A refined derived Torelli theorem for enriques surfaces, II: the non-generic case

We prove that two Enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their Kuznetsov components are equivalent. This improves and completes our previous result joint with Nuer where the same statement is proved for generic Enriques surfaces.


Introduction
The bounded derived category of coherent sheaves D b (X ) of an Enriques surface X has been widely investigated. Over the complex numbers, a very nice result by Bridgeland and Maciocia [5] shows that the derived category determines the surface up to isomorphism. This goes under the name of Derived Torelli Theorem as it can be viewed as a categorification of the usual Hodge-theoretic Torelli Theorem for Enriques surfaces.
In this paper, we want to work over any algebraically closed field K of characteristic different from 2 and, under this assumption, Enriques surfaces have a uniform definition: they are smooth projective surfaces X with 2-torsion dualizing sheaf and such that H 1 (X , O X ) = 0. From the geometric point of view, they are quotients of a K3 surface by a fixed-point-free involution. It is a fact [13] that the Derived Torelli Theorem above holds in any characteristic different from 2 as soon as the field is algebraically closed.
It was first showed in [28] that D b (X ) can be further decomposed into interesting pieces. As we will explain in Sect. 1.2, such a triangulated category always has a semiorthogonal decomposition where L = L 1 , . . . , L 10 is generated by 10 line bundles. If X is generic in moduli or generic in the divisor of the moduli space of Enriques surfaces parametrizing nodal Enriques surfaces (i.e. containing (−2)-curves), then the 10 line bundles are completely orthogonal. Otherwise, as clarified in Proposition 1.4, these line bundles organize themselves in completely orthogonal blocks and inside of each such block they differ by a rational curve. The residual category Ku(X , L) is called the Kuznetsov component of D b (X ). Note that, as it is expected and will be explained later (see Corollary 2.8), the definition of Ku(X , L) depends on the choice of L.
The study of the properties of the Kuznetsov component got a great impulse from [16] where the authors also investigated its relation to the derived category of generic Artin-Mumford quartic double solids. In our previous paper [22] we proved that for a generic Enriques surface X , its Kuznetsov component determines X up to isomorphism. As this refines the Derived Torelli Theorem mentioned above, we refer to it as the Refined Derived Torelli Theorem for Enriques surfaces. A natural question which remained open after [22] was whether such a result holds in general, that is for every Enriques surface.
In this paper we positively answer this question by means of the following theorem which is the main result of this paper.
Theorem A Let X 1 and X 2 be Enriques surfaces over an algebraically closed field K of characteristic different from 2. If they possess semiorthogonal decompositions where L i is as above, and there exists an exact equivalence F : Ku(X 1 , L 1 ) ∼ − → Ku(X 2 , L 2 ) of Fourier-Mukai type, then X 1 ∼ = X 2 .
This result sits in a very active research area. Statements as the above one are usually referred to as categorical Torelli theorems because, in analogy with the Hodge-theoretic Torelli theorems, they show that the geometry of a smooth projective variety can be reconstructed, up to isomorphism, from one relevant component of its derived category (possibly with some additional data). Such a component is meant to play the role of intermediate Jacobians or middle cohomologies and the additional data is the analogue of principal polarizations or special Hodge structures. In the case of Fano manifolds, many papers were devoted to this kind of problems including [1][2][3]14,23,26]. All these results heavily rely on the existence of Bridgeland stability conditions on the Kuznetsov component as proved in [2]. In general, we expect that there are no stability conditions on the Kuznetsov components of Enriques surfaces.
For this reason, in this paper we take a different perspective which is indeed the same as in [22]. Indeed, we prove Theorem A in Sect. 3.2 by showing that the given equivalence between the Kuznetsov components can be extended to the whole derived categories by adding the 10 line bundles one by one. Once we know that D b (X 1 ) and D b (X 2 ) are equivalent, we can invoke the Derived Torelli Theorem and conclude that X 1 and X 2 are isomorphic. It should be noted that Theorem A is actually a special instance of the more precise Theorem 3.3. This extension procedure is made possible by Proposition 3.2 which was proved in [22]. In turns, to make such a result applicable in the more complicated geometric setting of this paper, we need a complete classification of special objects in the Kuznetsov component: 3-spherical and 3-pseudoprojective objects. If X is generic and thus the 10 line bundles are orthogonal, we only get 3-spherical objects and their classification in [22] is considerably simpler. The hard part of this paper consists in dealing with 3-pseudoprojective objects which appear exactly because the line bundles in the semiorthogonal decomposition are no longer orthogonal. This is the content of Theorem 2.7 which is indeed the technical core of this paper. In order to make the paper self-contained we include in Sects. 1.1 and 1.2 some introductory material about semiorthogonal decompositions and the geometry of Enriques surfaces.
We conclude the introduction by pointing out that in [22] (see Remark 5.3 therein) we proposed a different approach to prove Theorem A. The strategy was to deform two Enriques surface X 1 and X 2 containing (−2)-curves and with an equivalence of Fourier-Mukai type Ku(X 1 , L 1 ) ∼ = Ku(X 2 , L) to generic unnodal Enriques surfaces and then apply the generic Refined Derived Torelli Theorem in [22]. Even though this strategy is still valid, a precise implementation would require a careful (and possibly complicated) comparison between the deformation theory of the Enriques surface and of its Kuznetsov component. We feel like that the approach in the present paper is more direct and conceptually clearer.

