SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Let X be a smooth scheme of finite type over a fied K, et E be a ocay free O X -bimodue of rank n, and et A be the non-commutative symmetric agebra generated by E. We construct an interna Hom functor, Hom GrA (, ), on the category of graded right A-modues. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of Hom GrA (O X, ). When X is a smooth projective variety, we prove a version of Serre Duaity for ProjA using the right derived functors of im Hom n GrA (A/A n, ). Contents 1. Introduction 3 1.1. Notation 5 2. Non-commutative Symmetric Agebras and P 1 -bundes 7 2.1. Preiminaries 7 2.2. The agebra A 9 2.3. Generators for D(ProjA) 12 3. Interna Tensor and Hom functors on GrA 15 3.1. The interna Tensor functor on GrA 15 3.2. The interna Hom functor on GrA 25 3.3. Adjointness of interna Hom and Tensor 30 3.4. Eementary properties of interna Hom 34 3.5. Torsion 36 4. Cohomoogy 39 4.1. Eementary properties of interna Ext 39 4.2. A is Gorenstein 40 4.3. Appication to Non-commutative P 1 -bundes 43 5. Serre duaity 46 5.1. Quotient categories and derived functors 46 5.2. Serre duaity for quotient categories 48 6. Compatibiities 51 7. Proofs 57 7.1. Proposition 3.11, associativity 57 7.2. Proposition 3.11, scaar mutipication 59 7.3. Proposition 3.13(1), is a morphism in GrA 62 7.4. Proposition 3.13(1), is a morphism in GrA 64 7.5. Theorem 4.4, d 1 is an epimorphism 66 Key words and phrases. Non-commutative geometry, Serre duaity, Non-commutative projective bunde. 2000 Mathematica Subject Cassification. Primary 14A22; Secondary 16S99. 1

2 ADAM NYMAN References 72

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 3 1. Introduction Athough the cassification of non-commutative surfaces is nowhere in sight, some casses of non-commutative surfaces are reativey we understood. For exampe, non-commutative deformations of the projective pane and the quadric in P 3 have been cassified in [1] and [20], respectivey. Non-commutative P 1 -bundes over curves have not been studied as extensivey as non-commutative P 2 s or non-commutative quadrics, but certainy pay a prominent roe in the theory of non-commutative surfaces. For exampe, certain noncommutative quadrics are isomorphic to non-commutative P 1 -bundes over curves [20]. In addition, every non-commutative deformation of a Hirzebruch surface is given by a non-commutative P 1 -bunde over P 1 [19, Theorem 7.4.1, p. 29]. Quotients of four-dimensiona Skyanin agebras by centra homogeneous eements of degree two provide important exampes of coordinate rings of non-commutative P 1 -bundes over P 1 [17, Theorem 7.2.1]. The purpose of this paper is to prove a version of Serre duaity for non-commutative P 1 -bundes over smooth projective varieties of dimension d over a fied K. Before describing this resut in more detai, we review some important notions from noncommutative agebraic geometry. If X is a quasi-compact and quasi-separated scheme, then ModX, the category of quasi-coherent sheaves on X, is a Grothendieck category. This eads to the foowing generaization of the notion of scheme, introduced by Van den Bergh in order to define a notion of bowing-up in the non-commutative setting. Definition 1.1. [18] A quasi-scheme is a Grothendieck category ModX, which we denote by X. X is caed a noetherian quasi-scheme if the category ModX is ocay noetherian. X is caed a quasi-scheme over K if the category ModX is K-inear. If R is a ring and ModR is the category of right R-modues, ModR is a quasischeme, caed the non-commutative affine scheme associated to R. If A is a graded ring, GrA is the category of graded right A-modues, TorsA is the fu subcategory of GrA consisting of direct imits of right bounded modues, and ProjA is the quotient category GrA/TorsA, then ProjA is a quasi-scheme caed the non-commutative projective scheme associated to A. If A is an Artin-Scheter reguar agebra of dimension 3 with the same hibert series as a poynomia ring in 3 variabes, ProjA is caed a non-commutative P 2. These definitions motivate the foowing Definition 1.2. [19] Suppose X is a smooth scheme of finite type over K, E is a ocay free O X -bimodue of rank 2 and A is the non-commutative symmetric agebra generated by E [19, Section 5.1]. Let GrA denote the category of graded right A-modues, et TorsA denote the fu subcategory of GrA consisting of direct imits of right-bounded modues, and et ProjA denote the quotient of GrA by TorsA. The category ProjA is a non-commutative P 1 -bunde over X. Athough no generay accepted definition of smooth non-commutative surface exists yet, it seems reasonabe to insist that such an object shoud be a noetherian quasi-scheme of cohomoogica dimension 2. Whie an intersection theory for noncommutative surfaces exists ([7], [12]), it has yet to be appied to the study of non-commutative P 1 -bundes over curves. Mori shows [12, Theorem 3.5] that if Y is a noetherian quasi-scheme over a fied K such that

4 ADAM NYMAN (1) Y is Ext-finite, (2) the cohomoogica dimension of Y is 2, and (3) Y satisfies Serre duaity then there is an intersection theory on Y such that the Riemann-Roch theorem and the adjunction formua hod. We prove a version of Serre duaity for non-commutative P 1 -bundes over smooth projective varieties of dimension d over K. In the course of the proof, we show that such a P 1 -bunde has cohomoogica dimension d + 1. Thus, the second and third items on the ist above are verified for non-commutative P 1 -bundes over smooth projective curves. The author is currenty using the techniques of this paper to prove the first item on the ist above hods for non-commutative P 1 -bundes. We now describe the main resuts of this paper. Suppose A is as in Definition 1.2, π : GrA ProjA is the quotient functor, ω is right adjoint to π, and τ : GrA GrA is the torsion functor. For any coection {C ij } i,j Z of O X -bimodues, et e i C denote the O X -bimodue C ij. j We prove the foowing version of Serre duaity (Theorem 5.20): Theorem 1.3. If X is a smooth projective variety of dimension d over K, there exists an object ω A in ProjA such that for 0 i d + 1 there is an isomorphism Ext i ProjA(π(O X e 0 A), M) = Ext d+1 i ProjA (M,ω A) natura in M. The prime denotes duaization with respect to K. To prove this theorem, we appy the Brown representabiity theorem foowing Jörgenson [9]. In order to appy the Brown representabiity theorem to prove our version of Serre duaity, we need to prove technica resuts (Theorem 4.13 and Theorem 4.16) regarding the cohomoogy of the functor (1) Hom ProjA (π(o(i) e m A), ). Our strategy for studying the cohomoogy of (1) is indirect. We construct a bifunctor, Hom GrA (, ) (Definition 3.7) whose eft input is a ocay free A A bimodue and whose right input and output are graded right A-modues. We prove the functor Hom GrA (, ) enjoys the expected properties: Hom GrA (A, ) = id GrA, Hom OX (L, Hom GrA (C, M) m ) = Hom GrA (L e m C, M) for an quasi-coherent O X -modue L, and τ( ) = im n Hom GrA (A/A n, ) (Propositions 3.15, 3.10 and 3.19, respectivey). Since A is not a sheaf of agebras, Hom GrA (, ) cannot be defined ocay. Instead, we use the natura categorica definition. We can use the first two properties above to find an aternative description for (1): Hom ProjA (π(o(i) e m A), ) = Hom GrA (O(i) e m A,ω( )) = Hom OX (O(i), Hom GrA (A,ω( )) m ) = Hom OX (O(i),(ω( )) m ) = Γ O( i) (ω( )) m

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 5 where Γ : ModO X ModΓ(X, O X ) is the goba sections functor. Since R i ω = (R i+1 τ)π for i 1 (Theorem 4.11), the cohomoogy of (1) is thus reated to the cohomoogy of τ. By the third property of Hom GrA (, ) above, the cohomoogy of (1) is thus reated to the cohomoogy of im Hom n GrA (A/A n, ). In order to compute the cohomoogy of im Hom n GrA (A/A n, ), we use the foowing theorem of Van den Bergh: Theorem 1.4. [19, Theorem 7.1.2] Let O X denote the trivia right A-modue. The sequence of O X -A bimodues (2) 0 Q e m+2 A E e m+1 A e m A O X 0 whose maps come from the structure of the reations in A, is exact. In order to use this theorem, we prove, using a variant of the first property of Hom GrA (, ) above, that each term in (2) to the eft of O X is Hom GrA (, M) m - acycic. We may thus use (2) to compute the cohomoogy of Hom GrA (O X, ). In fact, we prove that A is Gorenstein (Theorem 4.4): Theorem 1.5. Let L be a coherent, ocay free O X -modue. Then and Ext i GrA (O X, L e A) = 0 for i 2 { Ext 2 GrA (O X, L e A) j L Q = 2 if j = 2, 0 otherwise where Q 2 is the image of the unit map : O X A 2, 1 A 2, 1 (See Section 2.1 for the definition of ). Using this resut, we compute the cohomoogy of the imit im Hom n GrA (A/A n, ) which, in turn, aows us to prove the technica resuts regarding (1) we need to prove Serre duaity. 1.1. Notation. Suppose we are given a diagram of categories, functors, and natura transformations between functors: F X G Y Θ Z. F The horizonta composition of and Θ, denoted Θ is defined, for every object X of X, by the formua (Θ ) X = Θ F X G( X ) = G ( X ) Θ FX. Suppose A is an abeian category. If C is a ocaizing subcategory, we et π : A A/C denote the quotient functor, ω : A/C A denote the section functor and τ : A C denote the torsion functor. We et Ch(A), K(A) and D(A) denote the category of cochain compexes of objects of A, cochain compexes of objects of A with morphisms the chain homotopy equivaence casses of maps between compexes, and the derived category of A, respectivey. We wi sometimes just write Ch, K and D. We wi write Q : K D for the ocaization functor. If F : A B is a eft exact functor between abeian G

6 ADAM NYMAN categories whose derived functor exists, we wi write RF for the derived functor of F. Throughout, we et X denote a smooth scheme of finite type over a fied K, and we et ModX denote the category of quasi-coherent O X -modues.

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 7 2. Non-commutative Symmetric Agebras and P 1 -bundes 2.1. Preiminaries. We remind the reader of some definitions from [19]. A coherent O X -bimodue, E, is a coherent O X 2-modue whose support over both copies of X is finite. An O X -bimodue is a direct imit of coherent O X -bimodues. For i = 1,2 and 1 j < 3, et pr i : X 2 X and pr ij : X 3 X 2 denote the standard projections. Let d : X X X be the diagona map and et O = d O X. If E and F are O X -bimodues, define the tensor product by E OX F := pr 13 (pr 12E OX 3 pr 23F). If M is an O X -modue, define M OX E by pr 2 (pr 1M E). In what foows, we wi drop the subscript on the tensor product. Let O : E O E and O : O E E denote the eft and right scaar mutipication morphisms (see [14, Proposition 3.7, p. 35] for the definitions). A coherent O X -bimodue E is said to be ocay free of rank n if pr i E is ocay free of rank n for i = 1,2. We sha use the foowing resut without comment in the seque. Proposition 2.1. [19, p. 6] If E is a ocay free O X -bimodue of rank n, there exists a ocay free O X -bimodue E, the dua of E, of rank n and natura transformations E : id ModX ( E) E and E : ( E ) E id ModX such that ( E, E, E, E ) is an adjunction. Consider the composition of functors (3) O id ModX E ( E)( E ) (E E ) whose eft arrow is induced by scaer mutipication and whose right arrow is induced by the associativity isomorphism ([17, Propostion 2.5, p. 442]). By [19, Lemma 3.1.1, p.4], this morphism of functors corresponds to a morphism O E E. We wi often abuse notation by referring to this morphism as. Simiary, there is a morphism : E E O induced by E. the maps and are monomorphisms and epimorphisms, respectivey [19, Proposition 3.1.1, p. 7] Let E = E and et E = E. By Proposition 2.1, the compositions E E E id ModX EE E E E id ModX E and id ModX E E E E EE E E E id ModX equa E and E respectivey. As a consequence, one can show that the compositions and E E O 1 O E E (E E) O 1 E O E (E E ) E E E O = O E E E (E E ) = (E E) E E O E O E whose centra isomorphisms are associativity isomorphisms, equa E and E respectivey.

8 ADAM NYMAN Remark: In what foows, we wi abuse the notation we have introduced above. Suppose E and F are ocay free finite rank O X -bimodues. As an exampe of the abuse which foows, we wi write instead of E = O E E E E E (E F F ) E E E (E E ) E ((E ) E ) E ((E (F F )) E ) E. We wi aso use, without further comment, the fact that the category of O X - bimodues together with the tensor product forms a monoida category. Hence, the coherence theorem for monoida categories hods, so that any diagram whose arrows are associativity isomorphisms commutes. As another exampe of the notationa abuse we wi be guity of, if φ : E F, we wi write E E φ F E instead of E E φ E F E. In addition, we wi sometimes omit tensor product symbos, and wi write O X instead of O where no confusion arises. If E and F are ocay free, finite rank O X -bimodues, there is an isomorphism (E F) ( E) F of functors from ModX to ModX ([17, Propostion 2.5, p. 442]). Thus, there exists a canonica choice of unit, and counit,, making ( (E F),( E ) F,,) an adjunction, and there is a canonica isomorphism φ : ( F ) E (E F) (see Lemma 6.1 for more detais). The composition (F E ) ( F ) E φ (E F) whose eft arrow is the associativity isomorphism, induces an isomorphism (4) F E (E F) by [19, Lemma 3.1.1, p.4]. Definition 2.2. If g : C D is a morphism of coherent, ocay free O X -bimodues, the dua of g, g : D C, is the composition D D C C D g C D D C C. We wi need the foowing resut to prove Proposition 3.6. Lemma 2.3. Let i α O X Q E E be a factorization of the unit : O X E E, where Q is the image of in E E. Let δ : Q E E denotes the isomorphism Then the diagram Q E O 1 OQ E E i QQ E QQ E α 1 E. commutes. EE E δ EE Q E

