded derived category of coherent sheaves. If Y is of dimension n, then for any integer 0 ≤ d ≤ n, we define Then (Coh ≤d (Y), Coh ≥d+1 (Y)) is a torsion pair in the abelian category Coh(Y). For 0 ≤ d ′ < d, we form the quotient category Coh d,d ′ (Y) := Coh ≤d (Y)/Coh ≤d ′ −1 (Y), which is an abelian category. For a coherent sheaf F on Y, we write p(F) for its reduced Hilbert polynomial, and if F ∈ Coh ≤d (Y), we write p d,d ′ (F) for its reduced Hilbert polynomial as an element of Coh d,d ′ (Y) (see [HL,Section 1.6]).
For m ≥ 1, tilting with respect to the torsion pair (Coh ≤1 (X m ), Coh ≥2 (X m )) in Coh(X m ) gives a t-structure with heart on D b (X m ) (and in fact, on D(X m ) as well -see [Lo,Proposition 5.1]). The truncation functors associated to this t-structure will be denoted by τ ≤0 , and the cohomology functors denoted by H i A p m . We will drop the subscripts when the context is clear. On any Noetherian scheme Y, the cohomology functors with respect to the standard t-structure on D(Y) will always be denoted by We will use D ≤0 In summary, we have the following maps between the various schemes: and associated pullback and pushforward functors Since ι m is a closed immersion, it is a projective morphism. Hence we have the adjoint pair Lι * m ⊣ ι m * , i.e. Lι * m is the left adjoint, and ι m * the right adjoint [Huy,p.83]. Similarly, we have the adjoint pair Lι * m,m ′ ⊣ ι m,m ′ * for any 1 ≤ m ′ < m. Consistent with the definitions introduced in [AP] and [ABL], we will use the following notion of flatness for derived objects: Definition 1.2. Let S be a Noetherian scheme over k, and X a smooth projective three-fold over k. We say an object E ∈ D b (X × S ) is a flat family of objects in A p over S if, for all closed points s ∈ S , we have defined to be the union of the supports of the various cohomology H i (E • ), is d.
1.3. Stability Conditions. Let (X, H) be a smooth projective variety of dimension n with polarisation H. For any coherent sheaf F on X, we define its degree (with respect to H) as deg (F) = X c 1 (F) · c 1 (H) n−1 , and its slope as µ(F) = deg (F) rank (F) . Polynomial stability was defined on D b (X) by Bayer for any normal projective variety X [Bay,Theorem 3.2.2]. The particular class of polynomial stability conditions we will concern ourselves with for the rest of the paper consists of the following data, where X is a smooth projective three-fold: (1) the heart A p = Coh ≤1 (X), Coh ≥2 (X)[1] , and (2) a group homomorphism (the central charge) Z : The configuration of the ρ i is compatible with the heart A p , in the sense that for every nonzero E ∈ A p , we have Z(E)(m) ∈ H for m ≫ 0. So there is a unqiuely determined function φ(E)(m) (strictly speaking, a uniquely determined function germ) such that This allows us to define the notion of semistability on objects. We say that a nonzero object E is Z-semistable (resp. Z-stable) if for any nonzero subobject G ֒→ E in A p , we ) to denote this. Harder-Narasimhan filtrations for polynomial stability functions exist [Bay,Section 7].
By [Bay,Proposition 6.1.1], with respect to any polynomial stability function from the class above, the stable objects in A p with ch = (−1, 0, β, n) and trivial determinant are exactly the stable pairs in Pandharipande and Thomas' paper [PT], which are 2-term complexes of the form where F is a pure 1-dimensional sheaf and s has 0-dimensional cokernel. For this reason, and in line with calling a stability function as above a PT-stability function in [Bay], we call any polynomial stability condition satisfying the above requirements a PT-stability condition, and any nonzero object in A p semistable (resp. stable) with respect to it PTsemistable (resp. PT-stable).

Semistable Reduction in the Derived Category
Let us first recall how semistable reduction proceeds for flat family of sheaves in [Lan]. Suppose E ∈ Coh(X R ) is a flat family of sheaves on a projective variety X over the base Spec R, where R is a DVR over k. Suppose that the generic fibre E K is µ-semistable, while the central fibre E k is not. Then there is a unique maximal destabilising quotient sheaf E k ։ Q 0 in the category Coh(X k ). If we write I 0 := E, we can define I 1 to be the kernel of the composition E ։ E k ։ Q 0 , which is another flat family of sheaves on X over Spec R. This process of going from the family I 0 to the family I 1 via the short exact sequence We can now look at the central fibre of I 1 to see if it is µ-semistable. If it is not, we can perform another elementary modification to obtain a flat family I 2 over Spec R, and so on. It is the content of [Lan,Theorem (2)] (see also [HL,Theorem 2.B.1]) that this process will terminate after a finite number of steps, giving us a flat family of µ-semistable sheaves over Spec R. Our aim here is to show that this phenomenon happens in the more general setting of the derived category.
2.1. Elementary Modifications. The process of elementary modification works not only for flat families of coherent sheaves, but also for flat families of complexes in the derived category when we fix a t-structure: Proposition 2.1 (Elementary Modification in the Derived Category). Let R be a DVR over k. Given a flat family I ∈ D b (X R ) of objects in A p over Spec R, and a surjection Lι * I α ։ Q in A p , there exists a flat family J of objects in A p such that its generic fibre coincides with that of I, and we have an exact sequence in A p Proof. Suppose we have a surjection where ε is the adjunction map.