Semiorthogonal decompositions and Enriques categories
In this section we provide a short introduction to semiorthogonal decompositions of triangulated categories and we apply this to the derived categories of Enriques surfaces. In Sect. 1.3 we start discussing some preparatory material which will be used later for the classification of special objects in the Kuznetsov component of those surfaces.

Semiorthogonal decompositions and Serre functors
Let T be a triangulated category. A semiorthogonal decomposition of T is a sequence of full triangulated subcategories D 1 , . . . , D m of T such that: The subcategories D i are called the components of the decomposition. We say that two such distinct We will be mostly interested in the case where the inclusion functor α i : D i → T has left adjoint α * i and right adjoint α ! i . In this case, D i is called an admissible subcategory.
We need now to introduce another functor which is naturally defined in presence of a semiorthogonal decomposition satisfying the additional assumption that its components are admissible subcategories. The left mutation functor through D i is the functor L D i defined by the canonical distinguished triangle where η i denotes the counit of the adjunction. In complete analogy, one defines the right mutation functor through D i as the functor R D i defined by the canonical distinguished triangle where i is the unit of the adjunction.
The following is a very simple and well-known result.
Proof We have the following natural isomorphisms of exact functors Here the non-trivial isomorphism is the second one which follows from [20, Lemma 2.6] and the easy observation that We conclude this section by noting that, from now on, we will assume that all categories are linear over a field K. In this case, E ∈ T is exceptional if Hom(E, E[ p]) = 0, for all integers p = 0, and Hom(E, E) ∼ = K. A set of objects {E 1 , . . . , E m } in T is an exceptional collection if E i is an exceptional object, for all i, and Hom(E i , E j [ p]) = 0, for all p and all i > j.

Enriques surfaces
In the rest of the paper we work over an algebraically closed field K of characteristic different from 2. Let X be an Enriques surface. For later use, let us recall that its Serre functor S X is defined as S X (−) := (−) ⊗ ω X [2], where ω X is 2-torsion but non-trivial. Hence S 2 X = [4] but S X = [2]. Moreover the torsion part of the Néron-Severi group NS(X ) tor is 2-torsion as well and generated by the class of ω X . We set Num(X ) := NS(X )/NS(X ) tor . For a brief but very informative survey about the geometry of Enriques surfaces, one can have a look at [10]. A more complete treatment is in [9,11].
Let us now consider the bounded derived category of coherent sheaves D b (X ) of X . On one side we have the well-known Derived Torelli Theorem which is a summary of [ On the other hand we can study the structure of D b (X ) by means of the following result (which is mostly well-known to experts).
In view of the previous proposition we can state the following. It is worth discussing when the type of the semiorthogonal decomposition may be seen as an invariant associated to X .