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 9 Proof. It suffices to show the outer circuit in the diagram O X EE QQ α EE Q i 1 EE Q O 1 EE OQ commutes. The eft square commutes by functoriaity of the tensor product. To prove the right square commutes, it suffices to prove QQ i 1 OQ O 1 OQQ EE Q EE QQ i 1 EE OQ commutes, where we have used our convention that denotes the composition QQ Thus, it suffices to show : QQ EE QQ O 1 OQQ QQ EE QQ. Q i 1 O EE O 1 O 1 1 O OQ i 1 OO EE O commutes. Since O : OO O equas O : OO O, the diagram commutes by the functoriaity of. 2.2. The agebra A. We now review the definition of non-commutative symmetric agebra, and we study some of its basic properties. Definition 2.4. [19, Section 4.1] Let E be a ocay free O X -bimodue. A noncommutative symmetric agebra in standard form generated by E is the sheaf Z-agebra A ij with components i,j Z A ii = O A i,i+1 = E i, A ij = A i,i+1 A j 1,j /R ij for j > i + 1, where R ij A i,i+1 A j 1,j is the O X -bimodue j 2 Σ k=i A i,i+1 A k 1,k Q k A k+2,k+3 A j 1,j, and Q i is the image of the unit map O A i,i+1 A i+1,i+2, and A ij = 0 if i > j

10 ADAM NYMAN and with mutipication defined as foows: for i < j < k, A ij A jk = A i,i+1 A j 1,j A j,j+1 A k 1,k R ij R jk A i,i+1 A k 1,k = R ij A j,j+1 A k 1,k + A i,i+1 A j 1,j R jk by [14, Coroary 3.18, p.38]. On the other hand, R ik = Rij A j,j+1 A k 1,k + A i,i+1 A j 1,j R jk + A i,i+1 A j 2,j 1 Q j 1 A j+1,j+2 A k 1,k. Thus there is an epi ijk : A ij A jk A ik. If i = j, et ijk : A ii A ik A ik be the scaar mutipication map O : O A ik A ik. Simiary, if j = k, et ijk : A ij A jj A ij be the scaar mutipication map O. Using the fact that the tensor product of bimodues is associative, one can check that mutipication is associative. We define e k A k+n to be the sum of O X 2-modues e k A k+n+i and A n = i (e k A) k+n. We define A n simiary. If A ij = 0 for i > j, we write A n instead k of A n. Remark: Instead of writing ijk : A ij A jk A ik, we wi write. Furthermore, if M is a right A-modue, we wi sometimes write for every component of its mutipication. We wi use the fact that A ij is ocay free of rank j i + 1 [19, Theorem 7.1.2] without further comment. The proof of the foowing resut is simiar to the proof of [3, Proposition 7, p.18], so we omit it. Lemma 2.5. Let B, B, C and C be O X -bimodues such that B B and C C. Suppose, further, that either B/B is ocay free or C/C is ocay free. If (B C ) (B C) denotes the appropriate puback, the natura morphism is an isomorphism. B C (B C ) (B C) We wi need the foowing emma to prove Proposition 3.6 and Theorem 4.4. Lemma 2.6. If α : Q k A k,k+1 A k+1,k+2 denotes the incusion map, the composition is monic. α A k+1,k+2 A k Q k Ak A k,k+1 A k+1,k+2 Proof. We need to show that the puback of the diagram A k Q k α ker A k+1,k+2 A k A k,k+1 A k+1,k+2 A,k+1 A k+1,k+2

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 11 of incusions is trivia. Letting R = R k A k,k+1 A k+1,k+2, this is the same as showing, by Lemma 2.5, that the puback of the diagram R+A,+1 A k 2,k 1 Q k 1 A k+1,k+2 R R+A,+1 A k 1,k Q k R A,+1 A k,k+1 A k+1,k+2 R of incusions is trivia. Since tensoring with a ocay free O X -bimodue is exact, it suffice to show that the puback of the diagram (5) Q k 1 A k+1,k+2 A k 1,k Q k A k 1,k A k,k+1 A k+1,k+2 of incusions is trivia. However, since the sequence 0 A k 1,k Q k +Q k 1 A k+1,k+2 A k 1,k A k,k+1 A k+1,k+2 Ak 1,k+2 0 whose eft arrow is incusion is exact, A k 1,k Q k + Q k 1 A k+1,k+2 is ocay free of rank 4. Thus, since the puback of (5), P, is the kerne of a short exact sequence 0 P Q k 1 A k+1,k+2 A k 1,k Q k Q k 1 A k+1,k+2 + A k 1,k Q k 0 P is ocay free of rank 0, i.e., P = 0 as desired. We now present a definition and Lemma which wi be usefu in the proof of Lemma 3.18. Reca that the objects of GrA are sums M i of O X -modues with i Z a graded right A-modue structure. Definition 2.7. Let M be an object of GrA. The submodue of M generated by M i, M i, is the graded A-modue with M i k = im ik and with mutipication defined as foows. On the diagram im ik A kj M i A ik A kj M i A ij M k A kj = M k A kj M j cok ij whose eft vertica is induced by incusion, whose bottom right-most horizonta is the cokerne of ij and whose other unabeed maps are mutipication maps, consider the path starting at the top center and continuing eft. Since the right square commutes, this path is 0. Since the upper eft horizonta is an epimorphism, the path starting at the upper eft is 0. By the universa property of the cokerne, there is a morphism M i k A kj = im ik A kj im ij denoted i jk. The coection {i kj } k j gives M i a right A-modue structure.

12 ADAM NYMAN Lemma 2.8. Let M be an object of GrA. For i j and for k Z, Let (φ ij ) k : im ik im jk be the canonica morphism induced by the associativity diagram M i A ij A jk M i A ik M j A jk M k whose arrows are mutipication. Then φ ij : M i M j is a morphism of right A-modues making {M i,φ ij } a direct system. Furthermore, the direct imit of this system is M. 2.3. Generators for D(ProjA). The purpose of this section is to construct a set of generators for the derived category of a non-commutative P 1 -bunde ProjA, D(ProjA). We wi ater prove (Theorem 4.16, Proposition 5.12) these generators are compact. In order to construct a set of generators for the category D(ProjA), we must first study generators of the category GrA. Definition 2.9. A ocay noetherian category is a Grothendieck category with a set of noetherian generators. Lemma 2.10. [16, Exercise 5.4, p.56] In a ocay noetherian category, every finitey generated object is noetherian. Lemma 2.11. If C is ocay noetherian, any object M of C is the direct imit of its noetherian subobjects. We remind the reader that an object in GrA is torsion if it is a direct imit of right-bounded objects. Lemma 2.12. If GrA is ocay noetherian, an essentia extension of a torsion modue in GrA is torsion. Proof. Suppose M is torsion, and et F be an essentia extension of M. Since GrA is ocay noetherian, F is a direct imit of its noetherian subobjects by Lemma 2.11. Let N be a noetherian subobject of F. Then N M is torsion, hence rightbounded since N is noetherian. Thus N n M = 0 for n >> 0, whence N is right-bounded. Lemma 2.13. If GrA is ocay noetherian and M is an object in GrA, M is torsion if and ony if every noetherian subobject is right bounded. Proof. Since GrA is ocay noetherian, any object M in GrA is the direct imit of its noetherian subobjects by Lemma 2.11. Thus, if every noetherian subobject is right bounded, M is torsion. Conversey, suppose M is torsion. Since TorsA is cosed under subquotients, every noetherian subobject N of M is torsion. Since N is torsion, it is a direct imit of its right bounded submodues. On the other hand, since N is noetherian, it is a direct imit of finitey many of its right bounded submodues. Thus, N is right bounded. Coroary 2.14. If GrA is ocay noetherian, M is an object of GrA which is not torsion and M Z is given, there exists an m M and a noetherian subobject N of M generated in degree m which is not torsion.

