Compatibility of t-structures in a semiorthogonal decomposition

We describe how to obtain a global t-structure from a semiorthogonal decomposition with compatible t-structures on every component. This result is used to generalize a well-known theorem of Bondal on full strong exceptional sequences.


Introduction
The notion of algebraic triangulated categories has raised great attention in the last decades. These particular triangulated categories are described in several equivalent ways: they admit an enhancement by dg categories or A ∞ -categories; alternatively, they are obtained as the stable category of a Frobenius category. The most important examples are derived categories (assuming they are categories). * Interestingly, many results can be extended from derived categories to algebraic triangulated categories. We focus on the following.
Theorem -Bondal. [3, Theorem 6.2] Assume that the bounded derived category D b (X) of coherent sheaves on a smooth manifold X is generated by a strong exceptional sequence E 1 , . . . , E n . Then D b (X) is equivalent to the bounded derived category D b (mod-A) of right finite-dimensional modules over the algebra A = End( n i=1 E i ). Bondal's result has been generalized to algebraic triangulated categories by Keller ([15,Theorem 8.7]). In particular, the statement below is a consequence of Keller's work.
Theorem - Keller-Orlov. [20, Corollary 1.9] Let T be an algebraic triangulated category. Assume that T has a full strong exceptional sequence E 1 , . . . , E n . Then the category T is equivalent to the derived category D b (mod-A), where A = End( n i=1 E i ) is the algebra of endomorphisms of the collection E 1 , . . . , E n .
A question may arise: is it possible to drop the algebraic requirement? At the moment, no answer has been found. As a matter of fact, it is incredibly hard to study the general case of triangulated categories; indeed, the definition of an exceptional object requires the category to be K-linear, with K a field, and the only known example of non-algebraic K-linear triangulated category is studied in [23].
Our aim is to generalize Bondal's result. For this reason, we deal with the construction of a global t-structure, starting with compatible t-structures on semiorthogonal components. Surprisingly, the result is not hard to prove and it follows from basic theory. As a corollary, a full strong exceptional sequence of length 2 gives a heart of dimension at most 1, so that Hubery's result [11,Theorem 3.2] can be applied without any additional requirement on the triangulated category. We obtain the following.
2.13. Corollary. Let K be a field. Any K-linear triangulated category T with a full strong exceptional sequence E 1 , E 2 such that dim K Hom(E 1 , E 2 ) < ∞ is algebraic. In particular, T ∼ = D b (mod-A), where A = End( 2 i=1 E i ). For a strong exceptional sequence with length greater than 2 we deal with realized triangulated categories, i.e. triangulated categories T admitting an exact functor real : D b (A ) → S for every heart A of a bounded t-structure on a full subcategory S of T . In particular, it has been proven that all algebraic triangulated categories are realized (see Example 5.2 for a discussion on examples of realized triangulated categories). By induction on the length of the exceptional sequence, we can prove the main result. 5.6. Theorem. Let K be a field and let T be a realized K-linear triangulated category with a full strong exceptional sequence E 1 , . . . , E n such that i Hom(X,Y [i]) is a finite-dimensional vector space for any X,Y ∈ T . Then T ∼ = D b (mod-A), where A = End( n i=1 E i ).
In Section 1, we recall some basic results on t-structures. Section 2 is devoted to the notion of compatible t-structures with respect to a semiorthogonal decomposition. Section 3 covers the needed knowledge on quivers, while Section 4 deals with filtered triangulated categories, introduced by Beilinson in [1,Appendix A]. In Section 5, we introduce the concept of realized triangulated category and state the main theorem. Appendix A generalizes a result on Yoneda extensions of exact categories.

Some basic results on bounded t-structures
In this section, we define t-structures and hearts, and state some classical results.
1.1. Definition. A t-structure on a triangulated category T is a full subcategory T ≤0 closed by left shifts, i.e. T ≤0 [1] ⊂ T ≤0 , and such that for any object E ∈ T there is a distinguished triangle A → E → B → A [1], where A ∈ T ≤0 and B ∈ T ≥1 := (T ≤0 ) ⊥ . We remember that for any full subcategory C ⊂ T , we write C ⊥ to mean the full subcategory whose objects are Y such that Hom(X,Y ) = 0 for any X ∈ C .
We will write T ≤i := T ≤0 [−i] and T ≥ j := T ≥1 [− j + 1] for any i, j integers. A t-structure is said to be bounded if Moreover, the t-structure is non-degenerate if i T ≤i = j T ≥ j = 0.
The heart (of the bounded t-structure T ≤0 ) is the additive category A := T ≥0 ∩ T ≤0 , and it is proven to be abelian. We define the homological dimension of A in T , denoted by dim T A , as the greatest integer n such that Hom(A, B[n]) = 0 for some A, B ∈ A .