Remark 1.6
If X is generic in moduli, then all possible semiorthogonal decompositions as in the statement above get much simplified. Indeed, X does not contain (−2)-curves and then one always gets 10 orthogonal blocks, each consisting of only one line bundle. Thus, for unnodal Enriques surfaces, all semiorthogonal decompositions as above are of the type given by the 10-uple {1, . . . , 1}.
The situation gets much more interesting if one looks at the generic Enriques surfaces in the divisor of the moduli space of such surfaces parametrizing nodal Enriques surfaces (i.e. surfaces containing (−2)-curves). This is very much related to the fact that a natural polarization (called Fano polarization) is ample and not just nef. For an extensive discussion on this, the reader can have a look at [22,Section 3.1]. In this paper we will be interested in the following. On the other hand, by Theorem 6.5.4 and Corollary 6.5.9 in [11], any generic nodal Enriques surface has a nef Fano polarization which is not ample and gives rise to a semiorthogonal decomposition as in the proposition above but where c = 9 and of type given by the 9-uple (1.4)

Some useful computations
In this section we discuss some preliminary computations concerning special objects in D b (X ), for an Enriques surface X with a semiorthogonal decomposition as in (1.4) and satisfying properties (a) and (b). We set The following rather technical result will be used later.

Lemma 1.9
In the above setting, we have that As for (2), when As L i 1 is exceptional, the statement holds by definition.
When n i ≥ 2, we first observe that we have the following isomorphisms where the first one follows from the fact that the admissible subcategories L i 's are orthogonal and thus S −1 Consider the commutative diagram where the rows are distinguished triangles By the octahedron axiom, we get the morphisms In particular, we have the distinguished triangle for every 1 ≤ j ≤ n i − 1. If we apply R L i j+2 ,...,L i n i to (1.7) and we take into account that for every 1 ≤ j ≤ n i − 1. If we apply the functor RHom(L i 1 , −) to (1.8) when j ≥ 2 (which means that n i ≥ 3), then we get the isomorphisms of graded vector spaces Finally, if we apply RHom(L i 1 , −) to (1.8), for j = 1, and we use (1.9) and the fact that then we get the isomorphisms of graded vector spaces For the penultimate isomorphism we used (1.6).
1 and the only isomorphim in the statement hold automatically. When

Classifying spherical and pseudoprojective objects
In this section we study and classify 3-spherical and 3-pseudoprojective objects in the Kuznetsov component of the special semiorthogonal decompositions in Sect. 1.2.

Spherical and pseudoprojective objects: construction and classification
We begin with a fairly general definition.
Definition 2.1 Let T a triangulated category that it is linear over a field K and with Serre functor S T .
Remark 2.2 Some comments on the choice of the names in the above definition are in order here. Spherical objects were introduced in the seminal paper [27] and our definition is exactly the same. The name is clearly motivated by the fact the the Ext-algebra of such an object is isomorphic to the cohomology algebra of an n-sphere.
The situation is slightly different for n-pseudoprojective objects as in (b) of the definition above. Our objects look very much like a variant of P n -objects described in [15]. The first key different is that in [15] the Ext-algebra of a P n -object is assumed to be isomorphic to the cohomology algebra of the complex projective space P n and it is then generated in degree 2. On the contrary, our n-pseudoprojective objects have non-trivial Ext's in odd degrees as well. The case of objects with Ext-algebra isomorphic to the cohomology algebra of P n but with generator in degree 1 has been studied in [17]. Indeed, our objects E in Definition 2.1 (b) look very much like P n [1]-objects in [17]. The key difference is that , with x in degree 1, only as graded vector spaces and not as algebras. Of course, we expect the objects studied in Theorem 2.7 to be P 3 [1]-objects but, at the moment, we have no control on their Ext-algebra and, as we will see, this is not relevant for our computations. For these reasons we prefer to adopt a new name and call them 3-pseudoprojective with the hope that future studies may turn them into P 3 [1]-objects.
In this paper we will be mainly interested in 3-spherical and 3-pseudoprojective objects. The former were extensively used in [22] to prove the Refined Derived Torelli Theorem in the generic case.
In the more general setting of the present paper, we want to show how one can possibly construct 3-pseudoprojective objects in the Kuznetsov component of an Enriques surface whose derived category is endowed with a semiorthogonal decomposition as in (1.4). To this extent, we set By (1.1), the object S i sits in the distinguished triangle

