Geometric collections and Castelnuovo–Mumford regularity

arXiv:math/0609561v1 [math.AG] 20 Sep 2006 GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY L. COSTA∗ , R.M. MIRÓ-ROIG∗∗ Abstract. The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf F on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to F and the second one to its cohomology. The main goal of the paper is to generalize CastelnuovoMumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf F on X with respect to σ. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on Pn and for a suitable geometric collection of coherent sheaves on Pn both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface Qn ⊂ Pn+1 (n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo-Mumford regularity of their extension by zero in Pn+1 . Contents 1. Introduction 2. Geometric collections and spectral sequences 3. Regularity with respect to geometric collections: definition and properties 4. Regularity of sheaves on Qn 5. Final comments References 1 3 11 16 23 24 1. Introduction The goal of this paper is to extend the notion of Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties with a geometric collection with the hope to apply it to study coherent sheaves on smooth projective varieties. As a main tool we use geometric collections of exceptional sheaves and helix theory. Exceptional bundles were first considered in [7] by Drezet and Le Potier, where they were used to determine the set of triples (r, c1 , c2 ) such that there exists a semistable sheaf Date: February 2, 2008. 1991 Mathematics Subject Classification. Primary 14F05; Secondary 18E30, 18F20. ∗ Partially supported by MTM2004-00666. ∗∗ Partially supported by MTM2004-00666. 1 2 L. COSTA, R.M. MIRÓ-ROIG 2 E on P with rank r and Chern classes c1 and c2 ; and to describe moduli spaces of stable vector bundles on P2 . In the succeeding papers [8], [9], [11], [14] and [23], the notion of exceptional bundle on P2 was extended to other manifolds X and even more from the category of vector bundles on X to the bounded derived category of coherent sheaves on X. Any exceptional collection (E0 , E1, · · · , Em ) gives rise to a bi-infinite collection, {Ei}i∈Z , called helix and defined recursively by left and right mutations (see Definition 2.12). Helix theory was introduced by Drezet and Le Potier in [7] and by Gorodentsev and Rudakov in [11], in connection with the problem of constructing exceptional bundles on Pn ; and helix theory got its further progress in the succeeding papers [2], [6], [9], [18], [19], [22] and [23]. Again helix theory was first developed for vector bundles on P2 and generalized later to any triangulated category where for any two objects E and F , Hom• (E, F ) has a structure of finite dimensional graded vector space over C. In this paper, we make an effort to link the abstract and general context of helix theory and exceptional collections to concrete examples, their applications, and the geometrical properties that we can derive. First of all, we recall the notion of geometric collection (see Definition 2.15) introduced by Bondal and Polishchuk in [3]. It is well known that the length of any full exceptional collection of coherent sheaves σ = (E0 , E1, · · · , Em) on a smooth projective variety X of dimension n is equal to the rank of the Grothendieck group K0 (X) which turns out to be greater or equal to n + 1. We call geometric collection any full exceptional collection of coherent sheaves of length n + 1. Geometric collections have nice properties: They are automatically full strongly exceptional collections, their strong exceptionality is preserved under mutations, any thread of an helix associated to a geometric collection is a full strongly exceptional collection, etc... We address the problem of determining smooth projective varieties with geometric collections and we prove that Pn , any quadric hypersurface Qn ⊂ Pn+1 (n odd) and any Fano 3-fold X with P ic(X) ∼ = Z and trivial intermediate Jacobian have geometric collections. Given a coherent sheaf F on a smooth projective variety X with a geometric collection, we derive two spectral sequences: A Beilinson-Kapranov type spectral sequence which converges to F (Theorem 2.21) and an Eilenberg-Moore type spectral sequence which abuts to the cohomology of F (Theorem 2.22). The existence of geometric collections allows us to generalize the notion of CastelnuovoMumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties with a geometric collection (Definition 3.1). We also prove that many of the main properties of the Castelnuovo-Mumford regularity are accomplished i by this new concept. Finally, given a smooth projective variety X ֒→Pn with a geometric collection σ and a coherent OX -module F it would be very interesting to compare the regularity of F with respect to σ, Regσ (F ), to the Castelnuovo-Mumford regularity Reg CM (i∗ F ) of its extension by zero i∗ F . In the last part of this work, we will address this problem and we will show by carefully analyzing the cases of coherent sheaves on quadric hypersurfaces Qn ⊂ Pn+1 (n odd) that, in general, they are very different and we write down a formula which relates both notions of regularity (Theorem 4.3). GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 3 Next we outline the structure of this paper. In section 2, we overview the basic facts on exceptional collections, geometric collections, mutations and helix theory; we give examples to illustrate all these concepts, and we describe the spectral sequences that we use in the sequel to develop the theory of regularity with respect to a geometric collection. In section 3, we give, using a Beilinson-Kapranov type spectral sequence, the promised definition of regularity with respect to a geometric collection, we prove that the Castelnuovo-Mumford regularity of a coherent sheaf F on Pn coincides with the regularity of F with respect to a suitable geometric collection of coherent sheaves on Pn and we show that the main basic properties of the Castelnuovo-Mumford regularity carry over to the new setting. In section 4, we consider odd dimensional quadric hypersurfaces Qn ⊂ Pn+1 and their geometric collection σ = (OQn , OQn (1), · · · , OQn (n − 1), Σ(n − 1)), where Σ is the Spinor bundle. We compute the right dual basis of any thread of the strict helix Hσ = {Ei }i∈Z associated to σ, we illustrate our results on the regularity of a sheaf with respect to σ for the case of sheaves F on Qn and we compare it with the regularity of its extension by zero in the embedding Qn ⊂ Pn+1 . This last results show that, in general, the i regularity with respect to σ of a coherent sheaf F on a smooth projective variety X ֒→Pm does not square with the Castelnuovo-Mumford regularity of its extension by zero i∗ F in Pm . We end the paper in §5 with some final comments and questions which naturally arise from this paper. 