Applying Lι * to the above exact triangle, we get another exact triangle y y r r r r r r r Taking the long exact sequence of cohomology with respect to the heart A p yields ). Now, if we write η ′ for the adjunction map Lι * ι * (Lι * I) → Lι * I, then the composition On the other hand, using the following exact triangle from Corollary [Lo,Corollary 5.8] we see that H 0 (η ′ ) is also an isomorphism. Hence H 0 (Lι * ε) itself is an isomorphism.

which induces a morphism of exact triangles
.
If we apply the functor H 0 to the above diagram, the left-most and right-most columns would vanish, while the middle horizontal maps become isomorphisms by Lemma [Lo,Lemma 5.6(a)]. Then, since α = H 0 (α) is surjective by hypothesis, we get surjectivity of Lι * ι * α. This, combined with the fact that H 0 (Lι * ε) is an isomorphism, which we just proved, implies α ′ is surjective. Consequently, Lι * J = H 0 A p (Lι * J) ∈ A p , meaning that J is again a family of objects in A p over Spec R. This completes the proof of the proposition.
Given a nonzero object E ∈ A p , we can consider its HN filtration with respect to PTstability (or, indeed, any polynomial stability) Recall that the left-most factor E 0 in the HN filtration is called the maximal destabilising subobject, and that it satisfies the following property [Lo,Lemma 3.1 For semistable reduction in the derived category, we start with a flat family I 0 = I of objects in A p over Spec R where R is any DVR over k. For any i ≥ 0, if Lι * I i is not semistable in A p , then the proposition above says we can define a new flat family I i+1 by the exact triangle in D b (X R ) where G i is the cokernel of a maximal destabilising subobject B i ֒→ Lι * I i . We also know from the proof of the proposition that we have the short exact sequence in A p which we can break up into the two short exact sequences in A p We will call I m the family obtained from I 0 after m elementary modifications.
Suppose the above process never lets us arrive at a semistable central fibre, i.e. suppose for all i ≥ 0, we have a maximal destabilising subobject B i ֒→ Lι * I i such that φ(B i ) ≻ φ(Lι * I i ). Then we prove an analogue of [Lan, Lemma 1]: Lemma 2.2. Using the notation above, for any 0 E ⊆ Lι * I i+1 , we have φ(E) φ(B i ). If equality holds, then E ∩ G i = 0.
Given objects A, B, I in the abelian category A p such that A, B ⊂ I, we can construct objects A ∨ B and A ∩ B satisfying the following three properties: • A ∨ B is a subobject of I.
• We have an exact sequence Proof of lemma. Take any 0 E ⊂ Lι * I i+1 . Let E ∨ G i denote the following fibred product of subobjects of Lι * I i+1 in A p : We have the exact sequence where φ(Ẽ) φ(B i ) since B i is defined to be the maximal destabilising subobject. Then by the seesaw principle, φ( . LetẼ be as above. If E ⊆ G i , then E ∩ G i = E, and from (4) we get φ(Ẽ) = φ(B i ). Hence there is an injectionẼ ֒→ B i since B i is a maximal destabilising subobject. This implies B i =Ẽ, and so E = E ∩ G i = 0 from (4), contradicting E being nonzero. Thus we must have E G i .
We saw above that Z(E ∩ G i ) + Z(D) = Z(E) and φ(D) φ(B i ). By assumption, we , which forces E ∩ G i = 0 by the argument in the previous paragraph.
Suppose X is a smooth projective three-fold over k. In showing the valuative criterion for universal closedness for PT-semistable objects, we need to take a PT-semistable object Then we need to extend I 0 K to an R-flat family I 0 ∈ D b (X R ) of objects in A p , and then apply semistable reduction to I 0 . We now consider the properties the central fibre Lι * I 0 can have when we extend I 0 K to I 0 . Proposition 2.4. Let X be a smooth projective three-fold over k. If E K is a torsion-free sheaf on X K that is semistable in Coh 3,1 (X K ), then there is an R-flat coherent sheaf E on X R such that j * E = E K , and ι * E is a torsion-free sheaf semistable in Coh 3,1 (X k ).
Proof. Take any R-flat coherent sheaf E on X R that restricts to E K on X K . By [HL,Theorem 2.B.1], replacing E by a subsheaf if necessary, we can assume that ι * E is also semistable in Coh 3,1 . Suppose ι * E is not torsion-free. Then it has a maximal torsion subsheaf T ⊂ ι * E, which is necessarily 0-dimensional. And we can run semistable reduction on the family E; the details are as follows: Define E ′ as the kernel of the composition E ։ ι * ι * E ։ ι * (ι * E/T ). Pulling back the short exact sequence 0 → E ′ → E → ι * (ι * E/T ) → 0 to X k and taking the long exact sequence of Tor, we get Let T ′ be the maximal torsion subsheaf of ι * E ′ . Suppose T ′ is nonzero. Then we must have (ι * E/T ) ∩ T ′ = 0, or else ι * E/T would have a torsion subsheaf, contradicting the maximality of T . Continuing this process, we obtain a sequence of R-flat coherent sheaves If we do not arrive at some n with T (n) = 0, then because all the T (n) are 0-dimensional, we have T (n) T for some T for n ≫ 0. Then, by the same argument as in the proof of [HL,Theorem 2.B.1], this produces a nonzero 0-dimensional subsheaf of E K , a contradiction. The proposition follows.