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 13 Proof. Since M is not torsion, Lemma 2.13 impies that M has a noetherian subobject N which is not right bounded. If, for a m M, N m were torsion, then since N m is noetherian, N m woud be right bounded. This contradicts the fact that N is not right bounded. The foowing emma is a variant of [17, Lemma 3.4, p. 450]. Lemma 2.15. Let M be an O X -modue, et N be a graded A-modue and et u : A kk e k A be the incusion morphism. Then there is an isomorphism Hom GrA (M e k A, N) Hom OX (M, N k ) given by sending f Hom GrA (M e k A, N) to the composition M = M A kk id u M e k A f N The inverse is given by sending g Hom OX (M, N k ) to the composition M e k A g e ka N k e k A N. Lemma 2.16. Let N be an object of ModX. If N is noetherian, N e k A is a finitey generated graded A-modue. Proof. Suppose there is an epimorphism i I M i N e k A of graded A-modues with I countabe (the notation M i here is the ith A-modue, not the ith graded piece of a graded A-modue M). If we identify N with N A kk and M i with its image in N e k A, we find that i I M i N = N. Since N is noetherian, there exists a finite set I I such that I M i N = N. Thus, the restriction i I M i N e k A is an epimorphism of graded A-modues, as desired. Lemma 2.17. Let X be a scheme such that ModX is ocay noetherian with noetherian generators {M i } i I. If GrA is ocay noetherian, {M i e k A} i I,k Z is a set of noetherian generators of GrA. Proof. By Lemmas 2.16 and Lemma 2.10, M i e k A is noetherian. By Lemma 2.15, (6) Hom OX (M i, N k ) = Hom GrA (M i e k A, N). Thus, if N = 0, then the eft hand side of (6) is nonzero for some i I,k Z since {M i } i I generates ModX. Thus, the right hand side of (6) is nonzero for some i I,k Z. Definition 2.18. A category B satisfies (Gen) with a set of objects {B i,m } i,m Z (in B) if, for any C in D(B) with h n C 0 then Hom D(B) (B i,m { n}, C ) 0 for i,m >> 0, where { } is the suspension functor in D(B). The foowing resut is a variation of [9, Lemma 2.2, p.712].

14 ADAM NYMAN Proposition 2.19. If GrA is ocay noetherian then ProjA satisfies (Gen) with the coection {π(o X (i) e m A){n}} i,m,n Z. Proof. Let C D(ProjA) have h n C 0. Given M,I Z, we caim there exist m M and i I such that Hom D(ProjA) (O X ( i) e m A{ n}, C ) 0. To prove the caim, we first note that C is a compex over ProjA so D = ωc is a compex over GrA and C = πωc = πd. Since π is exact and h n C 0, h n D is not torsion. By Coroary 2.14 there exists an m M and a noetherian submodue N of h n D generated in degree m which is not torsion. Let ψ be the composition (Z n D ) m δ (h n D ) m (h n D /N) m. whose eft arrow, δ, is the quotient map. The image of the kerne of ψ in (h n D ) m is N m. Since X is noetherian, kerψ is the direct imit of its coherent submodues, and the image of some such submodue, M, under δ must generate a non-torsion A-submodue N of N. By [5, Coroary 5.18, p. 121] there exists an integer i 0 such that for i i 0 there is an epimorphism O X ( i) M N m. k The image of some summand must generate a non-torsion A-submodue N of N. Thus, there is a nonzero morphism (7) O X ( i) (Z n D ) m (h n D ) m whose image equas N m. Tensoring (7) by e m A gives a map of compexes O X ( i) e m A{ n} D whose induced map in cohomoogy has non-torsion image.

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 15 3. Interna Tensor and Hom functors on GrA Definition 3.1. Let BimodA A denote the category of A A-bimodues. Specificay: an object of BimodA A is a tripe (C = {C ij } i,j Z, { ijk } i,j,k Z, {ψ ijk } i,j,k Z ) where C ij is an O X -bimodue and ijk : C ij A jk C ik and ψ ijk : A ij C jk C ik are morphisms of O X 2-modues making C an A-A bimodue. A morphism φ : C D between objects in BimodA A is a coection φ = {φ ij } i,j Z such that φ ij : C ij D ij is a morphism of O X 2-modues, and such that φ respects the A A-bimodue structure on C and D. Let BimodO X A denote the fu subcategory of BimodA A consisting of objects C such that for some n Z, C ij = 0 for i n (we say C is eft-concentrated in degree n). Let B denote the fu subcategory of BimodA A whose objects C = {C ij } i,j Z have the property that C ij is coherent and ocay free for a i,j Z. Let GrA denote the fu subcategory of B consisting of objects C such that for some n Z, C ij = 0 for i n (we say C is eft-concentrated in degree n). In what foows, we usuay omit indices on mutipication morphisms. 3.1. The interna Tensor functor on GrA. Definition 3.2. Let C be an object in BimodA A and et M be a graded right A- modue. We define M A C to be the Z-graded O X -modue whose k-th component is the coequaizer of the diagram (8) (M A m ) C mk m M C m M m C mk M C = M C k = m M m C mk. Let C be an object in BimodA A and et D be an object in BimodO X A eft-concentrated in degree n. We define D A C to be the Z-graded O X 2-modue whose k-th component is the coequaizer of the diagram (9) (D n A m ) C mk m D C m D nm C mk D C = D n C k = m D nm C mk. Since each component of D A C is a quotient of a direct imit of O X -bimodues, each component is an O X -bimodue. Proposition 3.3. If M is an object in GrA, C is an object in BimodA A and D is an object in BimodO X A, then M A C and D A C inherit a graded right A-modue structure from the right A-modue structure of C, making A C : GrA GrA and A C : BimodO X A GrA functors.

