Tilting on non-commutative rational projective curves

arXiv:0905.1231v2 [math.AG] 9 Jul 2009 TILTING ON NON-COMMUTATIVE RATIONAL PROJECTIVE CURVES IGOR BURBAN AND YURIY DROZD Dedicated to Helmut Lenzing on the occasion of his 70th birthday Abstract. In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy 2 = x3 + x2 z. Contents 1. Introduction 2. Auslander sheaf of orders 3. Auslander–Reiten translation and τ -periodic complexes 4. Serre quotients and perpendicular categories 5. Tilting on rational projective curves with nodal and cuspidal singularities 5.1. Construction of a tilting complex 5.2. Description of the tilted algebra 5.3. Dimension of the derived category of a rational projective curve 6. Coherent sheaves on Kodaira cycles and gentle algebras 7. Tilting exercises on a Weierstraß nodal cubic curve 8. Some generalizations and concluding remarks 8.1. Tilting on other degenerations of elliptic curves 8.2. Tilting on chains of projective lines 8.3. Non-commutative curves with nodal singularities 8.4. Configuration schemes of Lunts References 2000 Mathematics Subject Classification. Primary 14F05, Secondary 14H60, 16G99. 2 3 8 12 18 18 26 29 30 32 36 36 36 37 38 39 2 IGOR BURBAN AND YURIY DROZD 1. Introduction By a result of Beilinson [6] it is known that the derived category of representations of the ( ~ = • Kronecker quiver Q 6 • over a field k is equivalent to the derived category of coherent sheaves on the projective line P1k . This article grew up from an attempt to find a similar “geometric” interpretation of other finite-dimensional algebras. Let X be a singular reduced projective curve of arbitrary geometric genus, having only nodes or cusps as singularities. We introduce a certain sheaf of OX –orders (called the Auslander sheaf) A = AX and study the category Coh(A) of coherent left modules on the ringed space (X, AX ). It turns out that the global dimension of Coh(A) is equal to two and the original category of coherent sheaves Coh(X) can be embedded into Coh(A) in two natural but different ways. Namely, we construct a pair of fully faithful functors F, I : Coh(X) → Coh(A), where F is right exact and I is left exact such that their images in Coh(A) are closed under extensions. The functor F has a right adjoint functor G : Coh(A) → Coh(X), which is exact. We show that G yields an equivalence between the category VB(A) of the locally projective coherent A–modules and the category TF(X) of the torsion free coherent sheaves on X. Moreover, we prove that Coh(X) is equivalent to the localization of Coh(A) with respect to a certain bilocalizing subcategory of torsion A–modules and G can be identified with the canonical localization functor.

It turns out that the derived functor LF : D − Coh(X) → D − Coh(A) is also fully faithful. Moreover, if X is
a rational curve of arbitrary arithmetic genus then the derived category D b Coh(A) has a tilting complex. The corresponding tilted algebra ΓX = EndDb (A) (H• ) is precisely the algebra described in [15, Appendix A.4]. In particular, ΓX
has
global dimension equal to two and the derived categories D b Coh(A) and D b mod − ΓX are equivalent. As a corollary we show that the dimension of the derived category
D b Coh(X) is at most two, confirming a conjecture posed by Rouquier [38]. Combining this derived equivalence with
the embedding LF, we obtain an exact fully faithful functor b Perf(X) → D mod − ΓX . If X is either a chain or a cycle of projective lines, then the corresponding tilted algebra ΓX belongs to the class of the so-called gentle algebras. They are known to be of derivedtame representation type. This gives an alternative proof of the tameness of Perf(X), obtained for the first time in [13]. If X is a Kodaira cycle of projective lines, then the
image of the triangulated category Perf(X) inside of D b mod − ΓX can be characterized in a very simple way. Let

Lν : D b mod − ΓX −→ D b mod − ΓX be the derived Nakayama functor then the image of Perf(X) is equivalent to the full subcategory of the complexes P • satisfying the property Lν(P • ) ∼ = P • [1]. 2 If E ⊆ P is a singular Weierstraß cubic curve over an algebraically closed field k given by the equation zy 2 = x3 + x2 z, then the corresponding algebra ΓE is the path algebra of the following quiver with relations: b a • c 4* • 4* • d ba = 0, dc = 0. TILTING ON NON-COMMUTATIVE CURVES 3 We explicitly describe the objects of D b mod − ΓE ) corresponding to the line bundles of degree zero and to the structure sheaves of the regular points under the embedding Perf(E) → D b mod − ΓE ). We hope that the results of this article will find applications to the homological mirror symmetry for degenerations of elliptic curves [25] and to the theory of integrable systems, in particular to the study of solutions of Yang-Baxter equations [32, 14]. Acknowledgement. Parts of this work were done during the authors stay at the Mathematical Research Institute in Oberwolfach within the “Research in Pairs” programme from March 8 – March 21, 2009. The research of the first-named author was supported by the DFG project Bu-1866/1-2, and the research of the second-named author was supported by the INTAS grant 06-1000017-9093. The first-named author would like to thank Bernhard Keller, Daniel Murfet and Catharina Stroppel for helpful discussions of the results of this article. 2. Auslander sheaf of orders In this section we introduce an interesting class of non-commutative ringed spaces supported on some algebraic curves over an algebraically closed field k. Let X be a reduced algebraic curve over k having only nodes or cusps as singularities. This means that for any bx is isomorphic point x from the singular locus Sing(X) the completion of the local ring O 2 3 to kJu, vK/uv (node) or to kJu, vK/(u − v ) (cusp). Let O = OX be the structure sheaf of X and K = KX the sheaf of rational functions on X. Consider the ideal sheaf I of the singular locus Sing(X) (with respect to the reduced scheme structure) and the sheaf F = I ⊕ O. Definition 2.1. The Auslander sheaf on a curve X with only nodes or cusps as singularities is the sheaf of O–algebras A = AX := EndX (F). π e −→ X be the normalization of X and Proposition 2.2. In the notations as above, let X e O := π∗ (OXe ). Then we have: • The ideal sheaf I of the singular locus of X is equal to the conductor ideal sheaf e e AnnX (O/O). In particular, I is an ideal sheaf in O. ∼ ∼ e e • Moreover, we have: I = HomX (O, O), O = EndX (I) and     e J O K K ∼ A = EndX (F) = ⊆ Mat2 (K) = . e O K K O In other words, A is a sheaf of O–orders. e Proof. Let J = AnnX (O/O) be the conductor ideal. Then the sheaf O/J is supported at the set of points of X where X is not normal. Since X is a reduced curve, it is exactly the singular locus
of X, hence J ⊆ I. Moreover, x ∈ X is either a nodal or cuspidal point, then O/J x ∼ = kx . This implies that J = I. By the general properties of the conductor ideal it follows that the morphism of O– e O) mapping a local section r ∈ H 0 (U, I) to the morphism ϕr ∈ modules I → HomX (O, e O), given by the rule ϕr (b) = rb, is an isomorphism. In a similar way, the HomU (O, 4 IGOR BURBAN AND YURIY DROZD e −→ EndX (I), given on the level of local sections by the rule b 7→ ψb , where morphism O ψb (a) = ba, is an isomorphism too.  In what follows we shall use the following standard results on the category of coherent sheaves over a sheaf of orders on a quasi-projective algebraic variety. Theorem 2.3. Let X be a connected quasi-projective algebraic variety over an algebraically closed field k of Krull dimension n and A be a sheaf of orders on X. Let QCoh(A) be the category of quasi-coherent left A-modules and Coh(A) be the category of coherent left A-modules. Then we have: (1) The category Qcoh(A) is a locally Noetherian Grothendieck category and Coh(A) is its subcategory of Noetherian objects. (2) For any quasi-coherent A–modules H′ and H′′ we have the following local-to-global
′ ′′ spectral sequence: H p X, ExtqA (H′ , H′′ ) =⇒ Extp+q A (H , H ). (3) If the variety X is projective, then Coh(A) is Ext–finite over k. (4) Assume the ringed space (X, A) has “isolated singularities”, i.e. Ax ∼ = Matn×n (Ox ) for all but finitely many x ∈ X. Then we have:



gl. dim QCoh(A) = gl. dim Coh(A) = sup gl. dim Ax − mod = sup gl. dim Abx − mod , x∈X x∈X where Abx := lim Ax /mtx Ax is the radical completion of the ring Ax and mx is the ←− maximal ideal of Ox .

(5) The canonical functor D ∗ Coh(A) → D*coh QCoh(A) is an equivalence of trian

gulated categories, where D*coh QCoh(A) is the full subcategory of D ∗ QCoh(A) consisting of complexes with coherent cohomologies and ∗ ∈ {∅, b, +, −}. Comment on the proof. (1) We refer to [16, Chapitre IV] for the definition and general properties of the locally Noetherian categories. Let H′ and H′′ be quasi-coherent A–modules. Then HomA (H′ , H′′ ) is a quasi-coherent O–module. Moreover, we have isomorphisms of bifunctors

HomA (H′ , H′′ ) ∼ = H 0 X, HomA (H′ , H′′ ) ∼ = HomX O, HomA (H′ , H′′ ) . The spectral sequence (2) is just the spectral sequence of the composition of two left exact functors. Note that ExtqA (H′ , H′′ ) is a coherent O–module for all q ≥ 0 provided both sheaves H′ and
H′′ are coherent A–modules. If X is projective, this implies that H p X, ExtqA (H′ , H′′ ) is finite dimensional over k. Hence, Coh(A) is Ext–finite in this case, what proves (3). Let A be a left Noetherian ring. By a result of Auslander [3, Theorem 1]  gl. dim(A) = sup pr.dim(A/I) | I ⊆ A is a left ideal . In particular, the global dimension of the category of all A–modules is equal to the global dimension of the category of Noetherian
A–modules. Quite analogous observations show
that gl. dim QCoh(A) = gl. dim Coh(A) . If (X, A) has isolated singularities, then for any coherent A–modules H′ and H′′ we have:
 kr. dim Supp(ExtiA (H′ , H′′ ) ≤ max 0, n − i . TILTING ON NON-COMMUTATIVE CURVES 5
n+j n+j This implies that ExtA (H′ , H′′ ) ∼ = H 0 ExtA (H′ , H′′ ) . Let A be an order over a Noetherian local ring O and r be the radical of A. Then we have:  gl. dim(A) = pr.dim(A/r) = sup m ≥ 0 | Extm A (A/r, A/r) 6= 0 , b For (5), we just as in [3, § 3] or [41, Chapter IV.C]. In particular, gl. dim(A) = gl. dim(A). refer to [8, Theorem VI.2.10 and Proposition VI.2.11], [23, 1.7.11] or [21]. The following well-known lemma plays a key role in our theory of non-commutative ringed spaces. Lemma 2.4. Let A be a commutative Noetherian ring, T = A ⊕ M be a finitely generated A-module and Γ = EndA (T ). Then the A-module T is a projective left Γ-module and the
canonical ring homomorphism A → EndΓ (T ), a 7→ t 7→ at is an isomorphism. In other words, T has the double centralizer property. Proof. First note that Γ has a matrix presentation   A M∨ Γ= M EndA (M ) ∼ Γ · eA as a left projective Γ-module. and 1Γ = 1A + 1M = eA + eM . In particular, T = Hence, HomΓ (ΓeA , ΓeA ) ∼ e Γe = A, and the canonical morphism A → EndΓ (T ) is an = A A isomorphism.  Next, we shall frequently use the following standard result from category theory. Lemma 2.5. Let F : C −→ D be a functor admitting a right adjoint functor G : D −→ C. Let φ : 1C → G ◦ F be the natural transformation given by the adjunction. Then for any objects X, Y ∈ Ob(C) the following diagram is commutative:
F / HomD F(X), F(Y ) HomC (X, Y ) QQQ QQQ QQQ Q (φY )∗ QQQ( kk5 kkk k k k kkk ∼ = kkk
HomC X, GF(Y ) In particular, F is fully faithful if and only if the natural transformation φ is an isomorphism. The main result of this section is the following theorem. Theorem 2.6. Let X be a reduced algebraic curve over k having only nodal or cuspidal singularities, F = I ⊕ O and A = EndX (F). Then the following properties hold. (1) The functor G = HomA (F, − ) : Coh(A) → Coh(X) is exact and has the left adjoint functor F = F ⊗O
− and the right adjoint functor H = HomO (F ∨ , − ). (2) We have: gl. dim Coh(A) = 2. (3) The functors G and H yield mutually inverse equivalences between the category VB(A) of locally projective coherent left A-modules and the category of coherent
torsion free sheaves TF(X). In particular, the derived functor RG : D b Coh(A) →
D b Coh(X) is essentially surjective. 6 IGOR BURBAN AND YURIY DROZD (4) The canonical transformation of functors φ : 1Coh(X) → G ◦ F is an isomorphism. In particular, the functor F is fully faithful. (5) Moreover, the canonical morphism ψ : 1D− (Coh(X)) → RG ◦ LF is an isomor