Notation.
Given a map f : A → B in a triangulated category, its cone will be denoted by C( f ). When there is no need to make f explicit, we will write C(A → B). Lemma 3.2]. Let T be a triangulated category. A heart (of bounded t-structure) on T is an additive category A satisfying the following properties: 1. For any two objects A, B ∈ A , Hom(A, B[n]) = 0 for every n < 0. 2. Given an object E ∈ T , we can find integers k 1 > · · · > k m and a filtration PROOF. Let E be in the intersection of all T ≤i . Since T ≤0 is bounded, E must be in T ≤ j ∩ T ≥h for some j, h. Then E is in T ≥h , but also in T ≤h−1 . By definition, So Hom(E, E) = 0, therefore E is a zero object. In the same way one proves that also i T ≥i = 0.
The cohomology functors give rise to an exact sequence The filtration in the definition proves that C ′ ∼ = H 0 (C ′ ) ∼ = C, so we can choose C ′ to be C with the same map appearing in the distinguished triangle.
Then the cohomology functors show that there is an exact sequence

Semiorthogonal decompositions and t-structures
After recalling the notion of semiorthogonal decomposition, we define compatibility between t-structures with respect to such decomposition. In Theorem 2.7 we show how this situation gives rise to a global t-structure. As an application of the result, we study exceptional sequences and state Corollary 2.13, which generalizes Bondal's theorem [3, Theorem 6.2] for exceptional sequences of length 2.
2.1. Definition. Let T be a triangulated category. A semiorthogonal decomposition is a sequence of full triangulated subcategories T 1 , T 2 , . . . , T n such that In this situation, we will write T = T 1 , T 2 , . . . , T n . 2.3. Definition. Let T be a triangulated category. Given two full subcategories X and Y of T , we define X * Y to be the full subcategory of T whose objects are This construction gives rise to an operation * between full subcategories of T .
2.5. Example. Let T be a triangulated category. Given a semiorthogonal decomposition T = T 1 , . . . , T n , we can write T = T n * · · · * T 2 * T 1 . If we consider a t-structure T ≤0 on T , we have T = T ≤0 * T ≥1 .
2.6. Definition. Let T = T 1 , T 2 be a semiorthogonal decomposition, T any triangulated category. Assume that T i has a t-structure T ≤0 2 ) = 0. Denoted by A 1 and A 2 the hearts of T ≤0 1 and T ≤0 2 respectively, the relative dimension of A 1 and A 2 in T is the number Notice that, whenever the set above is nonempty, rdim T (A 1 , A 2 ) ≥ 0 by compatibility. The reason why we have chosen the value −1 in case the set is empty will become clear reading the statement of Theorem 2.7.

