Nef divisors for moduli spaces of complexes with compact support

In Bayer and Macrì (J Am Math Soc 27(3):707–752, 2014), the first author and Macrì constructed a family of nef divisors on any moduli space of Bridgeland-stable objects on a smooth projective variety X. In this article, we extend this construction to the setting of any separated scheme Y of finite type over a field, where we consider moduli spaces of Bridgeland-stable objects on Y with compact support. We also show that the nef divisor is compatible with the polarising ample line bundle coming from the GIT construction of the moduli space in the special case when Y admits a tilting bundle and the stability condition arises from a θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}-stability condition for the endomorphism algebra. Our main tool generalises the work of Abramovich–Polishchuk (J Reine Angew Math 590:89–130, 2006) and Polishchuk (Mosc Math J 7(1):109–134, 2007): given a t-structure on the derived category Dc(Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_c(Y)$$\end{document} on Y of objects with compact support and a base scheme S, we construct a constant family of t-structures on a category of objects on Y×S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y \times S$$\end{document} with compact support relative to S.


Motivation
In recent years, a number of authors have applied wall-crossing techniques for Bridgeland stability conditions in order to systematically study the birational geometry of moduli spaces; see Sect. 1.5 for more background. The Positivity Lemma of [13] provides a clear, geometric link between the stability manifold and the moveable cone of the moduli space by producing a family of nef divisors on any moduli space of Bridgeland-stable objects on a smooth projective variety X .
Rich and interesting wall-crossing structures have also been observed in semi-local settings, including, for example, the resolution of singularities Y → SpecR of an affine singularity, with many interesting examples coming from geometric representation theory or the study of algebras that are finite over their centre. The main goal of this paper is to extend the machinery of [13] to such settings. In fact our approach works more generally, replacing X by any separated scheme Y of finite type over k.

The main result
Let k be an algebraically closed field, and let Y be a separated scheme of finite type over k. Let D(Y ) denote the bounded derived category of coherent sheaves on Y , and D c (Y ) the full subcategory of objects with proper support. As we explain in more detail at the beginning of Sect. 5, the usual notion of numerical stability conditions is not well-suited for the category D c (Y ) (and the frequently used replacement, the category D Z (Y ) of complexes supported on a proper subvariety Z ⊂ Y does not lead to nice moduli spaces).
We therefore propose to use a variant of the definition of numerical K -group: we define K num c (Y ) as the quotient of the Grothendieck group of D c (Y ) by the radical of the Euler pairing with perfect complexes on Y . We prove in Lemma 5.1.1 that

Additional background and outlook
In the projective setting, the link between wall-crossing for stability conditions and birational geometry of moduli spaces has led to a large number of results over the last five years. This was initiated with striking examples for abelian surfaces [2] and P 2 [4], and then exploited systematically for abelian [45,46,61] and K3 surfaces (see [12] in the smooth case, and [47] for singular O'Grady-type moduli spaces; see also [31] for a survey and more applications), for Enriques surfaces [50], for P 2 [15,[23][24][25][26]28,41,60], with the story now essentially completed in [42], and for other rational surfaces [8]; it has also led to results for general surfaces [9,27]. In many cases, there is a complete description of the movable cones of the moduli spaces, along with its chamber decomposition coming from associated minimal models.
On the other hand, a number of authors have studied stability conditions on quasiprojective ('local') Calabi-Yau varieties Y , see [19,22,35,56] and [11,16,57,58] for crepant resolutions of two-and three-dimensional canonical singularities, respectively, and [5] for higher-dimensional symplectic resolutions of singularities naturally associated to algebraic groups. Our goal is to provide in this context the machinery that is used in the projective setting.
Even in the case where Y is a projective crepant resolution of C 3 /G for a finite abelian subgroup G ⊂ SL 3 (C), a rich wall-crossing picture emerges by considering Y itself as a moduli space parametrising skyscraper sheaves of points. Indeed, a simple reinterpretation of [29] (along the lines of our Sect. 7) says that any (projective) birational model of Y appears as a moduli space of Bridgeland-stable objects; more generally, this result holds for any projective crepant resolution of a Gorenstein, affine toric 3-fold by [34]. We anticipate that this result can be generalised significantly, both by allowing for more general Y , and by considering different moduli spaces on Y . We also hope that it will simplify the study of the space of stability conditions itself: typically, one of the crucial steps is the systematic understanding of walls of the geometric chamber in Stab(D c (Y )), where skyscraper sheaves of points are all semistable, some of them being strictly semistable. Our results provide a nef line bundle on Y , whose associated contraction should govern the wall-crossing behaviour to a large extent.

Running assumptions and notation
All our schemes are assumed to be Noetherian schemes over an algebraically closed field k. In addition, we assume from Sect. 2.4 onwards that our schemes are separated and of finite type over k. Given a product of schemes Y × S, we write p : Y × S → Y and q : Y × S → S for the projections to the first and second factor.
For any scheme X , let D(X ) denote the bounded derived category of coherent sheaves on X , let D perf (X ) denote the full subcategory of perfect complexes on X , and write D(Qcoh(X )) for the unbounded derived category of quasi-coherent sheaves on X . To avoid a proliferation of R and L, we omit these symbols in our derived functors except in writing R .

Derived category with left-compact support
In this section, we define what it means for a complex of coherent sheaves on a product Y × S to have 'left-compact support'. We also study the basic properties, and compare this notion to that of an object on Y × S having proper support over S. The latter part of this section follows closely the work of Abramovich-Polishchuk [6] and Polishchuk [52] in defining sheaves of t-structures over a base.

Compact and left-compact support
For any Noetherian scheme Y over k, we may identify D(Y ) with the full subcategory of D(Qcoh(Y )) of bounded complexes with coherent cohomology. Definition 2.1. 1 The support of a quasi-coherent sheaf G is the locus Supp(G) = {y ∈ Y | G y = 0} of points with non-zero stalk. The support of an object F ∈ D(Qcoh(Y )) is the union of the supports of its cohomology sheaves.
Since localisation is exact, we could equivalently define where F y is the complex of stalks of F at the local ring at y. In addition, if F ∈ D(Y ), then by Nakayama's Lemma Supp(F) = {y ∈ Y | i * y F = 0}, (2.2) where i y is the inclusion of the spectrum Spec k(y) of the residue field of y. Also note that for F ∈ D(Y ), the support Supp(F) is closed. Write D c (Y ) ⊂ D(Y ) for the full subcategory of objects that have proper support. Following convention, we refer to such objects as having 'compact support'. We note the following easy properties of support.