phism and the derived functors LF : D − Coh(X) −→ D − Coh(A) and LF :
Perf(X) −→ D b Coh(A) are fully faithful. Proof. (1) First note that F is endowed with a natural left module structure over the sheaf of algebras EndX (F). Moreover, consider the global sections     1 0 0 0 (1) e1 = and e2 = 0 0 0 1 of the sheaf A. Then we have: F ∼ = A · e2 , hence F is locally projective, viewed as a left A–module. This implies that the functor G is exact. The fact that the functor F is left adjoint to G is obvious. To see that G possesses a e ⊕ O and the sheaf of algebras right adjoint functor, consider the coherent sheaf F ∨ ∼ =O ∨ B = EndX (F ). Since F is a torsion free coherent O–module, it is locally Cohen-Macaulay. Moreover, X is a Gorenstein curve, hence the contravariant functor TF(X) → TF(X), mapping a torsion free sheaf H to H∨ , is an anti-equivalence. This shows that   e O e O op B=A = I O and B op = A. In particular, the category of the right B-modules is isomorphic to the category of the left A-modules. Let Coh(B op ) be the category of coherent right B-modules, then we have a functor HomX (F ∨ , − ) : Coh(X) → Coh(B op ) having a left adjoint functor F ∨ ⊗B − : Coh(B op ) → Coh(A). The sheaf F ∨ is locally projective as a right B-module. Moreover, one easily sees (for instance, as in Proposition 5.3) that F ∨ ≃ HomA (F, A), so we have the following commutative diagram of categories of functors: = Coh(B op ) LLL LLL LL ∨ F ⊗B − LL& / Coh(A) s sss s s s ysssHomA (F , − ) Coh(X) Hence, the functor H is right adjoint to G. (2) By part (4) of Theorem 2.3, the global dimension of Coh(A) can be computed locally. cx ∼ It is clear that Ax ∼ = Mat2 (Ox ) for a smooth point x ∈ X. Note that O = kJu, vK/uv if the 2 3 singularity is a node and kJu, vK/(u − v ) if it is a cusp. By a result of Bass [5, Corollary 7.3], the indecomposable maximal Cohen-Macaulay modules over the ring O = kJu, vK/xy are kJuK, kJvK and O itself; whereas for O = kJu, vK/(u2 − v 3 ) ∼ = kJt2 , t3 K they are O and kJtK.
Hence, in both cases we have: CM(O) = add m ⊕ O . By a result of Auslander and b = EndO (m ⊕ O) has global dimension two, which proves Roggenkamp [4], the algebra A the claim. See also Remark 2.7. (3) Let x ∈ X be an arbitrary point of the curve X, O = Ox and F = Fx and B = EndO (F ). Note that F ∼ = F ∨ . Then we have a pair of adjoint functors H̄ = HomO (F, − ) : TILTING ON NON-COMMUTATIVE CURVES 7 mod − O → mod − B and Ḡ = F ⊗B − : mod − B → mod − O. By a result of Bass [5, Corollary 7.3], any maximal Cohen-Macaulay O-module is a direct summand of F n for some n ≥ 0, so the functors H̄ and Ḡ induce mutually inverse equivalences between the category CM(O) of the maximal Cohen-Macaulay modules and the category pro(B) of projective right B-modules. In particular, the canonical transformations of functors given by the adjunction ζ̄ : 1pro(B) → H̄ ◦ Ḡ and ξ̄ : Ḡ ◦ H̄ → 1CM(O) are isomorphisms. Since the functors G and H form an adjoint pair, we have natural transformations of functors ζ : 1VB(B) → H ◦ G and ξ : G ◦ H → 1TF(X) . For any torsion free sheaf H on the curve X we have a morphism of O-modules ξH : G ◦ H(H) → H. Moreover, for any point x ∈ X we have: (ξH )x = ξ̄Hx , hence ξH is an isomorphism for any H. This implies that ξ is an isomorphism of functors. In a similar way, ζ is an isomorphism of functors, too. Hence, the categories TF(X) and VB(A) are equivalent.
In order to show that G is essentially surjective, first note that any object in D b Coh(A) has a finite resolution by a complex whose terms are locally projective A-modules. Since the locally Cohen-Macauly O–modules are precisely the torsion free O–modules, any object
in D b Coh(X) is quasi-isomorphic to a bounded complex whose terms are torsion free coherent O–modules. Since G establishes an equivalence between VB(A) and TF(X) and

is exact, any object in D b Coh(X) has a pre-image in D b Coh(A) . (4) Let φ : 1Coh(X) → G ◦ F be the natural transformation of functors given by the adjunction. Since G is exact and F right exact, the composition G ◦ F is right exact, too. By Lemma 2.4 we know that the canonical morphism of sheaves of O–algebras φO : O → EndA (F) is an isomorphism. It implies that for any locally free coherent O–module E the canonical morphism φE : E → GF(E) is an isomorphism. Let N be a coherent sheaf on X. Since X is quasi-projective, we have a presentation f E1 → E0 → N → 0, where E0 and E1 are locally free. This gives a commutative diagram with exact rows E1 f / E0 /N φE0 φE1  G ◦ F(E1 ) G◦F(f )  / G ◦ F(E0 ) /0 φN  / G ◦ F(N ) / 0, where φE0 and φE1 are isomorphisms. Hence, φN is an isomorphism for any coherent sheaf N on the curve X and φ is an isomorphism of functors. The fact that F is fully faithful follows from Lemma 2.5. (5) The derived functors LF and RF form again an adjoint pair, see for example [24, Lemma 15.6]. Since G is exact, the adjunction morphism ψ : 1D− (Coh(X) → RG ◦ LF coincides with Lφ. Since φ is an isomorphism, the natural transformation
Lφ is an isomorphism,
too. Lemma 2.5 implies that the derived functor LF : D − Coh(X) −→ D − Coh(A) and
b its restriction on the category of perfect complexes LF : Perf(X) −→ D Coh(A) are fully faithful.  Remark 2.7. Let O be either kJx, yK/xy or kJx, yK/(y 2 − x3 ), R be the normalization of O and I = m be the maximal ideal of O, which in this case is the conductor ideal. 8 IGOR BURBAN AND YURIY DROZD Consider the following O–order: A = EndO (I ⊕ O) =  R I R O  , which is the Auslander algebra of the ring O. Then A is isomorphic to the completion of the path algebra of the following quiver with relations: b− a− * 1 ◦j a+ 2 ◦ t b+ 4 ◦3 b+ a− = 0, a+ b− = 0 if the singularity is a node and to the completion of the path algebra b+ a * 1 9◦j b− 2 ◦ a2 = b− b+
if the singularity is a cusp. Since gl. dim(A) = pr.dim A/rad(A) , to compute the global dimension of A it suffices to compute the projective dimension of the simple A–modules. • If A is nodal, the projective resolutions of the simple A–modules are (a+ b+ ) – 0 → P1 ⊕ P3 −−−−→ P2 → S → 0, b+ a− a+ b− – 0 → P3 −→ P2 −−→ P1 → S1 → 0, – 0 → P1 −−→ P2 −→ P3 → S3 → 0. • If A is cuspidal, the projective resolutions of the simple A–modules are b+ – 0 → P1 −→ P → S2 → 0, ”2 “ b− a (b+ a) – 0 → P1 −−−−→ P2 ⊕ P1 −−−→ P1 → S1 → 0. We conclude this section by the following easy observation.
Proposition 2.8. Let D be the full subcategory of the derived category D b Coh(A) consisting of the complexes G • such that for any point x ∈ X the localization Gx• ∈
Ob D b (Ax − mod) has a finite projective resolution by objects from add(Fx ). Then D is triangulated, idempotent complete and F : Perf(X) −→ D is an equivalence of categories.

Proof. First note that the image of Perf(X) under functor F : D − Coh(X) → D − Coh(A) belongs to D. Consider the pair of the natural transformations 1D− (X) −→ G ◦ F and φ F ◦ G −→ 1D− (A) . By Theorem 2.6, the natural transformation φ is an isomorphism. Moreover, ψ|D is an isomorphism, too. Hence, F and G are quasi-inverse equivalences between the categories Perf(X) and D.  ψ 3. Auslander–Reiten translation and τ -periodic complexes We first recall the following important result about the existence of the Serre functor
in the derived category D b Coh(A) , where A is the Auslander sheaf of orders attached to a reduced projective curve X with at most nodal and cuspidal singularities. TILTING ON NON-COMMUTATIVE CURVES 9 Theorem 3.1. Let X be a reduced projective curve having only nodes or cusps as singularities, F = I ⊕ O and A = EndX (F). Consider the A–bimodule   I I ωA := HomX (A, ωX ) ∼ = e O ⊗ ωX , O L where ωX is the canonical sheaf of X. Then the endofunctor τ :
G • 7→ ωA ⊗ G • is the Auslander–Reiten translation in the derived category D b Coh(A) . This means that for
• • b any pair of objects G , H of D Coh(A) we have bifunctorial isomorphisms


Hom b H• , G • ∼ G • , τ (H• ) , = D Ext1 b D (A) D (A) where D = Homk ( − , k) is the duality over the base field. Proof. Since X is a Gorenstein curve and the sheaf A is Cohen-Macaulay as an O–module, we have a quasi-isomorphism of the following complexes of A–bimodules: RHomX (A, ωX ) ∼ = ωA . Hence, Theorem 3.1 is a special case of [28, Theorem A.4] and [42, Proposition 6.14].  w Corollary 3.2. Let X be a Kodaira cycle of projective lines, O − → ωX be an isomorphism given by a no-where vanishing regular differential 1-form w ∈ H 0 (ωX ). Then we have an injective morphism of A–bimodules θ = θw : ωA → A yielding a natural transformation of exact endofunctors θ : τDb (A) −→ 1Db (A)
b of the derived category D
Coh(A) . In particular, the category Dθ defined as the full subcategory of D b Coh(A) consisting of all
objects G • such that θG • is an isomorphism, is a triangulated subcategory of D b Coh(A) . f g h Proof. Let G1• − → G2• − → G3• − → G1• [1] be a distinguished triangle in Dbcoh (A). Since θ is a natural transformation of exact functors, we have a commutative diagram τ (G1• ) τ (f ) / τ (G • ) 2 f  θG • 1 τ (g) / τ (G • ) 3 g  θG •  G1• / G• 2 τ (h) / τ (G • )[1] 1 θG • [1] θG • 2 / G• 3 3 h  1 / G • [1]. 1 This implies that if G1• and G2• are objects of Dθ , then G3• belongs to Dθ , too.  The main goal of this section is to establish other descriptions of the category Dθ . To do this, we consider the local case first. Lemma 3.3. Let O be a nodal singularity, R its normalization and I = annO (R/O) the conductor ideal. Let       R I I I I A = EndO (I ⊕ O) = , ωA = HomO (A, ωO ) = , F = A · e2 = R O R O O be the Auslander algebra of O, its dualizing module and the indecomposable projective module corresponding to the simple A–module of projective dimension one. Let P • be an object of the derived category D b (A − mod), then the following conditions are equivalent: 10 IGOR BURBAN AND YURIY DROZD L ∼ P •. (1) we have an isomorphism ωA ⊗A P • = (2) the complex P • is quasi-isomorphic to a bounded complex of modules with entries from add(F ). L (3) we have an isomorphism (A/ωA ) ⊗A P • ∼ = 0. Proof. We first consider the case when O is complete. Then O = kJx, yK/xy and R = L kJxK × kJyK. By [43, Lemma 6.4.1], the functor τ = ωA ⊗A − is an auto-equivalence of D b (A − mod). Next, the category D b (A − mod) is Krull–Schmidt and there are exactly three indecomposable A–modules:       kJxK kJyK I Px = , Py = and F = . kJxK kJxK O θ From the exact sequence 0 → ωA − → A → A/ωA → 0, for any complex P • from D b (A − mod) we get a distinguished triangle L θ L • L P ωA ⊗A P • −→ P • −→ (A/ωA ) ⊗A P • −→ ωA ⊗A P • [1]. This implies that τ |Hotb (add(F )) −→ 1Hotb (add(F )) is an isomorphism of functors. On the other hand, we have:     xkJxK ykJyK τ (Px ) = Ix := and τ (Py ) = Iy := . kJxK kJyK θ Note that we have the following short exact sequences: y 0 −→ Py −→ F −→ Ix −→ 0 and x 0 −→ Px −→ F −→ Iy −→ 0. For a complex P • from D b (A − mod) we define its defect d(P • ) as follows:  
L • n • d(P ) = sup n ∈ Z H A/ωA ⊗A P 6= 0 .

