A structure theorem for P^1-Spec k-bimodules

A STR UCTURE THEOREM F OR P 1 − Spec k -BIMODULES A. NYMAN Abstract. Let k b e an algebraically closed field. Using the Eilenberg-W a tts theorem ov er sc hemes [4], w e determine the structure of k -l inear right exact direct limit and coherence preserving functors from the categ ory of quasi- coheren t shea ve s on P 1 k to the categ ory of vecto r spaces ov er k . As a conse- quence, we c haract erize those functors whic h are integral t ransforms . 1. Introduction Throughout, we work ov er a n algebr aically clo s ed field k . F or any k -algebra R , we let M o d R denote the categ ory of right R -modules , and for any k -scheme X , we let Qcoh X denote the category of quasi-coherent sheaves on X . The purpose o f this note is to prov e the following (Theo rem 4.1): Theorem. If F : Qcoh P 1 k → Mo d k is a k -line ar right exact dir e ct limit and c oher- enc e p r eserving functor then ther e exists a c oher ent torsion she af T and nonne gative inte gers n , n i such that F ∼ = ⊕ ∞ i = − n H 1 ( P 1 , ( − )( i )) ⊕ n i ⊕ H 0 ( P 1 , − ⊗ T ) . Our motiv ation fo r determining the structure o f such functors is t wofold. Our first motiv a tio n comes from the c haracter ization of direct limit pres erv- ing rig ht exact functor s b et ween mo dule ca tegories of k - algebras carr ied out in- depe ndently b y Eilenber g [1] a nd W atts [6]. The c haracteriz a tion (known as the Eilenberg-W atts theorem) says that if R and S are k -algebra s and F : Mod R → Mo dS is a k -linea r right exa ct direct limit prese r ving functor, then F is an integral transform, i.e. is isomorphic to − ⊗ R M , where M is a k -central R − S -bimodule. In [4], k -linea r direct limit preserving right exact functors b etw een catego ries of quasi-coher en t sheav es on s c hemes ar e studied. The main r esult in [4] is that such functors ar e almost integral transfor ms. T o describ e the result more pre cisely , we int ro duce so me notation. W e let X b e a quasi-co mpact and separated sch eme and we let Y be a se pa rated scheme. W e let Bimo d k ( X − Y ) denote the categ ory of k -linear right exact direct limit pr eserving functors from Qcoh X to Qcoh Y and we let W : Bimo d k ( X − Y ) → Qcoh X × Y , denote the E ile nber g-W atts functor (see [4, Section 5 ] for details). W e let pr 1 , 2 : X × Y → X , Y denote standard pro jection maps and we reca ll a Definition 1. 1. A k -line ar funct or F : Qcoh X → Qcoh Y is t otal ly glob al if for every op en immersion u : U → X with U affine, F u ∗ = 0 . The E ilen b erg-W a tts theo rem ov er schemes is the following Date : No vem b er 13, 2018. 2010 Mathematics Subje ct Classific ation. Primary 18A25; Seconda ry 14A22. 1 2 A. NYMAN Theorem 1.2. [4, Section 6] If F is an obje ct in Bimo d k ( X − Y ) then ther e exists a natur al t r ansformation Γ F : F − → pr 2 ∗ (pr ∗ 1 − ⊗ O X × Y W ( F )) such that k er Γ F and cok Γ F ar e total ly glob al. F urthermor e, if X is affine, t hen Γ F is an isomorphism. The res ult leaves op en the questio n of w he ther there is a more precise descr iption of the structur e of such functor s when X is not affine. Our structure theorem addresses this question in case X = P 1 k and Y = Sp ec k . It fo llo ws immediately from the structure theorem that a c oherence pres erving functor F in Bimo d k ( P 1 k − Spec k ) is an integral trans fo rm, i.e. Γ F is an iso morphism, if a nd only if F is exact on short exact sequences of v ector bundles (Corolla r y 4.2). Our seco nd motiv a tion for proving the structure theorem is rela ted to non- commutativ e algebra ic geometry . The ob jects o f s tudy in this sub ject are non- commutativ e spaces, which ar e certain ab elian categories which serve a s general- izations of ca tegories o f q uasi-coherent sheaves on quasi- compact and separated schemes. Ma ps betw een such spaces are taken to b e eq uiv alence classes of adjoint pairs of functors b etw een the spaces (see [5, Definition 2.3] for a more precise dis- cussion). The fa ct that this notion o f map is so g eneral has the benefit of increased flexibility but also has the drawbac k of deviating too fa r from its commutativ e ori- gin. In order to obtain a more res tricted notion of map in the no n- comm utative setting, it may thus be instructive to study the question o f when a map betw een categorie s of q uasi-coherent sheaves on schemes is isomor phic to ( f ∗ , f ∗ ) for s o me morphism f o f s c hemes. It follows from o ur structure theorem that a coherence pre- serving functor F in Bimo d k ( P 1 k − Sp ec k ) is iso morphic to f ∗ where f : Sp ec k → P 1 k is a morphism of schemes if and only if F is exact on shor t ex act sequences of vector bundles and dim k ( F ( O ( n ))) = 1 for so me n (Corolla ry 4.3). Notation and Conventions : In addition to the nota tion introduced a bove, we let Funct k ( Qcoh X, Qco h Y ) deno te the a b elian category of k - line a r functors fro m Qcoh X t o Qcoh Y . W e routinely use the fact that in this catego r y , a kernel o f a natural transformation Υ : F → G is the functor which assig ns to an o b ject M the kernel of Υ M and which assig ns to a morphism φ : M → N the induced morphism k er Υ M → k er Υ N . A cokernel o f Υ can b e defined similarly . W e denote the full sub category of Bimo d k ( X − Y ) consisting of c oherence preserv ing functors by bimo d k ( X − Y ). W e will routinely inv oke Γ F , W ( F ), ker Γ F and cok Γ F from Theorem 1.2 without explicit reference to Theor em 1 .2. W e write P 1 for P 1 k . All unadorned tensor pro ducts are ov er O P 1 , and we write O ( i ) fo r O P 1 ( i ). W e shall abuse notation by identifying Qcoh (Spec k ) with the catego ry Mo d k . Other nota tio n and co n ven tions will b e in tro duced lo cally . 2. The structure of W ( F ) Although some of the prop erties of the Eilenberg- W atts functor W will play a crucial role in the pr oo f o f o ur main result, the details of the construction o f W , which are somewha t c o mplicated, will no t b e needed in what follows. F or this reason, we refer the r eader to [4, Section 5] for the definition of W . Our main goa l A STR UCTURE THEOREM FOR P 1 − Sp ec k -BIMODULES 3 in this section is to prov e that if F ∈ bimo d k ( P 1 − Sp ec k ) then W ( F ) is coherent torsion. Prop osition 2.1. If F ∈ bimo d k ( P 1 − Spec k ) then W ( F ) is c oher ent torsion. Pr o of. W e claim tha t, to prove the prop osition, it s uffices to prov e that if p ∈ P 1 is a closed p oint, U is the co mplemen t of p in P 1 and u : U → P 1 is inclusion, then F ( u ∗ u ∗ O P 1 ) is finite dimensional. T o pr ov e the claim, we fir st note that u ∗ W ( F ) ∼ = W ( F u ∗ ) by [4, P rop osition 5.2], where w e have abused notation by ident ifying P 1 × Sp ec k Spec k with P 1 . By the