2. Geometric collections and spectral sequences Let X be a smooth projective variety defined over the complex numbers C and let D = D b (OX -mod) be the derived category of bounded complexes of coherent sheaves of OX -modules. For any pair of objects A, B ∈ Obj(D) we introduce the following notation: M ExtkD (A, B) Hom• (A, B) := k∈Z • and if V is a graded vector space and A an object of D, then the tensor product can be constructed as M (2.1) V•⊗A = V α ⊗ A[−α]. α We will use the dualization defined for graded vector spaces by the rule (V ו )p = (V • )−p . (2.2) A covariant cohomological functor Cov• is called linear if for any β satisfies M M (2.3) Covβ (V • ⊗ A) = V α ⊗ Covβ (A[−α]) = V α ⊗ Covβ−α (A). α α Definition 2.1. Let X be a smooth projective variety. (i) An object F ∈ D is exceptional if Hom• (F , F ) is a 1-dimensional algebra generated by the identity. (ii) An ordered collection (F0 , F1, · · · , Fm ) of objects of D is an exceptional collection if each object Fi is exceptional and Ext•D (Fk , Fj ) = 0 for j < k. 4 L. COSTA, R.M. MIRÓ-ROIG (iii) An exceptional collection (F0, F1 , · · · , Fm ) of objects of D is a strongly exceptional collection if in addition ExtiD (Fj , Fk ) = 0 for i 6= 0 and j ≤ k. (iv) An ordered collection of objects of D, (F0 , F1 , · · · , Fm ), is a full (strongly) exceptional collection if it is a (strongly) exceptional collection (F0 , F1 , · · · , Fm ) and F0 , F1 , · · · , Fm generate the bounded derived category D. Example 2.2. (1) (OPr , OPr (1), OPr (2), · · · , OPr (r)) and (OPr , Ω1Pr (1), Ω2Pr (2), · · · , ΩrPr (r)) are full strongly exceptional collections of coherent sheaves on Pr . (2) Let Gr(k, n) be the Grassmannian of k-dimensional subspaces of the n-dimensional vector space and let S be the tautological k-dimensional bundle on X. Denote by Σα S the space of the irreducible Ps representations of the group GL(S) with highest weight αα= (α1 , . . . , αs ) and |α| = i=1 αi . Denote by A(k, n) the set of locally free sheaves Σ S on Gr(k, n) where α runs over Young diagrams fitting inside a k × (n − k) rectangle. Set ρ(k, n) := ♯A(k, n). By [14]; Proposition 2.2 (a) and Proposition 1.4, A(k, n) can be totally ordered in such a way that we obtain a full strongly exceptional collection (E1 , . . . , Eρ(k,n) ) of locally free sheaves on Gr(k, n). (3) Let Qn ⊂ Pn+1 , n ≥ 2, be the quadric hypersurface. By [15]; Proposition 4.9, if n is even and Σ1 , Σ2 are the Spinor bundles on Qn , then (Σ1 (−n), Σ2 (−n), OQn (−n + 1), · · · , OQn (−1), OQn ) is a full strongly exceptional collection of locally free sheaves on Qn ; and if n is odd and Σ is the Spinor bundle on Qn , then (Σ(−n), OQn (−n + 1), · · · , OQn (−1), OQn ) is a full strongly exceptional collection of locally free sheaves on Qn . Remark 2.3. The existence of a full strongly exceptional collection (F0 , F1 , · · · , Fm ) of coherent sheaves on a smooth projective variety X imposes a rather strong restriction on X, namely that the Grothendieck group K0 (X) = K0 (OX − mod) is isomorphic to Zm+1 . Definition 2.4. Let X be a smooth projective variety and let (A, B) be an exceptional pair of objects of D. We define the left mutation of B, LA B, and the right mutation of A, RB A, with the aid of the following distinguished triangles in the category D: (2.4) LA B → Hom• (A, B) ⊗ A → B → LA B[1] (2.5) RB A[−1] → A → Homו (A, B) ⊗ B → RB A. Remark 2.5. If we apply Hom• (A, ·) to the triangle (2.4) and we apply Hom• (·, B) to the triangle (2.5) we get the following orthogonality relations: Hom• (A, LA B) = 0 and Hom• (RB A, B) = 0. Definition 2.6. Let X be a smooth projective variety and let σ = (E0 , · · · , En ) be an exceptional collection of objects of D. A left mutation (resp. right mutation) of σ is defined as follows: for any 1 ≤ i ≤ n a left mutation Li replaces the i-th pair of consequent GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 5 elements (Ei−1 , Ei) by (LEi−1 Ei , Ei−1 ) and a right mutation Ri replaces the same pair of consequent elements (Ei−1 , Ei) by (Ei , REi Ei−1 ): Li σ = LEi−1 σ = (E0 , · · · , LEi−1 Ei , Ei−1 , · · · En ) Ri σ = REi−1 σ = (E0 , · · · , Ei, REi Ei−1 , · · · , En ). Notation 2.7. It is convenient to agree that R(j) Ei = R(j−1) REi = REi+j · · · · · · REi+2 REi+1 Ei and similar notation for compositions of left mutations. According to these notations, mutations satisfy the following relations: Li Ri = Ri Li = Id Li Lj = Lj Li for |i − j| > 1 Li+1 Li Li+1 = Li Li+1 Li for 1 < i < n. Proposition 2.8. Let X be a smooth projective variety and let σ = (E0 , · · · , En ) be an exceptional collection of objects of D. Then any mutation of σ is an exceptional collection and if σ generates the category D, then the mutated collection also generates D. Proof. See [2]; Assertion 2.1 and Lemma 2.2.  Remark 2.9. In general a mutation of a strongly exceptional collection is not a strongly exceptional collection. For instance, let X = P1 × P1 be a smooth quadric surface in P3 and denote by OX (a, b) = OP1 (a) ⊠ OP1 (b). By [4]; Proposition 4.16, (see also [5] σ = (OX , OX (1, 0), OX (0, 1), OX (1, 1)) is a full strongly exceptional collection of line bundles on X. Using the exact sequence 0 → OX (−1, 1) → Hom(OX (0, 1), OX (1, 1)) ⊗ OX (0, 1) → OX (1, 1) → 0 we get that LOX (0,1) OX (1, 1) = OX (−1, 1). But, since Ext1 (OX (1, 0), OX (−1, 1)) = C the mutated exceptional collection of line bundles on X Lσ = (OX , OX (1, 0), OX (−1, 1), OX (0, 1)) is no more a strongly exceptional collection of line bundles on X. We will come back to the problem of whether strongly exceptionality is preserved under mutations. Definition 2.10. Let X be a smooth projective variety. Given any full exceptional collection σ = (E0 , · · · , En ) the collection (L(n) En , L(n−1) En−1 , · · · , L(1) E1 , E0 ) will be called left dual base of σ and the collection (En , R(1) En−1 , · · · , R(n) E0 ) will be called right dual base of σ. 6 L. COSTA, R.M. MIRÓ-ROIG Remark 2.11. Given a full exceptional collection σ = (E0 , · · · , En ), its corresponding right and left dual basis are uniquely determined up to a unique isomorphism and they satisfy the following orthogonality conditions: Homα (R(j) Ei , Ek ) = 0, Homα (Ek , L(j) Ei ) = 0 for all α, i, j and k except Homk (R(k) En−k , En−k ) = Homn−k (En−k , L(n−k) En−k ) = C. The definition of helix and the first results about helices appeared in [7] and [11]. Let us recall its definition. Definition 2.12. Let X be a smooth projective variety. A helix of period n + 1 is an infinite sequence {Ei}i∈Z of objects of D such that for any i ∈ Z, (Ei , · · · , Ei+n ) is an exceptional collection and En+1+i = R(n) Ei . Any exceptional collection of objects of D, σ = (E0 , · · · , En ), induces a unique helix by the rule (2.6) En+i = R(n) Ei−1 and E−i = L(n) En−i+1 , i > 0. In that case we say that the helix is generated by σ and that the collection Hσ := {Ei}i∈Z is the helix associated to σ. Each collection σi = (Ei , Ei+1, · · · , Ei+n ) is called a thread of the helix and it is clear that a helix is generated by any of its thread. Example 2.13. Let σ = (OPr , OPr (1), OPr (2), · · · , OPr (r)) be the full exceptional collection of line bundles on Pr = P(V ) given in Example 2.2, (1). The helix associated to σ is given by Hσ = {OPr (i)}i∈Z . Indeed, denote by Ei = OPr (i), 0 ≤ i ≤ r. By definition, for any i > 0, Er+i = R(r) Ei−1 and E−i = L(r) Er−i+1. Using the exterior powers of the Euler sequence (2.7) 0 −→ ∧k−1 TPr −→ ∧k V ⊗ OPr (k) −→ ∧k TPr −→ 0 we deduce that for any k > 0 R(k) E0 = REk REk−1 · · · RE1 E0 = ∧k TPr . So, in particular we get Er+1 = R(r) E0 = ∧r TPr = OPr (r + 1). More in general, repeating the process and using once more the exact sequence (2.7) we get Er+i = R(r) Ei−1 = ∧r TPr (i − 1) = OPr (r + i). Analogously, we deduce that E−i = OPr (−i) and hence we get that the helix associated to σ is given by Hσ = {OPr (i)}i∈Z . Notice that in this case, any thread σi = (OPr (i), OPr (i + 1), OPr (i + 2), · · · , OPr (i + r)) of the helix is again a full strongly exceptional collection of line bundles. Moreover, we have that for any i ∈ Z, Ei = Ei+r+1 ⊗ KPr , where KPr = OPr (−r − 1) is the canonical line bundle. This last observation is indeed a more general fact: GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 7 Remark 2.14. Let X be a smooth projective variety of dimension m and let σ = (E0 , · · · , En ) be a full exceptional collection of objects of D. Then the helix Hσ associated to σ has the following property of periodicity: for any i ∈ Z, Ei = Ei+n+1 ⊗ KX [m − n] where KX is the canonical line bundle on X and the number in square brackets denotes the multiplicity of the shift of an object to the left viewed as a graded complex in D. Bondal and Polishchuk introduced in [3] the notion of geometric collection as the exceptional collection σ of objects of D that generates a geometric helix, i.e., an helix Hσ = {Ei}i∈Z such that for any k > 0 and any i ≤ j, Extk (Ei, Ej ) = 0. Then they proved that full geometric collections are exactly full exceptional collections of length equal to the dimension of the variety plus one (see [3]; Proposition 3.3). We find it more convenient to use this latter property as the definition: Definition 2.15. Let X be a smooth projective variety of dimension n. We call geometric collection to any full exceptional collection of coherent sheaves on X of length n + 1 and we call strict helix to the helix generated by it. Remark 2.16. (1) Notice that since all full strongly exceptional collections of coherent sheaves on X have the same length and it is equal to the rank(K0 (X)) ≥ n+ 1, the length of a geometric collection is the minimum possible. (2) The existence of a geometric collection on a smooth variety X imposes a strong restriction on X, namely that its Grothendieck group K0 (X) ∼ = ZdimX+1 and X is forced to be a Fano variety (see [3]; Theorem 3.4). Geometric collections have a nice behavior. For instance, geometric collections are automatically strongly exceptional collections of coherent sheaves and its strongly exceptionality is preserved under mutations (Recall that, in general, strongly exceptionality is not preserved under mutations as we have showed in Remark 2.9). More precisely we have: Proposition 2.17. Let X be a smooth projective variety of dimension n and let σ = (E0 , · · · , En ) be a geometric collection of coherent sheaves on X. Then, (i) Any mutation of the collection σ consists also of sheaves, i.e. complexes concentrated in the zero component of the grading. (ii) The collection σ is a full strongly exceptional collection of coherent sheaves. (iii) Any mutation of σ is a full strongly exceptional collection of coherent sheaves. (iv) Any thread (Ei, Ei+1 , · · · , En+i) of the helix Hσ associated to σ is a full strongly exceptional collection of coherent sheaves on X. Proof. It follows from [2]; Assertion 9.2, Theorem 9.3 and Corollary 9.4.  Example 2.18. According to Example 2.2, there exists a geometric collection of coherent sheaves on Pr and on Qn ⊂ Pn+1 , n odd, and hence both varieties have strict helices. On the other hand, there are no geometric collections of coherent sheaves on Qn for n even and on Gr(k, n) = Gr(k, V ), 2 ≤ k ≤ n − 2. 8 L. COSTA, R.M. MIRÓ-ROIG Proposition 2.19. Let X be any smooth Fano threefold with P ic(X) ∼ = Z and trivial intermediate Jacobian. Then, X has a geometric collection. Proof. According to the classification of Fano threefolds X with P ic(X) ∼ = Z and trivial intermediate Jacobian ([13]; Table 3.5), there exist four kinds of such manifolds: The projective space P3 , a smooth quadric Q3 ⊂ P4 , the manifold V5 ⊂ P6 and the family of manifolds V22 ⊂ P12 . The cases X ∼ = Q3 follow from Example 2.18, (1) and = P3 and X ∼ ∼ (2), respectively. The case X = V5 is due to Orlov ([20]) and the case X ∼ = V22 is due to Kuznetsov ([16];Theorem 3).  Remark 2.20. Restricting ourselves to the strict helix area Hσ , σ = (E0 , · · · , En ) all the theory can be pulled down from the triangulated category D = D b (OX -mod) of bounded complexes of coherent sheaves into the category of coherent sheaves Coh(X). For instance, for any i < j the canonical distinguished triangles (2.8) LEi Ej → Hom• (Ei, Ej ) ⊗ Ei → Ej → LEi Ej [1] (2.9) REj Ei [−1] → Ei → Homו (Ei , Ej ) ⊗ Ej → REj Ei , turn to usual triples of coherent sheaves (2.10) 0 → LEi Ej → Hom(Ei , Ej ) ⊗ Ei → Ej → 0 (2.11) 0 → Ei → Hom∗ (Ei , Ej ) ⊗ Ej → REj Ei → 0. So, without loss of generality, we could define the mutations inside a strict helix in terms of sheaves. Theorem 2.21. (Beilinson-Kapranov type spectral sequence) Let X be a smooth projective variety of dimension n with a geometric collection σ = (E0 , E1 , · · · , En ) and let Hσ = {Ei }i∈Z be the associated strict helix. Then for any thread σi = (Ei, · · · , Ei+n ) of the helix Hσ and any coherent sheaf F on X there is a spectral sequence with E1 -term (2.12) i E1pq = Extq (R(−p) Ei+n+p , F ) ⊗ Ei+p+n situated in the square 0 ≤ q ≤ n, −n ≤ p ≤ 0 which converges to ( F for i = 0 i E∞ = 0 for i 6= 0. Proof. First of all notice that since the helix Hσ is strict, the thread σi = (Ei, · · · , Ei+n ) also generates the category D. We write Vk• for the graded vector spaces Vk• = Hom• (R(n−k) Ek+i, F ) and we consider the complex • L• : 0 → V0• ⊗ Ei → V1• ⊗ Ei+1 → · · · → Vn−1 ⊗ Ei+n−1 → Vn• ⊗ Ei+n → 0 where the tensor product is defined as in (2.1). The right mutations produce a canonical right Postnikov system of the complex L• , which naturally identifies F with the canonical GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 9 right convolution of this complex. Then, for an arbitrary linear covariant cohomological functor Φ• , there exists an spectral sequence with E1 -term i E1pq = Φq (Lp ) situated in the square 0 ≤ p, q ≤ n and converging to Φp+q (F ) (see [15]; 1.5). Since Φ• is a linear functor, it follows from (2.3) that M M (2.13) Φq (Lp ) = Φq (Vp• ⊗ Ei+p ) = Vpl ⊗ Φq−l (Ei+p ) = Vpα ⊗ Φβ (Ei+p ). α+β=q l In particular, if we consider the covariant linear cohomology functor which takes a complex to its cohomology sheaves and acts identically on sheaves, i.e. ( F for β = 0 Φβ (F ) = 0 for β 6= 0 on any sheaf F , in the square 0 ≤ p, q ≤ n, we get i E1pq = Vpq ⊗ Ei+p = Extq (R(n−p) Ep+i, F ) ⊗ Ei+p where the last equality follows from the definition of Vk• . Finally, if we call p′ = p − n we get the spectral sequence with E1 -term i ′ E1p q = Extq (R(−p ) Ep′ +n+i, F ) ⊗ Ep′ +n+i ′ in the square 0 ≤ q ≤ n, −n ≤ p′ ≤ 0 and which converges to ( F for i = 0 i E∞ = 0 for i 6= 0.  Theorem 2.22. (Eilenberg-Moore type spectral sequence) Let X be a smooth projective variety of dimension n with a geometric collection σ = (E0 , E1 , · · · , En ) and let Hσ = {Ei }i∈Z be the associated strict helix. Then for any thread σi = (Ei, · · · , Ei+n ) of the helix Hσ and any pair of coherent sheaves F , G on X there is a spectral sequence with E1 -term M i pq Extα (R(−p) Ei+n+p , F ) ⊗ Extβ (G, Ei+p+n ) (2.14) E1 = α+β=q situated in the square 0 ≤ q ≤ n, −n ≤ p ≤ 0 which converges to ( Extp+q (G, F ) for i = 0 i = E∞ 0 for i 6= 0. Proof. We follow step by step the proof of Theorem 2.21 but in this case in (2.13) we take as Φ• the covariant functor Hom• (G, ·) and in such a way we get the spectral sequence M i pq Extα (R(−p) Ei+n+p , F ) ⊗ Extβ (G, Ei+p+n ) E1 = α+β=q situated in the square 0 ≤ q ≤ n, −n ≤ p ≤ 0 which converges to Φp+q (F ) = Extp+q (G, F ).  10 L. COSTA, R.M. MIRÓ-ROIG We will end this section with technical results that will be used in next sections. Lemma 2.23. Let X be a smooth projective variety of dimension n and let σ = (E0 , · · · , En ) be a geometric collection of coherent sheaves on X. For any i < j and any invertible sheaf F , it holds: (a) (LEi Ej )∗ = REi∗ Ej∗ ; (b) (REj Ei )∗ = LEj∗ Ei∗ ; (c) (REj Ei ) ⊗ F ∼ = LEi ⊗F (Ej ⊗ F ). = REj ⊗F (Ei ⊗ F ) and (LEi Ej ) ⊗ F ∼ Proof. (a) According to Remark 2.20; (2.10), LEi Ej is given by the aid of the exact sequence (2.15) 0 → LEi Ej → Hom(Ei , Ej ) ⊗ Ei → Ej → 0. Dualizing this exact sequence we get 0 → Ej∗ → Hom(Ej∗ , Ei∗) ⊗ Ei∗ ∼ = Hom∗ (Ei , Ej ) ⊗ Ei∗ → (LEi Ej )∗ → 0 which according to Remark 2.20; (2.11) gives that (LEi Ej )∗ = REi∗ Ej∗ . (b) The proof is analogous to the proof of (a). (c) Since Hom(Ei , Ej ) ∼ = Hom(Ei ⊗ F , Ej ⊗ F ), it is enough to tensor by F the exact sequences 0 → LEi Ej → Hom(Ei , Ej ) ⊗ Ei → Ej → 0 and 0 → Ei → Hom∗ (Ei , Ej ) ⊗ Ej → REj Ei → 0.  Corollary 2.24. Let X be a smooth projective variety of dimension n with canonical line bundle K and let σ = (E0 , · · · , En ) be a geometric collection of coherent sheaves on X. Assume that τ = (F0 , · · · , Fn ) is the right dual base of σ. Then, for any integer λ, the right dual base of σλ(n+1) = (Eλ(n+1) , Eλ(n+1)+1 , · · · , Eλ(n+1)+n ) is τλ(n+1) = (F0 ⊗ (K ∗ )⊗λ , F1 ⊗ (K ∗ )⊗λ , · · · , Fn ⊗ (K ∗ )⊗λ ). Proof. By definition of right dual base we have Fj = R(j) En−j 0 ≤ j ≤ n. On the other hand, by Remark 2.14 Eλ(n+1)+i = Ei ⊗ (K ∗ )⊗λ . Therefore, applying Lemma 2.23, we get R(j) Eλ(n+1)+n−j = R(j) (En−j ⊗ (K ∗ )⊗λ ) = (R(j) En−j ) ⊗ (K ∗ )⊗λ = Fj ⊗ (K ∗ )⊗λ .  GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 11 3. Regularity with respect to geometric collections: definition and properties In this section, using the so-called Beilinson-Kapranov spectral sequence, we generalize the notion of Castelnuovo-Mumford regularity for coherent sheaves on a projective space to coherent sheaves on a smooth projective variety with a geometric collection of coherent sheaves. We establish for coherent sheaves on Pn the agreement of the new definition of regularity with the old one and we prove that many formal properties of CastelnuovoMumford regularity continue to hold in our more general setup. Let X be a smooth projective variety of dimension n and let σ = (E0 , E1 , · · · , En ) be a geometric collection on X. Associated to σ we have a strict helix Hσ = {Ei }i∈Z ; and for any collection σi = (Ei, Ei+1 , · · · , Ei+n ) of n + 1 subsequent sheaves (i.e. for any thread of the helix Hσ ) and any coherent OX -module F a spectral sequence (See Theorem 2.21) (3.1) i E1pq = Extq (R(−p) Ei+p+n , F ) ⊗ Ei+p+n situated in the square 0 ≤ q ≤ n, −n ≤ p ≤ 0 which converges to ( F for i = 0 i E∞ = 0 for i 6= 0. Definition 3.1. Let X be a smooth projective variety of dimension n with a geometric collection σ = (E0 , E1 , · · · , En ) and let F be a coherent OX -module. We say that F is m-regular with respect to σ if Extq (R(−p) E−m+p , F ) = 0 for q > 0 and −n ≤ p ≤ 0. So, F is m-regular with respect to σ if −n−m E1pq = 0 for q > 0 in (3.1). In particular, if F is m-regular with respect to σ the spectral sequence −n−m E1pq collapses at E2 and we get the following exact sequence: (3.2) 0 −→ L−n −→ · · · −→ L−1 −→ L0 −→ F −→ 0 where Lp = H 0 (X, (R(−p) E−m+p )∗ ⊗ F ) ⊗ E−m+p for −n ≤ p ≤ 0. Definition 3.2. Let X be a smooth projective variety of dimension n with a geometric collection σ = (E0 , E1, · · · , En ) and let F be a coherent OX -module. We define the regularity of F with respect to σ, Regσ (F ), as the least integer m such that F is m-regular with respect to σ. We set Regσ (F ) = −∞ if there is no such integer. Remark 3.3. It would be nice to characterize the sheaves F on X with Regσ (F ) = −∞. Example 3.4. Let V be a C-vector space of dimension n + 1 and set Pn = P(V ). We consider the geometric collection σ = (OPn , OPn (1), · · · , OPn (n)) on Pn and the associated strict helix Hσ = {OPn (i)}i∈Z (see Example 2.13). Using the exterior powers of the Euler sequence 0 −→ ∧k−1 TPn −→ ∧k V ⊗ OPn (k) −→ ∧k TPn −→ 0 12 L. COSTA, R.M. MIRÓ-ROIG we compute the right dual basis of any