16 ADAM NYMAN Proof. We ony prove the first caim. We construct a map (10) ij : (M A C) k A kj (M A C) j, i.e., we construct a map from the coequaizer of (11) to the coequaizer of (12) ((M A m ) C mk ) A kj (M m C mk ) A kj m m = (M C k ) A kj (M m C mk ) A kj = m ((M A m ) C mj M m C mj m m = M C j = m M m C mj. We think of (11) and (12) as the top and bottom, respectivey, of a cube whose vertica arrows are induced by mutipication maps. In order to construct a map (10), it suffices, by the universa property of coequaizers, to prove that the vertica faces of this cube commute. The diagram ((M A m ) C mk ) A kj (M m C mk ) A kj m m (M A m ) C mj M m C mj m m whose arrows are induced by mutipication, commutes by functoriaity of. In addition, the diagram ((M A m ) C mk ) A kj (M C k ) A kj m (M A m ) C mj M C j m whose arrows are induced by mutipication, commutes by the associativity of A Abimodue mutipication. Thus, by the universa property of coequaizers, there exists a map ij : (M A C) k A kj (M A C) j. The fact that is associative and compatibe with scaar mutipication foows from the fact that the right A mutipication on C is associative and compatibe with scaar mutipication. The proof of the functoriaity of A C and M A is straightforward, and we omit it. We now estabish some important properties of A. Lemma 3.4. If L is an object of ModX and D is an object in GrA, there is an isomorphism of right A-modues (L D) A C L (D A C).

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 17 natura in C. Proof. The proof foows from the associativity of the ordinary tensor product of bimodues, and we omit the detais. Proposition 3.5. If M is an object in GrA, mutipication induces a natura isomorphism M A A M in GrA. If C is an object in BimodA A, mutipication induces a natura isomorphism e i A A C e i C. Proof. The proof of the second resut is simiar to the proof of the first resut, so we omit it. We first prove M : M m A mk M k is the coequaizer of k m k (13) k (M A m ) A mk m α= M A k m M m A mk β=m A = k M A k = k m M m A mk which wi prove that mutipication induces an isomorphism M A A M. We wi then prove that this isomorphism is natura. Part 1: We show M : M m A mk M k is the coequaizer of (13). k m k In order to show M : M m A mk M k is the coequaizer of (13), it m suffices to prove that given a diagram (14) k (M A m ) A mk m β k k M A k α k k D k k M = m M m A mk M k M k f g D k k of right A-modues such that fα = fβ, there exists a unique g making the right square commute. Notice that the right square in (14) commutes by associativity of right A-modue mutipication. Let δ k denote the composition M k O 1 M k A kk m M m A mk whose right arrow is incusion, and et δ = δ k. It is easy to see that M δ = id M. Let γ : (M A m ) A mk M m A mk k m k m denote the map β + α δ M β. Part 1, Step 1: We prove the map γ is an epimorphism. To show that γ is an epi, it suffices to construct a map ζ : k m M m A mk k k (M m A m ) A k m such that γζ = id. Let ζ k be induced by the composition A mk O 1 A mk A kk A m A k

18 ADAM NYMAN whose right composite is incusion of a summand. Now, γζ = ( β + α δ M β)ζ = βζ + αζ δ M βζ. We notice that (βζ) k equas the composition 1 O M m A mk m M m (A mk A kk ) M m (A m A k ) m m m M m A mk whose second arrow is incusion in a direct summand. This composition is the identity map. Next, we compute αζ. Since the diagram m M m A mk O 1 m M m (A mk A kk ) m M m (A m A k ) M M A kk α k M k O 1 M k A kk M A k whose top horizontas compose to give ζ k and whose bottom horizontas compose to give δ k, commutes, αζ = δ M. Thus, γζ = βζ = id as desired. Thus γ is an epi. Part 1, Step 2: We prove (15) k (M A m ) A mk m γ β α k M m A mk m = M m A mk k m commutes. Using computations from Step 1, we have M m A mk δ M id k m (δ M id)γ = δ M ( β + α δ M β) ( β + α δ M β) = δ M β + δ M α δ M δ M β + β α + δ M β = δ M β + δ M α δ M β + β α + δ M β = δ M α δ M β + β α since M δ = id. By the commutivity of the right square in (14) δ M α = δ M β, i.e. δ M (α β) = 0. This estabishes the commutivity of (15). Part 1, Step 3: We prove that the map g = fδ makes the right square in (14) commute. We show that g M = f, i.e. fδ M = f, or f(δ M id) = 0. The fact that f(δ M id) = 0 foows from the commutivity of (15) and the fact that γ is an epi. These resuts were estabished in Step 2 and Step 1, respectivey. Part 1, Step 4: We prove g = fδ : M D is unique such that g M = f. Suppose g M = f. Then g = g M δ = fδ = g. We may concude from Part 1 that mutipication M : M m A mk M k k m k induces an isomorphism M A A M

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 19 Part 2: We show that the isomorphism M A A M constructed in Part 1 is natura in M. Let φ : M N be a morphism in GrA and consider the induced diagram k M m A mk m ψ φ m A mk k m N m A mk k m M A A N A A M φ whose bottom verticas are induced by mutipication, whose midde horizonta is induced by the top horizonta, and whose top verticas are the cokernes of the diagram defining A. Notice that the composition of the eft verticas is M and the composition of the right verticas is N. We must show that the bottom square of this diagram commutes. We know the outer square commutes since mutipication is natura, and we know that the top square commutes since the midde horizonta is induced by the top horizonta. This impies that M m A mk k m ψ N M A A M N A A N commutes. Thus, the bottom square of the first diagram commutes since ψ is an epi. Proposition 3.6. Let L be a coherent, ocay free O X -modue and et N be an object of GrA. The incusion A 1 A induces an epi Hom GrA (N A A, L e A) Hom GrA (N A A 1, L e A). Proof. Let φ : N A A 1 L e A be given. It suffices, in ight of Proposition 3.5, to construct an A-modue morphism ψ : N L e A such that (16) N A A 1 L e A = ψ N A A 1 N commutes, where we have abused notation by etting : N A A 1 N denote the morphism induced by mutipication k m<k φ N m A mk N k. k By the definition of the graded tensor product, the map φ k corresponds to a map γ k : N m A mk L A k m<k

20 ADAM NYMAN such that (17) N A (N A m ) A mk m<k N m A mk m<k N A = commutes. N A k <k N m A mk = m<k γk L A,k Step 1: We construct the components of ψ. To define the kth graded component of ψ, notice that the diagram N k Q k N k (A k,k+1 A k+1,k+2 ) N k A k,k+2 γ k+1 A k+1,k+2 γ k+2 L A,k+1 A k+1,k+2 L A,k+2 whose right horizontas are mutipication maps and whose eft horizonta is induced by the incusion Q k A k,k+1 A k+1,k+2 commutes, and hence factors as (18) Let N k Q k N k A k,k+1 A k+1,k+2 N k A k,k+2 γ k+1 A k+1,k+2 γ k+2 ker L A,k+1 A k+1,k+2 L A,k+2. R = k 1 Σ A,+1 A i 1,i Q i A i+2,i+3 A k,k+1 A k+1,k+2. i= By [14, Coroary 3.18, p.38], there is a unique isomorphism ζ : ker L R + L A,+1 A k 1,k Q k L R making the diagram ker L A,k+1 A k+1,k+2 ζ L R+L A,+1 A k 1,k Q k L R L A,+1 A k,k+1 A k+1,k+2 L R whose right vertica is the canonica isomorphism from [14, Coroary 3.18, p.38] and whose horizontas are incusions, commute. Letting R A,+1 A k 1,k Q k denote the appropriate puback, the canonica morphism k 2 Σ i= A,+1 A i 1,i Q i A i+2,i+3 A k 1,k Q k R A,+1 A k 1,k Q k is an isomorphism by Lemma 2.5. Hence, there are isomorphisms (19) R + A,+1 A k 1,k Q k R = A,+1 A k 1,k Q k R A,+1 A k 1,k Q k = A k Q k.

