On the Abuaf-Ueda flop via non-commutative crepant resolutions

The Abuaf-Ueda flop is a $7$-dimensional flop related to $G_2$ homogeneous spaces. The derived equivalence for this flop is first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence in which we use tilting bundles. Our proof also show the existence of non-commutative crepant resolutions of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over our non-commutative crepant resolution.

1. Introduction 1.1. The Abuaf-Ueda flop. First of all we give the construction of the flop we study in this article. Let us consider the G 2 Dynkin diagram ≡ ≡ . Then by the classification theory of homogeneous varieties, projective homogeneous varieties of the semi-simple algebraic group of type G 2 corresponds to a marked Dynkin diagram. The one × ≡ ≡ corresponds to the G 2 -Grassmannian G = Gr G2 . Another one ≡ ≡ × corresponds to the 5-dimensional quadric Q ⊂ P 6 . The last one × ≡ ≡ × corresponds to the (full) flag variety F of type G 2 . There are projections F → G and F → Q, and both of them give P 1 -bundle structures of F.
Let us consider the Cox ring of F where O F (a, b) (resp. V ∨ (a,b) ) is a line bundle on F (resp. the dual of an irreducible representation of G 2 ) that corresponds to the dominant weight (a, b). Put C a,b := H 0 (F, O F (a, b)) and C n := a∈Z C n+a,a .
Using them we can define a Z-grading on C by This grading corresponds to a G m -action on Spec C that is obtained by a map G m → (G m ) 2 , α → (α, α −1 ) and the natural (G m ) 2 -action on Spec C coming from the original bi-grading. Then we can take the geometric invariant theory quotients The projective quotients Y + and Y − are the total spaces of rank two vector bundles on G and Q respectively. The affinization morphism φ + : Y + → X and φ − : Y − → X are small resolution of the singular affine variety X and they contract the zero-sections. Furthermore we can show that the birational map Y + Y − is a 7-dimensional simple flop with a interesting feature that the contraction loci are not isomorphic to each other.
The author first learned this interesting flop from Abuaf. Later the author noticed that the same flop was found by Ueda independently [Ued16]. Thus the author would like to attribute this new flop to both of them, and would like to call this flop the Abuaf-Ueda flop.
When there is a flop Y + Y − between two smooth varieties, it is important to compare their derived categories. According to a famous conjecture due to Bondal and Orlov [BO02], we expect that we have a derived equivalence D b (Y + ) D b (Y − ). In the case of the Abuaf-Ueda flop, Ueda proved that the derived equivalence using the theory of semi-orthogonal decomposition and its mutation. However, since there are many other methods to construct an equivalence between derived categories, it is still interesting problem to prove the derived equivalence using other methods.
1.2. Results in this article. The main purpose of this article to construct tilting bundles on both sides of the flop Y + Y − , and construct equivalences between the derived categories of Y + and Y − using those tilting bundles. A tilting bundle T * on Y * ( * ∈ {+, −}) is a vector bundle on Y * that gives an equivalence between two derived categories. In particular, if we find tilting bundles T + and T − with the same endomorphism ring, then we have an equivalence D b (Y + ) D b (Y − ) as desired.
The advantage of this method is that it enables us to study a flop from the point of view of the theory of non-commutative crepant resolutions (= NCCRs) that is first introduced by Van den Bergh [VdB04]. In our case, an NCCR appears as the endomorphism algebra End Y * (T * ) of a tilting bundle T * . Via the theory of NCCRs, we also study the Abuaf-Ueda flop from the moduli-theoretic point of view.
Recall that Y + and Y − are the total spaces of rank two vector bundles on G and Q respectively. If there is a variety Z that gives a rational resolution of an affine singular variety and that is the total space of a vector bundle on a projective variety W admitting a tilting bundle T , it is natural to expect that the pull back of T via the projection Z → W gives a tilting bundle on Z. Indeed, in many known examples, we can produce tilting bundles in such a way [BLV10,H17a,WZ12].
However, in our case, we cannot obtain tilting bundles on Y + or Y − as a pull back of known tilting bundles on G or Q. Thus the situation is different from previous works. Nevertheless, by modifying bundles that are obtained from tilting bundles on the base G or Q, we can find tilting bundles on Y + and Y − . Namely, tilting bundles we construct are the direct sum of indecomposable bundles that are obtained by taking extensions of other bundles obtained from G or Q. We can also check that they produce derived equivalences 1.3. Related works. If we apply a similar construction to Dynkin diagrams A 2 or C 2 , then we have the four-dimensional Mukai flop or the (five-dimensional) Abuaf flop [Seg16] respectively. Therefore this article is a sequel of papers [H17a,Seg16,H17b].