2.7.
Theorem. Let T be a triangulated category with a semiorthogonal decomposition T = T 1 , T 2 . Given two compatible t-structures T ≤0 1 and T ≤0 2 on T 1 and T 2 respectively, the full subcategory defined by are bounded (respectively non-degenerate), then T ≤0 is bounded (respectively non-degenerate). 2. Let A 1 , A 2 and A be the hearts associated to T ≤0 1 , T ≤0 2 and T ≤0 respectively. Then 3. The following equality holds true whenever at least one of the two hearts A 1 , A 2 is nonzero: it is clear that also T ≤0 is closed by left shifts. We aim to show that T = T ≤0 * T ≥1 , where T ≥1 := (T ≤0 ) ⊥ . Notice that Since T 1 , T 2 is a semiorthogonal decomposition and compatibility holds, we have . The two distinguished triangles introduced give rise to the following hom-exact sequences: 2 ) = 0. The first exact sequence concludes that Hom(B, A) = 0. Finally, A ∈ T ≥1 . Conversely, if A ∈ T ≥1 , then there exists a distinguished triangle with A ≤0 ∈ T ≤0 and A ≥1 ∈ T ≥1 2 * (T ≥1 1 [1]). Notice A ≤0 → A must be zero because A ∈ T ≥1 . Since A ≥1 cannot have a direct summand in T ≤0 , we get that A ≤0 = 0. In particular, A = A ≥1 ; as wanted, T ≥1 2 * (T ≥1 1 [1]) = T ≥1 . 1. First, we deal with boundedness. Let A ∈ T . From the semiorthogonal decomposition, we get a distinguished triangle i (here we use the closure by right shifts). We conclude that To prove non-degeneracy when T ≤0 1 and T ≤0 2 are non-degenerate, let C ∈ j T ≤ j . By Remark 2.2, we have C = C(E → F) for E ∈ j T ≤ j 1 and F ∈ j T ≤ j 2 . By hypothesis, both intersections are zero, so C ∼ = 0 as wanted. The proof of j T ≥ j = 0 is analogous For any A ∈ A we can find two distinguished triangles, according to the fact that A ∈ T ≤0 and A[−1] ∈ T ≥1 = T ≥1 2 * (T ≥1 1 [1]). Then Remark 2.2 proves that A is exactly as described in the statement.
For any m, we consider the long exact sequence [1]. By considering the first and the last term, we can create two exact sequences associated to To conclude, it suffices to show that dim T A ≥ ℓ.
We have two cases. If ℓ is realized by the homological dimension of A 1 or A 2 , we notice that there is a sequence (a n ) ⊂ Z such that a n → +∞ and Hom(A n 1 [1], B n 2 [a n ]) = 0 for any a n and some A n 1 [1], B n 2 ∈ A . Since item 2 tells us that A 1 [1], A n 1 [1], B 2 , B n 2 ∈ A , in both cases dim T A cannot be less than ℓ. If ℓ = 0, then ℓ is also equal to the homological dimensions of A 1 or A 2 , and this possibility has already been addressed.
2.8. Remark. As already used in the last part of the proof, the constructed t-structure may not behave as wanted. For instance, using the notation of the statement, A 1 is not contained in A : we need to consider its shift A 1 [1].
One may think this shifting could be easily adjusted, but the requirement needed is incredibly strong. The first idea it comes to mind is to consider the t-structure T ≤1 to be compatible, no shift will be involved, and in particular A 1 , A 2 ⊂ A . However, requiring T ≤1 1 and T ≤0 2 to be compatible implies that Hom(A 1 , A 2 ) = 0, which is generally too restrictive.
2.9. Remark. Theorem 2.7 is incredibly linked to torsion pairs (for an introduction of the concept, we refer to [10, Section I.2]). Let T be a triangulated category with a semiorthogonal decomposition T 1 , T 2 and a t-structure T ≤0 such that T ≤0 . As a matter of fact, T ≤0 # gives rise to a heart which is a tilted version of the heart A of T ≤0 . This is simply true by picking the couple F = A ∩ T 1 and T = A ∩ T 2 , which is a torsion pair by [17, Exercise 6.5]. First of all, we recall that any recollement gives rise to a semiorthogonal decomposition. We consider T = T 1 , T 2 a semiorthogonal decomposition and let T ≤0 i a t-structure on T i for i = 1, 2. Then, under their respective assumptions, from Theorem 2.7 we get the global t- In other words, the new result gives a tilted version of the old statement (see Remark 2.9).
Moreover, the two theorems deal with different situations. Indeed, although it is possible to construct a left adjoint i * to the inclusion i * : T 1 → T (i.e. T 1 is left admissible) and a right adjoint j * to the inclusion j ! : T 2 → T (i.e. T 2 is right admissible) by [3, Lemma 3.1], in general a left (respectively right) admissible subcategory does not need to be right (respectively left) admissible. Conversely, a recollement does not ensure that the compatibility requirement is satisfied, since T ≥1 2 is not necessarily equal to T ≥1 ∩ T 2 . Concerning our studies, Theorem 2.7 is to be preferred because it computes the homological dimension of the obtained heart; this is crucial, especially for Corollary 2.13.
The definition of compatible t-structures so that Theorem 2.7 holds can be generalized to semiorthogonal decompositions of any length, but the requirement may result unnatural since we need to consider some shifting.
2.11. Definition. Let T = T 1 , . . . , T m and assume T i has a t-structure T ≤0 With this notion of compatibility, an analogous of Theorem 2.7 can be obtained by recursion. With the same notation of the definition above, if A i is the heart of T ≤0 i , the heart A ⊂ T built via Theorem 2.7 is described as 2.12. Example -Exceptional sequence. Let K be a field and consider a K-linear triangulated category T . We recall that an exceptional object is an object E ∈ T such that ) is a finite-dimensional vector space for any A, B ∈ T ‡ . By [12, §1.4], it is known that such a full exceptional sequence gives rise to a semiorthogonal decomposition given by We will use the notation E 1 , . . . , E m to indicate exceptional sequences. Notice that on each T i we can consider a bounded t-structure with heart If the full exceptional sequence is also strong § , i.e. Hom(E i , E j [n]) = 0 for any i, j and n = 0, the above t-structures are compatible: indeed, taking k > i, Moreover, the t-structure induced on T is bounded.
PROOF. Theorem 2.7 and Example 2.12 prove that T has a heart of dimension at most 1. By [11,Theorem 3.2], T is algebraic. We conclude by [20, Corollary 1.9].
2.14. Example. By the previous corollary, D b (P 1 K ) is the unique K-linear triangulated category with a full strong exceptional sequence E 1 , E 2 such that dim K Hom(E 1 , E 2 ) = 2.