The following is a strengthening of the d = 0 case of [Lo,Theorem 4.1]: Proposition 2.5. Let X be a smooth projective three-fold over k. Given any object

there exists a 3-term complex E • of R-flat coherent sheeaves with R-flat cohomology on X R such that:
• the generic fibre j Proof. The proof of [Lo,Theorem 4.1] still works, except that we now extend ker (s K ) in the proof to an R-flat family of torsion-free sheaves that are semistable in Coh 3,1 using Proposition 2.4.
Hence if we start with a PT-semistable object I 0 K ∈ A p (X K ) of nonzero rank, we can extend it to an R-flat family I 0 of objects in A p using Proposition 2.5, and apply elementary modifications starting from I 0 . The following proposition describes what happens when we do not arrive at a semistable central fibre after a finite number of modifications: Proposition 2.6. Suppose I 0 is an R-flat family of objects of nonzero rank in A p , where the generic fibre j * (I 0 ) is PT-semistable, and the central fibre Lι . Suppose that we do not arrive at a PT-semistable central fibre Lι * I j after a finite number of elementary modifications starting from I 0 . Then the morphisms B i+1 → B i and G i+1 → G i induced by Lι * I i+1 → Lι * I i eventually become isomorphisms. Say they are isomorphic to B and G. Then we have is a degree-2 polynomial, then Z(B i+1 )(m) could be of degree 2, 0 or 3; but if Z(B i )(m) is a degree-0 polynomial, then Z(B i+1 )(m) can only be of degree 0 or 3. Note that, since H −1 (Lι * I 0 ) is nonzero and torsion-free, if H −1 (B 0 ) is nonzero then it is also torsion-free, in which case B 0 has 3-dimensional support. By Corollary 2.3, none of the subsequent B i can be 2-dimensional. Hence all B i are 0-dimensional or 3dimensional. One consequence of this is that H −1 (Lι * I i ) is torsion-free for all i: for, if not, then H −1 (Lι * I i ) has a 2-dimensional subsheaf T for some i, and T [1] ֒→ Lι * I i would be destabilising, and has a larger φ than B i , a contradiction.
is a decreasing sequence (by Corollary 2.3) in 1 r! Z bounded from below, and so must eventually become constant. We show that, in fact, the phase φ(B i ) itself must eventually become constant.
Suppose B i is 0-dimensional for 0 ≤ i ≤ i 0 , and 3-dimensional for i ≥ i 0 + 1 (we allow i 0 + 1 to be 0). Recall the short exact sequences (2), (3) in A p for i ≥ 0: . Repeating the argument in the previous paragraph, we get that Since all the Lι * I i have the same Hilbert polynomial, this implies that all H −1 (Lι * I i ) have the same reduced Hilbert polynomial p 3,1 .
for all i ≥ i 0 + 1. These inequalities, together with the conclusions of the last two paragraphs, imply that p 3, Suppose this injection has cokernel C. From the long exact sequence of cohomology of 0 → B i+1 → B i → C → 0, and the fact that the cohomolgy of B i becomes constant at each degree for i ≫ 0 (compare Hilbert polynomials to see this), we get that each cohomology of C must be zero for large enough i. In other words, for i ≫ 0, the map

This induces isomorphisms on the cokernels of the inclusions
Let us fix objects B, G that are isomorphic to B i , G i for all i ≫ 0, respectively. Now, if we take the injection f 1 : G i ֒→ Lι * I i+1 from (1) and compose it with the surjection f 2 : Lι * I i+1 ։ G i+1 from (2), the kernel of the composition f 2 f 1 : Hence the map f 1 ( f 2 f 1 ) −1 splits the short exact sequence (2), and so Lι * I i B i ⊕G i B⊕G for i ≫ 0.

Constructing an Inverse System.
For the rest of this section, assume the setup of Proposition 2.6. That is, start with a PT-semistable object I 0 K ∈ A p K , extend it to a flat family I := I 0 of objects in A p over Spec R, assume that the central fibre of I 0 is not PTsemistable, and that a finite number of semistable reduction does not yield a PT-semistable central fibre. This gives us an infinite sequence of flat families · · · → I 2 → I 1 → I 0 in D b (X R ). We would now like to show that, if R is a complete DVR, then this would imply that j * I 0 is PT-unstable, a contradiction.

The Cohomology of Various Objects.
Using the techniques developed in [Lo], we now compute the cohomology of various objects. Definition of Q m . By Proposition 2.6, we can replace I 0 by I m for m ≫ 0 if necessary, and assume that all the B i and G i are isomorphic, to some B and G, respectively. We can pull back the composition I m → I m−1 → · · · → I 0 = I from D(X R ) to D(X m ), obtaining a morphism in A p m which we will denote by θ m : In particular, we obtain the exact triangle in D(X R ) for every m ≥ 1: Modifying the proof of [AP,Lemma 2.4.1] slightly, we get the following lemma that says Q m lives in D(X m ): Proof. We adapt the proof of [AP,Lemma 2.4.1] to our situation. Let F • , G • be bounded complexes of coherent sheaves on X m , X n , representing F, G, respectively. We can choose a bounded complex P • of torsion-free O X R -modules, and a surjective morphism of complexes of O X R -modules q : P • → F • that is a quasi-isomorphism, and a chain map p : Define K • := ker (q) in the abelian category Kom(X R ). Then K • is acyclic (i.e. all cohomology are 0), and multiplication by π n is injective on each K i . Set P • := P • /π n K • (quotient in Kom(X R )). Then q factors through q : Hence q, p can both be considered as chain maps over X m+n , and so α is the pushforward of a morphism in D b (X m+n ).