pro of of [4, Pr opo sition 2.2], W ( F u ∗ ) has k - mo dule structure F ( u ∗ u ∗ O P 1 ). Ther e fore, the finite dimensionality o f F ( u ∗ u ∗ O P 1 ) would imply the finite dimensio nalit y of u ∗ W ( F ). Since p is a rbitrary , the fact that W ( F ) is c o herent, and hence tor s ion, w ould follow. W e now pr oc e e d to pr ov e that F ( u ∗ u ∗ O P 1 ) is finite dimensional, which will establish the result. T o this end, b y [3, Coro lla ry 1.9], there is a short exact sequence 0 → O P 1 → u ∗ u ∗ O P 1 → H 1 p ( O P 1 ) → 0 where H 1 p : Qcoh P 1 → Qcoh P 1 is the functor which sends a quasi-co he r en t sheaf F to the first c o homology of P 1 with coefficie n ts in F a nd supp ort in p . Applying the right exac t functor F to this se q uence yields an exact s e q uence F ( O P 1 ) → F ( u ∗ u ∗ O P 1 ) → F ( H 1 p ( O P 1 )) → 0 . Thu s, to pro ve that F ( u ∗ u ∗ O P 1 ) is finite dimensiona l, it suffices to pro ve that F ( H 1 p ( O P 1 )) is z e r o. T o prov e that F ( H 1 p ( O P 1 )) v a nishes, we first recall that if I is the ideal shea f of p and we let O n := O P 1 / I n for n ≥ 1, then H 1 p ( O P 1 ) ∼ = lim → E xt 1 O P 1 ( O n , O P 1 ) where the dir ect system is induced by the canonical quotient maps O n → O m for n ≥ m ([3, Theorem 2.8]). Th us, the prop osition will follow if we ca n show that F (lim → E xt 1 O P 1 ( O n , O P 1 )) ∼ = lim → F ( E xt 1 O P 1 ( O n , O P 1 )) is zero. T o prov e this final fact, we first leav e it as a stra ight forward exercise to show that lim → E xt 1 O P 1 ( O n , O P 1 ) is is omorphic to the direct limit of the direct s ystem ( O i , µ ij ) wher e for i < j , µ ij : O i → O j is the inclusio n o f O i as the kernel of the canonica l quotient O j → O j − i and µ ii is the iden tit y map. W e next no te that since there are epimor phisms O j → O i for all i ≤ j a nd O P 1 → O i for all i and since F is right exact, dim k F ( O i ) ≤ dim k F ( O j ) ≤ dim k F ( O P 1 ) fo r all i ≤ j . It follo ws fro m this that there exists a natural n umber n 0 such that dim k F ( O i ) = dim k F ( O j ) for all i, j ≥ n 0 . Ther efore, for i, j such that n 0 ≤ i ≤ j , the canonical quotient map O j → O i induces an isomorphism F ( O j ) → F ( O i ), and thus, in this case, the canonical inclusio n µ j − i,j induces the zero map F ( O j − i ) → F ( O j ). It follows that lim → F ( O n ) is zero , so tha t F ( H 1 p ( O P 1 )) is z e r o. The result follows.  Corollary 2. 2. If F ∈ bimo d k ( P 1 − Sp ec k ) , t hen cok Γ F = 0 . Thus, ther e is a short exact se quen c e 0 → ker Γ F → F Γ F → H 0 ( P 1 , − ⊗ W ( F )) → 0 in F unct k ( Qcoh P 1 , Qcoh (Spec k )) . 4 A. NYMAN Pr o of. Since W ( F ) is coherent torsion b y Pro pos ition 2.1, the functor H 0 ( P 1 , − ⊗ W ( F )) is righ t exact and commutes with dir ect limits. Thus, a dia gram chase establishes that cok Γ F is righ t exa c t and comm utes with direct limits a s w ell. Since k is algebraically closed, it follows from [4, Co rollary 7.3] and [4, Theo rem 7.12] that there exist in tegers m, n i ≥ 0 such that cok Γ F ∼ = ⊕ ∞ i = − m H 1 ( P 1 , ( − )( i )) ⊕ n i . Thu s, if cok Γ F were nonzero, cok Γ F ( O ( i )) would hav e arbitrarily large dimension for negative i ∈ Z . On the other hand, H 0 ( P 1 , O ( i ) ⊗ W ( F )) has constant dimen- sion. Since ther e is a n epimorphis m H 0 ( P 1 , O ( i ) ⊗ W ( F )) → cok Γ F ( O ( i )), we conclude that cok Γ F = 0.  