Lemma 2.1.2 Let F, E ∈ D(Y )
, and let f : Y → Y and g : Y → Y be morphisms between Noetherian schemes. Then: Proof The first part is immediate from (2.1), the second from (2.2). For the third part, assume y / ∈ f (Supp(F)); then there is an open neighbourhood y ∈ U ⊂ Y with F| f −1 (U ) = 0, and the claim follows from flat base change.  Proof Given F ∈ D lc (Y × U ), assume that it is represented by a bounded complex of coherent sheaves. Let F c ⊂ j * F be the subcomplex with F = j * F c by the previous Lemma. By the assertions in the Lemma above, we obtain where the last inclusion is a case of Lemma 2.1.2(iii). By assumption, Supp(F) ⊆ Z × U for some proper subscheme Z ⊆ Y ; hence Supp(F c ) ⊂ Z × S, which implies the claim. Proof Following Rouquier [53,Remark 3.14], the pullback j * : D(Y × S) → D(Y × U ) is essentially surjective and has kernel given by the subcategory of complexes supported in Y × T . It follows that D(Y ×U ) is equivalent as a triangulated category to the quotient of D(Y ×S) by the subcategory of complexes whose support is contained in Y × T . Now restrict j * to D lc (Y × S) and apply Corollary 2.2.2 to see that j * : D lc (Y × S) → D lc (Y × U ) is essentially surjective and has kernel given by the subcategory of complexes (with left-compact support) whose support is contained in Y × T . It follows that D lc (Y ×U ) is equivalent to the quotient of D lc (Y × S) by the subcategory of complexes supported in Y × T .

Objects with proper support over the base
Let S, Y be Noetherian schemes. Recall that p : Y × S → Y and q : Y × S → S are the first and second projection morphisms.
is not quasi-isomorphic to the restriction of an object F ∈ D(Y × P 1 ) that has proper support over P 1 , because otherwise Lemmas 2.1.5 and 2.3.2 would imply that O ∼ = (id Y × j) * F has leftcompact support which is absurd.

Integral functors
It is well known that when S and Y are smooth projective varieties, an object E ∈ D(Y × S) is the kernel for a pair of integral functors, sometimes denoted We present here the natural extension of this statement to complexes with compact support.
Proof Part (i) follows from [55, Tag 08E0]. For part (ii), since Y is separated over k and F has proper support over k, [55,Tag 01W6] implies that F has proper support over Y , so f * F ∈ D(Y ) by part (i). To see that f * F has proper support, it is enough by Lemma 2.1.2(iii) to show that f (Supp(F)) is proper. This follows from [55,Tag 03GN].
If in addition E is S-perfect, then we obtain a second integral functor Proof For the first claim, given F ∈ D c (S), we need to show that p * (E ⊗ q * F) ∈ D c (Y ). Since the projection q is flat, we have q * F ∈ D(Y × S). The support of E ⊗ q * F ∈ D(Y × S) is closed and contained in which is proper. Therefore, E ⊗q * F ∈ D c (Y × S), and the claim follows from Lemma 2.4.1.
To construct the functor E , let F ∈ D perf (Y ). Then p * F ∈ D(Y × S) is perfect. By [54,III,Proposition 4.5] (applied with f = id Y ×S and g = q-note that perfect is the same as "of finite amplitude with respect to the identity morphism"), it follows that On the other hand, [54,III,Proposition 4.8] shows that q * (E ⊗ p * F) is S-perfect, and therefore perfect.

On t-structures
Let D be a triangulated category. Recall that a t-structure on D is a pair of full subcategories More generally, we write D [−∞,b] := D b and D [a,∞] := D a , and refer to the subcategory D [a,b] for any interval [a, b] that may be infinite on one side. The heart of the t-structure is the abelian category D [0,0] . The inclusions D n → D and D n → D admit right-and left-adjoints τ n : D → D n and τ n : D → D n respectively. For F ∈ D and n ∈ Z, the corresponding truncation triangle is the exact triangle  [52] say that a t-structure is 'nondegenerate' if it is bounded in the sense defined above (and hence nondegenerate in the sense defined above).
Given triangulated categories D, D each equipped with a t-structure, a functor

Sheaves of t-structures over the base
Let S, Y be Noetherian schemes. We continue to write p : Y × S → Y and q : Y × S → S for the two projections.
This notion generalises that of a 't-structure on D(Y × S) that is local over S' [52] and that of a 'sheaf of t-structures on Y over S' when Y and S are projective [6]. To justify the terminology, we extend [ by sheafification, so with respect to the standard t-structures we have that Hom S (F, G)| U i j ∈ D 1 (Qcoh(U i j )) and hence Hom S (F, G) ∈ D 1 (Qcoh(S)). Since Hom • (F, G) = (Hom(F, G)) = (S, Hom S (F, G)), and since is left exact, it follows that Hom 0 (F, G) = 0. It remains to define the truncation functors. By boundedness of the t-structure on each D lc (Y × U i ) and an induction argument, we need only prove that for any F ∈ D 0 lc (Y × S), the left truncation H 0 (F) exists, as does a morphism H 0 (F) → F whose cone lies in  We now show that (2.3) induces the given t-structure on each D lc (Y × U i ). The restriction from Y × S to Y × U i is t-exact, so we need only show for any i and any Uniqueness Next, we show that the t-structure (2.3) is the unique t-structure on D lc (Y × S) over S which induces the given t- On the other hand, if we truncate any G ∈ D [a,b] lc (Y × S) with respect to the second t-structure, then ( τ a−1 G)| Y ×U i = τ a−1 (G| Y ×U i ) = 0 for each i. The uniqueness of gluing from [6, Corollary 2.1.11] implies that τ a−1 G = 0, and similarly we have Sheafify It remains to show that our given t-structure on D lc (Y ×S) extends uniquely to a sheaf of t-structures over S. To construct the associated t-structure over U ⊂ S, replace S by U and U i by U ∩U i in the construction and proof of uniqueness above. One easily verifies the sheaf property by applying (2.3) for S = i U i and analogously for U = i U ∩U i simultaneously, along with the sheaf property for the given t-structures on each U i .

Remark 2.6.3
The following rephrasing of the uniqueness result in the above theorem is useful in practice: Given a sheaf of t-structures on D lc (Y × S) over S and an object