Remark 2.3 (i)
By Lemma 1.2, we get the distinguished triangle In other words, since Hom(L i 1 , S X (S −1 L (L i 1 ))) ∼ = K, the object S i [1] is isomorphic to the cone of the unique non-trivial morphism L i We can now discuss some additional properties of the objects S i 's. Proof If n i = 1, then the result is simply [22,Lemma 4.8]. Thus we can assume n i ≥ 2. In this case, we can first compute the action of the Serre functor using again an argument very close to the one in [22,Lemma 4.8] and which consists of the following chain of isomorphisms where the first isomorphism is [20, Lemma 2.6], the second one is Remark 2.3 (i), the third one follows from the the fact that S 2 X ∼ = [4] and the fourth one follows from the observation that ζ ! K (S X (L i 1 )) = 0 (this by (1.1) and noticing that S X (L i 1 ) ∈ K ⊥ ). Finally, the last isomorphism is a consequence of Lemma 1.9 (3). This yields property (ii) in Definition 2.1 (b).
To conclude that S i is 3-pseudoprojective, we first apply Hom(S i , −) to (2.1). Hence, since S i ∈ K = ⊥ (S X (L)), we have an isomorphism of graded vector spaces By Lemma 1.9 and Serre duality, . Finally, if we apply the functor RHom(−, L i 1 ) to the distinguished triangle (2.1), then we get where the last vanishing is due to the fact that S −1 L (L i 1 ) is in L i which is orthogonal to L j to which L j 1 belongs. By (2.1), these two observations imply RHom(S i , S j ) = 0, when i = j, and thus the claim follows.
As we learnt from the previous lemma, we need to distinguish when n i = 1 and n i ≥ 2 in the semiorthogonal decomposition (1.4). Thus it is convenient to reformulate here and keep it in mind for the rest of the paper the setup we are working in: . . , L i n i } is an exceptional collection; and (c) If there is j ∈ {1, . . . , c} such that n j = 1, then there is d ∈ {1, . . . , c} such that n j ≥ 2 for all 1 ≤ j ≤ d while n j = 1 for all d < j ≤ c.

Remark 2.6
Since the components L i 's are orthogonal, the special choice in (c) above for the ordering of these components does not cause loss of generality.
The goal of the rest of this section is to prove the following result which is the key for our Refined Derived Torelli Theorem and answers the open question in [22,Remark 4.11].

Theorem 2.7 In Setup 2.5, if F is an object in
Furthermore, all these 3-spherical and 3-pseudoprojective objects are not isomorphic.
Before moving to the proof of this result, we want to discuss an easy application that shows that the Kuznetsov component of an Enriques surface is not, in general, intrinsic to the surface, up to equivalence.

Corollary 2.8 If X is a generic nodal Enriques surface, then there exist two semiorthogonal decompositions
for i = 1, 2, as in Setup 2.5 and such that L 1 and L 2 consist of 10 exceptional line bundles but Ku(X , L 1 ) Ku(X , L 2 ).
Proof Consider the two distinct semiorthogonal decompositions on D b (X ), for X a generic nodal Enriques surface, described in Example 1.7. Assume that there is an exact equivalence Ku(X , L 1 ) ∼ = Ku(X , L 2 ). Then the two Kuznetsov components would contain, up to shifts and isomorphisms, the same number of 3-spherical and 3-pseudoprojective objects. But this contradicts Theorem 2.7.

Remark 2.9
If one could prove that, given a generic nodal Enriques surface, the Kuznetsov components of the two semiorthogonal decompositions in Example 1.7 have no non-trivial semiorthogonal decompositions, then Corollary 2.8 would yield another counterexample to the Jordan-Hölder property of semiorthogonal decompositions. Roughly, such a property predicts that if X is a smooth projective variety then the semiorthogonal decompositions of D b (X ) are essentially unique, up to reordering of the components and equivalence. It is worth recalling that we already know counterexamples to such a property [4,18].

Proof of Theorem 2.7
It is clear that the 'if' part in (1) and (2) and the last part of the statement in Theorem 2.7 are the content of Lemma 2.4. Thus it remains to prove the hard implication in (1) and (2): if F is either a 3-spherical or 3-pseudoprojective object in Ku(X , L), then, up to shift, it is isomorphic to one of the S i 's. The proof is split in several steps.