W e end this s ection by describing a situation we shall enco un ter several times throughout this note. Supp ose F ∈ Funct k ( Qcoh X , Qcoh (Spec k )) is right exact. If (2-1) 0 → M → N → P → 0 is a short exact sequence in Qcoh X , G is an ob ject in Qcoh X a nd Υ : F → H 0 ( X, − ⊗ O X G ) is a natural transformatio n, then w e hav e a comm utative (but not necessa r ily exact) dia g ram (2-2) F ( M ) → F ( N ) → F ( P ) → 0   y   y   y 0 → H 0 ( X, M ⊗ O X G ) → H 0 ( X, N ⊗ O X G ) → H 0 ( X, P ⊗ O X G ) induced by Υ. The top row is alwa ys ex act, while the bo tto m row is e x act in case P is flat. If the bottom row is exac t, the Snake Lemma implies that there is a n exact seque nce (2-3) ker Υ( M ) → ker Υ( N ) → k er Υ( P ) → cok Υ( M ) → co k Υ( N ) → cok Υ( P ) in Qcoh (Sp ec k ). 3. The structure of ker Γ F Throughout this section, supp ose F ∈ bimo d k ( P 1 − Sp ec k ) is such that W ( F ) is nonzero. Since, by Pro p ositio n 2.1, W ( F ) is coherent torsion, it ha s finite supp ort. F or the r emainder of this note, we le t p ∈ P 1 be a closed p oin t outside the suppo rt of W ( F ) a nd we let q ∈ P 1 be a clo sed p oint in the suppor t of W ( F ). Let A = k [ x 0 , x 1 ] deno te the p olynomia l ring with the usual grading, let [ − ] denote the shift functor in the ca tegory of g r aded A mo dules, and let f , g : A [ − 1] → A denote multiplication by linea r for ms whose cor resp onding morphisms in Qco h P 1 , α 0 : O ( − 1 ) → O and β 0 : O ( − 1 ) → O , hav e co kernels k ( p ) and k ( q ), r espectively . F or any morphism δ 0 : O ( − 1) → O corr esp o nding to multiplication by a linear fo r m h : A [ − 1] → A , we define δ i : O ( i − 1 ) → O ( i ) to b e the morphism corre s ponding to the i th shift of h . Lemma 3 .1. (1) The morphism ker Γ F ( α i ) is epic. Thus, ther e is a nonne g- ative inte ger m such that dim k (ker Γ F ( O ( i ))) = m for al l su fficiently lar ge i . A STR UCTURE THEOREM FOR P 1 − Sp ec k -BIMODULES 5 (2) The image of the morphism ker Γ F ( β j − 1 ) c ont ains (ker Γ F ( α j )) − 1 (im(k er Γ F ( β j ))) . Ther efor e, if ker Γ F ( β i ) is epic then ker Γ F ( β j ) is epic for al l j ≤ i . Pr o of. The sequence (2 - 3) in the case (2 -1) is 0 → O ( i − 1) α i → O ( i ) → k ( p ) → 0 , Υ = Γ F , and G = W ( F ), ha s an exact subsequence ker Γ F ( O ( i − 1)) ker Γ F ( α i ) → ker Γ F ( O ( i )) → 0 since k er Γ F is to tally global b y Theore m 1.2. Therefore, ker Γ F ( α i ) is epic. The second statement in (1) follows fro m this. T o prov e (2), we no te that in the case (2-1) is (3-1) 0 − → O ( j − 2 ) ( α j − 1 , − β j − 1 ) − → O ( j − 1) ⊕ 2 β j + α j − → O ( j ) − → 0 , Υ = Γ F , and G = W ( F ), the sequence (2- 3) has a n exact s ubsequence (3-2) ker Γ F ( O ( j − 2)) (ker Γ F ( α j − 1 ) , − ker Γ F ( β j − 1 )) − → ker Γ F ( O ( j − 1 )) ⊕ 2 ker Γ F ( β j )+ker Γ F ( α j ) − → ker Γ F ( O ( j )) − → 0 by Corolla ry 2.2. If a ∈ (ker Γ F ( α j )) − 1 (im(ker Γ F ( β j ))), then there exists a b ∈ ker Γ F ( O ( j − 1)) such that ker Γ F ( α j )( a ) = ker Γ F ( β j )( b ). Ther e fo re, by exactnes s of (3-2), ( b, − a ) is in the image of (ker Γ F ( α j − 1 ) , − ker Γ F ( β j − 1 )), so that a is in the image o f ker Γ F ( β j − 1 ). The firs t pa rt of (2) follows. T o prov e the seco nd part of (2), w e note that if ker Γ F ( β i ) is epic, then the first pa rt of (2) implies that (ker Γ F ( α i )) − 1 (im(ker Γ F ( β i ))) = ker Γ F ( O ( i − 1)) s o that ker Γ F ( β i − 1 ) is epic a s well.  