In particular, d(P • ) = −∞ if and only if P • ∈ Ob Hotb add(F ) . By the

definition of the functor τ it is clear that d(P • ) ≥ d τ (P • ) and d(P • ) = d τ (P • ) if and only

if P • ∈ Ob Hotb add(F ) . Since D b (A − mod) is a Krull–Schmidt category, this shows the equivalence (1) ⇐⇒ (2). The equivalence (2) ⇐⇒ (3) easily follows from existence of minimal projective resolutions over A. Now we consider the general case, when O is not necessary complete. The implications (2) =⇒ (3) =⇒ (1) are clear in this case as well. In order to show the implication
(1) =⇒ (2), it is sufficient to show that a complex P • ∈ Ob D b (A − mod) is quasiisomorphic to a bounded complex of modules with entries from add(F ) if and only if
b − mod) is quasi-isomorphic to a bounded complex of modules with entries Pb • ∈ Ob D b (A from add(Fb ). By a result of Bass, see [5, Corollary 7.3], any indecomposable torsion free O–module is isomorphic either to O or to a direct summand of Rm for some m ≥ 1. Since the category of torsion free O–modules is equivalent to the category of projective A–modules, any indecomposable projective module is isomorphic either to F or to a direct summand TILTING ON NON-COMMUTATIVE CURVES 11 of P m for some m ≥ 1, where P = A · e1 . We assume all entries of the complex P • are projective. Then we have decompositions: P n = P1n ⊕ P2n for all n ∈ Z, where homotopic P1n ∈ add(F ) and P2n ∈ add(P ). In the set of complexes of projective modules
P • • to P , consider a representative Q with the smallest possible number n∈Z rkO (P2n ) . Let n be the biggest index for which P2n 6= 0. Then we have:   « „ αn−1 βn−1 γn−1 δn−1 ( αn βn )   −→ . . .  . Q = . . . −→ Q1n−1 ⊕ Q2n−1 −−−−−−−−−→ Qn1 ⊕ Qn2 −−−−−→ Qn+1 1 • b is Krull-Schmidt. If Q b • is quasi-isomorphic to a complex, The category of complexes over A b bn−1 → Q b n is surjective. Since whose entries belong to add(F ) then the morphism δ̂n−1 : Q 2 2 n−1 the completion functor is faithfully flat, the morphism δn−1 : Q2 → Qn2 is surjective, too. Since both modules are projective, δn−1 is the projection on a direct summand. This implies that the complex Q• is homotopic to a complex of the form
−→ . . . . . . . −→ Q1n−1 ⊕ Q̄2n−1 −→ Qn1 −→ Qn+1 1
Contradiction. Hence, Q• belongs to Hotb add(F ) , as wanted.  From now on, let X be a Kodaira cycle of projective lines. We fix a no-where vanishing regular differential form w ∈ H 0 (ωX ) identifying ωA with a sheaf of two-sided A-ideals. Hence, we have a short exact sequence of A–bimodules 0 → ωA → A → A/ωA → 0 and for any complex G • from Dbcoh (A) there is a distinguished triangle L θG • τ (G • ) −→ G • −→ (A/ωA ) ⊗ G • −→ τ (G • )[1]. The following theorem is the main result of this section. Theorem 3.4. Let X be a Kodaira cycle of projective lines, I be the ideal sheaf of the singular locus of X, F = I ⊕ O and A = EndX (F) be the Auslander sheaf of X. For an object G • of the derived category Dbcoh (A) the following conditions are equivalent: (1) we have: τ (G • ) ∼ = G•; (2) the morphism θG • is an isomorphism; L (3) we have: (A/ωA ) ⊗ G • ∼ = 0; (4) G • is an object of the category Dθ introduced in Corollary 3.2; (5) G • is an object of the category D introduced in Proposition 2.8. Proof. The equivalences (2) ⇐⇒ (3) ⇐⇒ (4) and the implication (2) =⇒ (1) are obvious.
L L Note that (A/ωA ) ⊗ G • ∼ = 0 in D b (Ax − = 0 in D b Coh(A) if and only if (A/ωA )x ⊗ Gx• ∼ mod) for all x ∈ X. Hence, the equivalence (3) ⇐⇒ (5) follows from Lemma 3.3. The implication (1) =⇒ (3) can be shown in a similar way.  Combining Proposition 2.8 and Theorem 3.4, we obtain the following corollary. Corollary 3.5. Let X be a Kodaira cycle of projective lines and A be its Auslander sheaf. Then the image of the functor F : Perf(X) → Dbcoh (A) is the category of complexes G • such L that τ (G • ) ∼ = G • , where τ = ωA ⊗ − is the Auslander-Reiten translate in Dbcoh (A). 12 IGOR BURBAN AND YURIY DROZD 4. Serre quotients and perpendicular categories Let X be a reduced curve over k having only nodal singularities and A be its Auslander sheaf. The main goal of this section is to construct two different but natural embeddings of Coh(X) into Coh(A) such that its image will be closed under extensions. In Theorem 2.6 it was shown that the functor F : Coh(X) → Coh(A) is fully faithful. The next proposition characterizes the image of the functor F. Proposition 4.1. Let M be a coherent left A–module. Then there exists a coherent O–module N such that F(N ) ∼ = M if and only if M has a locally projective presentation F ⊗O E1 −→ F ⊗O E0 −→ M −→ 0, where E0 and E1 are locally free coherent O–modules. Proof. One direction is clear: if N is a coherent O–module then it has a locally free presentation E1 → E0 → N → 0 inducing a locally projective presentation F ⊗O E1 → F ⊗O E0 → F ⊗O N → 0. Other way around, let E0 and E1 be locally free coherent O-modules such that f F ⊗O E1 −→ F ⊗O E0 −→ M → 0 is an exact sequence of coherent left A-modules. Since the functor F is fully faithful, there exists a morphism of O–modules g : E1 → E0 such that f = F(g). Put N := coker(g). Then we have a commutative diagram F ⊗ O E1 F(g) / F ⊗ O E0 ∼ = / F ⊗O N /0 /M / 0, ∼ =  F ⊗ O E1 f  / F ⊗ O E0 implying that M ∼ = F ⊗O N .  Proposition 4.2. The category Im(F) is closed under extensions in Coh(A). Proof. Let M′ and M′′ be two coherent left A modules belonging to the image of F and 0 −→ M′ −→ M −→ M′′ −→ 0 be an exact sequence in Coh(A). Then M also belongs to the image of F. Indeed, by the assumption there exists coherent OX -modules N ′ and N ′′ such that M′ ∼ = F ⊗O N ′ and M′′ ∼ = F ⊗O N ′′ . Take any locally free presentation E1′ → E0′ → N ′ → 0 of the coherent sheaf N ′ . By Serre’s vanishing theorems (see [20, Section III.5]) there exists an ample line bundle L and a natural number n ≫ 0 such that k • the evaluation morphism f ′′ = ev : HomX (L⊗−n , N ′′ ) ⊗ L⊗−n → N ′′ is an epimorphism;
• we have the vanishing: H 1 HomA (F, F ⊗O N ′ ) ⊗ L⊗n = 0. k ′′ Set E0′′ := HomX (L⊗−n , N ′′ ) ⊗ L⊗−n and observe
that the coherent left A-module F ⊗ E0 i ′′ is locally projective. Hence, ExtA F ⊗O E0 , − = 0 for i ≥ 1. Moreover, for any coherent
left A-module G we have an isomorphism of functors HomA (G, − ) ∼ = H 0 HomA (G, − ) TILTING ON NON-COMMUTATIVE CURVES 13 from the category of coherent A-modules to the category of finite dimensional vector spaces over k, where HomA (G, − ) : Coh(A) → Coh(X). This induces the short exact sequence:

0 −→ H 1 HomA (G, G ′ ) −→ Ext1A (G, G ′ ) −→ H 0 Ext1A (G, G ′ )
coming from the standard local-to-global spectral sequence H p ExtqA (G, G ′ ) =⇒ ′ ′′ Extp+q A (G, G ). Since the A–module F ⊗O E0 is locally projective, it implies that
Ext1A (F ⊗O E0′′ , F ⊗O N ′ ) ∼ = = H 1 HomA (F ⊗O E0′′ , F ⊗O N ′ ) ∼
∨ 1 ′ ′′ ∼ = 0. = H HomA (F, F ⊗O N ) ⊗O E 0 Hence, we can lift the epimorphism 1 ⊗ f ′′ : F ⊗O E0′′ → M′′ to a morphism f¯′′ : F → M, yielding a morphism f¯ = (f¯′ , f¯′′ ) : F ⊗O (E0′ ⊕ E0′′ ) → M: F ⊗O E0′ 1⊗f ′ f¯′  $ z /M / M′ 0 f¯′′ F ⊗O E0′′ 1⊗f ′′  / M′′ /0 By 5-lemma, f¯ is an epimorphism. In a similar way, we construct a presentation E1′′ → E0′′ → N ′′ → 0, inducing a presentation F ⊗O (E1′ ⊕ E1′′ ) → F ⊗O (E0′ ⊕ E0′′ ) → M → 0.  Summing up, Theorem 2.6 and Proposition 4.2 imply that the category of coherent sheaves Coh(X) is equivalent to a full subcategory of Coh(A), which is closed under taking cokernels and extensions. It turns out that if the category Coh(X) can be embedded into Coh(A) in a completely different way. Recall that for any singular point x ∈ X the algebra A = Abx is isomorphic to the radical completion of the path algebra of the following quiver with relations: b− a− (2) * 1 •j a+ 2 ◦ t b+ 4 •3 b+ a− = 0, a + b− = 0 Let T be the full subcategory of the category of torsion coherent A–modules, supported at the singular locus of X and corresponding to the simple A–modules labeled by bullets. Then T is a semi-simple abelian category. Moreover, T is a Serre subcategory of Coh(A) and for any T ′ , T ′′ ∈ Ob(T) we have: Ext1A (T ′ , T ′′ ) = 0. Remark 4.3. Although the category T is semi-simple, the second extension group Ext2A (T ′ , T ′′ ) is not necessarily zero for a pair of objects T ′ , T ′′ ∈ Ob(T). Indeed, for the two simple A–modules S1 and S3 from T we have: Ext2A (S1 , S3 ) = k = Ext2A (S3 , S1 ). Definition 4.4. Following Geigle and Lenzing [18], the perpendicular category T⊥ of the Serre subcategory T is defined as follows: 
T⊥ = G ∈ Ob Coh(A) HomA (T , G) = 0 = Ext1A (T , G) for all T ∈ Ob(T) . In particular, T⊥ is closed under taking kernels and extensions inside of Coh(A). 14 IGOR BURBAN AND YURIY DROZD Proposition 4.5. Let Tsim be the set of the simple objects of T. The perpendicular category T⊥ has the following description: 
T⊥ = G ∈ Ob Coh(A) HomA (T , G) = 0 = Ext1A (T , G) for all T ∈ Tsim . Next1, the category VB(A) of locally projective left A–modules is a full subcategory of T⊥ . Proof. First note that for any objects G ∈ Ob(Coh(A)), T ∈ Ob(T) and i ∈ Z, the sheaves ExtiA (T , G) are torsion
O–modules. In particular, we have the isomorphisms: 0 Exti (T , G) . Hence, for any i ∈ Z, the vanishing of Exti (T , G) is ExtiA (T , G) ∼ H = A A equivalent to the vanishing of ExtiA (T , G). By induction on the length we get: G ∈ Ob(T⊥ ) ⇐⇒ HomA (T , G) = 0 = Ext1A (T , G) for all T ∈ Tsim . This implies the first part of the statement. In order to show that VB(A) is a full subcategory of T⊥ , it is sufficient to prove that for any indecomposable projective A–module P and any simple module S ∈ Tsim we have: HomA (S, P ) = 0 = Ext1A (S, P ). This vanishing easily follows from the explicit form of the projective resolution of S given in Remark 2.7.  For a coherent A–module H, consider its maximal subobject tD (H) and the canonical short • be an injective resolution of exact sequence 0 → tD (H) → H → H̄ → 0. Next, let H̄ → IH̄
H̄. Choose a distinguished triangle in D b Coh(A) determined by the canonical evaluation morphism of complexes of sheaves evH̄ : M
ev u v • • e −→ (3) H̄ −→ H HomDb (A) T , IH̄ [1] ⊗k T −−H̄ → IH̄ [1]. T ∈Tsim Obviously, this triangle corresponds to a representative of the class of the universal extension sequence M u v e −→ (4) 0 −→ H̄ −→ H Ext1A (T , H̄) ⊗k T −→ 0. T ∈Tsim Inspired by the work of Geigle and Lenzing [18, Section 2], we get the following result. e can be extended to a functor J : Coh(A) → Theorem 4.6. The correspondence H 7→ H ⊥ T . This functor J is left adjoint to the inclusion I : T⊥ → Coh(A). Moreover, the I P functor T⊥ → Coh(A)/ T defined as the composition T⊥ −→ Coh(A) −→ Coh(A)/ T is an equivalence of categories. Here Coh(A)/ T is the Serre quotient category and P is the corresponding projection functor. In particular, the perpendicular category T⊥ is abelian and the functor J is right exact. Proof. Denote ET (H̄) := ⊕T ∈Tsim Ext1A (T , H̄) ⊗k T . We check first that for any coherent e given by the universal extension sequence A–module H, the corresponding A–module H ⊥ e be a non-zero (4) belongs to T . Let S be an arbitrary element of Tsim and f : S → H morphism. Since HomA (S, H̄) = 0, the morphism g := v∗ (f ) = vf is non-zero too. Since 1The first-named author would like to thank Catharina Stroppel for drawing his attention to this fact. TILTING ON NON-COMMUTATIVE CURVES 15 the category T is semi-simple, the morphism g : T → ET (H̄) can be identified with the inclusion morphism of a direct summand. By of the evaluation morphism,

the definition
evH̄ ◦ g 6= 0. But on the other hand, evH̄ ◦ g = evH̄ ◦ v ◦ f = 0. Contradiction.
e = Ext1 S, ET (H̄) = 0, the short exact sequence (4) Since HomA (S, H̄) = HomA (S, H) A induces an exact sequence u∗ e −→ 0. 0 −→ HomA (S, ET (H̄)) −→ Ext1A (S, H̄) −→ Ext1A (S, H)