Step 1: reduction to one component
We first prove that we can simplify our computation and reduce to the case L = L i , for some i = 1, . . . , c. For now, we can indifferently assume that F is either a 3-spherical or a 3-pseudoprojective object in Ku(X , L).
First of all, note that the semiorthogonal decomposition in Setup 2.5 can be conveniently rewritten as By (1.1) and Lemma 1.2, we have a distinguished triangle denotes the embedding of the corresponding admissible subcategory, then we set We now want to prove some relevant properties of G. Note that, by assumption, S Ku(X ,L) (F) ∼ = F [3]. Thus, by [20, Lemma 2.6], we have the isomorphisms where for the second one we use (2.3). The third isomorphism follows from the fact that S 2 X (F) ∼ = F [4] and S Ku(X ,L) (F) ∼ = F [3]. The fourth is a consequence of ζ ! L (F) = 0 while the penultimate is a simple computation. Here ζ L : L → D b (X ) is the embedding of the admissible subcategory L.
Furthermore, if we apply the equivalence S X to (2.3), we get the distinguished triangle If we apply RHom(G, −) to (2.5) and we use Serre duality, we get the isomorphism of graded vector spaces RHom(G, G) ∼ = RHom(G, S X (F)). Furthermore, if we apply RHom(−, S X (F)) to (2.5) and we take into consideration that, by assumption, then we get the isomorphisms of graded vector spaces Here we used that Hom(G, G) has dimension at least 2.
Denote by Q the collection of torsion sheaves Q i j,l for all 1 ≤ i ≤ c and 1 ≤ j < l ≤ n i , defined in (1.5). Since both F and S X (F) are in Q ⊥ = ⊥ Q (here we use Remark 1.8), by (2.5) both G and S X (G) are orthogonal to Q as well. Thus for every Q ∈ Q. We now prove the following result.

) The composition of non-trivial morphisms E → E[3] and E[3] → L i 1 [t + 3] is nontrivial.
Proof Let us first prove (1). Since the L i 's are orthogonal, we may write E = F 1 ⊕ · · · ⊕ F c , for F i in L i . For the same reason, Hence, by Lemma 1.9 (1), Since S L (E) ∼ = E and the L i 's are orthogonal, we must have , for every t ∈ Z, and since In view of what we have just proved, we can simplify the notation and assume L = L i and set n := n i , L j := L i j and Q j,l := Q i j,l . First, we can note that, if n = 1, then E = L[t] ⊕ L[t + 3], for some t ∈ Z. Thus (2) and (3) hold true automatically and we may assume n ≥ 2 from now on.
Let us prove (2) under this assumption. We set E 1 := E ∈ L and write it as an extension where now E 2 ∈ L 2 , . . . , L n . Inductively, for all j = 1, . . . , n − 1, we then define where E k ∈ L k , . . . , L n . If we apply the functor RHom(−, L j+1 ) to (2.8) we get the exact sequence (2.9) Now consider the distinguished triangle where the first map is the unique (up to scalar) non-trivial one, and apply RHom(E j , −) to it. Note that, by construction, the object E j is obtained with a finite number of extensions involving E and elements in L 1 , . . . , L j−1 . On the other hand, RHom(L s , Q j, j+1 ) = 0, when s ≤ j − 1, while, by assumption (c), we have RHom(E, Q j, j+1 ) = 0. Therefore we get an isomorphism of graded vector spaces Recall that Hom(L j , L j+1 ) ∼ = K with generator the non-trivial morphism in (2.10).
If we plug this into (2.9), we get the isomorphism and, iterating the argument, we get for all j = 1, . . . , n − 1. In particular, (2.8) gets the form On the one hand, we have On the other hand, this information plugged into (2.13) for j = n − 1 yields that RHom(E, E n−1 ) has non-trivial component concentrated in degree −m, . . . , m as well.  (2).
By (2.4), (2.6) and (2.7), the assumptions (a)-(c) of Lemma 2.10 for the object G. Thus G is in L i , for some 1 ≤ i ≤ c. The other parts of Lemma 2.10 will be use later in the proof.
Step 2: reduction to the pseudoprojective case (i ≤ d) Here ζ L i and ζ Ku(X ,L),L i are the embeddings of the corresponding admissible subcategories. Since condition (i) in Definition 2.1 is clearly satisfied, this implies that F is 3-spherical or 3-pseudoprojective in the larger category Ku(X , L),L i as well. Assume now i > d. By Lemma 2.10 (2) applied to G = S X (ζ ! S X (L) (F)) ∈ L i , we must have G ∼ = L i [t]⊕ L i [t +3] for some t ∈ Z. In particular, by (2.6), the object F is 3-spherical. By [22,Proposition 4.10], F ∼ = S i [k] for some k ∈ Z. Therefore, from now on, we may assume i ≤ d.