Lemma 2.6.4 For any schemes Y and S, suppose that D lc (Y × S) has a sheaf of t-structures over S. For
Proof The functor is well-defined by Lemma 2.1.5. Let S = i U i be an open cover such that is t-exact. Then we obtain by restriction a sheaf of t-structures on D lc (Y × S) over S.
Towards this goal, let T ⊆ S be a closed subset. Consider the subcategory of objects with left-compact support whose support lies over T . Our proof follows closely that of [6, Theorem 2.1.4], beginning with two results on the category D lc (Y × S) T . First we recall the following Lemma (the proof of which does not require Y to be smooth or projective): For any i ∈ Z, we write H i t for the i-th cohomology functor with respect to the t-structure on D lc (Y × S) listed as an assumption in Theorem 2.6.5. Lemma 2.6.7 Under the assumptions of Theorem 2.6.5, let T ⊆ S be a closed subset The given t-structure is bounded, so F is a finite extension of only finitely many cohomology sheaves For the opposite implication, let L denote an ample line bundle on S. Replacing L by a suitable power, we may assume that T is the common zero-locus of sections f 1 , . . . , f n of L. We apply Lemma 2.6.6 to obtain d for which f = f i 1 · · · f i d : F −→ F ⊗ q * L d is the zero map for all such sequences. By assumption, tensoring with q * L is t-exact for the t-structure in question, so it commutes with taking cohomology H i t . Therefore, by definition of a t-structure on D lc (T × S), this proves our claim. Lemma 2.6.8 Under the assumptions of Theorem 2.6.5, let A denote the heart of the given t-structure. Then for every closed subset T ⊆ S, the subcategory D lc (Y ×S) T ∩A of the abelian category A is closed under subobjects, quotients and extensions.
Proof Tensoring with q * L is a t-exact functor on D c (Y × S), so it is exact on A. Given a short exact sequence 0 → E → F → G → 0 in A, a diagram chase shows that if all maps F → F ⊗ q * L d as in Lemma 2.6.6 vanish, then so do all such maps E → E ⊗ q * L d and G → G ⊗ q * L d . Conversely, if all the maps E → E ⊗ q * L d and G → G ⊗ q * L d vanish, then so do all the maps F → F ⊗ q * L 2d given by sequences of length 2d. The result follows from Lemma 2.6.6.
Proof of Theorem 2.6.5 Let U ⊆ S be an open subset. For any interval [a, b] that may be infinite on one side, consider the subcategory We proceed in two steps.
Step 1: To verify that (2.4) defines a bounded t-structure, clearly . We next check that there are no nontrivial morphisms between any F 0 ∈ D 0 lc (Y × U ) and G 0 ∈ D 1 lc (Y × U ). Proposition 2.2.3 implies that if a morphism F 0 → G 0 does exist then it's obtained from a diagram of the form T . Thus all cohomology sheaves of τ 1 t (F ) lie in D lc (Y × S) T , and hence so does τ 1 t (F ) by Lemma 2.6.7. This object is the cone of g : τ 0 t (F ) → F which then lies in the localising class, and therefore the diagram (2.5). The t-structure on D lc (Y × S) shows that this map is zero, so the original morphism from F 0 to G 0 is zero as required. To check condition (iii) from the definition of a t-structure in Sect. 2.5, let E 0 ∈ D lc (Y × U ) and , which shows the t-structure on D lc (Y × U ) is also bounded.
Step 2: We verify that the t-structure from Step 1 defines a sheaf of t-structures over S. For any open subsets U ⊆ U ⊆ S, we have the following commutative diagrams of pullback functors , so j * is right exact. Left-exactness is similar, so the t-structure on D c (Y × S) induces a sheaf of t-structures over S.
Setting L = O S in Theorem 2.6.5 immediately gives: Corollary 2.6.9 Let S be an affine scheme. Then every bounded t-structure on D lc (Y × S) determines by restriction a sheaf of t-structures on D lc (Y × S) over S.

Sheaf of t-structures on D c (Y × P r ) over P r
In this section we construct a sheaf of t-structures on D c (Y × P r ) over P r following the work of Abramovich-Polishchuk [6, Theorem 2.3.6].

On resolution of the diagonal
Let Y be any scheme. For r 0, let p : Y × P r → Y denote the first projection and For F, G ∈ D(Qcoh(Y ×P r )), write F G := π * 1 F⊗π * 2 G ∈ D(Qcoh(X )). Let q : Y × P r → P r denote the second projection, O(1) = q * O P r (1) the relative hyperplane bundle and := q * P r the relative cotangent bundle. For F ∈ D(Qcoh(Y × P r )) and i ∈ Z, write F(i) := F ⊗ O(i).
The relative version of resolution of the diagonal ⊆ X by Orlov [51] is the resolution by the projection formula and flat base change. By Proposition 2.4.2, this functor restricts to an integral transform For any fixed j ∈ Z, if we break the above resolution into short exact sequences as described in [32, Proof of Corollary 8.29], any object F ∈ D c (Y × P r ) can be reconstructed by taking successive cone operations on the collection for every 0 i r and j ∈ Z.
The next two results record several useful consequences of these observations.

Lemma 3.1.2 For any m r + 1, there is an exact sequence
The observations above for Y = Spec k and j = r show that the sheaf O P r (m) ∈ D(P r ) can be reconstructed by taking successive cone operations using the objects for 0 i r . Our assumption on m gives m − r + i > 0, so Manivel [43] implies that the higher cohomology groups of i (m − r + i)) vanish; hence i,r (O(m)) = V m i ⊗O(r −i). Substituting these sheaves for 0 i r into the above cone operations yields (3.4).
Proof For (i), the 'only if' direction follows from Lemma 2.1.5(i) and Corollary 2.1.6. Conversely, the object F ∈ D(Y × P r ) can be reconstructed by taking successive cone operations on the collection r, our assumption on F ensures that the objects i,0 (F) from (3.3) are trivial for 0 i r , so F ∼ = 0 after taking cones. For (iii), Lemma 2.1.5(i) and Corollary 2.1.6 imply that we obtain a functor Each φ i is fully faithful by the projection formula. The approach of Orlov [51, §2] shows that the sequence of subcategories on the right-hand side of (3.5) is semiorthogonal. As for generation, consider F ∈ D c (Y × P r ). For 0 i r , the object , so after taking cones we have that F is contained in the right side of (3.5), as required.
Proof One direction is immediate from Lemma 2.1.5(i) and Corollary 2.1.6. For the other direction, it is enough to show that

A family of t-structures
From now on we work under the following assumption: For any interval [a, b] that may be infinite on one side, we obtain a subcategory The following purely categorical result, which combines [52, Lemma 3.1.1-3.1.2], enables us to glue t-structures of subcategories arising in a semi-orthogonal decomposition.
Then we obtain a t-structure on D by setting By tensoring with O(−n), it suffices to prove the result for n = 0. In this case, we show that Lemma 3.2.2 applies to semiorthogonal decomposition of D c (Y × P r ) from (3.5).

Corollary 3.2.4 Let
(i) If m r + 1, then p * F(m) can be reconstructed by taking successive cone operations using the objects V m Proof Pull the resolution of O P r (m) from Lemma 3.1.2 back along q : Y × P r → P r , tensor with F and pushforward along p : Y ×P r → Y to obtain a resolution of p * F(m) in terms of To construct the morphism of cohomology functors, note that the inclusion D 0

and hence a morphism
as required.

On graded S-modules in an abelian category
For V = H 0 (P r , O(1)), the symmetric algebra of V is a graded k-algebra S = m 0 S m generated by r +1 variables of degree one. We now recall several categorical notions and results from [6, Section 2.2], where the abelian category of interest is the heart A of the t-structure on D c (Y ) given by Assumption 3.2.1.
A graded S-module in A is a collection M = {M n | n ∈ Z} of objects in A and a collection of morphisms {ϕ m,n : S m ⊗ M n → M m+n | m, n ∈ Z, m 0} satisfying the obvious associativity condition, such that ϕ 0,n is the identity for each n ∈ Z. We for an object M in A and a fixed i ∈ Z, and where the morphisms The main result we require is the following [6, Theorem 2.2.2].

Theorem 3.3.1 The category of graded S-modules of finite type in A is abelian and Noetherian.
We record the following examples for later use. (i) Let 0. If ⊕ n M n is a graded S-module of finite type, then so is m M n . It follows that the graded S-module m S n ⊗ F is of finite type in A.
(ii) For m 0, tensor the Euler exact sequence on P r by O P r (m) and apply the global sections functor to see that obtained as the strand of the Koszul complex for M in degree d is exact for d 0.
Proof The proof is contained in [6, Step 5 of Proof of Proposition 2.3.3].