thread σi = (OPn (i), OPn (i + 1), · · · , OPn (i + n)) of the helix Hσ and we get (OPn (n + i), R(1) OPn (i + n − 1), · · · , R(j) OPn (i + n − j), · · · , R(n) OPn (i)) = (OPn (n + i), TPn (i + n − 1), · · · , ∧j TPn (i + n − j), · · · , ∧n TPn (i)). Therefore, for any coherent sheaf F on Pn our definition reduces to say: F is m-regular with respect to σ if Extq (∧−p T (−m + p), F ) = H q (Pn , Ω−p (m − p) ⊗ F ) = 0 for all q > 0 and all p, −n ≤ p ≤ 0. We now compute the regularity with respect to σ = (E0 , E1 , · · · , En ) of the sheaves Ei . Proposition 3.5. Let X be a smooth projective variety of dimension n with a geometric collection σ = (E0 , E1 , · · · , En ) and let Hσ = {Ei}i∈Z be the associated strict helix. Then, for any i ∈ Z, Regσ (Ei) = −i. Proof. First of all we will see that Regσ (Ei ) ≤ −i. By Remark 2.5, we have Extq (R(−p) Ei+p , Ei) = Extq (REi · · · REi+p+2 REi+p+1 Ei+p , Ei) = 0 for q > 0 and −n ≤ p ≤ 0. So, Ei is (−i)-regular with respect to σ or, equivalently, Regσ (Ei ) ≤ −i. Let us now see that Ei is not (−i − 1)-regular with respect to σ. To this end, it is enough to see that there is q > 0 and there is p, −n ≤ p ≤ 0, such that Extq (R(−p) Ei+1+p , Ei) 6= 0. To prove it, we write i = αn + j with 0 ≤ j < n, α ∈ Z, we consider the thread σαn = (Eαn , Eαn+1 , · · · , Eαn+j = Ei , · · · , Eαn+n ) and we construct its right dual basis (see Definition 2.10) (Eαn+n , R(1) Eαn+n−1 , · · · , R(n−j) Eαn+j , · · · , R(n) Eαn ). It follows from Remark 2.11 that Extn−j (R(n−j) Eαn+j , Eαn+j ) = Extn−j (REαn+n REαn+n−1 · · · REαn+j+1 Eαn+j , Eαn+j ) = C. We consider the exact sequence 0 −→ REαn+n−1 · · · REαn+j+1 Eαn+j −→ Hom∗ (REαn+n−1 · · · REαn+j+1 Eαn+j , Eαn+n ) ⊗ Eαn+n −→ REαn+n REαn+n−1 · · · REαn+j+1 Eαn+j −→ 0 and we apply the contravariant functor Hom(., Eαn+j ). Since Extq (Eαn+n , Eαn+j ) = 0 for q > 0, we get Extn−j−1(REαn+n−1 · · · REαn+j+1 Eαn+j , Eαn+j ) = C. We repeat the process using the consequent right mutations and we get Extn−j−k (REαn+n−k · · · REαn+j+1 Eαn+j , Eαn+j ) = C for 0 ≤ k ≤ n − 1 − j. In particular, Ext1 (REi+1 Ei, Ei ) = C which implies that Ei is not (−i − 1)-regular and we conclude that Regσ (Ei ) = −i.  GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 13 In [17], Lecture 14, D. Mumford defined the notion of regularity for a coherent sheaf over a projective space. Let us recall it. Definition 3.6. A coherent sheaf F on Pn is said to be m-regular in the sense of Castelnuovo-Mumford if H i (Pn , F (m − i)) = 0 for i > 0. We define the CastelnuovoMumford regularity of F , Reg CM (F ), as the least integer m such that F is m-regular. Notice that such an m always exists by the ampleness of OPn (1) ([12]; Chap. III, Proposition 5.3). Let us now establish for coherent sheaves on Pn the agreement of regularity definition in the sense of Definition 3.1 with Castelnuovo-Mumford definition. Proposition 3.7. A coherent sheaf F on Pn = P(V ) is m-regular in the sense of Castelnuovo-Mumford if and only if it is m-regular with respect to the geometric collection σ = (OPn , OPn (1), · · · , OPn (n)). Hence, we have Regσ (F ) = Reg CM (F ). Proof. According to Definitions 3.1 and 3.6, and Example 3.4 we have to see that H q (Pn , F (m − q)) = 0 for all q > 0 (3.3) if and only if H q (Pn , F ⊗ ΩpPn (m + p)) = 0 for all q > 0 and 0 ≤ p ≤ n. (3.4) Let us first see that (3.3) implies (3.4). By [17], Lecture 14, the equalities (3.3) are equivalent to H q (Pn , F (t)) = 0 for all q > 0 and t ≥ m − q. (3.5) Since ΩnPn (n) ∼ = OPn (−1), we deduce from (3.5) that H q (Pn , F ⊗ ΩnPn (n + t)) = 0 for all q > 0 and all t ≥ m − q + 1. (3.6) Using the exact cohomology sequence (3.7) q+1 · · · −→ H q (Pn , V ⊗F (t)) −→ H q (Pn , ΩPn−1 (Pn , ΩnPn (n)⊗F (t)) −→ · · · n (n−1)⊗F (t+1)) −→ H associated to 0 −→ ΩnPn (n) −→ ∧n V ⊗ OPn −→ ΩPn−1 n (n) −→ 0 together with the equalities (3.6) and (3.5) we deduce (3.8) H q (Pn , F ⊗ ΩPn−1 n (n − 1 + t)) = 0 for all q > 0 and all t ≥ m − q + 1. Going on using the exact cohomology sequence (3.9) · · · −→ H q (Pn , ∧i+1 V ⊗F (t)) −→ H q (Pn , ΩiPn (i)⊗F (t+1)) −→ H q+1(Pn , Ωi+1 Pn (i+1)⊗F (t)) −→ · · · associated to i+1 0 −→ Ωi+1 V ⊗ OPn −→ ΩiPn (i + 1) −→ 0 Pn (i + 1) −→ ∧ we deduce (3.10) H q (Pn , F ⊗ ΩiPn (i + t)) = 0 for all q > 0 and all t ≥ m − q + 1 14 L. COSTA, R.M. MIRÓ-ROIG which obviously implies (3.4). Let us prove the converse. To this end we consider the Eilenberg-Moore type spectral sequence (see Theorem 2.22) E1pq = M  α Ext (R (−p) α+β=q β Ei+n+p , F ) ⊗ Ext (G, Ei+n+p )  0 ≤ q ≤ n, −n ≤ p ≤ 0 pq p+r,q−r+1 dpq , E∞ ⇒ Extp+q (G, F ) r : Er −→ Er and we apply it to the case i = −m − n and G = OPn (t − m). So, we have E1pq =  M  (−p) 0 ≤ q ≤ n, H α (Pn , ΩPn ⊗ F ) ⊗ H β (Pn , OPn (p − t)) −n ≤ p ≤ 0. α+β=q (−p) By (3.4), H α (Pn , ΩPn ⊗ F ) = 0 for all α > 0 and all p, −n ≤ p ≤ 0 and by Bott’s formulas a non-zero cohomology group of line bundles on Pn , H β (Pn , OPn (p − t)), corresponds only to β = 0 or n and in the latter case p − t ≤ −n − 1 (i.e. p ≤ t − n − 1). pq Thus, E1pq 6= 0 forces p + q ≤ t − 1. Therefore, E∞ = 0 for p + q ≥ t and so t t n 0 = Ext (OPn (t − m), F ) = H (P , F (m − t)) for t ≥ 1 or, equivalently, F is m-regular in the sense of Castelnuovo-Mumford.  Let us now prove that the main formal properties of Castelnuovo-Mumford regularity over projective spaces remain to be true in the new setting. Proposition 3.8. Let X be a smooth projective variety of dimension n with a geometric collection σ = (E0 , E1, · · · , En ) and let F be a coherent OX -module. If F is m-regular with respect to σ then the canonical map Hom(E−m , F ) ⊗ E−m ։ F is surjective and F is k-regular with respect to σ for any k ≥ m as well. Proof. The first assertion immediately follows from the exact sequence (3.2) of F . To prove the second assertion, it is enough to check it for k = m + 1. Since F is m-regular with respect to σ we have (3.11) Extq (R(−p) E−m+p , F ) = 0 for all q > 0 and all − n ≤ p ≤ 0. In order to prove that F is (m + 1)-regular with respect to σ we have to check that (3.12) Extq (R(−p) E−m−1+p , F ) = 0 for all q > 0 and all − n ≤ p ≤ 0. To this end we apply the contravariant functor Hom(., F ) to the exact sequence 0 −→ E−m−1 −→ Hom∗ (E−m−1 , E−m ) ⊗ E−m −→ RE−m E−m−1 −→ 0 and we get the exact sequence · · · −→ Extq (RE−m E−m−1 , F ) −→ Hom∗ (E−m−1 , E−m) ⊗ Extq (E−m , F ) −→ Extq (E−m−1 , F ) −→ Extq+1 (RE−m E−m−1 , F ) −→ · · · . Since by (3.11), Extq (E−m , F ) = Extq (RE−m E−m−1 , F ) = 0 for all q > 0, we get that Extq (E−m−1 , F ) = 0 for all q > 0. Using again (3.11) and the exact sequence GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 15 0 −→ RE−m−1 E−m−2 −→ Hom∗ (RE−m−1 E−m−2 , E−m) ⊗ E−m −→ RE−m RE−m−1 E−m−2 −→ 0 we get that Extq (RE−m−1 E−m−2 , F ) = 0 for all q > 0. Going on and using the consequent right mutations, we get for all i, 1 ≤ i ≤ n − 1, Extq (RE−m−1 RE−m−2 · · · RE−m−i E−m−i−1 , F ) = 0 for all q > 0. Hence, it only remains to see that Extq (RE−m−1 RE−m−2 · · · RE−m−n E−m−n−1 , F ) = 0 for all q > 0. The vanishing of these last Ext’s groups follows again from (3.11) taking into account that, by Definition 2.10 RE−m−1 RE−m−2 · · · RE−m−n E−m−n−1 = R(n) E−m−n−1 = E−m .  