Step 3: the final isomorphism
First note that, for i as in Step 2 and since S X (F) ∈ ⊥ L, by Lemma 2.10 and (2.5), we have the isomorphisms up to shifting F (and thus G).
If ψ : F → L i 1 is the unique, up to scalars, non-trivial morphism, we set and get the distinguished triangle We first prove the following result.
Let k 1 be a non-trivial element in Hom(C, F [1]). Note that since L i 1 is exceptional, up to scalars, the morphism k 1 realizes L i 1 as the extension of F and C in (2.15) yielding the distinguished triangle ψ [1] −−→ L i 1 [1]. (2.16) Let k 2 be the non-trivial generator of Hom(S X (F), C).
On the other hand, as S X (F) ∈ S X (S −1 L (L i 1 )) ⊥ , by (2.5) we get where the second isomorphisms is by Serre duality while the penultimate is obtained by applying S L and using that S L (G) = G by (2.4). Let k 3 be a non-trivial generator of the vector space Hom(S X (S −1 L (L i 1 )), F [1]). Finally, we consider the isomorphisms where the first isomorphism is due to the fact that S X is an equivalence. For the second one, we apply RHom(−, S L (L i 1 )) to (2.5) and use that S X (F) ∈ ⊥ L. For the third one, we apply S L and use (2.4). The last isomorphism is due to Lemma 2.10 (2). We set k 4 to be the non-trivial generator of Hom(S X (F), S X (S −1 L (L i 1 ))).
Let us prove that k 3 • k 4 is a non-trivial morphism. By Lemma 2.10 (3), the composition of the unique (up to scalars) non-trivial morphisms is non-trivial. Call it k. If we apply S −1 L to the whole composition and since G ∼ = S −1 L (G), then the composition of non-zero morphisms is non-trivial. Denote by h the morphism from F to G in (2.5). Since S(F) ∈ ⊥ L, the composition of h with the non-trivial morphisms G → G [3] and are non-trivial, for t = 0, 3. Indeed, it is enough to apply RHom(−, G [3]) and RHom(−, S L (L i 1 )[t]) to (2.5). Combining these two remarks we see that the composition of the non-trivial morphisms is non-trivial because it factors as Since Hom(F, S X (F) [3]) ∼ = Hom(F, S X (F) [4]) = 0, the composition of non-trivial morphisms F → F [3] h [3] − − → G [3] is non-zero and it coincides with the non-trivial morphism F → G [3] above, up to a scalar. Therefore, the composition of non-trivial morphisms F → F [3] h [3] − − → G [3] is non-zero and the same is true for the composition of non-trivial morphisms Since the composition in ( In particular, Hom(S X (S −1 L (L i 1 )), C) ∼ = K and there exists a non-trivial map g : S X (S −1 L (L i 1 )) → C such that Let us now produce the morphism f . Consider the diagram where the first row is (a rotation of) the distinguished triangle (2.5), the second column is (2.15) while the third column is a distinguished triangle whose morphism G → L i 1 is the unique (up to scalars) non-trivial morphism from Lemma 2.10 (2). Since S X (F) ∈ ⊥ L, the top squares in the diagram commute and by the octahedron axiom, the bottom row is a distinguished triangle as well.
If we apply RHom(−, L i 1 ) to the triangle in the third column of (2.20) and we take into account that Hom(G, when t = 0 or −1. Now, if we apply RHom(−, S X (S −1 L (L i 1 ))) to the distinguished triangle in the bottom row of (2.20) and we use the vanishing above, then we get an isomorphism In particular, Hom(C, S X (S −1 L (L i 1 ))) ∼ = K and there a non-trivial f : To summarize, all morphisms we introduced so far fit in the following diagram where the two triangles are commutative. Note now that the composition k 3 • f is non-trivial. Indeed, if not, we would have 0 = k 3 • f • k 2 = k 3 • k 4 , contradicting (2.18). Thus there is 0 = λ ∈ K such that k 3 := λk 3 = 0 sits in the diagram where the two triangles are commutative.
Assume now 0 = f • g. By the commutativity of the diagram above which contradicts (2.19). Thus f • g = 0 as we want.
In conclusion we get the following sequence of isomorphisms [1].
Note that for the first isomorphism follows from (2.15), the second one from Lemma 2.11 while the last one is a consequence of the definition of S i and of the isomorphism This concludes the proof of Theorem 2.7.