Recently, Kanemitsu [Kan18] classified simple flops of dimension up to eight, which is a certain generalization of the theorem of Li [Li17]. It is interesting to prove the derived equivalence for all simple flops that appear in Kanemitsu's list using tilting bundles, and we can regard this article as a part of such a project.
This flop is also related to certain (compact) Calabi-Yau threefolds which are studied in [IMOU16a,IMOU16b,Kuz18]. Let us consider the (geometric) vector bundle Y + → G over G. Then as a zero-locus of a regular section of this bundle we have a smooth Calabi-Yau threefold V + in G. Similarly, we can construct another Calabi-Yau threefold V − in Q. Papers [IMOU16b,Kuz18] show that Calabi-Yau threefolds V + and V − are L-equivalent, derived equivalent but NOT birationally equivalent to each other. (L-equivalence and non-birationality is due to [IMOU16b], and derived equivalence is due to [Kuz18].) As explained in [Ued16], we can construct a derived equivalence with a certain nice property.
1.4. Open questions. It would be interesting to compare the equivalences in this article and the one constructed by Ueda. It is also interesting to find Fourier-Mukai kernels that give equivalences. In the case of the Mukai flop or the Abuaf flop, the structure sheaf of the fiber product Y + × X Y − over the singularity X gives a Fourier-Mukai kernel of an equivalence (see [Kaw02,Nam03,H17b]). Thus it is interesting to ask whether this fact remains to hold or not for the Abuaf-Ueda flop.
Another interesting topic is to study the autoequivalence group of the derived category. Since we produce some derived equivalences that are different to each other in this article, we can find some non-trivial autoequivalences by combining them. It would be interesting to find an action of an interesting group on the derived category of Y + (and Y − ) that contains our autoequivalences.
Acknowledgements. The author would like to express his sincere gratitude to his teachers Professor Michel Van den Bergh and Professor Yasunari Nagai for their advice and encouragement. The author is grateful to Professor Roland Abuaf for letting him know about this interesting flop. The author would like to thank Professor Shinnosuke Okawa for letting him know about the paper [Ued16]. The author also would like to thank Professors Hajime Kaji and Yuki Hirano for their interest and comments. A part of this work was done during the author's stay in Hasselt University. The author would like to thank Hasselt University for the hospitality and excellent working condition. This work is supported by Grant-in-Aid for JSPS Research Fellow 17J00857.

Preliminaries
2.1. Tilting bundle and derived category. First we prepare some basic terminologies and facts about tilting bundles.
Definition 2.1. Let Y be a quasi-projective variety and T a vector bundle (of finite rank) on Y . Then we say that T is partial tilting if Ext ≥1 Y (T, T ) = 0. We say that a partial tilting bundle T on Y is tilting if T is a generator of the unbounded derived category D(Qcoh(Y )), i.e. if an object E ∈ D(Qcoh(Y )) satisfies RHom(T, E) 0 then E 0.
If we find a tilting bundle on a projective scheme (over an affine variety), we can construct a derived equivalence between the derived category of the scheme and the derived category of a non-commutative algebra obtained as the endomorphism ring of the bundle.
Proposition 2.2. Let Y be a projective scheme over an affine scheme Spec(R). Assume that Y admits a tilting bundle T . Then we have the following derived equivalence These equivalences coming from tilting bundles are very useful to construct equivalences between the derived categories of two crepant resolutions.
Lemma 2.3. Let X = Spec R be a normal Gorenstein affine variety of dimension greater than or equal to two, and let φ : Y → X and φ : Y → X be two crepant resolutions of X. Put U := X sm = Y \ exc(φ) = Y \ exc(φ ). Assume that there are tilting bundles T and T on Y and Y , respectively, such that Then there is a derived equivalence The existence of a tilting bundle on a crepant resolution does not hold in general. For this fact, see [IW14,Theorem 4.20]. In addition, even in the case that a tilting bundle exists, it is still non-trivial to construct a tilting bundle explicitly. The following lemma is very useful to find a tilting bundle.
be a collection of vector bundles on a quasi-projective scheme Y . Assume that In particular, this assumption implies that E i is a partial tilting bundle for any i.
Then there exists a tilting bundle on Y .