Since the category T is semi-simple, we have: dimk HomA (S, ET (H̄)) = dimk Ext1A (S, H̄) . e = 0. Hence, H e belongs to T⊥ From the dimension reasons we conclude that Ext1A (S, H) ⊥ as stated. The functor T → Coh(A)/ T is fully faithful by [16, Chapitre III]. Moreover, by [18, Proposition 2.2] this functor is an equivalence of categories. In particular, the category T⊥ is abelian. e can be extended to a functor Coh(A) → T⊥ . Now we check that the assignment H 7→ H ′ ′′ Let f : H → H be a morphism in Coh(A). It is easy to see that f maps tD (H′ ) to w tD (H′′ ). For any object H we fix a representative of the cokernel H − → H/ tD (H). Then we obtain the induced map f¯ : H̄′ → H̄′′ such that the following diagram is commutative: / tD (H′ ) 0 w′ / H′ / H̄′ f¯ f   / tD (H′′ ) 0 /0 w ′′ / H′′  / H̄′′ / 0. Moreover, the assignment f 7→ f¯ is functorial: gf = ḡf¯ and 1̄H = 1H̄ . Next, functoriality of the evaluation morphism and axioms of triangulated categories imply there exists a e′ → H e ′′ making the following diagram commutative: morphism f˜ : H H̄′ u′ f¯ / e′ H v′ / ET (H̄′ ) v′′   u′′  / e ′′ H / ET (H̄′′ ) / H̄′ [1] f¯[1] f¯∗ f˜ H̄′′ evH̄′ evH̄′  / H̄′′ . e ′′ ) = 0 for all S ∈ Tsim , such a morphism f˜ is unique. So, we obtain a Since HomA (S, H functor J : Coh(A) → T⊥ . It is easy to see that we have an isomorphism of functors ξ : J ◦ I → 1T⊥ . Moreover, there is a natural transformation ζ : 1Coh(A) → I ◦ J, where for a coherent sheaf H the w u e From the short exact morphism ζH is defined to be the composition H − → H̄ − → H. e sequences defining the sheaves H̄ and H we conclude that for any object H′′ from the perpendicular category T⊥
the morphisms w∗′ and u′∗ are the isomorphisms. Hence, the morphism HomA J(H′ ), H′′ → HomA (H′ , H′′ ) given by the composition ′ ′ w∗ u∗ e ′ , H′′ ) −→ HomA (H HomA (H̄′ , H′′ ) −→ HomA (H′ , H′′ ) is an isomorphism. This shows that J is left adjoint to the embedding I : T⊥ → Coh(A). Since the category T⊥ is abelian, the functor J is right exact.  16 IGOR BURBAN AND YURIY DROZD Remark 4.7. By [18, Proposition 2.2] we also know that the Serre subcategory T is localizing. This means that the canonical functor P : Coh(A) → Coh(A)/ T has a right adjoint functor e J : Coh(A)/ T → Coh(A). Theorem 4.8 implies P has also a left adjoint functor, hence T is even a bilocalizing subcategory. Moreover, the image of e J belongs to ⊥ e the perpendicular category T and there is an isomorphism of functors J ◦ P ∼ = J. Note that the exact functor G = HomA (F, − ) : Coh(A) → Coh(X) vanishes on the category T. Using the universal property of the Serre quotient category, we obtain an exact functor E : Coh(A)/ T −→ Coh(X) such that E ◦ P = G. The main result of this section is the following theorem. Theorem 4.8. The functor E : Coh(A)/ T → Coh(X) is an equivalence of abelian categories. Moreover, the functors G ◦ I : T⊥ −→ Coh(X) and J ◦ F : Coh(X) −→ T⊥ are mutually quasi-inverse equivalences of abelian categories. Proof. By Remark 4.7, the second statement implies the first one. Next, recall that we have two adjoint pairs of functors: I - T⊥ l J G Coh(A) m - Coh(X). F Let F ◦ G −→ 1Coh(A) , 1Coh(X) −→ G ◦ F, 1Coh(A) −→ I ◦ J and J ◦ I −→ 1T⊥ be the morphisms given by the adjunction. Then the functors J ◦ F and G ◦ I also form an adjoint pair, whose adjunction morphisms are: φ η G(ζ)F µ : 1Coh(X) −→ GF −−−→ GI ◦ JF φ ζ ξ J(η)I and ν : JF ◦ GI −−−→ JI −→ 1T⊥ . ξ In order to show that JF and GI are mutually inverse equivalences of categories, it is sufficient to
show the morphisms µG and νH are isomorphisms for arbitrary objects G ∈ Ob Coh(X) and H ∈ Ob(T⊥ ). From the construction of the functors F, G, I and J it
is
clear that the morphism of Ox –modules µG x and the morphism of Ax –modules νH x are isomorphisms provided x is a smooth point of the curve X. Let x ∈ X be a singular point. Recall that the completion functor is faithfully flat f (see for example [2, Theorem 10.17]), hence a morphism of Ox modules M − → N is an ˆ f b c− bx –modules. Let isomorphism if and only if the morphism M →N is an isomorphism of O bx = kJu, vK/uv, A be the Auslander algebra of O, F = Fbx and T̄ be full subcategory O=O of A−mod whose objects correspond to the torsion sheaves from the category T supported at x. Let F̄ = F ⊗O − : O − mod → A − mod, Ḡ = HomA (F, ) : A − mod → O − mod, Ī : T̄ → A − mod be the inclusion functors and J̄ be the right adjoint to Ix . Then we have TILTING ON NON-COMMUTATIVE CURVES 17 a commutative diagram of categories and functors I (5) G - T⊥ l J  T̄ Ī ⊥ l - Coh(A) m Coh(X). F - Ḡ  A − mod m J̄ -  O − mod. F̄ The vertical arrows correspond to the composition of the localization functor with the functor of the radical completion. Thus, we have to show the natural transformations of functors µ̄ : 1O−mod −→ ḠĪ ◦ J̄F̄ and ν̄ : J̄F ◦ ḠĪ −→ 1T̄⊥ . are isomorphisms. By Lemma 2.4, the canonical map
O = HomO (O, O) −→ HomA F̄(O), F̄(O) = HomA (F, F ) ⊥ is an isomorphism of algebras. By Proposition 4.5, the module F belongs to T̄ . Our ⊥ next goal is to show F is a projective generator of T̄ . Indeed, by [18, Proposition 2.2] ⊥ we know the category T̄ is equivalent to the Serre quotient category A − mod/T̄. Let P̄ : A − mod −→ A − mod/T̄ be the canonical functor. Then P̄(F ) is a generator
of A − mod/T̄, i.e. any object in A − mod/T̄ is a quotient of an object from add P̄(F ) . To prove this, it is sufficient to show that for
any projective A–module Q the object P(Q) is the quotient of an object from add P̄(F ) . In the notations of Remark 2.7, we have the following short exact sequence in A − mod: b+ a− 0 −→ P3 −→ P2 −−→ P1 −→ S1 −→ 0, yielding the short exact sequence 0 → P(P3 ) → P(P2 ) → P(P1 ) → 0 in the quotient category A − mod/ T. In the same way, we have an exact sequence 0 → P(P1 ) → P(P2 ) → P(P3 ) → 0. f Finally, we check that P̄(F ) is projective in A − mod/T̄. Indeed, assume P̄(X) → P̄(F ) is an epimorphism in A − mod/T̄. By the definition of the Serre quotient category [16], such a morphism is represented by a diagram in A − mod Y _ g /Q OO p i  X f /F where i is a monomorphism with cokernel belonging to T̄, p is an epimorphism whose kernel belongs to T̄ and g is a morphism in A − mod. Since F has no subobjects from T̄, the morphism p is an isomorphism. A morphism P̄(g) : P̄(Y ) → P̄(F ) is an epimorphism in A − mod/T̄ if and only if the cokernel of g belongs to T̄. But F has no proper quotients belonging to T̄. Hence, g is an epimorphism in A − mod. Since F is projective, the morphism g splits, i.e there exists j : F → Y such that gj = 1F . But then f ◦ P(j) = 1P̄(F ) , hence P̄(F ) is projective, as wanted. 18 IGOR BURBAN AND YURIY DROZD ⊥ This implies that F = J̄F̄(O) is a projective generator in T̄ , hence the functor J̄F̄ : O − ⊥ mod → T̄ is a exact. Moreover, the canonical morphism HomO (O, O) −→ HomA (F, F ) is an isomorphism. Hence, J̄F̄ is an equivalence of categories and its adjoint functor ḠĪ is an equivalence, too.  Remark 4.9. Although T⊥ and Coh(X) are equivalent abelian categories
and the functors I and F are fully faithful, the full subcategories I(T⊥ ) and F Coh(X) of the category Coh(A) are different. To show this, it is sufficient to consider the local situation. Let O = kJu, vK/uv and A be the corresponding Auslander algebra. Consider the O–module Ou = v kJuK. It has a presentation O − → O → Ou → 0. The functor F is right exact, moreover, it induced an equivalence between the category add(O) and the category add(P2 ) = add(F ). b+ b− This implies that Xu = F(Ou ) is given by the presentation P2 −−−→ P2 → Xu → 0. It is ⊥ then easy to see that HomA (S3 , Xu ) = k. Hence, Xu does not belong to T̄ . 5. Tilting on rational projective curves with nodal and cuspidal singularities Let X be a reduced rational projective curve with only nodes or cusps as singularities and A be its Auslander sheaf of orders on X. The main goal of this section is to show that the derived category Dbcoh (A) has a tilting complex and is equivalent to the derived category of finite dimensional right modules over a certain finite dimensional algebra ΓX . 5.1. Construction of a tilting complex. Let X = n [ Xi , where all components Xi are i=1 π e −→ X be the normalization map. Then X e= irreducible and X e= the normalization of Xi and we have: O n M i=1 n [ i=1 fi , where X fi ∼ X = P1 is fi , where O fi = π∗ (O f ), 1 ≤ i ≤ n. O Xi Lemma 5.1. In the notations as above, consider the locally projective A–module P = A · e1 , where e1 ∈ H 0 (A) is the idempotent given by the equation (1). Then for any pair of line bundles L1 and L2 on the curve X of the same multidegree we have: P ⊗O L1 ∼ = P ⊗O L2 . e is a union of projective lines, we have: π ∗ L1 ∼ Proof. Since X = π ∗ L2 . The projection e ⊗X L1 ∼ e ⊗X L2 ∼ formula implies that π∗ π ∗ L1 ∼ = O = O = π∗ π ∗ L2 , hence P ⊗O L1 ∼ = P ⊗O L2 .  In what follows, we shall    e O P= = e O use the notation     e2 e1 O O e2 ⊕ · · · ⊕ e1 ⊕ O O en O en O  = P1 ⊕ P2 ⊕ · · · ⊕ Pn . For a vector m = (m1 , m2 , . . . , mn ) ∈ Zn and a line bundle L ∈ Pic(X) of multi-degree m we denote P(m) = P ⊗X L ∼ = P1 (m1 ) ⊕ P2 (m2 ) ⊕ · · · ⊕ Pn (mn ). For m = (m, m, . . . , m) we shall use the notation: P(m) = P(m). TILTING ON NON-COMMUTATIVE CURVES 19 Let H be a coherent left A–module and e1 , e2 ∈ H 0 (A) be the idempotents given by (1). Then, as O–module, H it splits into the direct sum H = e1 · H ⊕ e2 · H = H1 ⊕ H2 , e where H1 is an e1 Ae1 = O–module with the induced O–module and H2 is an  structure  H1 . Obviously, a left e2 Ae2 = O–module. Using these notations, we write H = H2 A–module H is torsion free as an O–module if and only if both O–modules H1 and H2 are torsion free. Next, we shall need the following standard technique from the theory of lattices over orders. Let O be a reduced local ring and Q = Q1 × Q2 × · · · × Qn be its total ring of fractions, where Qi is a field for all 1 ≤ i ≤ n. Let A be an order over O, then we have: Q(A) := Q ⊗O A ∼ = Mats1 ×s1 (Q1 ) × Mats2 ×s2 (Q2 ) × · · · × Matsn ×sn (Qn ) := Q1 (A) × Q2 (A) × · · · × Qn (A). Recall that the ring Matsi ×si (Qi ) is Morita-equivalent to Qi for all 1 ≤ i ≤ n. For an A–module M consider the Q(A)–module Q(M ) = Q ⊗O M . We say M is torsion free if the canonical morphism of A–modules M → Q(M ) is injective. In that case, we identify M with its image in Q(M ). Lemma 5.2. In the notations as above, A–modules,  ⊕m1  Q1  Q1      Q(M ) =  ..  ⊕  .   Q1 and    Q(N ) =   Q1 Q1 .. . Q1 ⊕l1        ⊕  let M and N be two Noetherian torsion free Q2 Q2 .. . Q2 Q2 Q2 .. . Q2 ⊕m2     ⊕l2        ⊕ ··· ⊕      ⊕ ··· ⊕   Qn Qn .. . Qn Qn Qn .. . Qn ⊕mn     ⊕ln     . Then there is an isomorphism of O–modules c : S(M, N ) → HomA (M, N ), where S(M, N ) is defined as follows: o n f = (f1 , f2 , . . . , fn ) ∈ Matl1 ×m1 (Q1 ) × Matl2 ×m2 (Q2 ) × · · · × Matln ×mn (Qn ) f · M ⊆ N and c(f ) · m = f · m for m ∈ M and f ∈ S(M, N ). Proof. First note that the morphism c is well-defined and injective. To prove surjectivity, note that Q(A) is injective as A–module. By [3, Theorem 1] it is sufficient to show that α for any exact sequence 0 → I − → A and any morphism β : I −→ Q(A) there exists a morphism γ : A → Q(A) such that γβ = α. To prove it note that Q ⊗O Q ∼ = Q and Q ⊗O − is an exact functor, hence the morphism α factorizes through Q(I) and we have 20 IGOR BURBAN AND YURIY DROZD a commutative diagram I α   β Q(I)     /A  / Q(A) { Q(A) Since the category of Q–modules is semi-simple, we get the factorization we need. Hence, for any A–module N the module Q(N ) is injective over A. Let g : M → N be a morphism of A–modules. By the injectivity of Q(N ) there exists a morphism ḡ : Q(M ) → Q(N ) making the following diagram commutative: M / Q(M ) g ḡ  N  / Q(N ). But then ḡ ∈ S(M, N ) and g = c(ḡ).  Our next aim is to transfer this technique to the case of sheaves of A–modules, where A is the Auslander order attached to a projective curve with only nodal or cuspidal singularities. Let Ki be the sheaf of rational functions on the irreducible component Xi and K be the sheaf of rational functions on X. Then we have: K ∼ = K1 × K2 × · · · × Kn . Let Qi be the field of rational functions on the component Xi , then the category of coherent Ki –modules is equivalent to the category of finite dimensional vector spaces over Qi . Let A be the Auslander sheaf of X and H be a torsion free coherent A–module. Then the canonical morphism of A–modules  mn m2  m1   ′  Kn K2 K1 H ⊕ ··· ⊕ ⊕ −→ K(H) := K ⊗O H = H= Kn K2 K1 H′′ is a monomorphism. In what follows, we consider a torsion free A–module H as a submodule of K(H). Proposition 5.3. In the notations as above, let G be a torsion free A–module and   l  l l Kn n K2 2 K1 1 . ⊕ ··· ⊕ ⊕ K(G) = Kn K2 K1 Consider the sheaf S(H, G) associated with the following presheaf: n o