Quivers
In order to study exceptional sequences of length greater than 2, we will need some basic knowledge on quivers. Here we will give a brief introduction, mostly following [3, Section 5].

Definition.
A quiver is a quadruple Q = (Q 0 , Q 1 , s,t), where Q 0 is a set of vertices, Q 1 a set of arrows between vertices and s,t : Q 1 → Q 0 are the maps indicating source and target respectively. A quiver is finite if Q 0 and Q 1 are finite. It is ordered if the vertices are ordered and for every arrow a, s(a) ≤ t(a).
A path p of length n is a sequence of arrows a 1 , . . . , a n ∈ Q 1 such that t(a i ) = s(a i+1 ). Moreover, with the same notation, t(p) := t(a n ) and s(p) := s(a 1 ). We also allow paths of length 0: such paths are in correspondence with the vertices. Let p, q be two paths. Then the composition of paths q • p is defined to be the concatenated path whenever s(q) = t(p). ‡ In fact, it suffices to require this property for A,B ∈ {E 1 ,... ,E m }. § This condition can be weakened.
Let K be a field. The path algebra KQ is the K-vector space with basis the paths. The product is described as follows: where λ , µ ∈ K and p, q are paths. In particular, paths of length 0 are idempotents in KQ.
If S ⊂ KQ is any subset, (Q, S) is called quiver with relations and its associated path algebra is given by KQ/ S , where S is the ideal generated by S. Now, let us consider A = KQ/ S the path algebra associated to the quiver with relations (Q, S). A left A-module is a vector space V over K with the left action of the algebra A. This is also called representation of a quiver. Using the paths of length 0, which are associated to the vertices of Q, then V , as a vector space, decomposes into a direct sum i∈Q 0 V i , where V i is the vector space associated to the vertex i. Moreover, for every path p ∈ A, we get a linear operator When dealing with right A-modules, one can consider the opposite quiver Q op where s,t are swapped with respect to Q. In other words, arrows go in the other direction, analogously to what happens with the notion of opposite category. As one expects, left modules associated to In case the quiver Q is finite and ordered, let X 1 , . . . , X n be the vertices and p i the idem- Let us denote with S i the representation for which G j S i = δ i j K, where δ i j is the Kronecker delta, and all arrows are represented by the zero morphisms. Notice that for each right A-module V we can create a filtration is a direct sum of copies of S i . Projective modules are P i = p i A and the decomposition A = n i=1 P i holds. As a matter of fact, Hom(P i , P j ).
These isomorphisms allow to interpret the arrows of a quiver as morphisms between projective modules. In particular, being A the path algebra of an ordered quiver, Hom(P i , P j ) = 0 for i > j. Furthermore, it is possible to consider the exact sequence for every i = 1, . . . , n. Notice that P 1 = S 1 .
Let T be a K-linear algebraic triangulated category with a full strong exceptional sequence E 1 , . . . , E n . Then A = End( n i=1 E i ) is the path algebra of an ordered and finite quiver with relations. In particular, the equivalence F : T → D b (mod-A) obtained in [20,Corollary 1.9] is such that F(E i ) = P i , the projective modules of the path algebra A.

Filtered enhancements
In this section, we explore the definition of filtered triangulated categories and give a fairly simple result that has not been found in the literature, namely if a triangulated category admits a filtered enhancement, then every full triangulated subcategory admits a filtered enhancement in a natural way (see Proposition 4.4). Main reference is [1, Appendix A]. In Remark 4.6, we discuss the relation of filtered enhancements with realization functors.
A triangulated category T admits a filtered enhancement if there exists a filtered triangulated category F such that T ∼ = F (≤ 0) ∩ F (≥ 0) in the sense of triangulated categories. With an abuse of notation, we will always assume that T = F (≤ 0) ∩ F (≥ 0).