Proof of the main result
In this section, after a quick discussion about Fourier-Mukai functors and the way we can extend them from admissible subcategories to larger subcategories, we prove our main result. As we will see the main theorem follows from the more precise statement in Sect. 3.2.

Fourier-Mukai functors and their extensions
Let us start with a short introduction to the theory of Fourier-Mukai functors. In complete generality, assume that X 1 and X 2 are smooth projective varieties over K with admissible subcategories for i = 1, 2. An exact functor F : A 1 → A 2 is of Fourier-Mukai type if there exists an object E ∈ D b (X 1 × X 2 ) such that there is an isomorphism of exact functors The exact functor E is defined as where p i : X 1 × X 2 → X i is the ith natural projection.

Remark 3.1 (i) Suppose we are given a Fourier-Mukai functor
(ii) It should be noted that, when F : A 1 → A 2 is an equivalence, [21, Conjecture 3.7] should imply that F is of Fourier-Mukai type in the above sense. This expectation is motivated by what is known for full functors between the bounded derived categories of smooth projective varieties (see [6,7,24,25]).
In general, one should not expect to be able to extend an equivalence between admissible subcategories to the whole triangulated categories. Nonetheless, this is possible under some compatibility assumptions as illustrated in the following result which we proved in [22].  (3) Ẽ | A 1 ,E is an equivalence onto A 2 , F .

Proof of Theorem A
Let X 1 and X 2 be smooth Enriques surfaces over K admitting admissible subcategories L and M which fall under Setup 2.5. More precisely, their derived categories D b (X 1 ) and D b (X 2 ) admit semiorthogonal decompositions We are now going to prove the following result which is actually a more precise version of Theorem A. such that E n 1 (L 1 j ) ∼ = M 1 j . By (3.1), (3.2) and Theorem 2.7, the admissible subcategory M 1 1 , . . . , M 1 n 1 must coincide with M 1 and thus n 1 = n 1 . At each step, the assumptions (a) and (b) of Proposition 3.2 are satisfied, so we can proceed further by induction on c and deal with the other components as we mentioned at the beginning of this proof. In conclusion, in a finite number of step, we get an equivalence D b (X 1 ) ∼ = D b (X 2 ) and, by Theorem 1.3, an isomorphism X 1 ∼ = X 2 . This proves (2) in the statement while properties (1) and (3) are automatic by construction.
Clearly, Theorem A has a trivial converse. Indeed, assume we are given an isomorphism f : X 1 → X 2 and a semiorthogonal decomposition for D b (X 2 ) as in Setup 2.5. Since the exact functor f * : D b (X 2 ) → D b (X 1 ) is an equivalence, we can take on D b (X 1 ) the semiorthogonal decomposition as in Setup 2.5 which is the image of the given one on D b (X 2 ) under f * . The exact functor, F := ( f * ) −1 | Ku(X 1 ,L 1 ) : Ku(X 1 , L 1 ) ∼ − → Ku(X 2 , L 2 ) is an exact equivalence of Fourier-Mukai type by construction. Of course, the two semiorthogonal decompositions on X 1 and X 2 have the same type and, in the argument, we can exchange the roles of X 1 and X 2 .