Proof. We use an induction on n. If n = 1, the statement is trivial. Let n > 1. Choose generators of Ext 1 Y (E 1 , E 2 ) as a right End Y (E 1 )-module, and let r be the number of the generators. Then we can take the corresponding sequence Indeed, if we apply the functor Ext i Y (E 1 , −) to the sequence above, we have the long exact sequence . Now δ is surjective by construction, and hence we have Y (E 1 , F ) = 0. Applying the derived functor Ext i Y (E 2 , −) to the same short exact sequence, we have Ext ≥1 (E 2 , F ) = 0 from the assumption that there is no former Ext ≥1 . Thus we have Ext ≥1 Y (F, F ) = 0. One can also show that Ext ≥1 Y (F, E 1 ) = 0, and therefore E 1 ⊕ F is a partial tilting bundle. Put Then it is easy to see that the new collection {E i } n−1 i=1 satisfies the assumptions (i), (ii) and (iii). Note that the condition (i) holds since the new collection Thus we have the result by the assumption of the induction. 2.2. Geometry and representation theory. Next we recall the representation theory and the geometry of homogeneous varieties we need. We also explain the geometric aspect of the Abuaf-Ueda flop in the present subsection.
2.2.1. Representation of G 2 . In the present subsection, we recall the representation theory of the semi-simple algebraic group of type G 2 . We need the representation theory when we compute cohomologies of homogeneous vector bundles using Borel-Bott-Weil theorem in Section 3.
Let V = {(x, y, z) ∈ R 3 | x + y + z = 0} be a hyperplane in R 3 . Then the G 2 root system in V is the following collection of twelve vectors in V .
An easy computation shows that ).
The lattice L = Z π 1 + Z π 2 in V generated by π 1 and π 2 is called the weight lattice of G 2 , and a vector in this lattice is called a weight. We call an weight of the form aπ 1 + bπ 2 for a, b ∈ Z ≥0 a dominant weight. The set of dominant weights plays a central role in the representation theory because they corresponds to irreducible representations. Let α ∈ ∆ be a root. Then we can consider the reflection S α defined by the root α. That is a linear map S α : V → V defined as The Weyl group W is defined by a subgroup of the orthogonal group O(V ) generated by S α for α ∈ ∆: . It is known that W is generated by two reflections S α1 and S α2 defined by simple roots. Using this generator, we define the length of an element in W as follow. The length l(w) of an element w ∈ W is the smallest number n so that w is a composition of n reflections by simple roots. In the case of G 2 , the Weyl group W has twelve elements. The Table 1 shows all elements in W and their length. In that table, we denote S αi k · · · S αi 2 S αi 1 by S i k ···i2i1 for short. Let ρ be the half of the sum of all positive weights. It is known that ρ also can be written as ρ = π 1 + π 2 . Using this weight, we can define another action of the Weyl group W on the weight lattice L that is called dot-action. The dot-action is defined by In our G 2 case, the dot-action is the following affine transform.
2.2.2. Geometry of G 2 -homogeneous varieties. Next we recall the geometry of G 2homogeneous varieties.
The G 2 -Grassmannian G = Gr G2 is a 5-dimensional closed subvariety of Gr(2, 7), and has Picard rank one. The Grassmannian Gr(2, 7) admits the universal quotient bundle Q of rank 5 and G is the zero-locus of a regular section of the bundle Q ∨ (1). Since det(Q ∨ (1)) O Gr(2,7) (4) and ω Gr(2,7) O Gr(2,7) (−7), we have ω G O G (−3). Thus G is a five dimensional Fano variety of Picard rank one and of Fano index three. We denote the restriction of the universal subbundle on Gr(2, 7) to G by R. The bundle R has rank two and det(R) O G (−1). It is known that the derived category D b (G) of G admits a full strong exceptional collection (see [Kuz06]). In particular, the variety G admits a tilting bundle The other G 2 -homogeneous variety of Picard rank one is the five dimensional quadric variety Q = Q 5 . On Q there are two important vector bundles of higher rank. One is the spinor bundle S on Q. The spinor bundle S has rank 4 and appear in a full strong exceptional collection Lemma 2.5. For the spinor bundle S on Q, we have (2) There exists an exact sequence This lemma should be well-known but we give the proof here for convenience.
Proof. To show this lemma, we use the theory of mutations of an exceptional collection. For the detail, see [H17b,Appendix B].