Corollary 3. 2. If m = 0 in L emma 3.1(1), t hen k er Γ F ( β i ) is epic for al l i . It fol lows that t her e exist inte gers n, n i ≥ 0 such that ker Γ F ∼ = ⊕ ∞ i = − n H 1 ( P 1 , ( − )( i )) ⊕ n i . Pr o of. Either ker Γ F ( O ( n )) = 0 for all n (in which cas e the last result is c le ar) or there ex ists n 0 minimal such that ker Γ F ( O ( i )) = 0 for all i ≥ n 0 . Therefor e, ker Γ F ( β i ) is trivially epic for a ll i ≥ n 0 . The first result now follows immedia tely from Lemma 3 .1(2). T o prov e the second result, we prov e tha t k er Γ F is admissible [4, Definition 7.1]. T o this end, we note that it is ha lf-exact on vector bundles by the exactness o f (2-3) in ca se (2-1) is an exact sequence of vector bundles, Υ = Γ F and G = W ( F ). W e next note that ker Γ F ( α i ) is epic for all i b y Lemma 3.1 (1) while ker Γ F ( β i ) is epic for all i by the arg umen t in the first para graph. Since p is an arbitrar y point outside the supp ort of W ( F ) while q is an arbitra ry p oint in the suppor t of W ( F ), and since any nonzero map δ : O ( i − 1) → O ( i ) has c okernel k ( r ) for some clo s ed po in t r ∈ P 1 , it follows that k er Γ F ( δ ) is epic. Ther efore, since k is algebr aically closed, ker Γ F ( γ ) is epic for all nonzero γ ∈ Hom O P 1 ( O ( m ) , O ( n )). It remains to c heck that ker Γ F commutes with direct limits. This follows fr om the fact that b oth F and H 0 ( P 1 , − ⊗ W ( F )) comm ute with direct limits. Therefore, [4, Theor em 7.12] implies the seco nd result.  6 A. NYMAN F or the remainder o f this section, we will prov e that m from Lemma 3.1(1) must equal 0. Our strateg y will b e to use the fact that if m > 0, w e can embed a functor which is no t totally global into ker Γ F , th us obta ining a contradiction. Our strateg y is made p oss ible by the fact that na tural transfor ma tions Ω : F → G between dir e c t limit pres erving functors in F unct k ( Qcoh P 1 , Qcoh (Spec k )) such that G is totally global may b e constructed inductiv ely . The precise result (used implicitly in the pro of of [4, Prop osition 7.6]) is given by the following Lemma 3.3. L et F and G b e dir e ct limit pr eserving k -line ar functors fr om Qcoh P 1 to Qco h Y such that G is total ly glob al. Su pp ose for al l n ∈ Z , morphisms Ω O ( n ) : F ( O ( n )) → G ( O ( n )) ar e define d such that the diagr am F ( O ( i )) F ( ψ ) → F ( O ( i + 1)) Ω O ( i )   y   y Ω O ( i +1) G ( O ( i )) → G ( ψ ) G ( O ( i + 1)) c ommut es for al l i ∈ Z and al l ψ ∈ Hom O P 1 ( O ( i ) , O ( i + 1)) . Then ther e is a unique natur al tr ansformation Ω : F → G such t hat Ω O ( n ) = Ω O ( n ) for al l n . F u rthermor e, (1) if Ω O ( n ) is monic for al l n and F vanishes on c oher ent torsion mo dules, then Ω is m onic, and (2) if Ω O ( n ) is epic for al l n then Ω is epic. Pr o of. By [4, Lemma 7.5], to construct Ω, it suffices to co nstruct a natural tr ansfor- mation Ω : F | coh P 1 → G | coh P 1 where coh P 1 denotes the full sub categor y of Qcoh P 1 consisting of cohere n t O P 1 -mo dules. Uniqueness follows fro m [4 , Lemma 7.5 ]. W e will see from the construction of Ω below that if Ω O ( n ) is monic for all n and F v anishes on cohe r en t to rsion mo dules, then Ω is monic so that (1) follows from [4, Lemma 