Proposition. [1, Proposition A.3].
Let F be a filtered triangulated category. Then the following assertions hold true: 1. The inclusion i ≤n : F (≤ n) → F has a left adjoint σ ≤n , and the inclusion i ≥n : F (≥ n) → F has a right adjoint σ ≥n . In particular, these adjoints are exact (see, for instance, [12, Proposition 1.41]). 2. There is a unique natural transformation δ : σ ≤n → σ ≥n+1 [1] such that, for any X ∈ F , is a distinguished triangle. Up to unique isomorphism, this is the only distinguished triangle A → X → B → A [1] with A ∈ F (≥ n + 1) and B ∈ F (≤ n).
PART OF THE PROOF. We want to prove the first two isomorphisms of item 3, since it is the only part of the statement not considered in [1]. Being the reasoning analogous, let us focus just on the first isomorphism. Let X ∈ F . If m ≥ n, then F (≤ m) ⊃ F (≤ n). We recall that σ ≤m i ≤m ∼ = id because the inclusion i ≤m is fully faithful. Since σ ≤n (X) ∈ F (≤ m), we simply have that σ ≤m σ ≤n (X) ∼ = σ ≤n (X) by the natural isomorphism mentioned before. We conclude that σ ≤m σ ≤n ∼ = σ ≤n . Let m ≤ n, so that F (≤ m) ⊂ F (≤ n). Then, by adjunction, we have the following isomorphisms for any X ∈ F and Y ∈ F (≤ m): In particular, σ ≤m σ ≤n is left adjoint to i ≤m . Since adjoints are determined up to a natural isomorphism, σ ≤m σ ≤n ∼ = σ ≤m as wanted.

Remark.
By item 2 of Proposition 4.2, we also have the following isomorphisms: Let us set gr n := σ ≤n σ ≥n . This is not the definition used in [1], but it will come in handy in the proof of the following statement.

Proposition.
Let T be a triangulated category admitting a filtered enhancement F . Then any full triangulated subcategory S of T has a filtered enhancement given by the full subcategory G of F with objects {X ∈ F | s −n gr n (X) ∈ S ∀n}.
PROOF. First of all, we would like to show that G is a triangulated subcategory of F . Notice that the shift functor of F obviously restricts to G since s −n gr n is exact, being composition of exact functors. Let us consider X → Y with X,Y ∈ G . This gives a distinguished triangle is a distinguished triangle in T , with s −n gr n (X) and s −n gr n (Y ) objects of S . This suffices to conclude that s −n gr n (Z) ∈ S , so that Z ∈ G . Next, we set G (≤ 0) := G ∩ F (≤ 0) and G (≥ 0) := G ∩ F (≥ 0). We would like to prove that the autoequivalence s : F → F can be restricted to G . Let X ∈ G . Then, by Remark 4.3, we have s −n gr n (sX) = s −n σ ≤n σ ≥n s(X) ∼ = s −n σ ≤n sσ ≥n−1 (X) So we can restrict s and create an exact autoequivalence s : G → G , called s as well by an abuse of notation. Of course, the restriction of α : id F → s gives us the required natural transformation and fcat5 is ensured. We set G (≤ n) and G (≥ n) via s as described in Definition 4.1. Being s an equivalence, we have the following and analogously G (≤ n) = G ∩ F (≤ n). This immediately shows that fcat1,2,3,6 hold. As fcat5 has already been dealt with, it remains to show fcat4. In order to do that, we recall the distinguished triangle in item 2 of Proposition 4.2. Therefore, the statement is reduced to establish that the images of σ ≤n and σ ≥n are in G (≤ n) and G (≥ n) respectively, so that these functors are adjoints to the inclusions as in F . Let X ∈ G and consider σ ≤m . By item 3 of Proposition 4.2 and Remark 4.3 the following isomorphisms hold: s −n gr n (σ ≤m X) = s −n σ ≤n σ ≥n σ ≤m (X) ∼ = s −n σ ≤n σ ≤m σ ≥n (X) ∼ = s −n σ ≤m σ ≤n σ ≥n (X) ∼ = σ ≤m−n s −n σ ≤n σ ≥n (X).
The reason why filtered enhancements become of great interest is their relation with realization functors.

Definition. Let T be a triangulated category. Given a heart (of a bounded t-structure)
A ⊂ T , we call realization functor (of A in T ) an exact functor real : D b (A ) → T such that real |A = id A . 4.6. Remark. In [1,Appendix], it is proven that every triangulated category with a filtered enhancement admits a realization functor for any heart. However, some authors point out that an additional requirement, called fcat7, may be necessary to provide the result (see [21, Appendix A] for further details).
For the sake of completeness, let us state this new axiom using the same notation of Definition 4.1.
fcat7 Given any morphism f : can be extended to a 3 × 3-diagram whose rows and columns are distinguished triangles. Once ensured that F satisfies fcat7, it is easy to prove that also G as defined in Proposition 4.4 fulfills fcat7. This will be key in what follows.

Realized triangulated categories
This section revolves around the unconventional notion of realized triangulated categories. After the definition, we will give some large classes of examples studied in the literature and prove a crucial result, Proposition 5.3. As an application, the generalization of Bondal's theorem [3, Theorem 6.2] is ensured for realized triangulated categories.   Moreover, under such circumstances, real is an exact equivalence.