. ThatQ m has bounded cohomology follows from Q m itself having bounded cohomology, and the exactness of pushforward ι m * . Lemma 2.9. With respect to the t-structure with heart A p m , and for r ≥ 0, m ≥ 1, (a) H i Lι * m I r = 0 whenever i 0. (b) H i Lι * m Q r = 0 whenever i 0, −1. In particular, Lι * m I r ∈ A p m for all m ≥ 1, r ≥ 0. Proof. By assumption, Lι * I 0 ∈ A p , so (a) follows from Proposition [Lo,Proposition 5.10]. As for the cohomology of Q r , we start with the exact triangle which can be pulled back to . Then we can take the long exact sequence of cohomology with respect to the heart A p m , and use part (a) to conclude (b).
Lemma 2.10. The objectQ m ∈ D b (X m ) lies in the heart A p m . Proof. From the long exact sequence of cohomology of (6) with respect to A p r when r = m, we see that H i This follows from [Lo,Corollary 5.8]. (5), we obtain the following commutative diagram, where each straight line is exact at the middle term. Because the I j are flat families, the morphisms labelled γ m , γ m−1 and µ m are surjective in A p : Lemma 2.12. For m > 1, the morphism µ m := H 0 Lι * (Q m → Q m−1 ) is an isomorphism, and H 0 Lι * Q m G for all m > 0.
Proof. We are assuming that the central fibres I i have stabilised, i.e. Lι * I i B ⊕ G for each i, and that the pullback of each elementary modification I m → I m−1 to D(X k ) looks like which takes B isomorphically onto B and G to 0. Hence γ m , γ m−1 are surjective with the same kernel. Thus µ m is an isomorphism for each m > 1. That the central fibres have stabilised also means Q 1 = ι * G. Therefore, H 0 Lι * Q 1 G. By the previous paragraph, H 0 Lι * Q m G for all m > 1.  (5), and the lower right corner of the diagram becomes If we further apply the functor H 0 Lι * m,m−1 , we obtain where we define the vertical map to be q m−1 . Here we are using the isomorphism of functors Repeating this construction for α m−1 , we obtain a commutative diagram Now we see that, if we can show that each q m−1 is an isomorphism, then we would have the inverse system described above.
By Lemma 2.12, q 1 is an isomorphism (since q 1 is just µ 2 in the notation of that lemma). To show that q m is an isomorphism for all m ≥ 1, we make use of the following: Lemma 2.13. IfQ m−1 is a flat family of objects in A p over Spec R/π m−1 , then q m−1 is an isomorphism.
Proof. Define K to be the kernel of q m−1 in A p m−1 , so that we have a short exact sequence Pulling back this exact triangle via Lι * m−1,1 , and then applying the functor H 0 A p , we get the exact sequence in A p are both isomorphic to G, the surjection between them is an isomorphism, forcing the cohomology H 0 Lι * m−1,1 K to be zero. By [Lo,Lemma 5.5], K = 0. That is, q m−1 is an isomorphism.
This lemma reduces the problem of constructing an inverse system to the problem of showing the flatness ofQ m over Spec R/π m for all m ≥ 1. We can solve the latter problem in two steps: (1) show that, for m ≥ 1, ifQ m is flat thenQ 2m is also flat; (2) show that, for m ≥ 3, ifQ m is flat, thenQ m−1 is also flat. SinceQ 1 G is trivially flat over Spec R/π, we will then have flatness for allQ m .

Flatness ofQ m in General.
Lemma 2.14. Let m > 2. Suppose A, B are objects in A p m−1 and A p 1 , respectively. Given a morphismβ : Proof. Since the truncation τ ≥0 is left adjoint to the embedding functor, we have a morphism of functors id → τ ≥0

This yields a commutative diagram
where η is the adjunction map and the right-most vertical map is the identity. Since H 0 Lι * m−1,1β = 0 by assumption, η • Lι * m−1,1β is the zero map since the outer edges of the diagram commute. However, η • Lι * m−1,1β andβ correspond to each other via the adjunction Henceβ itself is zero.
Lemma 2.15. Let m ≥ 2. Then we have the following short exact sequence in A p m :

Proof. We begin with the commutative triangle
Lι * m I m−1 $ $ r r r r r r r r r where the cohomology objects are computed using the flatness of the I j and Lemma 2.11. Define θ, β, β 1 as above.
Now we move our attention to the commutative triangle Applying the octahedral axiom to it gives

Note that all the objects in this diagram lie in D ≤0
A p m (X m ). Pulling back this entire diagram via Lι * m,1 , and taking cohomology with respect to A p , we obtain the following commutative diagram where the three straight lines passing through the triangle in the center are exact sequences. Define ε, ε ′ as above. Also, the sequence of seven terms that winds around that triangle is the long exact sequence of cohomology for the exact triangle Here is why H 0 Lι To understand the other cohomology objects of H 0 Lι * m,1 (ι m,m−1 * Q m−1 ), we look at Since multiplication by π m−1 induces the zero map on objects pushed forward from D(X m−1 ), such as ι m,m−1 * Q m−1 , we have Therefore H i Lι * m,1 ι m,m−1 * Q m−1 are isomorphic for all even i ≤ 0 and in particular are isomorphic to G. As a consequence, in figure (13), the map ε ′ must be an isomorphism. So ε must be the zero map, implying H 0 Lι * m,1 β 1 is also zero. Thus H 0 Lι * m,1 β itself is zero from the commutativity of diagram (12). Now, by [Lo,Lemma 5.7], β = ι m,m−1 * β for someβ :Q m−1 → ι m−1,1 * G. This gives us [Lo,Corollary 5.4] H 0 Lι * m−1,1β by [Lo,Lemma 5.6].