A sheaf of t-structures on D c (Y × P r )
We now use the family of t-structures from Proposition 3.2.3 to construct a 'limiting' t-structure on D c (Y × P r ) that is actually a sheaf of t-structures over P r . We continue to work under Assumption 3.2.1.
As a first step, we provide an application of the categorical results from the previous section by establishing a technical result that will be used in the proof of Propo- It remains to prove that p * G n (r + 1) ∈ A. We proceed in three steps.
Step 1: For 0 i r , by tensoring with O(i) and pushing forward, we get another exact triangle The left-hand object in (3.10) lies in , the exact triangle (3.10) is a standard truncation triangle and hence by uniqueness of objects in such triangles we have Examples 3.3.2 and Theorem 3.3.1 imply that the cokernel of this map cok(φ) = ⊕ j C j is a graded S(V )-module of finite type in A. Moreover, for any n r +1 and 0 i r , the equality V n+i Step 1 above. This completes Step 2.
Step 3: Deduce that p * G n (r + 1) ∈ A using a resolution by p * G n (i) = C n+i−r for 0 i r . The twist of the Koszul complex associated to the space As in the proof of Corollary 3.2.4, pull this resolution back along q : Y × P r → P r , tensor with G n and pushforward along p : of objects in A to p * G n (r + 1). To show that p * G n (r + 1) ∈ A, we need only show that the complex (3.11) has nonzero cohomology only in the right-hand position. For this, Step 2 enables us to rewrite this complex as which we recognise from (3.9) as forming part of the Koszul complex of degree n + 1 associated to the graded S(V )-module ⊕ n C n in A. We need only show that this latter complex is exact for n 0, but this is immediate from Lemma 3.3.3 because ⊕ n C n is of finite type.
The key observation in constructing the sheaf of t-structures is the following stabilisation result for the cohomology objects H i n (F) associated to the family of t-structures Proof We claim that it suffices to prove that the highest nonzero cohomology group of F stabilises. Indeed, boundedness of the t-structure , so the inclusions (3.7) and (3.8) imply that for all n 0 we To simplify the claim, shift F by b to obtain F ∈ D 0 0 and hence F ∈ D 0 n for n 0 by (3.7). Proving the claim is equivalent to proving that τ 0 . As a result, the claim is equivalent to requiring that each Moreover, this t-structure satisfies . Now (3.12) follows from (3.7) and (3.8), and hence D −1 ⊆ D 0 . Moreover, D 1 is right-orthogonal to D 0 , because for F ∈ D 1 = n 0 D 1 n , we have Hom(E, F) = 0 for all n 0 .7) and (3.8) respectively. Define for n 0. Proposition 3.4.2 gives N ∈ Z such that the cohomology groups stabilise for t-structures indexed by n N and hence Finally, tensoring by q * O P r (1) preserves the heart, so it's t-exact, and the result follows from Theorem 2.6.5.

The Noetherian property
Theorem 3.4.3 provides a bounded t-structure on D c (Y × U ) for each open subset U ⊆ P r . We now show that every such t-structure has a Noetherian heart. For this, associate to each F ∈ D [0,0] a graded S-module M(F) in the category A by setting       (−k i D) and hence assume that the injection F i → F extends to a morphism φ i : F i −→ F. The restriction to Y ×U is unchanged by this, as is the property of having left-compact support, so we obtain an increasing chain The sequence (3.15) stabilises by the case U = P r above, and restricting this chain to Y × U shows that (3.14) also stabilises.

Sheaf of t-structures over an arbitrary base
In this section we follow closely the approach of Polishchuk [52] in extending the construction of the sheaf of t-structures on D c (Y × P r ) over P r to an arbitrary base scheme S that is separated and of finite type. We then extend the work of Abramovich-Polishchuk [6] to show these these t-structures satisfy the open heart property.

Extending t-structures to the quasi-coherent setting
Let D be a triangulated category. A full subcategory P is a pre-aisle if P is closed under extensions and the shift functor X → X [1] for any X ∈ P. For any subcategory S ⊆ D, the pre-aisle generated by S is the smallest pre-aisle containing S, denoted p-a D [S]. A full subcategory P ⊆ D is an aisle if P = D 0 for some t-structure (D 0 , D 0 ) on D. Every aisle is a pre-aisle, but the converse is false in general; see [52, Remark of Section 2.1]. If we assume further that D is a triangulated category in which all small coproducts exist, then a pre-aisle P is cocomplete if it is closed under small coproducts. For any subcategory S ⊆ D, the cocomplete pre-aisle generated by S is the smallest cocomplete pre-aisle containing S, denoted by p-a D [[S]].

Lemma 4.1.1 Let Y and S be Noetherian schemes. Any t-structure
(4.1) Furthermore, for every interval [a, b] that may be infinite on one side, we have that

Sheaf of t-structures over an affine base
Given a Noetherian, bounded t-structure on D c (Y ), Theorem 3.4.3 gives a sheaf of t-structures on D c (Y × P r ) over P r . We now study the restriction of this t-structure to subcategories D lc (Y × U ), where U is an affine scheme of finite type over k. The main result of this section is the following: where p : Y × U → Y is the projection to the first factor. Moreover, p * : We prove this result in two stages. We first restrict along an open immersion Y × A r → Y × P r to prove the special case U = A r ; our proof runs parallel to that of [52, The projection formula shows that pullback along the first projection p : Y × P r → Y is t-exact, hence so is p * = (id Y × j) * • p * . It remains to show that the t-structure on D lc (Y × A r ) satisfies (4.4). We proceed in three steps: Step 1: We first claim that For the opposite inclusion, we need only show that p * D 0 c (Y ) ⊆ D 0 lc (Y × A r ), which follows from the t-exactness of p * shown above.
Step 2: We deduce (4.4) for A r by applying Lemma 2.5.2 to p * : D lc (Y × A r ) → D(Qcoh(Y )), and for this we must check that p * is t-exact and has trivial kernel. For left t-exactness, let F ∈ D 0 c (Y ). We know D 0 (Qcoh(Y )) is closed under small coproducts by (4.1). The projection formula gives Since the image of a preaisle under p * is a pre-aisle and since D 0 (Qcoh(Y )) is itself a pre-aisle, we deduce from Step 1 above that as required. For right t-exactness, use Lemma 4.1.1, adjunction and Step 1 to obtain

4] because Y × S is quasi-compact and separated. (iii)
• is right t-exact with respect to the given t-structure. For this, let F ∈ D lc (Y × S). The projection formula gives (  3) to the t-structure on D lc (Y × A r ) constructed in the special case above gives a t-structure on D lc (Y × U ), satisfying for any interval [a, b] that may be infinite on one side. This gives (4.4). Corollary 2.6.9 and Proposition 3.

imply that we obtain a sheaf of Noetherian t-structures on
. By (4.4) and the projection formula, we must show that p * p This completes the proof that p * is t-exact and hence concludes the proof of Proposition 4.2.1.