Proposition 3.9. Let X be a smooth projective variety of dimension n with a geometric collection σ = (E0 , E1 , · · · , En ), let F and G be coherent OX -modules and let (3.13) 0 → F1 → F2 → F3 → 0 be an exact sequence of coherent OX -modules. Then, (a) Regσ (F2 ) ≤ max{Regσ (F1 ), Regσ (F3 )}. (b) Regσ (F ⊕ G) = max{Regσ (F ), Regσ (G)}. Proof. (a) Let m = max{Regσ (F1 ), Regσ (F3 )}. Since, by Proposition 3.8, F1 and F3 are both m-regular with respect to σ considering the long exact sequence · · · → Extq (R(−p) E−m+p , F1 ) → Extq (R(−p) E−m+p , F2 ) → Extq (R(−p) E−m+p , F3 ) → · · · associated to (3.13) we get Extq (R(−p) E−m+p , F2 ) = 0 for any q > 0 and −n ≤ p ≤ 0, which implies that Regσ (F2 ) ≤ m. (b) It is enough to see that if F is m-regular with respect to σ and G is s-regular with respect to σ then F ⊕ G is t = max(s, m)-regular with respect to σ. Since, by Proposition 3.8, F and G are both t-regular with respect to σ and, moreover, the functor Ext’s is additive we get Extq (R(−p) E−t+p , F ⊕ G) = Extq (R(−p) E−t+p , F ) = Extq (R(−p) E−t+p , G) = 0 for q > 0 and −n ≤ p ≤ 0. Therefore, F ⊕ G is t-regular with respect to σ.  The following Example will show that in Proposition 3.9, (a) we can have strict inequality. Example 3.10. We consider the geometric collection σ = (OPn , OPn (1), · · · , OPn (n)) of locally free sheaves on Pn and we denote by H a hyperplane section on Pn . For any n ≥ r ≥ 0, we have the exact sequence 0 → OPn (r − 1) → OPn (r) → OH (r) → 0. 16 L. COSTA, R.M. MIRÓ-ROIG According to Proposition 3.5, Regσ (OPn (r)) = −r and Regσ (OPn (r − 1)) = −r + 1. So, Regσ (OPn (r)) < max{Regσ (OPn (r − 1)), Regσ (OH (r))}. In Proposition 3.7, we have seen that a coherent sheaf F on Pn is m-regular in the sense of Castelnuovo-Mumford if and only if it is m-regular with respect to the geometric collection σ = (OPn , OPn (1), · · · , OPn (n)). A more subtle problem is the following one. i Let X ֒→Pn be a smooth projective variety, let F be a coherent sheaf on X and let σ = (E0 , · · · , En ) be a geometric collection of coherent sheaves on X. How are related the regularity of F with respect to σ and the Castelnuovo-Mumford regularity of its extension i by zero i∗ F via the embedding X ֒→Pn ?. In next section, we will address this problem for the case of coherent sheaves on odd dimensional quadric hypersurfaces Qn ⊂ Pn+1 . 4. Regularity of sheaves on Qn Let n ∈ Z be an odd integer (n = 2t + 1) and let Qn ⊂ Pn+1 be a smooth quadric hypersurface. In [15], M. M. Kapranov defined the locally free sheaves ψi , i ≥ 0, on Qn and the Spinor bundle Σ on Qn to construct a resolution of the diagonal ∆ ⊂ Qn × Qn and to describe the bounded derived category D b (OQn − mod). In particular, he got that (Σ(−n), OQn (−n + 1), · · · , OQn (−1), OQn ) is a full strongly exceptional collection of locally free sheaves on Qn ([15]; Proposition 4.9) and hence, according to Definition 2.15, Qn has a geometric collection of locally free sheaves. Dualising each bundle of the above geometric sequence and reversing the order, we get that σ := (OQn , OQn (1), · · · , OQn (n − 1), Σ(n − 1)) is also a geometric collection of locally free sheaves on Qn . In this section we will give an elementary description of the locally free sheaves ψi and their basic properties (for more details the reader can look at [15]) and, for any coherent sheaf F on Qn , we will relate Regσ (F ) to the Castelnuovo-Mumford regularity of its extension by zero i∗ F in the embedding i : Qn ֒→ Pn+1 . From now on, we set Ωj := ΩjPn+1 and we define inductively ψj : ψ0 := OQn , ψ1 := Ω1 (1)|Qn and, for all j ≥ 2, we define the locally free sheaf ψj as the unique non-splitting extension (Note that Ext1 (ψj−2 , Ωj (j)|Qn ) = C): 0 −→ Ωj (j)|Qn −→ ψj −→ ψj−2 −→ 0. t+1 In particular, ψj+2 = ψj for j ≥ n and ψn = Σ(−1)2 , (n = 2t + 1). Before computing the right dual basis of any thread σj = (Ej , Ej+1, · · · , Ej+n ) of the helix Hσ = {Ej }j∈Z associated to σ, we collect in the following Lemma the cohomological properties of the Spinor bundles we need later. Lemma 4.1. Let n ∈ Z be an odd integer, let Qn ⊂ Pn+1 be a smooth quadric hypersurface and let Σ be the Spinor bundle on Qn . Then, (i) H i (Qn , Σ(t)) = 0 for any i such that 0 < i < n and for all t ∈ Z. GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 17 n+1 2 (ii) H 0 (Qn , Σ(t)) = 0(for all t < 0 and h0 (Qn , Σ) = 2 . 0 if i 6= j (iii) Exti (Σ(j), Σ) = for any 1 ≤ j ≤ n. C if i = j ( 0 if i 6= j (iv) H i (Qn , ψj (−j)) = for any 1 ≤ j ≤ n. C if i = j Proof. The assertions (i) and (ii) follows from [21]; Theorem 2.3. The assertion (iii) follows by induction on j, using (i) and (ii) and the long exact sequence · · · → Exti−1 (OQn (j)2 n+1 2 , Σ) → Exti−1 (Σ(j−1), Σ) → Exti (Σ(j), Σ) → Exti (OQn (j)2 n+1 2 , Σ) → · · · obtained by applying the functor Hom(·, Σ) to the exact sequence 0 → Σ(j − 1) → OQn (j)2 n+1 2 → Σ(j) → 0. The assertion (iv) follows from [15]; Proposition 4.11.  Proposition 4.2. Let n ∈ Z be an odd integer, let Qn ⊂ Pn+1 be a smooth quadric hypersurface and let Hσ = {Ei}i∈Z be the helix associated to σ = (OQn , OQn (1), · · · , OQn (n − 1), Σ(n − 1)). Let us denote by σk the thread (Ek , Ek+1, · · · , Ek+n). Then: (a) The right dual base of σ0 is (Σ(n − 1), ψn−1 (n), ψn−2 (n), · · · , ψ1 (n), ψ0 (n)). (b) For any j, 1 ≤ j ≤ n, the right dual base of the geometric collection σj = (OQn (j), · · · , OQn (n − 1), Σ(n − 1), OQn (n), OQn (n + 1), · · · , OQn (n + j − 1)) is ∗ (n + j − 1), (OQn (n + j − 1), ψ1∗ (n + j − 1), · · · , ψj−1 Σ(n + j − 1), ψn−j−1 (n + j), · · · , ψ0 (n + j)). (c) For any λ ∈ Z, the right dual base of the geometric collection σλ(n+1) = (OQn (λn), OQn (1 + λn), · · · , OQn (n − 1 + λn), Σ(n − 1 + λn)) is (Σ(n − 1 + λn), ψn−1 ((λ + 1)n), ψn−2 ((λ + 1)n), · · · , ψ1 ((λ + 1)n), ψ0 ((λ + 1)n)). (d) For any j, 1 ≤ j ≤ n and any λ ∈ Z, the right dual base of the geometric collection σj+λ(n+1) = (OQn (j + λn), · · · , OQn (n − 1 + λn), Σ(n − 1 + λn), OQn ((λ + 1)n), · · · , OQn (j − 2 + (λ + 1)n), OQn (j − 1 + (λ + 1)n)) is ∗ ((λ + 1)n + j − 1), (OQn ((λ + 1)n + j − 1), ψ1∗ ((λ + 1)n + j − 1), · · · , ψj−1 Σ((λ + 1)n + j − 1), ψn−j−1 ((λ + 1)n + j), · · · , ψ0 ((λ + 1)n + j)). 