Definition. A triangulated category T is called realized if for every heart
PROOF. We start with 1. ⇒ 2. Let E, F ∈ D b (A ) and considerẼ = real(E) andF = real(F). Then, by Proposition 1.3, there exist integers k 1 > · · · > k m , j 1 > . . . j n and filtrations Since real is an exact functor, their imagesẼ i ,Ẽ i ,F h andF h give the same filtrations. We consider the exact hom-sequences  (A, B[1]) is an isomorphism by Dyer's Theorem A.2, we know that f 2,A,B is injective. Moreover, since real is full, f 2,A,B is surjective, thus an isomorphism. The same reasoning proves that f n,A,B is an isomorphism for every n, concluding that T has all the Ext groups of A .

From these sequences, an induction on i and h proves that Hom
We now assume that real is fully faithful and prove that it is also an equivalence. Let E ∈ T . By Proposition 1.3, there are a sequence of integers k 1 > · · · > k m and a filtration We prove by induction on m that E is in the essential image of real. If m = 0, there is nothing to prove. If m > 0, then by induction hypothesis E m−1 = real(Ê m−1 ). Obviously, E m = real(Ê m ) because all shifts of the heart A are in the essential image of real. By the filtration, E = C(E m [−1] → E m−1 ). The map associated to this cone is the image of a unique map f :Ê m [−1] →Ê m−1 in D b (A ) because real is fully faithful. We consider its cone C( f ). Since real is exact, we obtain an isomorphism real(C( f )) ∼ = E.

Remark. As a corollary, it is immediately proven that not all hearts in a derived category have all the Ext groups. Indeed, in
gives a heart (this can be done by applying Theorem 2.7). As highlighted in [17, With a different approach, notice that does not factor through an object in A [1], and therefore Corollary A.13 proves that D b (P 1 ) does not have all the Ext groups of A .

Remark.
Let K be a field and consider a realized K-linear triangulated category T with a full strong exceptional sequence E 1 , . . . , E n . Then we can consider the heart A on T obtained according to Theorem 2.7 and Example 2.12, giving rise to a realization functor D b (A ) → T . One would like to prove that such functor is in fact an equivalence, so that [20,Corollary 1.9] can be applied to ensure the generalization of Bondal's result [3,Theorem 6.2]. However, when n > 2, it is not said that T has all the Ext groups of A ; for instance, if n = 3, does not necessarily factor through A [1]. In general, we would have f / ∈ Ext 2 A (E 1 [2], E 3 ) by Proposition A.7, item 1. For example, consider the quiver obtained by the following vertices and arrows: In order to resolve this issue, we recall what already discussed in Remark 2.9. If the length of the exceptional sequence is 2, the heart obtained by Theorem 2.7 is a tilt of mod-A, where A = End( 2 i=1 E i ). As we will see, the same idea can be used to prove the general case.
5.6. Theorem. Let K be a field and let T be a realized K-linear triangulated category with a full strong exceptional sequence E 1 , . . . , E n such that i Hom(X, PROOF. We will prove the statement by induction on n, the length of the exceptional sequence. The base case n = 2 is already taken care of by Corollary 2.13. If n > 2, we write T = T , E n . By induction hypothesis, there exists an exact equivalence ϕ : D b (mod-Ã) →T withÃ = End( n−1 i=1 E i ). We divide the proof in two parts: 1. The t-structures associated to ϕ(mod-Ã) and E n are compatible. By Theorem 2.7, we obtain a heart A on T . 2. T has all the Ext groups of A . Once both items are ensured, Proposition 5.3 can be applied, proving that T ∼ = D b (A ), and an application of [20, Corollary 1.9] will complete the proof.
From (3.2), every object X ∈ mod-Ã has an associated filtration where F k X/F k−1 X is a direct sum of copies of S k . Moreover, for each P k there is a short exact sequence 0 → F k−1 P k → P k → S k → 0 by (3.3). In particular, S 1 = P 1 . Let us deal with 1. In order to prove it, it suffices to show that Hom(ϕ(X), E n [m]) = 0 for every m ≤ −1 and X ∈ mod-Ã. This can be done by induction on k, requiring F k X = X. If k = 1, F 1 X is in fact a direct sum of copies of P 1 = ϕ −1 (E 1 ), so the claim holds.
If k > 1, notice that the short exact sequence 0 → F k−1 P k → P k → S k → 0 is associated to a distinguished triangle in T , so it gives rise to the hom-sequence By induction, Hom(ϕ(F k−1 P k ) [1], E n [m]) = 0, while Hom(E k , E n [m]) = 0 by hypothesis. Therefore, Hom(ϕ(S k ), E n [m]) = 0. We now consider X = F k X and the distinguished triangle obtained by the filtration. From the associated hom-sequence, Hom(ϕ(X), E n [m]) = 0 since the same holds for F k−1 X and X/F k−1 X, the last one being a direct sum of copies of S k .
It remains to prove item 2. According to Corollary A.14, we will prove by induction on m that Hom(ϕ(X), The cases m = 0, 1 are true since A is a heart. Let m > 1. By Proposition A.7, it holds that Ext m (E k [1], E n ) ⊂ Hom(E k [1], E n [m]) = 0, and therefore Ext m (E k [1], E n ) = 0. Let us consider the distinguished triangle F k−1 P k → P k → S k → F k−1 P k [1]. Applying Hom(ϕ(−), E n [m]), we get the following commutative diagram ) for every k (use, for instance, the five lemma). Now, we proceed by induction on the length of the filtration. If X = F 1 X, there is nothing to prove since F 1 X is a sum of copies of S 1 = E 1 , and therefore Hom(ϕ(F 1 X) [1], E n [m]) = 0 since m > 1. If X = F k X, we consider the short exact sequence 0 → F k−1 X → X → X/F k−1 X → 0. Then we get the following diagram: To show that f k is an isomorphism, it suffices to apply the five lemma whenever g k is a monomorphism. More strongly, we claim that g k is an isomorphism. The idea is exactly the one seen above with the diagram (5.7). In order to prove that for any Y ∈ A , and conclude by item 3 of Proposition A.7. This is in fact true. Indeed, notice that for any X ∈ mod-Ã because E k is projective in ϕ(mod-Ã). Furthermore, as remarked before since any Y ∈ A is the extension of a direct sum of copies of E n and an object ϕ(X)[1] ∈ ϕ(mod-Ã).