First, by taking dual of the collection above, we have another exceptional collec- On the other hand, by applying a functor (−) ⊗ O Q (1) to the original collection, Then by mutating O Q (3) to the left end, we have another collection Therefore we have S ∨ S(1) from a basic fact about exceptional collections. By taking det, we have det S O Q (−2). Let us show (2). From the exceptional collections above we have for some a ∈ Z, where L O Q is the left mutation over O Q . By definition of a left mutation, we have an exact triangle Since S(1) and L O Q (S(1)) are (some shifts of) sheaves, the integer a should be a = −1 and we have an exact triangle By computing the rank of bundles, we have dim C Hom Q (O Q , S(1)) = 8.
Another important vector bundle on Q is the Cayley bundle C. The Cayley bundle C is a homogeneous vector bundle of rank two, and det C O Q (−1). Historically, this bundle was first studied by Ottaviani [Ott90]. Later we will see that the variety Y − that gives one side of the Abuaf-Ueda flop is the total space of C(−2).
The G 2 -flag variety F is a 6-dimensional variety of Picard rank two. There is a projection p : F → G, and via this projection, F is isomorphic to the projectivization of the universal subbundle R(1) (with some line bundle twist): Similarly, via a projection q : F → Q, we have For homogeneous vector bundles on homogeneous varieties, we can compute their sheaf cohomologies using the Borel-Bott-Weil theorem.
Theorem 2.6 (Borel-Bott-Weil). Let E be a homogeneous vector bundle on a projective homogeneous variety Z that corresponds to a weight π. Then one of the following can happen.
(i) There exists an element w of the Weyl group W such that w·π is a dominant weight. (ii) There exists w ∈ W such that w · π = π. Furthermore, (I) In the case of (i), we have In the case of (ii), we have Note that we use the dot-action in this theorem. We also note that the condition (ii) is equivalent to the condition (ii') in our case: On the G 2 -Grassmannian G, a homogeneous vector bundle corresponding to a weight aπ 1 + bπ 2 exists if and only if b ≥ 0, and that bundle is Sym b (R ∨ )(a). On the five dimensional quadric Q, a homogeneous vector bundle corresponding to a weight aπ 1 + bπ 2 exists if and only if a ≥ 0, and that bundle is Sym a (C ∨ )(a + b). On the flag variety F, a line bundle O F (aH +bh) corresponds to a weight aπ 1 +bπ 2 . Thus we can compute the cohomology of these bundles using the Borel-Bott-Weil theorem.
2.2.4. Geometry of the Abuaf-Ueda flop. We explain the geometric description of the Abuaf-Ueda flop. First as explained in [Ued16], Y + is the total space of a vector bundle R(−1) on G. Since det(R (−1)) The other side of the flop Y − is also a total space of a vector bundle of rank two on Q. The bundle is C(−2). Note that det(C(−2)) O Q (−5) ω Q .
Let G 0 ⊂ Y + and Q 0 ⊂ Y − be the zero-sections. Then the blowing-ups of these zero-sections give the same variety and exceptional divisors of p : Y → Y + and q : Y → Y − are same, which we denote by E. There is a morphism Y → F, and via this morphism, Y is isomorphic to the total space of O F (−H − h). The zero-section F 0 (via this description of Y ) is the exceptional divisor E.
Thus we have the following diagram

Tilting bundles and derived equivalences
3.1. Tilting bundles on Y + . First, we construct tilting bundles on Y + . Recall that the derived category D b (G) has an exceptional collection where R is the universal subbundle. Pulling back this collection, we have an collection of vector bundles on Y + that is The direct sum of these vector bundles gives a generator of D(Qcoh(Y + )) by the following Lemma 3.1. However, the following Proposition 3.2 shows that the direct sum of these vector bundles is NOT a tilting bundle on Y + . Lemma 3.1. Let π : Z → W be an affine morphism and E ∈ D(Qcoh(W )) is a generator. Then the derived pull back Lπ * (E) is a generator of D(Qcoh(Z)).
Proposition 3.2. We have Proof. Here we prove (4) and (5) only. Other cases follow from similar (and easier) computations. Let a ≥ −2 and i ≥ 1. Since there are irreducible decompositions we have ). The second term of this decomposition is zero by (1), and hence we have   by adjunction. To compute this cohomology, we use the following decomposition According to this irreducible decomposition, it is enough to compute the cohomology of the following vector bundles.
Thus the Borel-Bott-Weil theorem implies Using the Borel-Bott-Weil theorem in the same way, we can show that bundles of type (ii) and (iii) has no higher cohomology. This shows (4) and (5).