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 21 Thus, the morphism (20) N k Q k ker ζ L R + L A,+1 A k 1,k Q k L R L A k Q k whose rightmost arrow is L tensored with (19), gives a morphism N k Q k L A,k Q k. Tensoring this morphism with Q k gives a morphism N k Q k Q k L A k Q k Q k. Since Q k is invertibe, we thus get a morphism ψ k : N k L A,k. Step 2: Let Q = Q k, E = A k,k+1 and et α : Q E E We show the diagram denote incusion. N k Q ψ k Q α N k E E = N k E E γ k+1 E L A k Q α L A k E E L A,k+1 E commutes. Consider the diagram N k Q α N k EE γ k+1 ker ζ LR+LA,+1 A k 1,k Q LR LA,k+1 E = LA,+1 E LR LA,+1 A k 1,k Q R A,+1 A k 1,k Q LA kq α = LA,k+1 E LA k EE whose two midde two horizontas are induced by incusion, whose upper eft vertica is the eft vertica in (18), whose ower eft vertica and horizonta are induced by (19) and whose other unabeed isomorphisms are both the canonica isomorphisms. Since the eft coumn of this diagram composed with the bottom eft horizonta in the diagram equas ψ k tensored with Q k, it suffices to show the outer circuit of the diagram commutes. Since the top two squares of the diagram commute, it suffices to show the bottom circuit commutes. This foows from the commutivity of the

22 ADAM NYMAN diagram A,+1 A k 1,k Q R + A,+1 A k 1,k Q (21) A,k α A,k EE A,+1 E A,k+1 E whose unabeed arrows are canonica incusions. As the reader can check, the commutivity of (21) foows from the associativity of mutipication. Step 3: Keep the notation as in the previous step, and et δ : Q E E denote the isomorphism defined in Lemma 2.3. We show the diagram (22) N k E ψ k E = N k E γ k+1 L A k E L A,k+1 commutes. We first prove the rectange consisting of the diagram (23) ψ k E N E N QQ α E ψ k QQ E LA k E LA k QQ α N EE Q E = N EE Q E γ k+1 E Q E LA k EE Q E LA,k+1 E Q E to the eft of the diagram (24) N EE Q E γ k+1 E Q E δ N EE E γ k+1 E E N E γ k+1 LA,k+1 E Q E δ LA,k+1 E E LA,k+1 commutes. For, the eft square in (22) commutes by the functoriaity of the tensor product, the right rectange in (22) commutes by Step 2, and the squares in (23) commute by the functoriaity of the tensor product. The top row of the rectange consisting of (23) to the eft of (24) is the identity morphism by Lemma 2.3. Thus, in order to show (22) commutes, it suffices to show the bottom route in the rectange consisting of (23) to the eft of (24) is equa to the mutipication morphism : L A k E L A,k+1. Thus, it suffices to prove that the diagram (25) A k E A k QQ E A,k+1 α A,k+1 E E δ A k EE Q E A,k+1 E Q E

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 23 commutes. By functoriaity of the tensor product, the diagram A k EE Q E δ A k EE E A,k+1 E Q E δ A,k+1 E E commutes. Thus, to prove (25) commutes, it suffices to prove (26) A k E A k A k QQ E α A,k+1 E E A k EE Q E δ A k EE E commutes. By Lemma 2.3, the composition of the first three maps of the top route is induced by the counit : E EE E. Thus, it suffices to show the eft square in A k E A,k+1 A k EE E A k E A,k+1 E E A,k+1 commutes. The right square commutes by functoriaity of the tensor product. Since composed with the top route of the right square is : A k E A,k+1, whie composed with the bottom route of the right square is the top route of the eft square, the eft square commutes. Thus (26), and hence (22), commutes. Step 4: We show ψ is an A-modue morphism. Since (22) commutes by Step 3, the two right hand squares of N k 1 A k 1,k N k N k A k,k+1 A = k,k+1 N k A k,k+1 A k,k+1 γ k ψ k ψ k A k,k+1 A k,k+1 γ k+1a k,k+1 LA,k LA,k = LA,k A k,k+1 A k,k+1 LA,k+1 A k,k+1 commute. Since the bottom horizonta is monic by Lemma 2.6, in order to show the eft square commutes, it suffices to show that (27) N k 1 A k 1,k N k N k A k,k+1 A k,k+1 γ k γ k+1 A k,k+1 L A,k L A,k A k,k+1 A k,k+1 commutes. Since (17) commutes, the diagram N m A m,k 1 A k 1,k N m A mk γ k L A,k+1 A k,k+1 N k 1 A k 1,k γk L A k

24 ADAM NYMAN commutes. Thus N k 1 A k 1,k A k,k+1 N Nk A k,k+1 γ k A k,k+1 γ k+1 L A,k A k,k+1 L A,k+1 commutes. By tensoring this diagram on the right by A k,k+1 and precomposing on the eft with the diagram N k 1 A k 1,k N k 1 A k 1,k A k,k+1 A k,k+1 γ k γ k A k,k+1 A k,k+1 L A,k L A,k A k,k+1 A k,k+1 whose horizontas are counits, it is readiy seen that the eft hand square in (27) commutes. We note that ψ is an A-modue map since the eft hand square in (27) commutes and (22) commutes. Since A is generated in degree 1, the resut foows. Step 5: We show (16) commutes. We must show (28) N m A mk = N m A mk γk N k ψ k L A,k commutes. In order to prove (28) commutes, it suffices to show N m A m,k 1 A k 1,k N m A mk = N m A mk γk N k ψ k L A,k commutes, since the eft horizonta is an epi. Since A-modue mutipication is associative, the diagram N m A m,k 1 A k 1,k N m A mk N k 1 A k 1,k N k whose arrows are mutipication maps, commutes. Since (17) hods, the diagram (29) N m A m,k 1 A k 1,k N m A mk γ k N k 1 A k 1,k γk L A k