A. Yoneda extensions in a triangulated category
A necessary remark to prove Hubery's main result is that, for a heart A in a triangulated cate- [11,Section 3]). This appendix aims to generalize this observation, providing results on Yoneda extensions in any triangulated category. First of all, we want to recall a theorem by Dyer, as it will give the desired generality for Proposition A.7. For this reason, let us give the definition of exact category according to Quillen [22].
A.1. Definition. An exact category A is a full extension closed additive subcategory of an abelian category B. A conflation (or short exact sequence) is given by a short exact sequence in B contained in A .
A.2. Theorem -Dyer. [7]. Let A be a full extension closed additive subcategory of a triangulated category T such that Hom(A, B[−1]) = 0 for any A, B ∈ A .
Then A has a natural exact structure, given by defining is a distinguished triangle in T for some C → A [1]. This association gives rise to a natural isomorphism under the equivalence relation generated by identifying two exact sequences X, Y if there is a family of morphisms ψ = {ψ 1 , . . . , ψ n } satisfying the following commutative diagram If n = m = 0, the product is simply the composition of maps. The case n > 0 and m = 0 requires a more sophisticated definition. Let X 1 ∈ Ext 1 (K, B) and g : B → C. Then g · X 1 is described by the following commutative diagram where g · X 1 is the pushout of g and B → X 1 . Now, considering an n-extension X : 0 → B → X 1 → X 2 → · · · → X n → A → 0 and g : B → C, the Yoneda product is given by substituting 0 → B → X 1 with 0 → C → g · X 1 : g · X : 0 → C → g · X 1 → X 2 → · · · → X n → A → 0.
Dually, one can describe the case n = 0 and m > 0. The Yoneda product so defined behaves according to the composition of maps (up to shift) The structure of abelian group of Hom D b (A ) (A, B[n]) can be considered on Ext n (A, B) via the Baer sum, described as follows. Let X, Y ∈ Ext n (A, B). Consider the direct sum of the long exact sequences The (absolute) homological dimension of A , denoted by dim A , is the greatest integer n such that Ext n (A, B) = 0 for some A, B ∈ A .
A.6. Remark. Last definition can be generalized to any exact category A , where an n-extension is a sequence such that, for i = 1, . . . , n − 1, ξ i factor through an object C i ∈ A and for some C i ∈ A , i ∈ {1, . . ., n − 1}.
To X we can associate short exact sequences which are associated to distinguished triangles; therefore, we can consider a map We need to show that if (X, ξ ) and (Y, η) give the same n-extension, then the associated map A → B[n] obtained is the same. Without loss of generality, assume there is a family of morphisms ψ as in Definition A.4. Then for each i ∈ {0, . . . , n − 1} we have where ϕ i is obtained by the universal property of the kernel. In order to prove that the middle square is commutative, we notice that Since ϕ i+1 is the only one making the middle square commutative by the universal property of the cokernel, TR3 entails that also the right-hand square is commutative. We obtain a commutative diagram where ϕ n = id and ϕ 0 = id, so that the rows are in fact the same map. This gives the welldefinition of every f n,A,B . 1. Let us consider a map α : . To any C i [−1] → C i−1 , we can associate a cone, which is in A by Theorem A.2. Let us call such cone X i . We have the following short exact sequences: It is easy to notice that such exact sequence is associated to the map α : A → B[n] via f n,A,B . 