U 7→ f = f1 , . . . , fn ∈ Matl1 ×m1 K1 (U ) ⊕ · · · ⊕ Matln ×mn Kn (U ) f · H(U ) ⊆ G(U ) . Then the canonical morphism of O–modules c : S(H, G) → HomA (H, G) is an isomorphism. TILTING ON NON-COMMUTATIVE CURVES 21 Proof. First note the morphism c is well-defined. Moreover, for any point x ∈ X its stalk cx coincides with the morphism from Lemma 5.2 applied to the Ox –order Ax . Hence, c is an isomorphism of O–modules.  Corollary 5.4. Let G and H be a pair of coherent torsion free left A–modules and G be locally projective. Then for any n ≥ 0 we have an isomorphism of vector spaces:
ExtnA (G, H) = H n S(G, H) . Proof. Indeed, since ExtiA (G, H) = 0 for all
i ≥ 1, the local-to-global spectral sequence
i 0 0 ∼ ∼ implies that ExtA (G, H) = H HomA (G, H) = H S(G, H) .  Corollary 5.5. Let X be a curve with nodal or cuspidal singularities, A be its Auslander sheaf, G and H be two torsion free coherent A–modules such that       K1 K2 Kn ∼ K(G) = ⊕ ⊕ ··· ⊕ = K(H). K1 K2 Kn Then the O–module HomA (G, H) is isomorphic to the sheaf associated with the presheaf o n
f · H′ (U ) ⊆ G ′ (U ) . U 7→ f = f1 , . . . , fn ∈ K1 (U ) ⊕ · · · ⊕ Kn (U ) ′′ ′′ f · H (U ) ⊆ G (U ) In particular, for the locally projective A–modules P = A · e1 and F = A · e2 , where e1 , e2 ∈ H 0 (A) are the idempotents given by (1), we have: e∼ e∼ O = HomA (P, P), O ∼ = HomA (F, F), O = HomA (F, P) and I ∼ = HomA (P, F). The following proposition plays the key role in our approach to non-commutative rational projective curves. Proposition 5.6. Let X be a rational reduced projective curve with only nodal or cuspidal singularities and A be its Auslander sheaf of orders. Consider the torsion A–module S given by its locally free resolution     I I 0 −→ −→ −→ S −→ 0. I O Note that the first term of this short exact sequence is isomorphic to P(−2), the middle term is F and S = S1 ⊕ S2 ⊕ · · · ⊕ St , where the torsion module Si is supported at the singular point xi ∈ X and corresponds to the unique simple Abxi –module of projective dimension one. Then the complex H• := S[−1] ⊕ P(−1) ⊕ P =



= S1 ⊕ S2 ⊕ · · · ⊕ St [−1] ⊕ P1 (−1) ⊕ P1 ⊕ P2 (−1) ⊕ P2 ⊕ . . . Pn (−1) ⊕ Pn
is rigid in the derived category of coherent sheaves D b Coh(A) , i.e. for all i 6= 0 we have
HomDb (A) H• , H• [i] = 0. 22 IGOR BURBAN AND YURIY DROZD Proof. By Corollary 5.4 we have: 

ExtiA P(−1) ⊕ P, P(−1) ⊕ P = H i S P(−1) ⊕ P, P(−1) ⊕ P =

e O e (−1) ⊕ O⊕2 ⊕ O e (1) = 0 = H i X, Õ(−1) ⊕ Õ⊕2 ⊕ Õ(1) = H i X, e X X X for all i 6= 0. Now we check the torsion A–module S is also exceptional. Again,
using the local-toglobal spectral sequence, we have: ExtiA (S, S) = H 0 X, ExtiA (S, S) , i ≥ 0. Hence, the vanishing of ExtiA (S, S) can be checked locally. Using the projective resolution of the simple Abxi –module Sbxi given in Remark 2.7, we get the desired vanishing.
Next, S is torsion and P(n) is torsion free, hence we get: HomA S, P(n) = 0 for all
n ∈ Z. Since S has a locally projective resolution of length one, we have: ExtiA S, P(n) =
0 for i 6= 1. The local-to-global spectral sequence implies that ExtiA S, P(n) = 0 for all
i ∈ Z. Finally, it remains to note that ExtiA P(n), S = 0 for all n ∈ Z and i ≥ 0, so the
local-to-global spectral sequence implies again ExtiA P(n), S = 0.  Let D be a triangulated category admitting all set-indexed coproducts. Recall that an  object X ∈ Ob(D) is called compact if for an arbitrary family Yi i∈I of objects of D the canonical map ⊕i∈I HomD (X, Yi ) −→ HomD (X, ⊕i∈I Yi ) is an isomorphism. An object X compactly generates D if it is compact and n o X ⊥ := Y ∈ Ob(D) HomD (X, Y [n]) = 0 ∀ n ∈ Z = 0. Recall the following result of Keller [22]. Theorem 5.7. Let D be an algebraic triangulated category admitting all set-indexed co
products and X be a compact generator of D such that HomD X, X[n] = 0 for all n ∈ Z \ {0}. Let Γ = EndD (X) and Mod − Γ be the category of all right Γ–modules.
Then there exists an exact equivalence of triangulated categories
T : D −→ D Mod − Γ such that for an arbitrary object Y ∈ Ob(D) we have: H n T(Y ) = HomD (X, Y [n]), where HomD (X, Y [n]) is endowed with the natural structure of a right Γ = EndD (X)–module. Such an object X is called tilting and its endomorphism algebra Γ is the corresponding tilted algebra. In order to restrict the equivalence T on the derived category of Noetherian objects of a Grothendieck abelian category, we use the following result of Krause [26, Proposition 2.3]. Theorem 5.8. Let A be a locally Noetherian Grothendieck category of finite global dimension and N be its full subcategory of Noetherian objects. Let Dc (A) be the category of compact objects of D(A). Then the image of the canonical functor D b (N) → D(A) is equivalent to Dc (A). The following result was explained to the first-named author by Daniel Murfet. TILTING ON NON-COMMUTATIVE CURVES 23 Proposition 5.9. Let A be a locally Noetherian Grothendieck category of finite global dimension, N be its full subcategory of Noetherian objects and D = D(A) be the derived category of A. Let I : D − (N) → D be the canonical functor. Then an object X of the category D belongs to the image of I if and only if for every family of objects {Y }i∈I of objects of D such that ⊕i∈I Yi has bounded below cohomology, the canonical map M
M
Yi HomD X, Yi −→ HomD X, i∈I i∈I is an isomorphism. Proof. We follow the main steps of the proof of [26, Lemma 4.1]. First check that any object X of the category D − (N) has the stated property. Let {Yi }i∈I be a family of complexes from A with common lower bound for the non-vanishing cohomology. Then there exists n ∈ Z such that for all j < n and i ∈ I we have: H j (Yi ) = 0. Let τ≥n (X) be the truncation of X. By the assumption, the complex τ≥n (X) has bounded Noetherian cohomology. Since A has finite global dimension, the complex τ≥n (X) is compact in D and we have canonical isomorphisms M M
M

M Yi ). Yi ∼ HomD τ≥n (X), Yi ∼ HomD X, Yi ∼ = HomD (X, = HomD τ≥n (X), = i∈I i∈I i∈I i∈I Now, let X be an object of D such that the map ⊕i∈I HomD (X, Yi ) −→ HomD (X, ⊕i∈I Yi ) is an isomorphism for an arbitrary family of objects with a common lower bound for the non-vanishing cohomology. First we check there exists n ∈ Z such that for all m ≥ n
δj δj−1 we have: H m (X) = 0. Let X = · · ·
→ X j−1 −−−→ X j −→ X j+1 → . . . and assume H j (X) 6= 0. Let ker(δj ) := Z j (X), αj , Z j (X) → H j (X) be the canonical epimorphism
γj and H j (X) −→ E H j (X) be the injective envelope of H j (X). Note that the composition
j γj βj is non-zero.
Moreover, since E H (X) is injective, there exists a morphism ϕj : X j → E H j (X) such that ϕj αj = γj βj . Next, by the universal property of the kernel, there exists a morphism δ̃j−1 : X j−1 → Z j (X) such that αj δ̃j− = δj−1 . ... j−1 / Xj / X j−1 δ LLL O LLL αj LLL L%  δ̃j−1 ? δj / X j+1 / ... Z j (X)  βj φj H j  (X) _ γj 
E H j (X)
Note that ϕj δj−1 = γj βj δ̃j−1 = 0. As a result, we get a morphism X → E H j (X) [−j] inducing a non-zero map in cohomology. Hence, if X has unbounded cohomology to the 24 IGOR BURBAN AND YURIY DROZD
right, the morphism X → ⊕j∈Z+ E H j (X) [−j] can not factor through a finite set of indices in Z+ . This implies that the canonical map     M M