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 25 whose unabeed maps are mutipication maps, commutes. Thus, to prove (28) commutes, it is enough to show (30) N k 1 A k 1,k = N k 1 A k 1,k γk N k ψ k L A k commutes. Since, by Step 4, ψ is an A-modue morphism, the commutivity of (30) foows from Step 3. 3.2. The interna Hom functor on GrA. Definition 3.7. Let C be an object in B and et M be a graded right A-modue. We define Hom GrA (C, M) to be the Z-graded O X -modue whose kth component is the equaizer of the diagram (31) ΠM i C i ki β α ΠM j C j kj γ Π j (Π i (M j A ij ) C ki ) δ Π j (Π i M j (C ki A ij ) ) where α is the identity map, β is induced by the composition (32) M i Mi A ij A ij γ is induced by the dua (Definition 2.2) of and δ is induced by the composition C ki A ij Ckj, M j A ij, (M j A ij) C ij M j (A ij C ki) M j (C ki A ij ) whose eft arrow is the associativity isomorphism and whose right arrow is induced by the canonica map (4). If C is an object of GrA eft-concentrated in degree k, we define Hom GrA (C, M) to be the equaizer of (31). Remark 3.8. If C is an object in B and D is an object in BimodA A then we can define Hom GrA (C, D) to be the bigraded O X 2-modue whose,kth component is the equaizer of the diagram k ΠD i C i ki β α k ΠD j C j kj γ Π(Π(D j A ij ) C k j i ki ) Π(ΠD j (C ki A ij ) ) δ k j i where α is the identity map, β is induced by the composition γ is induced by the dua of D i Di A ij A ij C ki A ij Ckj, D j A ij,

26 ADAM NYMAN and δ is induced by the composition (D j A ij) C ij D j (A ij C ki) D j (C ki A ij ) whose eft arrow is the associativity isomorphism and whose right arrow is induced by the canonica map (4). Since we wi not use this object in what foows, we wi not study its properties. Lemma 3.9. Let F be an object of GrA eft-concentrated in degree k and et L be an O X -modue. Then Hom GrA (O X F, M) is the equaizer of the diagram (33) ΠHom OX (L F ki, M i ) i = ΠHom OX (L F kj, M j ) j Π(ΠHom OX (L F ki, M j A ij )) Π(ΠHom j i OX (L (F ki A ij ), M j )) = j i whose eft vertica is induced by the composition whose right vertica is induced by M i Mi A ij A ij F ki A ij Fkj, M j A ij, and whose bottom horizonta is induced by the composition of isomorphisms Hom OX (L F ki, M j A ij) = Hom OX ((L F ki ) A ij, M j ) = Hom OX (L (F ki A ij ), M j ) where the ast isomorphism is induced by the associativity of the tensor product. Proof. Suppose Π i f i Π i Hom OX (L F ki, M i ). Then Π i f i maps, via the top route, to the product Π j (Π i g ij ), where g ij is the composition L (F ki A ij ) f j L F kj Mj. Π i f i maps, via the bottom route, to Π j (Π i h ij ), where h ij is the composition f i A ij L (F ki A ij ) (L F ki ) A ij Mi A ij (M i (A ij A ij) A ij ) A ij Aij (M j A ij) A ij M j. Thus, Π i f i is in the equaizer of (33) if and ony if the diagram (34) f j (L F ki ) A ij L F kj f i A ij Mj M i A ij whose top eft arrow is the composition M i (A ij A ij ) A ij A ij Aij M j A ij A ij (35) (L F ki ) A ij L (F ki A ij ) L F kj

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 27 with eft arrow induced by associativity of tensor product, commutes for a i and j. However, since M i A ij (M i (A ij A ij ) A ij) M i A ij A ij Aij (M j A ij ) A ij commutes, Π i f i is in the equaizer of (33) if and ony if the diagram M j f i A ij (L F ki ) A ij Mi A ij L F kj fj M j whose eft vertica is the morphism (35), commutes for a i and j. The foowing resut provides some justification for Definition 3.7. Proposition 3.10. If F is an object of GrA which is eft-concentrated in degree k, and if L is an O X -modue, Hom OX (L, Hom GrA (F, M)) = Hom GrA (L F, M). In particuar, if Γ(X, ) : ModX ModΓ(X, O X ) is the goba sections functor, Γ(X, Hom GrA (F, M)) = Hom GrA (O X F, M). Proof. Since Hom OX (L, ) is eft exact, Hom OX (L, Hom GrA (F, M)) is isomorphic to the equaizer of the diagram (36) Π(Hom OX (L, M i F i ki )) = Π(Hom OX (L, M j F j kj )) Π(Π(Hom OX (L,(M j A ij ) F j i ki ))) Π(Π(Hom OX (L, M j (F ki A ij ) ))) = j i whose eft vertica is induced by the composition M i Mi A ij A ij whose right vertica is induced by the dua of F ki A ij Fkj, and whose bottom horizonta is induced by M j A ij, (37) (M j A ij) F ki M j (A ij F ki) M j (F ki A ij ). To compete the proof, we must show the equaizer of (36) is isomorphic to the equaizer of (33). We think of (36) and (33) as the top and bottom, respectivey, of a cube whose vertica arrows are adjointness isomorphisms. To show the equaizers of (36) and (33) are isomorphic, it suffices to show the vertica faces of this cube commute. By naturaity of adjointness isomorphisms, three of these vertica faces

28 ADAM NYMAN obviousy commute. To compete the proof, we must show that the diagram (38) Hom OX (L,(M j A ij ) F ki ) Hom O X (L, M j (F ki A ij ) ) Hom OX (L F ki, M j A ij ) Hom O X (L (F ki A ij ), M j ) whose eft and right arrows are adjointness isomorphisms, whose top arrow is induced by (37), and whose bottom arrow is the composition Hom OX (L F ki, M j A ij) = Hom OX ((L F ki ) A ij, M j ) = Hom OX (L (F ki A ij ), M j ) whose first isomorphism is the adjointness isomorphism and whose second isomorphism is induced by associativity, commutes. The diagram (38) can be rewritten as the diagram Hom OX ((L F ki ) A ij, M j ) Hom OX (L,(M j A ij ) F ki ) Hom OX (L (F ki A ij ), M j ) Hom OX (L, M j (F ki A ij ) ) whose top horizonta is the composition of adjointness isomorphisms Hom OX ((L F ki ) A ij, M j ) = Hom OX (L F ki, M j A ij) = Hom OX (L,(M j A ij) F ki), whose bottom horizonta is the adjointness isomorphism, whose eft vertica is induced by associativity, and whose right vertica is induced by (37). This commutes by Lemma 6.1, and the resut foows. Proposition 3.11. If M is an object in GrA and C is an object in B, Hom GrA (C, M) inherits a graded right A-modue structure from the eft A-modue structure of C, making Hom GrA (, ) : B op GrA GrA a bifunctor. Proof. We first show that Hom GrA (C, ) is a functor from GrA to GrA. We begin by constructing a map of O X -modues (39) ij : Hom GrA (C, M) i A ij Hom GrA (C, M) j. To this end, we note that Hom GrA (C, M) i is an O X -submodue of Π M C i. Thus, we have a map (40) Hom GrA (C, M) i A ij (Π M C i) A ij = Π M (C i A ij ) whose right composite is induced by associativity of the tensor product. The eft A-modue structure of C yieds a mutipication map A ij C j C i. Duaizing gives us a map (41) C i (A ij C j ) = C j A ij whose right composite is the canonica isomorphism. Tensoring (41) by A ij on the right and composing with the map induced by the counit Cj : C j A ij A ij Cj gives us a map (42) C i A ij (C j A ij) A ij C j.