Construction over an arbitrary base
We are now in a position to establish the first main result of this section following [ for any open affine U ⊆ S (4.7) where we abuse notation by writing p for the first projection from Y × U . Moreover

a,b] (Qcoh(Y )) for every i;
(ii) the functor p * : D c (Y ) → D lc (Y × S) is t-exact with respect to these t-structures.
(iii) Assume in addition that S is projective. Then this t-structure satisfies where L is any ample bundle on S and q : Y × S → S is the second projection.
Before the proof we present a compatibility result for open immersions of affine schemes. This extends to our setting a statement from the proof of [52, Theorem 3.3.6].

is t-exact with respect to the t-structures on both sides given by (4.4).
Proof We have all the elements in place to reproduce the proof of this statement from [52, Proof of Theorem 3.3.6] so we provide only an outline. Extend the t-structures on D c (Y ) and D lc (Y × U i ) to t-structures on D(Qcoh(Y )) and D(Qcoh(Y × U i )) by Lemma 4.1.1.
Step 1: For any interval [a, b] that may be infinite on one side, we prove an analogue of (4.4), namely that Step 2: We now claim that Step 1. Equation (4.2) and adjunction now imply so the right-hand side of (4.9) is contained in the left. The opposite inclusion fol- Step 3: On the one hand, Lemma 4.1.1 shows that D 0 lc (Y × U 1 ) is an aisle in D lc (Y × U 1 ) which extends to a t-structure on D(Qcoh(Y × U 1 )) that satisfies On the other hand, since pulling back along id Y × j commutes with extensions and left shifts, and respects coproducts (see [49, Proposition 1.21]), Step 2 combined with the analogue of (4.10) for U 2 gives is also an aisle in D lc (Y × U 1 ) which by (4.11) extends to the same t-structure on D(Qcoh(Y × U 1 )). Comparing (4.10) and (4.11), Remark 4.1.2 implies that these t-structures coincide as desired. Finally, to show that p * is t-exact, write p i :

The open heart property
Let Y and S be a separated schemes of finite type, and consider the sheaf of Noetherian t-structures on D lc (Y × S) over S from Theorem 4.3.1. For any subset V ⊆ S that is either open or closed, let A V denote the heart of the induced t-structure on D lc (Y × V ) and let H i V (F) denote the i-th cohomology of an object F ∈ D lc (Y × V ).

Lemma 4.4.1 Let T ⊂ S be an effective Cartier divisor. Any object F
Proof Let i : T → S denote the closed immersion, and let f ∈ H 0 (O S (T )) be a defining section for T . Lemma 2.6.4 implies that is an exact triangle in D lc (Y × S), it follows from the cone construction that   Proof The support of E is closed, so it suffices to prove that E| Y ×T = 0. Let f ∈ H 0 (O S (T )) be a defining section for T . Since the abelian category A S is Noetherian, there is a maximal S-torsion subobject E tor ⊂ E supported in T and a short exact sequence where F has no torsion subobject with support in T . By restricting to Y × T and applying Lemma 4.4.1, we obtain an exact triangle that (id ×i) * (F| Y ×T ) is the cokernel and hence also lies in A S . We obtain F| Y ×T ∈ A S by Lemma 4.2.2, so H −1 T (F| Y ×T ) = 0 which proves the claim. It remains to show that E tor | Y ×T = 0. Since F| Y ×T = 0, we may assume from the beginning that E is S-torsion with support in T . Let k 0 be the minimal value such that E is annihilated by f k . Lemma 4.4.1 and the assumption H 0

Proposition 4.4.3 (The open heart property) Let Y and S be separated schemes of finite type, and let T ⊂ S be a local complete intersection. Let F ∈ D lc (Y × S). If F| Y ×T ∈ A T , then there is an open neighborhood
Proof It suffices to prove the statement under the additional assumption that T is an effective Cartier divisor in S, as an induction on the codimension of T in S proves the general case. Let a, b ∈ Z be such that F ∈ D [a,b] lc (Y × S) with b > 0. We proceed in two steps: Step 1: We find an open neighbourhood T ⊆ U ⊆ S such that F| Y ×U ∈ D [a,b−1] lc (Y × U ), and hence by induction, shrinking U at each step if necessary, we deduce that F| Y ×U ∈ D [a,0] lc (Y × U ). For this, restrict to Y × T a truncation exact triangle for F to obtain an exact triangle (4.14) lc (Y × T ), so the long exact sequence in cohomology for (4.14) gives lc (Y × T ), and since F| Y ×T ∈ A T holds by assumption, the exact triangle (4.15) shows that the object These two statements force τ −1 F| Y ×T = 0 as claimed. To conclude, the support of τ −1 F is closed in S, so there exists an open neighbour-

Numerical Bridgeland stability conditions for compact support
The goal of this section is to provide the right setting for stability conditions for objects with compact support on a non-compact quasi-projective variety Y . Note that the Kgroup of D c (Y ) almost always has infinite rank (for example, skyscraper sheaves of points that do not lie on proper subvarieties of positive dimension have linearly independent classes), and yet the numerical K -group of D c (Y ) is not defined when Y is singular. Even when Y is smooth, the class of skyscraper sheaves of points is 0, so D c (Y ) is unlikely to admit numerical stability conditions (where one requires that the central charge factors via the numerical Grothendieck group). To get around these problems, many authors (see Sect. 1.5) have instead considered stability conditions on D Z (Y ), the derived category with objects supported on a proper subvariety Z ⊂ Y . However, this does not lead to moduli spaces of finite type: even the moduli space of skyscraper sheaves of points would be the completion of Y at Z .
We therefore propose to use a variant of the numerical Grothendieck group of D c (Y ), defined via the Euler pairing with perfect complexes.

Numerical Grothendieck groups
Let Y be a separated scheme of finite type over k. For any objects E ∈ D perf (Y ) and The Euler form between the Grothendieck groups of these categories is the bilinear form given by (5.1) The quotient of K (D perf (Y )) and K (D c (Y )) with respect to the kernel of χ on each factor defines the numerical Grothendieck groups K num perf (Y ) and K num c (Y ) respectively, and we use the same notation for the induced perfect pairing. Our interest lies in studying the category D c (Y ) when K num c (Y ) has finite rank. Here we present a sufficient condition for this to hold. Proof First assume that Y is smooth. We may choose a smooth projective completion

Stability conditions for compact support
We assume that the reader is familiar with the notion of stability condition as introduced in [17], in particular the notion of slicing. We note that typically the category D c (Y ) is decomposable into infinitely many factors; indeed, any closed point y ∈ Y that does not lie on a proper subvariety of positive dimension of Y gives rise to such a factor. Hence, instead of applying the notion of stability condition verbatim to the category D c (Y ), we restrict to the situation where K num c (Y ) is a finite rank lattice and we allow only central charges that factor through K num c (Y ). Definition 5.2.1 Assume that K num c (Y ) has finite rank. A numerical stability condition for compact support on Y is a pair (Z , P), where Z : K num c (Y ) → C is a group homomorphism and P is a slicing of D c (Y ), such that the following properties hold: (i) For any φ ∈ R and any non-zero E ∈ P(φ), we have Z ([E]) ∈ R >0 · e πiφ ; and (ii) There exists a quadratic form Q on K num c (Y ) ⊗ R such that: • for any φ ∈ R and any E ∈ P(φ), we have Q([E]) ≥ 0; and Let Stab(D c (Y )) denote the space of numerical stability conditions for compact support on Y .
The deformation results of Bridgeland [17] extend to this setting (see e.g. [14, Appendix A] for a discussion under the assumptions as formulated above); in particular, Stab(D c (Y )) is a complex manifold of dimension equal to the rank of K num c (Y ). Moreover, the results of [18, Section 9] carry over completely to give a wall-andchamber structure on Stab(D c (Y )) for any given class v ∈ K num c (Y ). More precisely, there exists a locally finite set of walls (real codimension one submanifolds) such that the set of σ -semistable objects of class v does not change as σ varies within a connected component of the complement of walls (called a chamber), and such that on every wall there exist strictly semistable objects that become unstable on one side of the wall.