Using Lemma 2.14, we concludeβ = 0 and thus β = 0. This means that, finally, the sequence of outer six terms in Figure (12) breaks up into two short exact sequences, one of them being the one we want.
Before we move on, let us define the objects R m,m−r and Q m,r . These definitions will help clarify our arguments. The objects R m,m−r . Since we have an exact sequence as in Lemma 2.15 for each m ≥ 2, we have a sequence of surjections Define R m,m−r ∈ A p m to be the kernel of this composition of surjections. On the other hand, if we apply the octahedral axiom to the commutative triangle .
That is, we have a short exact sequence Iterating this process (by looking at the output of the octahedral axiom applied to the com-positionQ m → ι m,m−(r−1) * Q m−(r−1) → ι m,m−r * Q m−r ), we obtain short exact sequences of the form (14) 0 → R m,m−(r−1) → R m,m−r → ι m,1 * G → 0 in A p m for each m > r ≥ 2. Comparing these with the short exact sequence of Lemma 2.15, we see that R m,m−r and ι m,r * Q r have the same Hilbert polynomial, both being r times the Hilbert polynomial of ι m,1 * G. The Objects Q m,r . For any 0 ≤ r < m, we can define the objects Q m,r by the exact triangle where the morphism I m → I r comes from composition of the elementary modifications.
The octahedral axiom gives us diagrams such as from which we see Q m,m−2 = ι 2 * Q m,m−2 for someQ m,m−2 ∈ D b (X 2 ) (Lemma 2.7). If we fix m and iterate this process, we would obtain Q m,r = ι m−r * Q m,r for someQ m,r ∈ D b (X m−r ). Since the heart of a t-structure is extension-closed, eachQ m,r lies in the heart A p m−r . It is easy to see thatQ m−r andQ m,r have the same Hilbert polynomial. Proof. Applying the octahedral axiom to the commutative triangle Applying H 0 Lι * m , we get a short exact sequence in A p where the rows are copies of the short exact sequence 0 → R 2m,m →Q 2m → ι 2m,m * Q m → that comes from the definition of R 2m,m , and the vertical maps are all multiplication by ±π m . The diagram (16) has two properties: (a) Each column is a chain complex in the abelian category A p 2m (i.e. the composition of two successive differentials is zero). In particular, the differentials in the first and third columns are all zero maps. (b) Consider each column of (16) as a complex in D(A p 2m ) sitting at degrees ≤ 0. Then the degree-i cohomology of the first, second the third column can be computed by applying H i A p 2m (ι 2m,m * Lι * 2m,m (−)) to R 2m,m ,Q 2m and ι 2m,m * Q m , respectively.
From the construction of R 2m,m , we know that R 2m,m is the pushforward of an object in A p m , i.e. R 2m,m ι 2m,m * R 2m,m for someR 2m,m (using Lemma 2.7). It is then clear that property (a) holds. To see property (b), we can use Postnikov systems (see [Orl,Section 1.3], for example). Consider the following Postnikov system in a triangulated category D where triangles marked with • are commutative, the triangles marked with * are exact triangles, and X is a chain complex in the category D (i.e. d i d i+1 = 0 for any i). Given such a Postnikov system in D, if we also have a t-structure on D with heart A, and X • is a complex with all the terms in A, then is the degree-i cohomology of X • considered as a chain complex in A, and H i A (Y 0 ) is the degree-i cohomology of Y 0 computed in D with respect to the t-structure with heart A. (To see this, apply cohomology functors H i A to the Postnikov system.) For our situation, we build a Postnikov system in D(X 2m ) of the above form as follows: be any 2-term complex of coherent sheaves representing an object of A p 2m . That is, F −1 , F 0 are coherent sheaves on X 2m , and d is a morphism of coherent sheaves. At some point, we will specify F • to be R 2m,m ,Q 2m or ι 2m,m * Q m ; however, we will not do so just yet. We can consider F • as an object in the derived category D(A p 2m ) via the canonical inclusion A p 2m → D(A p 2m ). Note that D(A p 2m ) = D(Coh(X 2m )) by [BV,Proposition 5.4.3], although we will not explicitly use this fact in this proof.
For −3 ≤ i ≤ 0, let X i = F • , and for −3 ≤ i ≤ −1, let d i be multiplication by π m . Then is a chain complex in the abelian category A p 2m . Let v −3 be multiplication by π m . For −2 ≤ i ≤ 0, let Y i be the cone of v i−1 , and let u i be the natural inclusion of X i into Y i . Also, for i = −2, −1, let v i be the projection of Y i onto X i = X i+1 followed by multiplication by π m . Note that, while the natural projection of Y i onto X i itself is not a chain map in Kom(X 2m ), it becomes a chain map when we post-compose it with multiplication by π m . This construction gives us a Postnikov system with where the dotted box means to 'take the total complex of the double complex inside,' a notation we adopt until the end of the proof of this proposition. Also, here we use the sign convention used in [Huy,Definition 2.15] in the definition for the mapping cone.