E H j (X) [−j] HomD X, E H j (X) [−j] −→ HomD X, j∈Z+ j∈Z+ is not an isomorphism. Contradiction. In remains to show that X has coherent cohomology. Let {Ei }i∈I be an arbitrary family of injective objects in A. Then we have: M
M
M M
Ei Ei ∼ Ei [−n] ∼ HomD X, HomA H n (X), = HomA H n (X), = A result of Rentschler [35] allows to conclude that i∈I i∈I i∈I i∈I H n (X) is Noetherian.  The following theorem is the main result of our article. Theorem 5.10. Let X be a reduced rational projective curve with nodal or cuspidal singularities, A be its Auslander sheaf of orders and H• = S[−1]⊕P(−1)⊕P be the rigid complex
from Proposition 5.6. Then H• is a tilting complex in the derived category D b Coh(A) . Proof. By Proposition 5.6, the complex H• is rigid. In order to apply Theorem 5.7, we have
to show that the right orthogonal of H• in the unbounded derived category D
Qcoh(A) is zero. Let C = D(H• ) be the smallest triangulated subcategory of D Qcoh(A)
containing H• . Our goal is to show that the right orthogonal of C inside of D Qcoh(A) is zero. First observe that the Euler sequences 0 −→ OP1 (m − 1) −→ OP1 (m)⊕2 −→ OP1 (m + 1) −→ 0 in the category of coherent sheaves on P1 induces the short exact sequences 0 −→ Pi (m − 1) −→ Pi (m)⊕2 −→ Pi (m + 1) −→ 0 in the category Coh(A) for all 1 ≤ i ≤ n and m ∈ Z. This implies that all locally projective A-modules Pi (m) belong to the triangulated category C. In particular, the locally projective A–module P(−2) belongs to the category C. The short exact sequence 0 −→ P(−2) −→ F −→ S1 ⊕ S2 ⊕ · · · ⊕ St −→ 0 implies that the locally projective A–module F belongs to C. Consider the torsion A–module T , which is the cokernel of the canonical inclusion morphism F → P. Since F and P belong to C, it follows that T is an object of C, too. t M ∼ Tpj . Let Tj be the Moreover, T is supported at the singular locus of X, hence T = j=1 finite-length module over Abpj corresponding to the torsion sheaf Tpj . We have: • if pj is a node then Tj is given by 1 kj 0 * k t 1 4k 0 TILTING ON NON-COMMUTATIVE CURVES 25 • if pj is a cusp then Tj is given by the quiver representation ( 01 ) ( 00 10 ) 6 k2 * k k ( 10 ) If the point pj is nodal then for any λ ∈ k∗ we have a short exact sequence 0 −→ Sj −→ Uj (λ) −→ Tj −→ 0, where Uj (λ) is the module ( 01 ) kj + k2 “ ” 0 λ s 4k. ( 10 ) ( 10 ) Similarly, for a cuspidal point pj the module Uj (λ) is given by the representation ( 00 01 ) ( 00 10 ) 2 6k k “ λ1 0 0 ” + k2 for some λ ∈ k. In particular, for any choice of λ1 , λ2 , . . . , λt ∈ k∗ , the torsion sheaf U t L corresponding to the module Uj (λj ) belongs to the category C. j=1 Let L ∈ Pic(X) be a line bundle of multidegree (1, 1, . . . , 1). Then for any m ≥ 1 there exists an exact sequence 0 −→ F ⊗O L −m −→ F ⊗O L −m+1 −→ t M Uj (λj ) −→ 0. j=1 In particular, for all m ≥ 0 the sheaf F ⊗O L−m belongs to C. Hence, for any m ≥ 0 the locally free A–module F ⊗O L−m belongs to C, too. Let G be an arbitrary coherent A–module. Then it is also a coherent O-module. Since L is an ample line bundle on X, by a theorem of Serre there exists m ≥ 0 such that the evalk ev uation morphism HomX (L−m , G) ⊗ L−m −→ G is surjective. In particular, there exists N ≥ 0 such that there exists an epimorphism of O–modules (L−m )N → G. Since the func1⊗ev can / / A ⊗O G / / G. tor A ⊗O − is right exact, we have epimorphisms A ⊗O (L−m )N
δn δn−1 Let G • =
· · · → G n−1 −−−→ G n −→ G n+1 → . . . be a complex from the category D Qcoh(A) . If G • ∼ 6 0 then there exists n ∈ Z such that Hn (G • ) 6= 0. Let Kn = ker(δn ). = Since any quasi-coherent A–module is a direct limit of its coherent submodules, there exists a coherent submodule Nn of Kn such that it is not a subobject of the sheaf im(δn−1 ) ⊆ Kn .

N f → G • in the derived category D Qcoh(A) Consider the morphism A ⊗ L−m [−n] − defined as the composition
N A ⊗ L−m [−n] −→ Nn [−n] −→ Kn [−n] −→ G • . 26 IGOR BURBAN AND YURIY DROZD
Since Hn (f ) 6= 0, the morphism f is non-zero in D b Qcoh(A) , too. This shows that C⊥ = 0, hence H• is a tilting complex.  Corollary 5.11. Theorem 5.10, Theorem 5.7 and Proposition 5.9 imply that there exists

an equivalence of triangulated categories T : D Qcoh(X) −→ D Mod − ΓX inducing equivalences of triangulated categories



D − Coh(A) −→ D − mod − ΓX and D b Coh(A) −→ D b mod − ΓX . In particular, we have exact fully faithful functors


D − Coh(X) −→ D − mod − ΓX and Perf(X) −→ D b mod − ΓX . 5.2. Description of the tilted algebra. Our next goal is to describe the tilted algebra EndDb (A) (H• ) as the path algebra of some quiver with relations. Recall our notation. Let e →X X be a rational projective curve with only nodes and cusps as singularities, π : X S n e= f e ∼ 1 its normalization, X i=1 Xi , where all Xi = P . Let  Sing(X) = p1 , p2 , . . . , pr , pr+1 , . . . pr+s be the singular locus of X, where p1 , . . . , pr are nodes and pr+1 , . . . , pr+s are cusps. Choose fi and for any pair (i, j) homogeneous coordinates (ui : vi ) on each irreducible component X such that 1 ≤ i ≤ n and 1 ≤ j ≤ r + s consider the set of points  fi = q (k) = (α(k) : β (k) ) 1 ≤ k ≤ mij , π −1 (pj ) ∩ X ij ij ij where mij = 0, 1 or 2. We additionally assume the coordinates are chosen in such a way (k) (k) that (αij : βij ) 6= (1 : 0) for all indices i, j, k such that pj is a cusp. Definition 5.12. The algebra ΓX attached to a rational projective curve X with nodal or cuspidal singularities, is the path algebra of the following quiver with relations: • It has 2n + r + s vertices: for each index 1 ≤ i ≤ n we have two points ai and bi and for each index 1 ≤ j ≤ r + s we have one point cj . • The arrows of ΓX are as follows. – For any index 1 ≤ i ≤ n we have two arrows ui , vi : ai → bi ; (k) – For any index 1 ≤ j ≤ r (nodal point) we have mij arrows wij : cj → ai ; – For any index r + 1 ≤ j ≤ r + s (cuspidal point) and the unique index i with ′ :c →a mij = 1 we have two arrows wij , wij j i • The relations are as follows: for any 1 ≤ i ≤ n, 1 ≤ j ≤ r (nodal point) and 1 ≤ k ≤ mij we have: (k)
(k) (k) βij ui − αij vi wij = 0 and for any 1 ≤ i ≤ n, r + 1 ≤ j ≤ r + s, mij = 1 (cuspidal point) we have: (1) (1)
(1) (1)
′ ′ . = 0, βij ui − αij vi wij = vi wij βij ui − αij vi wij Remark 5.13. The algebra ΓX is exactly the algebra defined in [15, Appendix A]. Since all paths in ΓX have the length at most two, we have: gl. dim(ΓX ) = 2. Proposition 5.14. The endomorphism algebra of the tilting complex H• from Theorem 5.10 is isomorphic to the algebra ΓX introduced in Definition 5.12. TILTING ON NON-COMMUTATIVE CURVES 27 ei yields a pair of distinguished Proof. A choice of homogeneous (ui : vi ) on X
coordinates 0 0 ∞ 0 sections zi , zi ∈ H OXei (1) , where zi (0 : 1) = 0 and zi∞ (1 : 0) = 0. They correspond
to a pair of distinguished morphisms ui , vi ∈ HomA Pi (−1), Pi , 1 ≤ i ≤ n and form a basis of this morphism space. In the course of the proof of Proposition 5.6 we have seen that the only non-trivial contributions to EndDb (Coh(A)) (H• ) come from:

HomA Pei (−1), Pi ∼ = HomP1 OP1 (−1), OP1 = k2 and 


Ext1A Sj , Pi (−1) ∼ = H 0 Ext1A Sj , Pi (−1) ∼ = k2 ∼ = Ext1A Sj , Pi .
Since the spaces Ext1A Sj , Pi (−1) can be computed locally, we carry out calculations over the complete ring Abpj , following the notations of Remark 2.7. 1-st case. Assume pj ∈ Sing(X) is nodal and its both preimages  (1) (2)  (1) (1) (2) (2) π −1 (pj ) = qij , qij = (αij : βij ), (αij : βij ) fi . Recall that in this case, the algebra A d belong to the same irreducible component X pj is given by the completion of the following quiver with relations: a− j * 1 •j b− j a+ j 2 • t b+ j − b+ j aj = 0, 4 •3 − a+ j bj = 0. (2) d ∼ (1) ⊕ P (3) , Fbp ∼ Moreover, we have: (P . Recall that the simple module i )pj = Pj j = Pj j (2) Sj = Sj has a projective resolution + + (3) (aj bj ) (1) Hence, Ext1Ab pj (2) 0 −→ Pj ⊕ Pj −−−−→ Pj −→ Sj −→ 0. (1) (3)
∼ 2 (1) (2) (1) Sj , Pj ⊕ Pj = k = wij , wij , where wij is given by the following morphism in the homotopy category: 0 / P (1) ⊕ P (3) j j / P (2) j ( 10 00 ) 0  / P (1) ⊕ P (3) j j /0 (2) and the morphism wij is defined in a similar way. This implies that any morphism from
Sj [−1] to Pi in the derived category D b Coh(A) factors through Pi (−1). Next, note that (k) (k) (k) (k) (k) the section βij ui − αij vi vanishes only at the point qij = (αij : βij ), k = 1, 2. Hence, we have the equalities: (6) (1) (1) (1) (βij ui − αij vi )wij = 0, (2) (2) (2) (βij ui − αij vi )wij = 0 28 IGOR BURBAN AND YURIY DROZD
(1) (1) (2) in the morphism space Hom(A) Sj [−1], Pi . Moreover, the morphisms (βij ui −αij vi )wij (2) (2) (1) (1) (3)
∼ 2 and (β ui −α vi )w are linearly independent. Since Ext1 Sj , P ⊕P = k , there ij ij bp A j ij are no other relations between (k) wij , j j ui and vi but those described in (6). ei . In this 2-nd case. Assume the point pj is cuspidal and π −1 (pj ) = qij = (αij : βij ) ∈ X case, Abpj is isomorphic to the completion of the path algebra b+ j * 1 9•j aj b− j + a2j = b− j bj . 2 • +
(1) (1) bj (2) Pi (−1) p ∼ (Pi )pj ∼ Pj and 0 → Pj −→ Pj → Sj → 0 is a projective = = j
∼ H 0 Ext1 (Sj , Pi ) = k2 . resolution of the rigid simple module Sj . Hence, Ext1A (Sj , Pi ) = A Since the homogeneous coordinates (ui : vi ) are chosen in such a way that qij 6= (1 : 0), the morphisms vi and βij ui − αij vi : Pi (−1) → Pi are linearly independent. Let ′ := (β u − α v ) b wij ij i ij i pj and wij := (vi )pj be the induced Apj –linear morphisms of the We have: (1) projective module Pj . Denote by the same letters the induced morphisms of complexes + (1) bj (2)
(1) from Pj −→ Pj [−1] to Pj in the homotopy category of projective Abpj –modules.
′ form a basis of Ext1 S , P (−1) = k 2 and we obtain the relations Then wij and wij j i A  ′ (βij ui − αij vi )wij = 0 ′ (βij ui − αij vi )wij = vi wij in the morphism space Ext1A (Sj , Pi ). As in the Case 1, it follows that there are no other ′ . relations between ui , vi , wij and wij 3-rd case. The case when pj is nodal and its preimages belong to different components of e is completely similar to the first case and is therefore left to the reader. X  Example 5.15. Let X be an irreducible nodal rational projective curve of arithmetic π → X its normalization. Assume that coordigenus two, p1 and p2 its singular points, P1 − nates on P1 are chosen in such a way that   π −1 (p1 ) = 0 = (0 : 1), ∞ = (1 : 0) and π −1 (p2 ) = (1 : 1), (λ : 1) with λ ∈ k \ {0, 1}. Then the algebra ΓX is the path algebra of the following quiver a2 b2 4* • tj • a1 u • v b1   • subject to the relations ua1 = 0, va2 = 0, (u − v)b1 = 0 and (u − λv)b2 = 0. It seems to be an interesting problem to study compactified moduli spaces of vector bundles on X in terms of representations of the algebra ΓX . TILTING ON NON-COMMUTATIVE CURVES 29 5.3. Dimension of the derived category of a rational projective curve. As a consequence of our approach, we obtain an upper bound of the dimension of the derived category of coherent sheaves of reduced rational projective curve with only nodal or cuspidal singularities. Let C be an idempotent complete triangulated subcategory and A, B be its two idempotent complete full subcategories closed under shifts. Following Rouquier [38], we denote by A ∗ B the full subcategory of C, whose objects are those objects X of C for which there exists a distinguished triangle A −→ X −→ B −→ A[1] with A ∈ Ob(A) and B ∈ Ob(B). For an object X ∈ Ob(C) we denote by hXi the smallest full subcategory of C, closed under taking shifts, direct sums and direct summands. Next, for any positive integer n we define subcategories hXin by the following rule: hXi1 = hXi, hXin+1 = hXi1 ∗ hXin . An object X ∈ Ob(C) is a strong generator if hXin = C for some positive integer n. Rouquier suggested the following definition of the dimension of a triangulated category C: o n dim(C) = inf n ∈ Z+ ∃ X ∈ Ob(C) : hXin+1 = C . He has also proven that the dimension of the derived category of coherent sheaves of a separated scheme X of finite type over a perfect field k is always finite, see [38,