The linearisation map with compact support
The goal of this section is to prove the main result. Let Y and S be separated schemes of finite type, and write p : Y × S → Y and q : Y × S → S for the first and second projection respectively.

The linearisation map
For any closed point s ∈ S, define Y s := Y × {s} and write i Y ×{s} : Y s → Y × S for the closed immersion. For E ∈ D(Y × S), we identify Y s ∼ = Y and let denote the derived pullback of E to Y s . Assume that S is separated and of finite type over k. Let N 1 (S) denote the vector space of real Cartier divisor classes modulo numerical equivalence; here numerical equivalence is taken with respect to proper curves C ⊂ S. Dually, N 1 (S) denotes the space of proper 1-cycles in S modulo numerical equivalence (with respect to Cartier divisor classes on S). Let [C] ∈ N 1 (S) denote the class of a 1-cycle. In fact, for a fixed family E, equation (6.1) defines a numerical Cartier divisor on S for any numerical stability condition for compact support on Y . The resulting map obtained by sending a stability condition σ to the divisor class E,σ is the linearisation map of the family E.
We present the proof of Theorem 6.1.4 in two stages. We first prove that the linearisation map is well-defined, postponing until the next subsection the proof of the positivity statements. Lemma 6.1.5 The assignment of (6.1) defines a numerical Cartier divisor class E,σ ∈ N 1 (S).

Lemma 6.1.6 Let S, Y be schemes of finite type. Let E ∈ D lc (Y × S) be S-perfect and let F ∈ D perf (Y ). For any proper subscheme i : T → S, we have
Proof Use the projection formula repeatedly to obtain as required.
Proof of Lemma 6.1.5 Note first that the integral functor E : D c (S) → D c (Y ) is well-defined by Proposition 2.4.2. Since the stability condition is assumed to be numerical, we can choose P i ∈ D perf (Y ) and a i ∈ R for 1 i m such that It is sufficient to show that for each i, there exists a Cartier divisor class L i on S, such that for all projective curves C ⊆ S. Proposition 2.4.2 gives E (P ∨ i ) := q * (E ⊗ p * P ∨ i ) ∈ D perf (S). We claim that the object E (P ∨ i ) has rank zero. Indeed, for any closed point s ∈ S, apply Lemma 6.1.6 to the closed immersion i : Spec k(s) → S to obtain Now apply Lemma 6.1.6 to the closed immersion i : C → S and deduce from Riemann-Roch that Since E (P ∨ i ) is perfect, it has a determinant line bundle L i by [39]. By the compatibility of the determinant construction with restriction to C we conclude L i .C = deg E (P ∨ i )| C and thereby also equation (6.3).

Positivity
We now establish the positivity statements from Theorem 6.1.4, and for this we follow closely the approach of [13,Section 3]. We continue to work under the assumptions of Theorem 6.1.4. In particular, D c (Y ) carries a Noetherian bounded t-structure with heart A. For any proper curve C ⊆ S, we obtain a sheaf of Noetherian t-structures on D lc (Y × C) over C by Theorem 4.3.1. Write A C for the heart of this t-structure. Proof of Theorem 6.1.4 For any σ ∈ Stab(D c (Y )), we may assume that Z σ (v) = −1 using the C-action on Stab(D c (Y )).
We first prove that the numerical divisor class E,σ ∈ N 1 (S) is nef. Let C be a proper curve in S. As in [13,Proposition 3.2], it is straightforward to show that the value of E,σ ([C]) from (6.1) is unchanged if we replace O C by any line bundle L on C. In particular, if L is of sufficiently high degree on C, Lemma 6.2.1 gives E (L) ∈ A and hence as required, because Z σ sends objects of A to the semi-closed upper half plane.
To prove the second statement, suppose first that E,σ · C = 0. For any smooth point c ∈ C and for any L ∈ Pic(C) of sufficiently high degree, applying E to the short exact sequence and invoking Lemma 6.2.1 gives a short exact sequence of objects in A. We have 0 = E,σ · C = Z σ ( E (L)) and Z σ (v) = −1, so both E (L) and E c have phase 1. Since E c is a quotient of E (L) in A, each Jordan-Hölder factor of E c is a Jordan-Hölder factor of E (L). The latter factors don't depend on the choice of the smooth point c ∈ C. Since k is an infinite field, [13,Lemma 3.7] implies that E c is S-equivalent to E c for any c, c ∈ C. For the other direction, assume E c is S-equivalent to E c for any two general closed points c, c ∈ C. The analogue of [13,Lemma 3.9] gives a filtration of E| Y ×C of length n, say, whose successive quotients are of the form p * F i ⊗ q * L i , where each L i ∈ Pic(C) and each F i ∈ A Y has phase 1. The projection formula and flat base change give which lies on the real axis. Therefore E,σ ([C]) = 0 as required.
Proof of Theorem 1.2.1 This is immediate from Lemma 5.1.1 and Theorem 6.1.4.

A geometric condition to ensure proper support
The goal of this subsection is to show that one of the assumptions of Theorem 6.1.4, namely that the universal family has proper support over S, holds for moduli spaces of simple objects when Y is semi-projective.
We continue to assume that all our schemes are separated and of finite type over k. We begin with two Lemmas, for which we make the same assumptions as in Proposition 6.3.1. Proof If Supp(E s ) is disconnected, we can write E s = E s ⊕ E s where the summands have disjoint support; this contradicts the assumption that E s has only k as endomorphism. Similarly, assume that τ (Supp(E s )) contains more than one point. Since X is affine, there exists a function on X whose pullback to Y is non-constant on the support of E s . Multiplication with this function would give a non-scalar endomorphism of E s which is absurd.
Let τ S := τ × id S : Y × S → X × S, and consider W := τ S (Supp(E)) as a topological subspace of X × S. Note that by Lemma 2.1.2(ii), the formation of W commutes with base change. The induced map of topological spaces q : W → S is bijective on closed points by Lemma 6.3.2.

Lemma 6.3.3 Assume additionally that S is irreducible. Then W is irreducible.
Proof Assume that W is reducible. Since S is irreducible, there has to be an affine curve in S, intersecting the images of at least two irreducible components of W under q. Without loss of generality, we may therefore assume that S itself is an affine curve. After base change to the normalisation, we may assume further that S is smooth.
Since q is injective, there is an irreducible component of W that maps to a point s 0 ∈ S. It follows that this component is a point, and is therefore a connected component of W ; consequently, , in contradiction to our assumption. Replacing S by an open subset if necessary, we may assume that s 0 ∈ S is the scheme-theoretic zero locus of a regular function f ∈ H 0 (O S ). Each cohomology sheaf H j (E 0 ) has a filtration 0 ⊂ ker f ⊆ ker f 2 ⊆ . . . whose successive quotients are isomorphic to the pushforward (i s 0 ) * F of a coherent sheaf F on Y . Restricting the short exact sequence Since E 0 is a successive extension of its (finitely many nonzero) cohomology sheaves H j (E 0 ), the same holds for the class of i * s 0 E 0 . However, this is absurd because i * Proof of Proposition 6.3. 1 We first claim that the bijective morphism q : W → S is a homeomorphism, for which it only remains to prove that q is closed. This can be checked after base change via any proper and surjective map S → S. Hence we may assume that S is normal and, by restricting to one of its connected components, also irreducible. By Lemma 6.3.3, W is irreducible. Let W be the reduced subscheme W ⊆ X × S; then the induced morphism q : W → S is a bijective map of varieties over k, with S normal. Since k is algebraically closed of characteristic zero, the dominant morphism q is birational. The original form of Zariski's main theorem implies that q is an open immersion, so it's an isomorphism, and hence q is a homeomorphism. Since the same arguments apply after base change, it follows that q is universally closed, and thus proper. Since τ S is proper, and Supp(E) is a closed subset of τ −1 S (W ), it follows that the support of E is proper over S.