Now, for the infinite complex [·
, the cohomology at all degrees i ≤ −1 are isomorphic by periodicity. On the other hand, for any i ≤ −1 we have (Y 0 ) by our Postnikov system above and (17) since we have a quasi-isomorphism between these two complexes, given by the chain map Then, where the first isomorphism follows because we only need the top four rows in the middle dotted box to compute the cohomology H −1 (A p 2m ). And so, overall, we obtain Taking F • to be R 2m,m ,Q 2m or ι 2m,m * Q m , we obtain property (b). Now, by properties (a) and (b), the long exact sequence of the double complex (16) From Lemma 2.16, we know ψ 0 is an isomorphism. So φ 0 is the zero map, and δ 1 is an isomorphism (since ι 2m,m * R 2m,m and ι 2m,m * Q m have the same Hilbert polynomial). Consequently, ψ 1 is the zero map. However, the double complex (16) is 1-periodic in the rows, and so all the ψ i are zero maps for i ≥ 1. That ψ 2 is zero means δ 2 is an isomorphism (by comparing Hilbert polynomials), hence φ 1 is the zero map. By the same periodicity argument, we get that all φ i are zero maps for i ≥ 1. As a result, we obtain H i Lι * by [Lo,Proposition 5.10]. Applying Lι * m,m−1 and taking the long exact sequence of cohomology with respect to A p m−1 , we get By [Lo,Lemma 5.9], for all odd integers i < 0, the objects H i Lι *  Q m for each m ≥ 1. Then we can use these isomorphisms and their compositions with the various Lι * m,m ′ c m to construct compatible isomorphisms that, together with the {Q m } m , give an inverse system that satisfy the hypotheses of: Proposition 2.20. [Lie,Prop 3.6.1] Let X → S be a flat morphism of finite presentation of quasi-separated algebraic spaces. Let (A, m, k) be a complete local Noetherian S -ring and  T  T  T  T  T  T  T  T  T  T  T  T  T  T  T . Now, by the following lemma, we have an isomorphism Lemma 2.22. [Lie,Lemma 4.1.1(2)] Let X → Spec A 0 be a proper flat algebraic space of finite presentation over a reduced Noetherian ring and I a finite A 0 -module. Given E, F ∈ D b p (X/A 0 ) and any i, if m ⊂ A 0 is a maximal ideal, then Even though the lemma is stated only for the case E = F in [Lie], the proof works in general.
Therefore, the commutative diagram above lifts to an element ϕ ∈ Hom X R (I 0 , Q R ). And ϕ restricts to the maps α m : Lι * m I 0 ։Q m (see the start of section 2.2.2).
2.3. Universal Closedness over an Arbitrary DVR. The goal of this section is to show the valuative criterion of universal closedness for PT-semistable objects. So far, we have learned that if R is a complete DVR, and E K ∈ A p K is PT-semistable with ch 0 (E K ) 0, then we can produce an R-flat object I 0 ∈ D b (X R ) that restricts to E K on X K . We have also learned, that if the central fibre Lι * I i does not become PT-semistable after a finite sequence of elementary modifications, we can produce a morphism ϕ : . Pulling back this exact triangle via j * gives On the other hand, we can also pull back the same triangle using Lι * to obtain Since Lι * ϕ is a surjection in A p , Lι * A is necessarily in A p . So by openness of the heart A p [Tod1, Lemma 3.14], j * A and j * Q R both lie in the heart A p K . Since Chern character is locally constant, Lι * I 0 and j * I 0 would have the same Chern character, as are Lι * Q R and j * Q R . This means that (19) yields a short exact sequence in A p K that destabilises j * I 0 E K , which is a contradiction. We have just proved: Theorem 2.23 (Valuative criterion for universal closedness). Let (R, π) be a DVR, and K its field of fractions. Given a PT-semistable object E K ∈ A p (X K ) such that ch 0 (E K ) 0, there exists E ∈ D b (X R ), a flat family of objects in A p over Spec R such that j * E E K and Lι * E is PT-semistable. Proof. Our proof above is for the case where R is a complete DVR. This is sufficient for us to define the moduli of PT-semistable objects and conclude that it is an Artin stack of finite type over k (see Proposition 3.3 and Section 4). Then, by [LMB,Remark 4,(7.4)] and [LMB,Theorem (7.10)], the result for an arbitrary DVR follows.
If we did not wish to use the existence of our moduli space as an Artin stack in showing the valuative criterion for an arbitrary DVR, we could prove the above theorem under an additional hypothesis, as follows: Consider the commutative diagram where R is an arbitrary DVR andR is its completion, and K andK are their respective fields of fractions: Suppose we are given a PT-semistable object E K in the heart A p (X K ). The additional hypothesis we need to add is, that ch 0 (E K ) 0 and ch 0 (E K ), ch 1 (E K ) are coprime. By [Lo,Theorem 4.5], we can extend E K to a flat family On the other hand, p * E is also a flat family overR, and Lι * (p * E) Lι * E. Moreover, j * (p * E) ∈ A pK by Lemma 2.26 below. Since j * (p * E) p * ( j * E), j * (p * E) is also PT-semistable by Proposition 2.27 below. Moreover, Lι * (p * E) also has E 0 as a maximal destabilising subobject in A p (X k ).
We can now apply elementary modifications to the family E. Since any flat family over Spec R has the same central fibre as the flat family obtained after the base change via p * , by our result over a complete DVR, Theorem 2.23, the elementary modifications applied to E ∈ D b (X R ) must produce a semistable central fibre after finitely many steps. This proves Theorem 2.23 again with the additional hypothesis.