Theorem b 7.38]. Moreover, if X is smooth of dimension n then n ≤ dim D Coh(X) ≤ 2n, see [38, Proposition 7.9 and Proposition 7.16]. By a recent result of Orlov [30], for a smooth projective curve X over a field k we have:

dim D b Coh(X) = 1. The case of the singular projective curves still remains open. However, our technique allows to deduce the following result. Theorem 5.16. Let X be a reduced rational projective curve with only nodal or cuspidal singularities. Let S = {p1 , p2 , . . . , pt } be the singular locus of X and X1 , X2 , . . . , Xn be the irreducible components of X. Let Oi = OXi be the structure sheaf of Xi , 1 ≤ i ≤ n and Oi (−1) = Oi ⊗ OX (−qi ), where qi ∈ Xi is a smooth point. Consider the coherent sheaf G= n  M i=1 t  M kpj . Oi (−1) ⊕ Oi ⊕ j=1


Then we have: hGi3 = D b Coh(X) . In particular, dim D b Coh(X) ≤ 2. Proof. Let
A be the Auslander sheaf of X. By Theorem 5.10, the derived category D b Coh(A) is equivalent to D b (mod − ΓX ). Moreover, gl. dim(ΓX ) = 2 and the equivalence T maps the tilting complex H• to the regular module ΓX . By [38, Lemma 7.1] it is
known that hΓX i3 = D b (mod − ΓX ). This implies that hH• i3 = D b Coh(A) . Consider now the exact functor G : Coh(A) → Coh(X). By Theorem 2.6, the de

rived functor G : D b Coh(A) → D b Coh(X) is essentially surjective, hence G(H• ) 3 = 30 IGOR BURBAN AND YURIY DROZD
D b Coh(X) . To conclude the proof, it remains to note that G(H• ) ∼ = n  M i=1 t  M kpj [−1]. Oi (−1) ⊕ Oi ⊕ j=1  6. Coherent sheaves on Kodaira cycles and gentle algebras In this section we discuss some corollaries from the results obtained in the previous section, in the case of Kodaira cycles of projective lines. To deal with left modules, we prefer to replace the tilted algebra ΓX by its opposite ΛX = Γop X. Proposition 6.1. Let E = En be a Kodaira cycle of n projective lines (in the case n = 1 it is an irreducible plane nodal cubic curve), A be the Auslander sheaf and Λ = ΛE be the opposite algebra of the corresponding tilted algebra. Then we have: (1) The algebra ΛE is gentle, see [1]. (2) The categories Perf(E) and Coh(E) are tame in the “pragmatic sense”2. Proof. The fact that the algebra Λ is gentle, follows from Proposition 5.14 and the definition of the gentle algebras. Moreover, the gentle algebras are [31,
37].
∼derived-tame b b → D Λ − mod , and Since we have fully faithful functors Perf(E) → D Coh(A) −

∼ b b → D Λ − mod , the categories Perf(E) and Coh(E) Coh(X) → Coh(A) → D Coh(A) −
are equivalent to full subcategories of a representation-tame category D b Λ − mod . This precisely means they are pragmatic-tame.  Example 6.2. Let E = E2 be a Kodaira cycle of two projective lines. Then the algebra ΛE is the path algebra of the following quiver ? • _@ ~~ @@@w21 ~ @@ ~ @ ~~ v ( ~ •h 6•@ @@ ~~ @@ ~~w ~ w12 @@  ~~ 22 w11 u1 • v1 u2 • v2 • subject to the relations: w11 v1 = 0, w22 v2 = 0, w12 u1 = 0 and
w21 u2 = 0. In particular, there exists a fully faithful functor Perf(E) → D b ΛE − mod . Let Λ be a finite-dimensional algebra over a field k. Then the Nakayama functor ν := D HomΛ ( − , Λ) : Λ − mod −→ Λ − mod ~ is the path algebra of a finite quiver with relations is right exact. Moreover, if Λ = k Q/ρ then ν(Pi ) = Ii , where Pi and Ii are the indecomposable projective and injective modules corresponding to the vertex i ∈ Q0 . If gl. dim(Λ) < ∞ then by a result of Happel [19], the derived functor S := Lν : D b (Λ − mod) −→ D b (Λ − mod) 2A more precise result implying the tameness in a “strict sense” was obtained in [13]. TILTING ON NON-COMMUTATIVE CURVES 31 is the Serre functor of the category D b (Λ − mod) and the functor τΛ = S[−1] is the Auslander-Reiten translate in D b (Λ − mod) (see also [34]). Corollary 6.3. Let E be a Kodaira cycle of projective lines, A be its Auslander sheaf and Λ be the opposite algebra of the corresponding tilted algebra. Consider the category Band(Λ), which is the full subcategory of D b (Λ − mod) whose objects are the complexes P • such that τΛ (P • ) ∼ = P • . Then Band(Λ) is triangulated and idempotent complete. Moreover, it is triangle equivalent to the category of perfect complexes Perf(E).
Proof. Since T : D b Coh(A) → D b (Λ − mod) is an equivalence of categories, we have an isomorphism of functors T ◦ τA ∼ = τΛ ◦ T. This implies that the category Band(Λ) is equivalent to the category Dθ introduced in Corollary 3.2. By Theorem 3.4 and Corollary 3.5 we get that Band(Λ) is equivalent to Perf(E). In particular, it is idempotent-complete.  Remark 6.4. Corollary 6.3 implies the following result on the shape of the Auslander– Reiten quiver of the gentle algebra Λ, attached to a cycle of projective lines E. Let P • be an indecomposable object of D b (Λ − mod) such that τΛn (P • ) ∼ = P • . Then n = 1 and P • is an object of Band(Λ). In other words, the Auslander-Reiten quiver of D b (Λ − mod) does not contain tubes of length bigger than one. Note that all indecomposable band complexes of D b (Λ − mod) in the sense of the work of Butler and Ringel [9] (see also [37]) are τ –periodic. However, their definition of bands and strings of the derived category of a gentle algebra is purely combinatorial. In particular, in certain cases algebra automorphisms map band complexes to string complexes. Moreover, it is not completely clear that their notion of bands and strings coincide with the corresponding notion of bands and strings used in our previous paper [12]. In the particular situation of a gentle algebra Λ which is the tilted algebra attached to a Kodaira cycle of projective lines, we say that an indecomposable object of D b (Λ − mod) is a band if and only if it belongs to Band(Λ). The indecomposable objects of D b (Λ − mod) which are not bands will be called strings. The description of indecomposable objects in D b (Λ − mod) [31, 37, 12] implies that strings do not have continuous moduli and are classified by discrete parameters. The same concerns the indecomposable objects of the
derived category D b Coh(E) which do not belong to Perf(E), see [13]. The interplay between various categories occurring in our construction can be explained by the following diagram:
T

F / D − ΛE − mod / D − Coh(AE ) D − Coh(E) O Db nn6 nnn n n n ) nnnn
Coh(E) ? Db O ? Perf(E) ∼ O O
Coh(AE ) ? T O ? / Dθ ∼ ?
/ D b ΛE − mod O ? / Band(ΓE ). Summing up, we get that the triangulated category Perf(E) is a full subcategory of two different derived categories. From one side, it is a subcategory of the derived category 32 IGOR BURBAN AND YURIY DROZD of coherent sheaves Coh(E), whose global dimension is infinity. From another side, it is a subcategory of the derived category of representations of the algebra ΓE , whose global dimension is two. The “complement” of Perf(E) in both categories is “small”: it consists of direct sums of complexes described by discrete parameters. 7. Tilting exercises on a Weierstraß nodal cubic curve Let E ⊆ P2 be a singular Weierstraß cubic curve over an algebraically closed field k given by the equation zy 2 = x3 + x2 z. By Corollary 5.11, there exists a fully faithful functor
T
F Perf(E) −→ D b Coh(A) −→ D b Λ − mod , where A = AE is the Auslander sheaf of orders and Λ = ΛE is the path algebra of the following quiver with relations: b a *4 1 c *4 2 3 ba = 0 = dc. d The algebra Λ is interesting from various perspectives. First of all, it is gentle, hence derived-tame. Next, by a work of Seidel [40, Section 3], it is related with the directed Fukaya category of a certain Lefschetz pencil.
Our next goal is to compute the complexes in D b Λ− mod corresponding to the images of certain perfect coherent sheaves on E under the functor T ◦ F. Let π : P1 → E be the normalization of E and s = (0 : 0 : 1) ∈ E be the singular point. Choose coordinates on P1 in such a way that π −1 (s) = {0, ∞}, where 0 = (0 : 1) and ∞
= (−1), O = (1 : 0). This choice yields two distinguished sections z , z ∈ Hom O 1 1 1 0 ∞ P P P
0 H OP1 (1) such that z0 (0) = 0 and z∞ (∞) = 0. Recall that   kJuK × kJvK (u, v) A := Abs ∼ = kJuK × kJvK kJu, vK/(uv) is isomorphic to the radical completion of the following quiver with relations: b− a− * α •j Let a+ β ◦ t γ 4• b+ a− = 0, a+ b− = 0. b+       kJuK kJvK (u, v) Pα = , Pγ = and Pβ = F = kJuK kJvK kJu, vK/(uv) be the indecomposable projective A–modules. We distinguish two locally projective A– modules     e I O P= and F = e e O O \s ∼ cs ∼ and for any n ∈ Z we have: P(n) = Pα ⊕ Pγ , whereas F = F . By Lemma 5.3, there are the following canonical isomorphisms: 


H 0 OP1 (1) ∼ = HomA P(−1), P . = H 0 S P(−1), P ∼ TILTING ON NON-COMMUTATIVE CURVES 33
Hence, we get two distinguished elements in HomA P(−1), P , which will be denoted by z0 and z∞ . They are characterized by the property that there exists isomorphisms \ s → Pα ⊕Pγ and t2 : P cs → Pα ⊕Pγ making the following diagrams commutative: t1 : P(−1) z0 \s P(−1) t1  Pα ⊕ Pγ “ x 0 0 id /P cs \s P(−1) / Pα ⊕ Pγ Pα ⊕ Pγ ” t1 t2   z∞ “ id 0 0 y ” /P cs t2  / Pα ⊕ Pγ . Let Sβ be the torsion A–module supported at the singular point s ∈ E and corresponding to the simple A–module Sβ . Then for any n ∈ Z canonical map 

H 0 Ext1A Sβ , P(n) −→ Ext1A Sβ , P(n) is an isomorphism. By Remark 2.7, the simple module Sβ has the following projective resolution: ( u, v ) 0 −→ Pα ⊕ Pγ −−−→ Pβ −→ Sβ −→ 0. Hence, Ext1A (Sβ , Pα ⊕ Pγ ) = k2 = hξ, ηi, where ξ and η are induced by the A–linear morphisms given by the matrices     1 0 0 0 ξ= , η= : Pα ⊕ Pγ −→ Pα ⊕ Pγ . 0 0 0 1 In particular,
z0 ξ = 0 and
z∞ η = 0. By the definition of the tilting equivalence T : D b Coh(A) → D b Λ − mod given by the tilting complex H• = Sβ [−1] ⊕ P(−1) ⊕ P, we have the following result.
Lemma 7.1. We have: T(P) ∼ = P1 , T P(−1) = P2 and T(Sβ ) ∼ = P3 [1], where Pi is the indecomposable projective Λ–module corresponding to the vertex i, 1 ≤ i ≤ 3. Our next goal is to compute the images of certain finite length objects in Coh(A). Let x = (λ : µ) ∈ P1 be an arbitrary point. Consider the torsion A–module Tx given by its locally projective resolution µz0 −λz1 0 −→ P(−1) −−−−−→ P −→ Tx −→ 0. Note that Tx is supported at the point π(λ : µ) ∈ E. In particular, if x 6∈ {0, ∞} then Tx is supported at a smooth point of E. For x ∈ {0, ∞} the sheaf Tx corresponds to the A–modules 1 T0 = k j * k t 4 0 and T∞ = 0 j * k t 1 4k. 0 0
Since ExtiA Sβ , P(n) = 0 for all n ∈ Z and i ∈ Z≥0 , we also have the vanishing ExtiA (Sβ , Tx ) = 0 for all i ≥ 0.