18 L. COSTA, R.M. MIRÓ-ROIG Proof. (a) By [15]; Proposition 4.11 and using the fact that the right dual basis of an exceptional collection is uniquely determined up to unique isomorphism by the orthogonality conditions described in Remark 2.11, we get that the right dual basis of the exceptional collection (Σ(−n), OQn (−n + 1), · · · , OQn (−1), OQn ) is ∗ , Σ∗ (1)). (OQn , ψ1∗ , · · · , ψn−1 In particular, we have R(j) OQn (−j) = ψj∗ for 0 ≤ j ≤ n − 1; and (4.1) R(n) Σ(−n) = Σ∗ (1). Let us now compute the right dual base of σ0 . Since R(0) Σ(n − 1) = Σ(n − 1) and (n) R OQn = OQn (n), according to the definition of right dual base we only need to compute R(j) O(n−j) for 1 ≤ j ≤ n−1. It follows from Lemma 2.23 and the fact that Σ∗ (−n+1) = Σ(−n), that R(j) O(n − j) = RΣ(n−1)OQn (n−1)···OQn (n−j+1) O(n − j) = (LΣ(−n)OQn (−n+1)···OQn (j−n−1) O(j − n))∗ = (LΣOQn (1)···OQn (j−1) O(j))∗ ⊗ O(n). By [10]; 2.8.1, (LΣOQn (1)···OQn (j−1) O(j))∗ ⊗ O(n) = (ROQn (−n+j+1),OQn (−n+j+2)···OQn OQn (−n + j))∗ ⊗ OQn (n) = (R(j) OQn (−j))∗ ⊗ OQn (n) = ψn−j (n) where the last equality follows from (4.1). Hence, R(j) O(n − j) = ψn−j (n) which finishes the proof of (a). (b) Applying Lemma 2.23; (c) to (4.1), we get that (4.2) R(p) OQn (n + j − 1 − p) = ψp∗ (n + j − 1) for 0 ≤ p ≤ j − 1. On the other hand, by Lemma 4.1 we have: - For any α ∈ Z and j ≤ t ≤ n + j − 1 Extα (Σ(n + j − 1), OQn (t)) = 0; ( and 0 α 6= j Extα (Σ(n + j − 1), Σ(n − 1)) = C α = j. - For any α ∈ Z, n − j − 1 ≤ t ≤ 0 and j ≤ γ ≤ n + j − 1 Extα (ψt (n + j), Σ(n − 1)) =(0; and C if n − α = t = γ − j Extα (ψt (n + j), OQn (γ)) = 0 otherwise. Since the right dual base of an exceptional collection is uniquely determined up to unique isomorphism by the orthogonal conditions given in Remark 2.11, it follows from (4.2), these last cohomological relations and from Lemma 4.1 that indeed ∗ (n + j − 1), (OQn (n + j − 1), ψ1∗ (n + j − 1), ψ2∗ (n + j − 1), · · · , ψj−1 Σ(n + j − 1), ψn−j−1 (n + j), · · · , ψ1 (n + j), ψ0 (n + j)) GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY is the right dual base of σj . (c) and (d) follows from (a) and (b), respectively, and Corollary 2.24. 19  Let F be a coherent sheaf on Qn and let i∗ F be its extension by zero in the embedding i Qn ֒→Pn+1 . We are now ready to compare Regσ (F ) to Reg CM (i∗ F ) and state the main result of this section. i Theorem 4.3. Let n be an odd integer and let Qn ֒→Pn+1 be a quadric hypersurface. We consider the geometric collection σ = (OQn , · · · , OQn (n − 1), Σ(n − 1)). For any coherent sheaf F on Qn we have: ⌊ nRegσ (F ) nRegσ (F ) ⌋ ≤ Reg CM (i∗ F ) ≤ ⌊ ⌋ + 1. n+1 n+1 σ (F ) ⌋ + 1. To this end, since for Proof. First of all, we will see that Reg CM (i∗ F ) ≤ ⌊ nReg n+1 all t ∈ Z and 0 ≤ q ≤ n, H n+1 (Pn+1 , i∗ F (t)) = 0, (4.3) H q (Pn+1 , i∗ F (t)) ∼ = H q (Qn , F (t)), we will see that if F is m-regular with respect to σ then H i (Qn , F (m + λ + 1 − i)) = 0, 1≤i≤n −m where λ := ⌊ n+1 ⌋. Write −m = λ(n + 1) + r with 0 ≤ r ≤ n. So, ( (λ − 1)(n + 1) + r + 1 if 0 ≤ r ≤ n − 1 −m − n = λ(n + 1) if r = n. Therefore, σ−m−n = (E−m−n , · · · , E−m ) is equal to (OQn ((λ − 1)n + r + 1), · · · , OQn ((λ − 1)n + n − 1), Σ((λ − 1)n + n − 1), OQn (λn), OQn (λn + 1), · · · , OQn (λn + r)), if 0 ≤ r ≤ n − 1 and is equal to (OQn (λn), OQn (λn + 1), . . . , OQn (λn + n − 1), Σ(λn + n − 1)) if r = n. We consider the Eilenberg-Moore type spectral sequence (see Theorem 2.22) E1pq = M  Extα (R(−p) Ei+n+p , F ) ⊗ Extβ (G, Ei+n+p ) α+β=q pq p+r,q−r+1 dpq , r : Er −→ Er  0 ≤ q ≤ n, −n ≤ p ≤ 0 E∞ ⇒ Extp+q (G, F ) and we apply it to the case i = −m − n and G = OQn (t − m − λ − 1). So, in the square 0 ≤ q ≤ n, −n ≤ p ≤ 0 we have  M  pq α β (−p) Ext (R E−m+p , F ) ⊗ Ext (OQn (t − m − λ − 1), E−m+p ) . E1 = α+β=q 20 L. COSTA, R.M. MIRÓ-ROIG Since F is m-regular with respect to σ, Extα (R(−p) E−m+p , F ) = 0 −n ≤ p ≤ 0. On the other hand, if 0 ≤ r ≤ n − 1, then  β  H (Qn , OQn (2 − t + p)) Extβ (OQn (t − m − λ − 1), E−m+p) = H β (Qn , Σ(−t + p + 1))  H β (Q , O (1 − t + p)) n Qn for all α > 0 and all p, if − n ≤ p ≤ −r − 2 if p = −r − 1 if − r ≤ p ≤ 0, and if r = n, then ( H β (Qn , OQn (1 − t + p)) if − n ≤ p ≤ −1 Extβ (OQn (t − m − λ − 1), E−m+p ) = H β (Qn , Σ(−t)) if p = 0. Therefore, applying Serre’s duality and Lemma 4.1, we get that the only non-zero Ext’s groups Extβ (OQn (t − m − λ − 1), E−m+p )) correspond to β = 0 or β = n and in the latter case −r ≤ p ≤ 0 and p ≤ t − 1 − n or −n ≤ p ≤ −r − 1 and p ≤ t − 2 − n. Thus, E1pq 6= 0 pq = 0 for p + q ≥ t and so for any t > 0, forces p + q ≤ t − 1. Therefore, E∞ Extt (OQn (t − m − λ − 1), F ) = H t (Qn , F (m + λ + 1 − t)) = 0. In order to prove the other inequality, we will see that if i∗ F is (m + λ)-regular in the −m sense of Castelnuovo-Mumford, then F is m-regular with respect to σ, where λ := ⌊ n+1 ⌋. If i∗ F is m-regular, then it is also (m + γ)-regular for all γ ≥ 0 and we have (4.4) 0 = H i (Pn+1 , i∗ F (t)) = H i (Qn , F (t)), for all t ≥ m + λ − i, i > 0. We write −m = λ(n + 1) + r with 0 ≤ r ≤ n. If r 6= n then σ−m−n is given by (OQn ((λ − 1)n + r + 1), · · · , OQn ((λ − 1)n + n − 1), Σ((λ − 1)n + n − 1), OQn (λn), OQn (λn + 1), · · · , OQn (λn + r)), and by Proposition 4.2 its right dual base is given by (OQn (λn + r), ψ1∗(λn + r), · · · , ψr∗ (λn + r), Σ(λn + r), ψn−r−2(λn + r + 1), · · · , ψ0 (λn + r + 1)), and if r = n, σ−m−n is equal to (OQn (λn), OQn (λn + 1), . . . , OQn (λn + n − 1), Σ(λn + n − 1)) and by Proposition 4.2 its right dual base is given by (Σ(n − 1 + λn), ψn−1 ((λ + 1)n), ψn−2 ((λ + 1)n), · · · , ψ1 ((λ + 1)n), ψ0 ((λ + 1)n)). So, according to Definition 3.1, we have to see that for any q > 0 (i) (ii) (iii) (iv) Extq (OQn (λn + r), F ) = H q (Qn , F (m + λ)) = 0 1≤α≤r Extq (ψα∗ (λn + r), F ) = Extq (ψα∗ , F (m + λ)) = 0 q q Ext (ψn−β (λn + r + 1), F ) = Ext (ψn−β , F (m + λ − 1)) = 0 r + 2 ≤ β ≤ n Extq (Σ(λn + r), F ) = Extq (Σ, F (m + λ)) = 0 if r 6= n and we have to see that (v) Extq (ψn−α ((λ + 1)n), F ) = Extq (ψn−α , F (m + λ)) = 0 1 ≤ α ≤ n (vi) Extq (Σ(λn + n − 1), F ) = Extq (Σ, F (m + λ + 1)) = 0 GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 21 if r = n. From (4.4) we immediately get (i). Equalities (ii), (iii) and (v) are consequence of the following stronger results (4.5) Extα (ψj , F (t)) = 0 for any α > 0, 0 ≤ j ≤ n, and t ≥ m + λ − α, Extα (ψj∗ , F (t)) = 0 for any α > 0, 1 ≤ j ≤ r, and t ≥ m + λ − α. and (4.6) We will prove (4.5) by induction on j. Analogously, we can prove (4.6) and we right it to the reader. If j = 