Now we can show that the bundle Σ is partial tilting and that a bunde
is a tilting bundle on Y + as in the proof of Lemma 2.4. We also note that the dual Σ ∨ of Σ is isomorphic to Σ(1). Indeed, the bundle Σ ∨ lies in the sequence The isomorphism R ∨ R(1) and the uniqueness of such a non-trivial extension imply that Σ ∨ Σ(1).
We can apply the same method to another collection and then we get another tilting bundle. As a consequence, we have the following.
Theorem 3.4. The following vector bundles on Y + are tilting bundles.
(1) T ♠ Note that the pair T ♠ + and T ♣ + are dual to each other, and the pair T ♥ + and T ♦ + are dual to each other.

3.2.
Tilting bundles on Y − . To find explicit tilting bundles on Y − , we need to use not only the Borel-Bott-Weil theorem but also some geometry of the flop. Recall that the derived category D b (Q) has an exceptional collection where S is the rank 4 spinor bundle on the five dimensional quadric Q. Pulling buck this collection by the projection π − : Y − → Q, we have a collection of vector bundles on Y − The direct sum of these vector bundles is a generator of D(Qcoh(Y )) by Lemma 3.1, but does NOT give a tilting bundle on Y − . First, we compute cohomologies of line bundles.
Proposition 3.5. ( Proof. Let a ≥ −4. We have the following isomorphism by adjunction A bundle (Sym k C ∨ )(2k + a) corresponds to a weight kπ 1 + (k + a)π 2 . This weight is dominant if and only if k + a ≥ 0, i.e.  Definition 3.6. Let P be the rank 2 vector bundle on Y − which lies in the following unique non-trivial extension One can show that the bundle P is partial tilting as in Lemma 2.4. Note that, by the uniqueness of such a non-trivial sequence, we have P ∨ P(1).
To prove this Proposition, we have to use the geometry of the flop. The following two lemmas are important.
Lemma 3.8. On the full flag variety F, there is an exact sequence of vector bundles Lemma 3.9. There is an isomorphism P Rq * (p * R(−h)).
Proof. From the lemma above, we have an exact sequence on Y Using projection formula and and this bundle lies in the exact sequence . Since the zero-section G 0 has codimension two in Y + , if the bundle R(1)| (Y+\G0) is split, the bundle R(1) is also split. This is contradiction.
Proof of Proposition 3.7. First, we have H ≥1 (Y − , P(a)) = 0 for all a ≥ 0, and H ≥2 (Y − , P(a)) = 0 for all a ≥ −2, by the definition of P and Proposition 3.5. Thus the non-trivial parts are the vanishing of H 1 (Y − , P(−1)) and H 1 (Y − , P(−2)). The first part also follows from the definition of P using the same argument as in the proof of Lemma 2.4.
Therefore we can compute the cohomology as follows.
To compute this cohomology, we use a spectral sequence This shows that there is an isomorphism of cohomologies Let us consider the exact sequence Hence there is an exact sequence we finally have the desired vanishing Now the results follow from Proposition 3.7.
Next we compute the cohomology of (the pull back of) the spinor bundle S. For this computation, we use the geometry of the flop again.
The following lemma is due to Kuznetsov.
Lemma 3.11. There is an exact sequence on the flag variety F
Using this lemma, we have the following.
Using this exact sequence, we can do the following computations.
Lemma 3.14. We have Proof. Since S ∨ S(1), it is enough to show that H ≥1 (Y − , S(a)) = 0 for a ≥ −1. By Lemma 2.5, for any a ∈ Z, there is an exact sequence for any i ≥ 1. Let us consider a spectral sequence for k ≥ 1 and hence we have This finishes the proof.
(2) follows from (1). Let us prove (1). Recall that Rq * (p * R) P(1). Therefore by the Grothendieck duality we have ). Let us consider the exact sequence ). Let us consider a spectral sequence ). Note that E k,0 2 = Ext k Y+ (R, R(2)) = 0 for k ≥ 1. Thus we have Combining all Ext-vanishings in the present subsection, we obtain the following consequence.
Theorem 3.17. The following vector bundles on Y − are tilting bundles. (1) (1) ⊕ P(−1) ⊕ P ⊕ S We note that these bundles are generators of D(Qcoh(Y − )) because they splitgenerate another generators respectively, that are obtained from tilting bundles on Q. We also note that the pair T ♠ − and T ♣ − are dual to each other, and the pair T ♥ − and T ♦ − are dual to each other.