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 29 Tensoring (42) on the eft with M, taking the product over a, and composing with (40) gives us a map (43) Hom GrA (C, M) i A ij Π (M C j). We must show this map factors through Hom GrA (C, M) j. Let (44) Π(M Ci )A ij Π(M (Cj A ij ))A ij = = Π m (M m C im )A ij Π m (M m (C jm A ij ))A ij ΠM Cj = ΠM m C m jm Π m Π (M m (C i A m ) )A ij Π m Π (M m ((C j A m ) A ij )A ij) Π m Π M m (C j A m ) be the diagram whose top and midde rows are induced by (42) and whose bottom row is induced by the composition (45) (C i A m ) ((A ij C j ) A m ) (C j A m ) A ij (whose right composite is the canonica isomorphism). Let (46) Π(M Ci )A ij Π(M (Cj A ij )A ij) ΠM Cj Π m Π ((M m A m )C i )A ij Π m Π ((M m A m )(C j A ij )A ij) Π m Π (M m A m )C j = = = Π m Π (M m (C i A m ) )A ij Π m Π ((M m (C j A m ) )A ij )A ij Π m Π M m (C j A m ) be the diagram whose top and midde rows are induced by (42), whose bottom eft horizonta is induced by (45), whose top verticas are induced by the composition M M A m A m M m A m, and whose bottom verticas are induced by the canonica isomorphisms (4). We eave it as an exercise for the reader to prove, using the universa property of equaizer, that, in order to prove (43) factors through Hom GrA (C, M) j, it suffices to show that the outer circuits of (44) and (46) commute. The upper squares in (44) commute since the vertica maps are identity maps. The right squares in (44) and (46), and the upper eft square in (46), commute by the functoriaity of the tensor product. Thus, in order to show that (43) factors through Hom GrA (C, M) j, we must show that the ower-eft squares in (44) and (46) commute, i.e. we must show (47) Cim Cjm A ij (C i A m ) (C j A m ) A ij

30 ADAM NYMAN whose maps are induced by C, commutes, and (48) A m C i A m (C j A ij ) (C i A m ) (C j A m ) A ij whose horizonta maps are induced by C and whose vertica maps are canonica isomorphisms, commutes. By Coroary 6.3, in order to show (47) commutes, it suffices to show the diagram Cim (A ij C jm ) (C i A m ) ((A ij C j ) A m ) whose arrows are induced by mutipication, commutes. This foows from the functoriaity of ( ). Expanding (48), we have a diagram A m C i A m (A ij C j ) A m (C j A ij ) (C i A m ) ((A ij C j ) A m ) (C j A m ) A ij whose right vertica is a composition of an associativity isomorphism with the canonica isomorphism (4). The eft square commutes by Coroary 6.3, whie the right square commutes by Coroary 6.5. We defer the proof that the map (39) is associative and compatibe with scaar mutipication to Section 7. We omit the routine proofs that Hom GrA (C, ) and Hom GrA (, M) are functoria, that Hom GrA (C,id M ) = id HomGrA (C,M) and that Hom GrA (C, ) is compatibe with composition. 3.3. Adjointness of interna Hom and Tensor. Definition 3.12. We say that F : A B A and G : B op A A are conjugate bifunctors if, for every C B, F(, C) has a right adjoint G(C, ) via an adjunction which is natura in M, N and C. φ : Hom A (F(M, C), N) Hom A (M,G(C, N)) Proposition 3.13. Let C be an object in B. (1) There exist natura transformations and : id GrA Hom GrA (C, A C) : Hom GrA (C, ) A C id GrA making ( A C, Hom GrA (C, ),,) : GrA GrA an adjunction. (2) There exists a unique way to make A : GrA B GrA a bifunctor such that A and Hom GrA (, ) are conjugate.

SERRE DUALITY FOR NON-COMMUTATIVE P 1 -BUNDLES 31 Proof. We construct an A-modue map M : M Hom GrA (C, M A C) natura in M. To begin, for each k we construct a map (49) M k (M A C) i Cki for a i. In order to construct (49), we construct a map (50) M k (M m C mi ) C m ki. The map (50) is just the composition M k Mk C ki C ki ( m M m C mi ) C ki whose right arrow is induced by incusion of a direct summand. Thus, we have a map for every i, and hence a map M k ( m M m C mi ) C ki (M A C) i C ki ψ k : M k Π i (M A C) i C ki. In order to show this map induces a map we must show the diagram M k ψ k M,k : M k Hom GrA (C, M A C) k, (51) Π(M i A C) i Cki Π(M j A C) j Ckj Π j Π i ((M A C) j A ij ) C ki = Π j Π i ((M A C) j (C ki A ij ) ) whose unabeed maps are those in (31), commutes. In order to prove this, it suffices to show the projection of (51) onto its i,j component, the eft square in M k = = M k M k (M k C ki )Cki (M k C kj )Ckj M k (C kj Ckj ) ((M k C kj )A ij )C ki = (M k C kj )(C ki A ij ) M k (C kj (C ki A ij ) )

32 ADAM NYMAN whose bottom right horizonta is induced by associativity, commutes. Since the outer circuit of this diagram equas the outer circuit of the diagram M k = = M k M k (C ki (A ij A ij )C ki ) M k((c ki A ij )(C ki A ij ) ) M k (C kj Ckj ) M k ((C kj A ij )C ki ) M k(c kj (C ki A ij ) ) = M k (C kj (C ki A ij ) ) whose unabeed arrows are canonica isomorphisms, it suffices to show each circuit in this diagram commutes. The right rectange commutes by Coroary 6.7, the bottom-eft square commutes by the functoriaity of the tensor product and the upper-eft square commutes by Lemma 6.2. The fact that is natura foows from a straightforward computation, which we omit. We defer the proof the is compatibe with A-modue mutipication to Section 7. We next construct a map M : Hom GrA (M, C) A C M natura in M. To begin, we construct a natura map k : (Hom GrA (C, M) A C) k M k. To construct k, it suffices to construct a map such that the diagram (52) δ k : m (Hom GrA (C, M)) m C mk M k M k ((Hom GrA (C, M)) A m ) C mk (Hom GrA (C, M)) m C mk m m = (Hom GrA (C, M)) C k = m (Hom GrA (C, M)) m C mk δ M k whose unabeed arrows are induced by mutipication, commutes. We define δ k as the map induced by the composition (Π i M i C mi) C mk M k C mk C mk M k which we ca γ k. To show (52) commutes, it suffices to show the diagram ((ΠM i C m i i ) A m) C mk (ΠM i Cmi ) C mk m i γ k (ΠM i C i i ) C k M k γk whose eft vertica is induced by mutipication and whose top horizonta is induced by the composition (53) C i A m (A m C mi ) A m = C mi A m A m C mi