We end this section with the proofs of Lemma 2.26 (the heart is preserved under base change) and Proposition 2.27 (PT-semistability is preserved under base change). To this end, we first characterise PT-semistable objects under a coprime assumption on the Chern character: Proposition 2.24. Let X be a smooth projective three-fold over k. Suppose E ∈ A p (X) has nonzero ch 0 (E), and ch 0 (E), ch 1 (E) are coprime. Then E is PT-
It remains to show that, if E is PT-semistable, then it is PT-stable. Suppose F ֒→ E is a subobject such that φ(F) = φ(E). Then we must have µ(H −1 (F)) = µ(H −1 (E)). As above, the coprime assumption forces H −1 (F) ֒→ H −1 (E) to be an isomorphism, and so the induced map H 0 (F) → H 0 (E) is injective. By φ(F) = φ(E), the inclusion H 0 (F) ֒→ H 0 (E) must be an isomorphism. As a result, the inclusion F ֒→ E is a quasi-isomorphism, i.e. F and E are isomorphic objects in D b (X). This means that there are no strictly PT-semistable objects in A p (X), so E is PT-stable.
We have the easy corollary Corollary 2.25. With the same hypotheses as above, if H −1 (E) is torsion-free and µsemistable (or, equivalently in this case, µ-stable), H 0 (E) is 0-dimensional, and E itself is PT-unstable, then any maximal destabilising subobject of E is 0-dimensional. Lemma 2.26 (Base change preserves the heart). Let L/K be a field extension. Let p : X L → X K be the induced morphism. If E ∈ A p (X K ), then p * E ∈ A p (X L ).
The following proof was suggested to me by Ziyu Zhang.
Proof. Take any F ∈ Coh ≥2 (X K ). Consider the torsion-filtration of p * F, where G i is the maximal subsheaf of p * F of dimension ≤ i; this filtration is unique [HL,p.3]. Take any σ ∈ Aut(L/K). Then for any i, we find that σ * G i is again a subsheaf of p * F of dimension at most i. By the maximality of G i , we have σ * G i ⊆ G i . However, σ * G i and G i have the same Hilbert polynomial, and so they must be equal. Thus each G i is invariant under σ * for all σ ∈ Aut(L/K), and so by descent theory for sheaves, there are subsheaves Proposition 2.27 (Base change preserves PT-semistability). Let k ⊂ K ⊂ L be field extensions, and p : X L → X K the induced map, where X is a smooth projective three-fold over k. Suppose E ∈ A p (X K ), with ch 0 (E) 0 and ch 0 (E), ch 1 (E) coprime. If E is PT-semistable, then p * E is also PT-semistable. Notation. Following [Tod1], we define, for a three-dimensional Noetherian scheme Y, the following subcategories of A p (Y) As mentioned in [Tod1], an object is torsion-free and Hom(O x , E) = 0 for any skyscraper sheaf O x . For any smooth projective three-fold Y, we also define the dualising functor Proof. Since p * preserves the Chern character, using Corollary 2.25, we just need to show that Hom(O x , p * E) = 0 for all closed point x ∈ X L , where O x denotes the skyscraper sheaf with value L supported at x.
Since E is PT-semistable, E ∈ A p 1/2 . Then D(E) ∈ A p 1/2 ⊂ A p by [Tod1,Lemma 2.17]. And so p * D(E) ∈ A p by Lemma 2.26. However, we have shown that p * E cannot be PT-unstable. Remark. The proof above takes a different track from the sheaf case. In the sheaf case [Lan], the argument is as such: suppose L/K is a field extension over k. Form the base extension where X K = X ⊗ k K and X L = X ⊗ k L. Given a µ-semistable sheaf E ∈ Coh(X K ), we show that p * E is µ-semistable by considering the maximal destabilising subobject E 0 ⊆ p * E.
Since the inclusion E 0 ֒→ p * E is invariant under Galois automorphisms of L/K, we can use descent theory for sheaves to conclude that E 0 descends to X K , i.e. is the pullback of a sheaf on X K . However, this argument relies on the uniqueness of HN filtrations for sheaf stability conditions. For polynomial stabilities in the derived category, we only have uniquness of HN filtrations up to isomorphism, which is not enough in order to make use of descent theory for sheaves. p 3,1 (H −1 (E)) = p 3,1 (E). And because φ(E 0 ) ≻ φ(E), there is a lower bound for ch 3 (E 0 ), say ch 3 (E 0 ) ≥ α. Then ch 3 (E/E 0 ) has an upper bound, namely n − α. Therefore, we have ch 3 (H −1 (E/E 0 )) = −(ch 3 (E/E 0 ) − ch 3 (H 0 (E/E 0 ))) ≥ α − n, i.e. there is a lower bound for ch 3 (H −1 (E/E 0 )).

Openness and
On the other hand, H −1 (E/E 0 ) and H −1 (E) have the same p 3,1 , while we have a quotient H −1 (E) ։ H −1 (E/E 0 ) in Coh 3,1 . By the semistability of H −1 (E) in Coh 3,1 , H −1 (E/E 0 ) is also semistable in Coh 3,1 . In particular, H −1 (E/E 0 ) is µ-semistable. Therefore, by [Mar,Theorem 4.8], the set Proof. Since Chern classes are locally constant for flat families of complexes, we may assume that ch 0 (E s ) 0 for all s ∈ S . By Proposition 3.1, we can assume that the fibre E s satisfies properties A, B and C for every s ∈ S . Since S sub is bounded, using the same argument as in [Tod1,Proposition 3.16] and [Tod2,Proposition 3.17], we know that there exists a scheme π : Q → S of finite type over S , a relatively perfect complex E 0 ∈ D(X ×Q) (see [Lie,Definition 2.1.1]) that is a flat family of objects in A p (X) over Q, and a morphism α : E 0 → E Q such that, for every q ∈ Q, the fibre α q : (E 0 ) q → (E Q ) q = E π(q) is an injection in A p (X) with φ(E 0 ) q ≻ φ(E Q ) q , and any maximal destabilising subobject E 0 ֒→ E s for some s ∈ S occurs as a fibre α q of α where q ∈ π −1 (s).