3.3. Derived equivalences. According to Lemma 2.3, in order to show the derived equivalence between Y + and Y − , it is enough to show that there are tilting bundles T + and T − on Y + and Y − respectively, such that they give the same vector bundle on the common open subset U of Y + and Y − , which is isomorphic to the smooth locus of X. Using tilting bundles that we constructed in this article, we can give four derived equivalences for the Abuaf-Ueda flop.
Lemma 3.18. On the common open subset U , we have the following. ( (2) follows from the isomorphism P(1) Rq * (p * R). Let us proof (3). To see this, we show that Rp * (q * S) Σ(1). By Lemma 3.11, there is an exact sequence by projection formula, we have an exact sequence 0 → R → Rp * (q * S) → R(2) → 0 on Y + . Note that this short exact sequence is not split. Thus the uniqueness of such a non-trivial sequence shows that the desired isomorphism Rp * (q * S) Σ(1).
Corollary 3.19. The pair of bundles T * + and T * − give the same bundle on the common open subset U for any * ∈ {♠, ♣, ♥, ♦}.
As a consequence, we have the following theorem.
Theorem 3.20. Let * ∈ {♠, ♣, ♥, ♦} and put Then we have derived equivalences that are quasi-inverse to each other. In many cases, an NCCR is constructed from a tilting bundle on a (commutative) crepant resolution using the following lemma.
Lemma 4.2. Let X = Spec R be a normal Gorenstein affine variety that admits a (commutative) crepant resolution φ : Y → X. Then for a tilting bundle T on Y , the double-dual (φ * T ) ∨∨ of the module φ * T gives an NCCR End Y (T ) End R (φ * T ) of R. If one of the following two conditions are satisfied, then (φ * T ) ∨∨ is isomorphic to φ * T , i.e. we do not have to take the double-dual.
(a) The tilting bundle T contains O Y as a direct summand.
The resolution φ is small, i.e. the exceptional locus of φ does not contain a divisor.
When we find an NCCR Λ = End R (M ) of an algebra R, we can consider the moduli spaces of modules over Λ.
In the following we recall the result of Karmazyn [Kar17]. Let Y → X = Spec R be a projective morphism and T a tilting bundle on Y . Assume that T has a decomposition T = n i=0 E i such that (i) E i is indecomposable for any i, (ii) E i = E j for i = j, and (iii) E 0 = O Y . Then we can regard the endomorphism ring Λ := End Y (T ) as a path algebra of a quiver with relations such that the summand E i corresponds to a vertex i.

Now we define a dimension vector d
Note that, since we assumed that E 0 = O Y , we have d T (0) = 1. We also define a stability condition θ T associated to the tilting bundle T by Then we can consider King's moduli space M ss Λ,d T ,θ T of θ-semistable (right) End Y (T )modules with dimension vector d T . It is easy to see that there is no strictly θ Tsemistable object with dimension vector d T (see [Kar17]), and thus the moduli space M ss Λ,d T ,θ T is isomorphic to a moduli space M s Λ,d T ,θ T of θ T -stable objects. In this setting, Karmazyn [Kar17] proved the following.  By assumption, there is a surjective morphism O ⊕r Y → T ∨ for some r. Now we have the following commutative diagram .  Then the main component M of M s Λ,d T ,θ T is isomorphic to Y . Proof. Since Y and M s Λ,d T ,θ T are projective over Spec R (see [VdB04]), the monomorphism Y → M s Λ,d T ,θ T is proper. A proper monomorphism is a closed immersion. Since Y dominants Spec R, the image of this monomorphism is contained in the main component M . Since Y and M are birational to Spec R, they coincide with each other. 4.2. Application to our situation. First, from the existence of tilting bundles, we have the following.  (1) A bundle Σ(a) is globally generated if and only if a ≥ 2.
(2) A bundle P(a) is globally generated if and only if a ≥ 2.
Proof. First we note that R(a) is globally generated if and only if a ≥ 1. Recall that Σ(a) is defined by an exact sequence 0 → R(a − 1) → Σ(a) → R(a + 1) → 0.
The proof for P(a) is similar.
Then we have the following as a corollary.
Corollary 4.9. The crepant resolution Y + of X = Spec C 0 gives the main component of the moduli space M s Λ+,d+,θ+ of representations of an NCCR Λ + of X of dimension vector d + with respect to the stability condition θ + .
Similarly we define a quiver with relations (Q − , I − ) with (Q − ) 0 = {0, 1, 2, 3, 4, 5} whose path algebra is Λ − = End Y− (T − ), using the order  Finally we remark that there is an isomorphism of algebras Note that this isomorphism does not preserve the order of vertices of the quivers that we used above.