0, it follows from (4.4) that Extα (ψ0 , F (t)) = H α (Qn , F (t)) = 0 for all α > 0 and t ≥ m + λ − α. Assume j = 1. Restricting to Qn the Euler sequence and applying the functor Ext(·, F (t)) we get the exact sequence n+2 · · · → Extα (OQ , F (t)) → Extα (ψ1 , F (t)) → Extα+1 (OQn (1), F (t)) → · · · . n By (4.4), Extα+1 (OQn (1), F (t)) = H α+1 (Qn , F (t − 1)) = 0 for t − 1 ≥ m + λ − α − 1 and n+2 Extα (OQ , F (t)) = 0 for all α > 0 and t ≥ m + λ − α. Hence n (4.7) Extα (ψ1 , F (t)) = 0 for all α > 0 and t ≥ m + λ − α. Before going ahead with the general case, let us prove the following Claim: For any p ≥ 1, α > 0 and t ≥ m + λ − α, Extα (Ωp (p)|Qn , F (t)) = 0. Proof of the Claim: We will prove it by induction on p. If p = 1, by (4.7), Extα (Ω1 (1)|Qn , F (t)) = Extα (ψ1 , F (t)) = 0 for any α > 0 and t ≥ m + λ − α. Assume it holds for p and let us see the case p + 1. Applying the functor Ext(·, F (t)) to the exact sequence (n+2) 0 → Ωp+1 (p + 1)|Qn → OQp+1 → Ωp (p + 1)|Qn → 0 n obtained by restricting to Qn the p-th exterior power of the dual of the Euler sequence on Pn+1 , we get the long exact sequence (n+2) · · · → Extα (OQp+1 , F (t)) → Extα (Ωp+1 (p+1)|Qn , F (t)) → Extα+1 (Ωp (p+1)|Qn , F (t)) → · · · . n By hypothesis of induction, Extα+1 (Ωp (p+1)|Qn , F (t)) = 0 for any t−1 ≥ m+λ−α−1 and (n+2) , F (t)) = 0 for any t ≥ m+λ−α. Hence, Extα (Ωp+1 (p+1)|Qn , F (t)) = by (4.4), Extα (OQp+1 n 0 for any t ≥ m + λ − α and α > 0 which finishes the proof of the Claim. Let us now prove (4.5) by induction on j. The first two cases have been already done. Assume (4.5) holds for j ≥ 1 and let us see that it holds for j + 1. Applying the functor Ext(·, F (t)) to the exact sequence 0 −→ Ωj (j)|Qn −→ ψj −→ ψj−2 −→ 0 we get the long exact sequence · · · → Extα (ψj−1 , F (t)) → Extα (ψj+1 , F (t)) → Extα (Ωj+1 (j + 1)|Qn , F (t)) → · · · . 22 L. COSTA, R.M. MIRÓ-ROIG It follows from the Claim that Extα (Ωj+1 (j + 1)|Qn , F (t)) = 0 for all α > 0 and t ≥ m + λ − α and by hypothesis of induction Extα (ψj−1 , F (t)) = 0 for all α > 0 and t ≥ m + λ − α. Hence we obtain Extα (ψj+1 , F (t)) = 0 for any α > 0 and t ≥ m + λ − α, which finishes the proof of (ii), (iii) and (v). Finally, t+1 using the fact that ψn = Σ(−1)2 , we deduce from (4.5) that (iv) and (vi) also hold.  Now we will show that Theorem 4.3 is optimal, i.e., both inequalities can be realized. First, we will compare Regσ (Ej ) to Reg CM (i∗ Ej ) where Ej is part of the helix Hσ = {Ej }j∈Z associated to σ and later we will compare Regσ (ψ1 (3 + λn)) to Reg CM (i∗ (ψ1 (3 + λn))). Proposition 4.4. Let n be an odd integer and let i : Qn ֒→ Pn+1 be a quadric hypersurface. Let Hσ = {Ej }j∈Z be the strict helix associated to σ = (OQn , · · · , OQn (n − 1), Σ(n − 1)). For any integer j ∈ Z, we write j = λ(n + 1) + r with 0 ≤ r ≤ n. Then, we have Regσ (Ej ) = −j Reg CM (i∗ Ej ) = −j + λ + 1. In particular, the following relation holds: Regσ (Ej ) + λ + 1 = Reg CM (i∗ Ej ). Proof. By remark 2.14, if 0 ≤ r ≤ n − 1, we have Ej = Eλ(n+1)+r = Er ⊗ (K ∗ )⊗λ = OQn (λn + r) and if r = n, then we have Ej = Eλ(n+1)+n = En ⊗ (K ∗ )⊗λ = Σ(λn + n − 1). Using the exact sequence 0 → OQn (−2) → OQn → i∗ OQn → 0 we get that H q (Pn+1 , i∗ OQn (t)) = 0 for all t ∈ Z, 1 ≤ q ≤ n − 1 and q = n + 1 H n (Pn+1 , i∗ OQn (t)) = 0 for all t ≥ −n + 1, and we conclude that Reg CM (i∗ OQn (s)) = −s + 1. In particular, if j = λ(n + 1) + r with 0 ≤ r ≤ n − 1, we have Reg CM (i∗ Ej ) = Reg CM (i∗ OQn (λn + r)) = −λn − r + 1 = −j + λ + 1 = Regσ (Ej ) + λ + 1 where the last equality follows from Proposition 3.5. So, it only remains the case j = λ(n + 1) + n. Using the isomorphism H q (Pn+1 , i∗ Σ(t)) ∼ = H q (Qn , Σ(t)) GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY and Lemma 4.1; (i) − (ii), we obtain that Reg λ(n + 1) + n, we have CM 23 (i∗ Σ(t)) = −t. Therefore, if j = Reg CM (i∗ Ej ) = Reg CM (i∗ Σ(λn + n − 1)) = −λn − n + 1 = −j + λ + 1 = Regσ (Ej ) + λ + 1 where again the last equality follows from Proposition 3.5.  Proposition 4.5. With the above notations, we have: (i) Reg CM (i∗ (ψ1 (3 + λn))) = −2 − λn. (ii) Regσ (ψ1 (3 + λn)) = −2 − λ(n + 1). Proof. (i) It follows from the fact that H i (Qn , ψ1 (1 − i)) = 0 for all i ≥ 1 and H 1 (Qn , ψ1 (−1)) ∼ = C. (ii) Since by Proposition 4.2 the right dual basis of σ2+λ(n+1)−n is (OQn (λn + 2), ψ1∗ (λn + 2), ψ2∗ (λn + 2), Σ(λn + 2), ψn−4 (λn + 3), · · · , ψ0 (λn + 3)) and the right dual basis of σ3+λ(n+1)−n is (OQn (λn + 3), ψ1∗ (λn + 3), ψ2∗(λn + 3), ψ3∗(λn + 3), Σ(λn + 3), ψn−5(λn + 4), · · · , ψ0 (λn + 4)) we easily check (taking into account that ψ0 (λn+4) ∼ = OQn (λn+4) and that H 1 (Qn , ψ1 (−1)) = C) that ψ1 (3 + λn) is (−2 − λ(n + 1))-regular with respect to σ but not (−3 − λ(n + 1))regular with respect to σ.  5. Final comments In this section, we will first gather the problems that arose from this paper and we will end with some final remarks. Problem 5.1. To characterize smooth projective varieties which have a geometric collection. Let X be a smooth projective variety of dimension n. We know that a necessary condition for having a geometric collection is that the rank of the Grothendieck group K0 (X) is n+1 and X is Fano. We would like to know if these conditions are also sufficient. The main goal of this paper was to generalize the notion of Castelnuovo-Mumford regularity for coherent sheaves on a projective space to coherent sheaves on other smooth projective varieties X. We have succeed provided X has a geometric collection of coherent sheaves. It would be nice to extend the definition to the case of smooth projective varieties which have a full strongly exceptional collection of sheaves (not necessarily geometric). Hence, we propose: Problem 5.2. Let X be a smooth projective variety and let F be a coherent sheaf on X. To extend the definition of regularity of F with respect to a geometric collection σ to regularity of F with respect to a full strongly exceptional collection. 24 L. COSTA, R.M. MIRÓ-ROIG It is well known that Beilinson’s theorem ([1]) and Castelnuovo-Mumford regularity of sheaves play a fundamental role in the classification of vector bundles on projective spaces. 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Rudakov et al. Helices and vector bundles: Seminaire Rudakov Lecture Note Series, 148 (1990) Cambridge University Press. [23] A.N. Rudakov, Exceptional vector Bundles on a Quadric, Math. Izv. 33 (1989) 115-138. GEOMETRIC COLLECTIONS AND CASTELNUOVO-MUMFORD REGULARITY 25 Facultat de Matemàtiques, Departament d’Algebra i Geometria, Gran Via de les Corts Catalanes 585, 08007 Barcelona, SPAIN E-mail address: costa@ub.edu, miro@ub.edu