On schemes admitting a tilting bundle
The goal of this section is to prove Theorem 1.4.1. To this end, we modify slightly the standard set-up (see Sect. 1.5 for references) for stability conditions for quiver algebras of finite global dimension: rather than working with the category of nilpotent representations, we work with representations that are finite-dimensional over k, but insist that the central charge factors via a variant of the numerical Grothendieck group, see Sect. 7.1; this is analogous to our set-up in Sect. 5. In fact, when Y admits a tilting bundle, we show that these notions yield a compatible notion of stability conditions in Sect. 7.2, a compatible notion of flat families in Sect. 7.3, and finally compatible nef and semiample line bundles in the sense of Theorem 1.4.1 in Sect. 7.4.

Stability conditions for quiver algebras
We first recall notation and some standard facts from the representation theory of quivers; see, for example, [7, or [3,Section 4.2].
Let Q be a connected quiver where both the vertex set Q 0 and the arrow set Q 1 are finite. The path algebra kQ is graded by path length, and the part in degree zero has a basis of orthogonal idempotents e i for i ∈ Q 0 . We do not require that Q is acyclic, so kQ may be infinite-dimensional as a k-vector space.
Our interest lies with associative k-algebras A that can be presented in the form A ∼ = kQ/I , where Q is a quiver and I ⊂ kQ is a two-sided ideal generated by linear combinations of paths of length at least one; we refer to any such algebra A as a quiver algebra. For each vertex i ∈ Q 0 , there is an indecomposable projective A-module P i := Ae i corresponding to paths in Q emanating from vertex i. In addition, our assumption on the ideal I ensures that each vertex of the quiver also gives rise to a one-dimensional simple A-module S i on which the class in A of every arrow of the quiver acts as zero. Examples of quiver algebras include finite-dimensional algebras [7], finitely generated graded algebras whose degree zero part is finite-dimensional semisimple [20,Appendix A], and algebras whose ideal of relations is defined in terms of a superpotential [21].
For a quiver algebra A, let D perf (A) and D fin (A) denote the bounded derived categories of perfect A-modules and finite-dimensional A-modules respectively, and let K perf (A) and K fin (A) respectively denote the Grothendieck groups of these categories. The Euler form Proof The vertex simple A-modules define classes [S i ] ∈ K num (A) for i ∈ Q 0 , and the indecomposable projective A-modules define classes [P i ] ∈ K num (A) ∨ . Since Ext k A (P j , S i ) = k for k = 0 and i = j; 0 otherwise , (7.1) it follows that i∈Q 0 Z[S i ] is a subgroup of K num (A) and that i∈Q 0 Z[P i ] is a subgroup of K num (A) ∨ . The statements (i)-(iv) are now clearly equivalent.
From now one we assume that any of the equivalent conditions in Lemma 7.1.1 holds. Fix once and for all a dimension vector v : The choice of dimension vector v ∈ K num (A) therefore determines a wall and chamber structure on the space v of stability parameters, where two generic parameters θ, θ ∈ v lie in the same chamber if and only if the notions of θ -stability and θ -stability coincide.
For any integral parameter θ ∈ v , King [37], and more generally, Van den Bergh [59], constructs the coarse moduli space M A (v, θ) of S-equivalence classes of θ -semistable A-modules of dimension vector v as a GIT quotient where X is an affine scheme, G = i∈Q 0 GL(v i ) /k × , and χ θ ∈ G ∨ is a character determined by θ .
for all semistable objects E, and with respect to some norm · on K num (A) ⊗ R ∼ = R Q 0 . We may choose the supremum norm on R Q 0 . Up to shift, any semistable object lies in the heart A ⊂ D fin (A), so its class in K num (A) is a non-negative linear combination of the classes [S i ] for the simple objects for i ∈ Q 0 . Setting C := min i∈Q 0 λ([S i ]), the claim becomes evident. Proof An object E ∈ D fin (A) of class v is σ θ,λ,ξ -semistable of phase in (0, 1] if and only if E lies in the heart A, and the phase of Z θ,λ,ξ (F) is smaller than the phase of Z θ,λ,ξ (E) for every proper nonzero submodule F ⊂ E. Since θ(v) = 0, we have Z θ,λ,ξ (E) ∈ R >0 · √ −1 + ξ , so this is equivalent to θ(F) 0. Thus, the σ θ,λ,ξ -(semi)stable objects in D fin (A) of class v are precisely the θ -(semi)stable A-modules of class v.
Let Stab(D fin (A)) denote the space of numerical stability conditions on D fin (A) that satisfy the support property with respect to K num (A). Combining the above results gives the following picture. such that for any fixed λ ∈ , ξ ∈ R, the wall-and-chamber structure on v is obtained by pulling back the wall-and-chamber structure on Stab(D fin (A)) with respect to v.
When v is primitive and θ generic, the map (7.2) gives an identification of fine moduli spaces; otherwise, the moduli stack of σ θ,λ,ξ -semistable objects of class v has M A (v, θ) as coarse moduli space, which, as noted above, is projective over an affine.

On schemes with a tilting bundle
Let Y be a smooth scheme that admits a projective morphism τ : Y → X = Spec R, and let E be a locally-free sheaf of finite rank on Y .
We begin with a few comments about our conventions concerning left-and rightmodules; in this paragraph all of our functors are underived. For any coherent sheaf F on Y , the space Hom(E, F) is a right End(E)-module and therefore a left End(E) opmodule; equivalently, and more geometrically, Hom(E, F) is a left module over the algebra

Also, since E is a left End(E)-module and hence a right A-module, so E ⊗ A M is well-defined for any left A-module M.
Recall that a tilting bundle on Y is a locally-free sheaf E of finite rank such that Ext k (E, E) = 0 for k > 0, and such that if F ∈ D(Y ) satisfies Hom(E, F) = 0, then F = 0.  Proof Note that K num c (Y ) is generated by simple sheaves. For any such sheaf F, the image τ (Supp F) ⊂ X is a point, so either F = O y for some y / ∈ Z , or Supp F ⊂ Z . In the former case, the numerical class [O y ] does not depend on y ∈ Y ; thus there is a finite set of sheaves F 1 , . . . , F n supported on Z whose classes generate K num c (Y ). After a further filtration, we may assume that each F j is scheme-theoretically supported on Z . After tensoring each by a sufficiently high power of an ample line bundle on Y , we may assume that Ext i (E, Present A = kQ/I as a quiver algebra. We claim that every nontrivial cycle in Q acts as zero on each M j . Indeed, assumption (i) ensures that every nontrivial cycle starting at vertex i corresponds to an endomorphism of E i that factors via at least one other summand E k ; assumption (ii) ensures that any such endomorphism acts as zero on E i | Z . It therefore acts as zero on Hom