2. In the case of Ext n and Ext m with n, m > 0, the Yoneda product is sent to composition with a reasoning similar to item 1. Therefore, it suffices to show it is true when either m or n is zero. First, we recall that f 1,A,B is exactly the map considered in Theorem A.2, which is a natural transformation for both entries. So (A.5) can be translated to g [1] f in T . Let us prove that f n,A,− is a natural transformation, the proof of f n,−,B being dual. For a general n-extension is associated to only one equivalence class of extensions, the trivial one, whenever f n−1,A,X is an isomorphism for any X ∈ A . Let us consider such that f n,A,B (X) = 0 and the associated factorization We have the following diagram, where the rows are distinguished triangles: Now we pick the map . Since f n−1,A,X 1 is a surjective, we get that A → X 1 [n − 1] is associated to an exact sequence Composing Y with 0 → B → X 1 ⊕ B → X 1 → 0, we have the following: (A.10) We want to prove there are maps Y i → X i+1 making every square of the diagram above commutative. It suffices to consider the sequences starting at X 1 and C 1 respectively (remember that C 1 is the image of X 1 → X 2 ). The Yoneda product of Y and g : X 1 → C 1 gives us g · Y, whose associated map because of the right-hand commutative square in (A.9). Since f n−1,A,C 1 is injective by assumption, we know that g · Y is in the same equivalence class of Therefore, we can assume, up to equivalence, that X is in fact With this assumption, (A.10) can be completed with maps Y i → X i+1 as wanted: the first morphism is given according to (A.5), while all the others are the identity. It remains to show that the equivalence class of is the one associated to 0, which is obvious because the diagram id id commutes. 4. Let g n,A,B as in the statement and assume by induction that g m,C,D = f m,C,D for any m < n and C, D ∈ A . We consider X ∈ Ext n (A, B) given by Such an extension can be split into two shorter extensions: X 1 : 0 → B → X 1 → coker(ξ 1 ) → 0 X 2 : 0 → coker(ξ 1 ) → X 2 → · · · → X n → A → 0.
PROOF. The only if part is obvious: if f n,A,B is an isomorphism, then the image of such map contains all morphisms A → B[n]: item 1 of Proposition A.7 concludes. Conversely, item 1 of Proposition A.7 shows that f n,A,B is surjective. By Theorem A.2, f 1,A,B is an isomorphism: we obtain that f 2,A,B is injective according to item 3 of Proposition A.7. An induction proves that this holds for every n.
Using Remark A.11 and Theorem A.2, we prove the last part of the statement.
A.14. Corollary. Let T be a triangulated category with a semiorthogonal decomposition T = T 1 , T 2 and two compatible t-structures T ≤0 1 and T ≤0 2 on T 1 and T 2 respectively. We denote with A i the heart associated to T ≤0 i . By Theorem 2.7, we obtain the heart We consider the following hypotheses: PROOF. Before starting the actual proof, let us remark that Ext m A (A, B) = Ext m A 2 (A, B) whenever A, B ∈ A 2 . Indeed, let X : 0 → B → X 1 → X 2 → · · · → X n → A → 0 be an extension in A with A, B ∈ A 2 and let σ 2 : T → T 2 be the right adjoint of the inclusion functor ι : T 2 → T . Then we get ισ 2 X : 0 B ισ 2 X 1 . . . ισ 2 X n A 0 Given A, B ∈ A , we consider two distinguished triangle A 2 → A → A 1 → A 2 [1] and B 2 → B → B 1 → B 2 [1] with A 2 , B 2 ∈ A 2 and A 1 , B 1 ∈ A 1 [1]. We obtain the following hom-exact sequences ) above is exactly the classical one, that associates to each X ∈ Ext n (A, B) the map given by the composition of the inverse of the quasi-isomorphism (0 → B → X 1 → · · · → X n → 0) → A (the left-hand complex is such that X n is at level 0) and the morphism (0 → B → X 1 → · · · → X n → 0) → B[n].
In particular, in the case of D b (A ) every f n,A,B is an isomorphism.
PROOF. This is a direct consequence of item 4 of Proposition A.7. The last sentence is a classical result; see, for instance, [13,Proposition XI.4.8].
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