If we can show that s 0 π(Q), then we can conclude that PT-semistability is an open condition. Suppose s 0 ∈ π(Q). Then we can find a smooth curve C, a closed point p ∈ C, and a map γ : C → S taking p to s 0 , making the diagram commute: where W denotes the generic point of SpecÔ C,p . Let W denote SpecÔ C,p . The restriction α| W : E 0 | W → E Q | W is then an injection in A p (X × W) with φ(E 0 | W ) ≻ φ(E Q | W ). Let B W denote a maximal destabilising subobject of E 0 | W with respect to PT-semistability. Since H −1 (E Q | W ) is torsion-free, either B W is 0-dimensional (hence a sheaf) or is 3-dimensional.
Then we have an injection α ′ W : B W ֒→ E Q | W where B W is PT-semistable. By the properness of quot scheme (if B W is 0-dimensional) or Theorem 2.23 (if B W is 3-dimensional, hence of nonzero rank), we can extend B W to a flat family B of PT-semistable objects over W. By [Tod1,Lemma 3.18], we can extend α ′ W to a morphism α ′ : B → E Q | W with nonzero central fibre, which gives a destabilising subobject of the central fibre of E Q | W , namely E s 0 , contradicting the PT-semistability of E s 0 . This shows s 0 π(Q), thus proving the proposition.
The property of being PT-stable is easily seen to be an open property, too: Proposition 3.4 (Openness of PT-stability). Let S be a Noetherian scheme over k, and E ∈ D b (X × Spec k S ) be a flat family of objects in A p over S with ch 0 0. Suppose s 0 ∈ S is a point such that E s 0 is PT-stable. Then there is an open set U ⊆ S containing s 0 such that for all points s ∈ U, the fibre E s is PT-stable. Proof. By Proposition 3.3, it suffices to assume that E is a family where each fibre E s is a PT-semistable object. Then we can consider the set S ′ := {E 0 : E 0 is a subobject of E s for some s ∈ S , and φ(E 0 ) = φ(E s )} as in the proof of Lemma 3.2. Since {ch(E 0 ) : E 0 ∈ S ′ } is bounded, and each E 0 ∈ S ′ is necessarily PT-semistable, the set S ′ itself is bounded. Exactly the same proof as in Proposition 3.3 would then yield the result.

Separatedness for PT-Stable Objects.
Note that, for objects F, G ∈ D b (X R ), the R-module Hom D b (X R ) (F, G) is finitely generated and (20) as in [Tod1,Lemma 3.18]. Since Coh(Y) is a full subcategory of D b (Y) for Y = X K or X R (being the heart of the standard t-structure), when F, G ∈ Coh(X R ), the above isomorphism becomes Hom Coh(X R ) (F, G) ⊗ R K Hom Coh(X K ) (F ⊗ R K, G ⊗ R K).
Proposition 3.5 (Valuative criterion for separatedness). Let X be a smooth projective threefold over k, and R an arbitrary DVR over k. Let E 1 , E 2 ∈ D b (X R ) be flat families of objects in A p over Spec R, and suppose j * E 1 j * E 2 in D b (X K ). Suppose both Lι * E 1 , Lι * E 2 are PTsemistable objects in A p (X k ) and at least one of them is PT-stable. Then the isomorphism j * E 1 j * E 2 extends to an isomorphism E 1 E 2 .
In any case, π m f : E 1 → E 2 restricts to an isomorphism on X k . Let M be the homotopy kernel of π m f , so we have an exact triangle in D b (X R ) That Lι * (π m f ) is an isomorphism means Lι * M = 0. And this implies M itself is zero by [Lie,Lemma 2.1.4]. That is, π m f is an isomorphism.

Moduli Spaces of PT-Semistable Objects
By Lieblich's result [ABL,Corollary A.4], there is an Artin stack locally of finite presentation D A p (X) → Spec k such that, for any scheme B over Spec k, the fibre of the stack over B is the category of complexes E ∈ D b (X B ) such that for each point b ֒→ B, the fibre E| b obtained by derived pullback lies in the heart A p (X b ). If we fix a Chern character ch = (−r, −d, β, n), then we have a closed substack D ch A p (X) whose fibres are categories of families of complexes in A p with the prescribed Chern character. Then, by openness of PT-semistability (Proposition 3.3) and openness of , we have open substacks A p (X) whose fibres are categories of families of PT-stable and PT-semistable objects, respectively, with Chern character ch.
By boundedness of PT-semistable objects [Lo,Proposition 3.4 [Tod1], we can consider the moduli functor from schemes over Spec k to sets M : (Sch/Spec k) → (Sets) that takes a scheme B over Spec k to the set of complexes E ∈ D b (X B ) such that, for each point b ∈ B, the fibre E| b obtained by derived pullback satisfies Hom(E| b , E| b ) = k and Ext −1 (E| b , E| b ) = 0. Inaba [Ina,Theorem 0.2] showed that theétale sheafification of this functor is represented by a locally separated algebraic space, in the sense of [Knu]. Fixing a Chern character ch, we obtain a subfunctor M´e t ch,PT s that takes a scheme B to the set of families of PT-stable objects over B with Chern character ch. As in the previous paragraph, by our results on boundedness, openness, separatedness (Proposition 3.5) and universal closedness, M´e t ch,PT s is a proper algebraic space of finite type. This completes the proof of Theorem 1.1.