Remark 7.2.3
If the morphism τ from Proposition 7.2.2 is not birational, then Y is forced to be projective, in which case the fact that any tilting bundle admits a splitting such that the equivalent assumptions of Lemma 7.1.1 hold was well known [38]. Otherwise, the typical situation where Proposition 7.2.2 applies is to resolutions of an isolated singularity.
The tilting equivalence identifies the space Stab(D fin (A)) with the space of numerical stability conditions on Y for compact support in the sense of Definition 5.2.1. For any class v ∈ K num c (Y ) and for any θ ∈ v , λ ∈ and ξ ∈ R, we abuse notation and also write σ θ,λ,ξ for the resulting stability condition on D c (Y ).
Assume now that the equivalent assumptions of Lemma 7.1.1 hold. We now compute explicitly the image of σ θ,λ,ξ under the linearisation map E determined by any flat family E. For this, let S be any separated scheme of finite type, and for any numerical Bridgeland stability condition (Z , P) for compact support on Y , let E ∈ D(Y × S) be a family of semistable objects of class v ∈ K num c (Y ). .
The result follows from the proof of Lemma 6.1.5.

Comparison of flat families
From now on we assume that Y is a smooth scheme, projective over an affine scheme, that carries a tilting bundle E such that A := End(E ∨ ) is a quiver algebra satisfying Let S be any separated scheme of finite type. Our next goal is to extend the functor from (7.3) to obtain a natural correspondence between flat families of certain Bridgeland-semistable objects over S on one hand, and flat families of King-semistable objects over S on the other.
First, let P → A denote the minimal projective resolution of A as an (A, A)bimodule. Following Butler-King [10], the term of P in degree l ∈ Z is where e 1 , . . . , e k are the orthogonal idempotents corresponding to the summands E ∨ 1 , . . . , E ∨ k of E ∨ , and V l i, j = Tor l A (S i , S j ) is a finite dimensional k-vector space. Set d l i, j := dim k V l i, j . Let F be a locally-free sheaf on S that is also a left A-module, and write F = i F i for the idempotent decomposition as an A-module. The left End(E)-module E = i E i becomes a right A-module, and the derived tensor product E ⊗ A F is represented by the complex E ⊗ A P ⊗ A F, whose term in degree l ∈ Z is the locally-free sheaf on Y × S given by where p : Y × S → Y and q : Y × S → S are the first and second projections. The maps of the complex E ⊗ A P ⊗ A F are morphisms of sheaves induced by the maps of the complex P. Note that E ⊗ A F ∈ D(Y × S) because A has finite global dimension. For (i), note first that q * (( p * E) ∨ ⊗ F) = F (E ∨ ), and that the derived pullback to s ∈ S is We have that F ∈ D(Y × S) is S-perfect and proper over S. Since E is locally-free, Proposition 2.4.2 implies that F (E ∨ ) ∈ D perf (S). By (7.6), the derived restriction of F (E ∨ ) to each closed point of S is concentrated in degree zero, hence so is F (E ∨ ). Thus, we've shown that q * (F ⊗ p * (E ∨ )) is a locally-free sheaf on S whose fibre over each closed point s ∈ S is the θ -semistable A-module Hom Y (E, F s ) of dimension vector v.
To complete the proof of (i), it remains to show that the A-module structure on each fibre comes from a k-algebra homomorphism A → End(q * (F ⊗ p * (E ∨ ))). For any open subset U ⊆ S, the space of sections of q * (F ⊗ p * (E ∨ )) over U is (Y ×U, p * E ∨ ⊗F). Note that A = End(E ∨ ) acts on the first factor whose restriction to any closed point s ∈ U recovers the A-module structure on the fibre over s by (7.6).
For (ii), the locally-free sheaf F has a fibrewise left A-module structure, so E ⊗ A F ∈ D(Y × S) as above. Since p * E i and q * F j are S-perfect for 1 i, j k, we have that E ⊗ A F is S-perfect by (7.5) and [54,III,Proposition 4.5]. For a closed point s ∈ S, we have that for all 1 i, j k, where (F j ) s denotes the fibre of F j over s ∈ S. The functors commute with direct sums, so just as in (7.5) above, for each l ∈ Z, the l-th terms of i * s (E ⊗ A F) and E ⊗ A F s coincide, where F s is the fibre of F over s ∈ S. Since the maps in each complex derive from those of P, it follows that Since each F s is a θ -semistable A-module of dimension vector v, Lemma 7.1.5 and Theorem 7.2.1 imply that i * s (E ⊗ A F) is σ θ,λ,ξ -semistable of class v. The proof that these operations are mutually inverse requires the fact that E ⊗ A E ∨ ∼ = O for the diagonal ⊂ Y × Y as in King [38]; we leave the details to the reader.

Remark 7.3.2
The assumption in Proposition 7.3.1 that F is proper over S is superfluous for a flat family of σ θ,λ,ξ -stable objects by Proposition 6.3.1.

Example 7.3.3
The flat family E ⊗ A F of Bridgeland-stable objects was first studied by King [38] in the case when S = Y and F = E ∨ ; [ibid.] would write our E ⊗ A F as F A op E.

Comparison of line bundles
For any class v ∈ K num (A) and any integral parameter θ ∈ v , the GIT construction produces an ample line bundle L(θ ) on the coarse moduli space M A (v, θ). Given a family of θ -semistable A-modules of dimension vector v over a scheme S, the induced morphism f : S → M A (v, θ) produces a semi-ample line bundle f * L(θ ). We now provide an alternative description of this line bundle using the linearisation map.
Given a flat family E ∈ D(Y × S) of σ θ,λ,ξ -semistable objects of class v with respect to A that is proper over a separated scheme S of finite type, we obtain by where θ = 1 i k θ i [P i ]. The GIT construction of M A (v, θ) = X/ / θ G shows that the θ -semistable locus X ss in X carries a universal family V of framed θ -semistable A-modules of dimension vector v, equipped with an idempotent decomposition V = 1 i k V i , such that holds G-equivariantly on X ss , where π : X ss → M A (v, θ) is the quotient map. Proposition 7.3.1 shows that E (E ∨ ) is a flat family of θ -semistable A-modules of dimension vector v on S. Let π S : S → S be the principal G-bundle corresponding to a choice of framing (up to a common rescaling) of each summand E (E ∨ i ). By the universality of V , it comes with a G-equivariant map f : S → X ss that induces the map f between the corresponding quotients, and that satisfies f * V i ∼ = π * S N ⊗ E (E ∨ i ) for all i and a fixed line bundle N ∈ Pic(S). Pulling back (7.8) along this map gives the following identity of G-equivariant line bundles on S: where the last identity used 1 i k θ i rk(V i ) = 1 i k θ i v i = 0. This descends to the identity (7.7) on S, as required.
When v is primitive and θ ∈ v is generic, let C ⊆ v denote the GIT chamber containing θ , let M := M A (v, θ) denote the fine moduli space and write T =  Since integral functors commute with direct sum, we have that T i = E (E ∨ i ) for all 1 i k because each T i is indecomposable. Thus, for any η = 1 i k η i [P i ] ∈ v , we have The result follows by comparing this with the numerical divisor class E (σ η,λ,ξ ) from (7.4). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.