i Next, note that HomA P(n), = k and Ext = 0 for all i > 0 and n ∈ Z. T P(n), T x x A
i This implies that H T(Tx ) = 0 for i 6= 0. In particular, the complex T(Tx ) is isomorphic
in D b Λ − mod to the stalk complex . . . −→ 0 −→ Nx −→ 0 −→ . . . 34 IGOR BURBAN AND YURIY DROZD
where Nx = H 0 T(Tx ) . Recall that for an arbitrary representation M of the quiver Λ we have: HomΛ (Pi , M ) = M (i), where M (i) is the dimension of M at the vertex i, 1 ≤ i ≤ 3. This allows to compute
the multi-dimension of the zero cohomology of T(Tx ): dim(Nx ) = 1, 1, 0 . Moreover, the right Γ–module Nx endowed with the natural structure of an EndDb (A) H• )–module has the following projective resolution:
(µz0 −λz∞ )∗
0 −→ HomDb (A) H• , P(−1) −−−−−−−−→ HomDb (A) H• , P −→ Nx −→ 0. Interpreting it in the terms of quiver representations, we obtain the following result. Proposition 7.2. For any x = (λ : µ) ∈ P1 we have: T(Tx ) = Nx [0], where Nx is the following representation of the algebra Λ: λ *4 k µ *4 k 0. As the next step, we compute the images under T of two other exceptional simple A– modules. Proposition 7.3. Let Sα and Sγ be the simple A–modules corresponding to the vertices α and γ, and Sα and Sγ be the corresponding torsion A–modules. Then we have: 1 T(Sα ) = k *4 0 *4 k 0 0 k and T(Sγ ) = k 1 1 Proof. First note that ExtiA (Sβ , Sα ) =H 4* k 0
ExtiA (Sβ , Sα ) =  1 3+ k. 0 k if i = 1, 0 otherwise. In a similar way, for any n ∈ Z we have:  

k if i = 0, i 0 i ∼ ExtA P(n), Sα = H ExtA P(n), Sα = 0 otherwise. This implies that T(Sα ) and T(Sγ ) are indecomposable stalk complexes. Their zero cohomology are Λ–modules Mα and Mγ , whose multi-dimension is the vector (1, 1, 1). However, there are precisely two indecomposable Λ–modules with this multi-dimension. Since HomA (T∞ , Sα ) = 0 = HomA (T0 , Sγ ) and T is an equivalence of categories, we have: HomΛ (N∞ , Mα ) = 0 = HomΛ (N0 , Mγ ). This implies that Mα and Mγ are given by the quiver representations as stated above.  Finally, we shall compute the image of the Jacobian Pic0 (E) in the derived category D b (Λ − mod). Proposition 7.4. The functor T◦F identifies the Jacobian Pic0 (E) ∼ = k∗ with the following b family of complexes in the derived category D (Λ − mod) ) ( “ ” λb n o (a,c) d • , = P3 −→ P2 ⊕ P2 −→ P1 Uλ ∗ λ∈k λ∈k ∗ TILTING ON NON-COMMUTATIVE CURVES 35 where the underlined term of Uλ• has degree zero. Moreover, for any λ ∈ k∗ the complex Uλ• is spherical in the sense of Seidel and Thomas [39] and we have: H 0 (Uλ• ) = S3 , H 1 (Uλ• ) = S1
Proof. Since T : D b Coh(A) → D b (Λ − mod) is an equivalence of categories, we have an isomorphism of functors T ◦ τA ∼ = τΛ ◦ T. Next, we know that T(P) = P1 . This implies that   I R := τA (P) = and T(R) = τΛ (P1 ) = S1 [−1], e O where S1 is the simple Λ–module corresponding to the vertex 1 (note that S1 is injective). Consider the torsion A–module T supported at the singular point s ∈ E and defined as       0 I I T = = coker −→ . e e I O/I O
Since HomA P(n), Sβ = 0 for all n ∈ Z, the canonical morphism HomA (T , Sβ ) −→ HomA (R, Sβ ) e is an isomorphism. Another canonical morphism HomA (T , Sβ ) −→ HomE (O/I, O/I) is 2 ∼ e an isomorphism as well, hence HomA (R, Sβ ) = k . Since (O/I)s = ks ×ks and (O/I)s = ks e are rings, the vector space HomE (O/I, ks ) has two distinguished basis elements w̄0 and e w̄∞ , which correspond to the non-trivial idempotents of the ring (O/I) s . Let w0 and w∞ be the corresponding elements of HomA (R, Sβ ). For any (λ, µ) ∈ k2 \ {0, 0} consider the short exact sequence w 0 −→ X −→ R −→ T −→ 0, where w = µw0 − λw∞ . The A–module X only depends on the ratio x = (λ : µ) ∈ P1 . We claim that X is a locally projective A–module precisely when x 6∈ {0, ∞}. Moreover, in this case we have: Xs ∼ = F . Indeed, X is locally projective at all smooth points of E. Let w̄ = µw̄0 − λw̄∞ : T → Sβ . Then Xs is isomorphic to the middle term of the following short exact sequence:     I I 0 −→ −→ −→ ker(w̄) −→ 0, I I + (λ, µ)O where we view (λ, µ) ∈ k × k as an element of the normalization kJuK × kJvK. Other way around, it is not difficult to show that for any line bundle Lx ∈ Pic0 (E) ∼ = k∗ the locally projective A–module X := F ⊗E Lx fits into a short exact sequence 0 −→ X −→ R −→ Sβ −→ 0. Summing up, for all non-zero morphisms w = µw0 − λw∞ ∈ HomA (R, Sβ ), where x = (λ : µ) ∈ / {0, ∞}, the mapping cone cone(w)[−1] is τA –periodic and isomorphic to a stalk complex X [0], where X is a locally projective A–module. Applying the functor T, we obtain a distinguished triangle T(w) T(X ) −→ S1 [−1] −−−→ S3 [1] −→ T(X )[1]. 36 IGOR BURBAN AND YURIY DROZD
Next, we have: HomDb (Λ) S1 [−1], S3 [1] ∼ = Ext2Λ (S1 , S3 ) = k2 . Note that S1 has the projective resolution “ b 0 0d ” (a c) 0 −→ P3⊕2 −−−−→ P2⊕2 −−−→ P1 −→ S1 −→ 0. Hence, any element w ∈ Ext2Λ (S1 , S3 ) is given by a morphism of Λ–modules (λ1, µ1) : P3⊕2 → P3 , where (λ, µ) ∈ k2 . Moreover, one can check that the complex cone(w) is τΛ –periodic if and only if (λ : µ) ∈ / {0, ∞}. In that case, cone(w) is isomorphic to the complex Uν• , where ν = − µλ .  8. Some generalizations and concluding remarks 8.1. Tilting on other degenerations of elliptic curves. Theorem 5.10 can be generalized to the case of curves with more complicated singularities. For example, let E ⊆ P2 be e→E a tachnode plane cubic curve given by the equation y(yz − x2 ) = 0. Again, let π : E e e be the normalization, O = π∗ (OEe ) and I = AnnE (O/O) be the conductor ideal. In this case, consider again the coherent sheaf F = I ⊕ O and the sheaf of O–orders A = EndE (F). Let A = P1 ⊕ P2 ⊕ F be the decomposition of A into a direct sum of indecomposable locally projective modules. Define the torsion A–module S via the short exact sequence 0 −→ P1 ⊕ P2 −→ F −→ S −→ 0.

Then the complex H• = S[−1] ⊕ P1 (−1) ⊕ P1 ⊕ P2 (−1) ⊕ P2 is rigid and the endomorphism algebra ΓE = EndDb (A) (H• ) is isomorphic to the path algebra ϕ v •h v2 •o u2 a2 •  a1 u1 /• v1 ( 6• subject to the relations ϕ2 = 0, v1 a1 = 0, v2 a2 = 0, u1 a1 ϕ = 0 and u2 a2 ϕ = 0. Note that in this case gl. dim(ΓE ) = ∞. However, similar to the proof of Theorem 5.10 one can show the complex H• is tilting in the sense of Theorem 5.7. Thus, we have a fully faithful functor Perf(E) → D b (mod − ΓE ). 8.2. Tilting on chains of projective lines. Let X be a chain of projective lines. In
[10] it was shown that D Qcoh(X) has a tilting vector bundle. In particular, we have
∼ = a triangle equivalence D ∗ Coh(X) −→ D ∗ (AX − mod), where ∗ ∈ {−, b} and AX is the opposite algebra of the corresponding tilted algebra. Composing this functor with the embedding obtained in Corollary 5.11, get an interesting fully faithful functor D − (AX − mod) −→ D − (ΛX − mod) which is worth to study in further details. Note that both algebras AX and ΛX are gentle, hence derived-tame. We hope that these geometric realizations of gentle algebras will contribute to a better understanding of the representation theory of gentle algebras. TILTING ON NON-COMMUTATIVE CURVES 37 Example 8.1. Let X = V (x0 x1 ) ⊂ P2 be a chain of two projective lines. Then AX is the path algebra of the following quiver ? • _@ ~~ @@@b ~ @@ ~~ c @ ~~ +• •k a d subject to the relations cd = 0 = dc. The algebra ΛX is the path algebra of another quiver u2 • v2 ( 6• a2 /•o a1 •h v u1 • v1 subject to the relations ui ai = 0, i = 1, 2. 8.3. Non-commutative curves with nodal singularities. Similar to constructions of Section 2 and Section 5, one can study a new class of derived-tame non-commutative curves which generalizes weighted projective lines of Geigle and Lenzing [17]. Their characteristic property is that the completion of their stalks are generically matrix algebras over kJtK, whereas the singular stalks are either hereditary orders or nodal algebras. The last class of kJtK–orders was introduced in [11].  Example 8.2. Let X = P1 , Z = 0, ∞ and I = IZ be the ideal sheaf of Z. Consider the following sheaf of OP1 –orders:   O I I A =  O O I . O I O
 cx ∼ Then for x 6= 0, ∞ the algebra A = Mat3 kJtK , whereas for x ∈ 0, ∞ completion of the path algebra of the so-called Gelfand quiver : a+ G = •1 t a− it is the b+ 4 •3 j * b− 2 • a− a+ = b− b+ . Let Si be the simple G–module corresponding to the vertex i = 1, 2 and Six be the corresponding A–module for x ∈ {0, ∞}. Let e1 ∈ H 0 (A) be the primitive idempotent corresponding to the left upper corner unit element and   O P = A · e1 =  O  O be the corresponding locally projective A–module. Similar to the proof of Proposition 5.6
• = S 0 ⊕ S 0 ⊕ S ∞ ⊕ S ∞ [−1] ⊕ and Theorem 5.10 one can show that the complex H 2 1 2 1
P(−1) ⊕ P is tilting in D b Coh(A) . Its endomorphism algebra Γ = EndDb (A) (H• ) is 38 IGOR BURBAN AND YURIY DROZD isomorphic to the path algebra of the quiver • • • OOO • @ ~ OOO @@ a ooo OOO @@ 2 b1 ~~~ ooooo ~ oo a1 OOOO@@ O' ~wo~ooo b2 • u v   • subject to the relations uai = 0, vbi = 0, i = 1, 2. The algebra Γ is derived equivalent to the path algebra of the quiver o • @OO oo~o~~ @@O@OOOOa4 o o @ OO oo ~ ooo ~~~ a2 a3 @@ OOOOO o o ' • • OwoOO • @ oo • ~ OOO @@ b o o ~ OOO @@2 b3 ~~ oooo OOO @@ ~~ ooo b b1 4 OO' ~wooo a1 • subject to the relations b1 a1 = b2 a2 and b3 a3 = b4 a4 . This algebra is a degeneration of the canonical tubular algebra Tλ = T (2, 2, 2, 2; λ), λ ∈ k \ {0, 1}, introduced by Ringel in [36]. The algebra T (2, 2, 2, 2; λ) has the following geometric interpretation. Consider the elliptic curve E ⊆ P2 given by the equation zy 2 = x(x − z)(x − λz), where λ ∈ k \ {0, 1}. Then the group Z2 acts on E by the rule (x : y : z) 7→ (x : −y : z). By a result of Geigle and Lenzing [17, Proposition 4.1 and Example 5.8], see also [33, Corollary 1.4], there is a derived equivalence
∼ = D b CohZ2 (E) −→ D b (mod − Tλ ).

Hence, the derived category D b Coh(A) can be viewed as a degeneration of D b CohZ2 (E) . 8.4. Configuration schemes of Lunts. Our construction of non-commutative curves attached to a nodal rational projective curve, is closely related with the category of coherent sheaves on a configuration scheme, introduced by Lunts in [27]. Let X be a union of projective lines intersecting transversally. Lunts introduces a category Coh(X ), constructs

b a fully faithful exact functor Perf(X) → D Coh(X ) and shows that D b Coh(X ) has a tilting sheaf, whose endomorphism algebra is isomorphic to the algebra ΓX introduced in Definition 5.12. Moreover, his approach can be generalized to higher dimensions, in particular to the case of the singular quintic threefold V (x0 x1 x3 x3 x4 ) ⊆ P4 important in the string theory. The construction of Lunts seems not to generalize straightforwardly to the case of arbitrary nodal rational projective curves. Moreover, one can check that for a union of projective lines X intersecting transversally, the category Coh(X ) is not equivalent to Coh(AX ). In other words, Coh(X
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