7.5]. Similarly , w e will see that if Ω O ( n ) is epic for all n then Ω is epic s o that (2) follows fro m [4, Lemma 7.5]. If T is coherent torsio n, we define Ω T : F ( T ) → G ( T ) to b e the zero map. Next, w e define Ω F when F is isomorphic to O ( n ). Let α : F → O ( n ) b e an isomorphism. Define Ω F := ( G ( α )) − 1 Ω O ( n ) F ( α ) . If β : F → O ( n ) is a nother isomor phism, then β = bα for so me 0 6 = b ∈ k , whence ( G ( β )) − 1 = b − 1 ( G ( α )) − 1 and F ( β ) = bF ( α ); th us the definition of Ω F do es not depe nd on the choice of α . Next, w e define Ω F for ar bitr ary F by writing F a s a direct sum of indecompo s- ables, say F = ⊕ F i , and defining Ω F := ⊕ Ω F i . T o show that Ω is a na tural transfor mation w e must show that (3-3) F ( F ) F ( ψ ) − → F ( G ) Ω F   y   y Ω G G ( F ) − → G ( ψ ) G ( G ) commutes for all F and G and all maps ψ : F → G . It suffices to chec k this w he n F and G are indecomp osable. The diagr a m commutes when G is torsion b ecause G ( G ) = 0. If G is torsion-fr ee and F torsion, then ψ = 0 so the dia gram commutes. A STR UCTURE THEOREM FOR P 1 − Sp ec k -BIMODULES 7 Thu s, the only remaining ca se is that when F ∼ = O ( i ) a nd G ∼ = O ( j ) w ith i ≤ j . The case i = j is stra igh tforward and we omit the verification in this ca s e. Thus, we may supp ose i < j . W rite ψ = β − 1 φα where α : F → O ( i ) and β : G → O ( j ) are isomorphisms and 0 6 = φ : O ( i ) → O ( j ). Since k is alg ebraically closed, we can write φ a s a comp osition φ j φ j − 1 · · · φ i +1 where each φ l : O ( l − 1) → O ( l ) is monic. Now Ω O ( j ) F ( φ j ) · · · F ( φ i +1 ) = G ( φ j ) · · · G ( φ i +1 )Ω O ( i ) and the fact that Ω G F ( f ) = G ( f )Ω F now follows from a straightforw ard computa- tion, which we o mit.  W e define a functor R q : Qcoh P 1 → Qcoh (Spec k ) by R q ( − ) := H 0 ( P 1 , (( − ) / H 0 q ( − )) ⊗ k ( q )) , where, H 0 q : Qcoh P 1 → Qcoh P 1 is the functor which sends a quasi-co herent sheaf to its subshe a f with supp ort at q . In the cas e that the integer m from Lemma 3.1(1) is grea ter than 0, w e will em b ed R q in ker Γ F in the pr oo f of P r op osition 3.6. Lemma 3 .4. The functor R q (1) is k -line ar, (2) c ommutes with dir e ct limits, and (3) R q ( T ) = 0 for al l c oher ent torsion mo dules T . F urthermor e, (4) R q ( α n ) is an isomorphism for al l n , and (5) R q ( β n ) = 0 for al l n . Pr o of. Since H 0 q is k - linear and since Funct k ( Qcoh X , Qcoh Y ) is ab elian, R q is a comp osition of k -linear functors, a nd (1) follows. T o prov e (2), we note that it suffices to prov e H 0 q commutes with direct limits. This is a straig h tforward exer cise, which we omit. Part (3) follows fro m the fact that the supp ort of T / H 0 q ( T ) do e s not include q . W e now prov e (4). Since H 0 q v anishes on vector bundles , R q ( α n ) = H 0 ( P 1 , α n ⊗ k ( q )). Therefore, it suffices to prove that α n ⊗ k ( q ) is a n isomorphism. T o this end, since p is dis jo in t from the suppor t o f k ( q ), it follo ws that − ⊗ k ( q ) is ex act when applied to 0 → O ( n − 1) α n → O ( n ) → k ( p ) → 0 and the result fo llo ws. T o prov e (5), we first note that, as in the pro of of (4), R q ( β n ) = H 0 ( P 1 , β n ⊗ k ( q )). Therefore, it suffices to prov e that β n ⊗ k ( q ) = 0. T o this end, we apply − ⊗ k ( q ) to the short ex a ct sequenc e 0 → O ( n − 1) β n → O ( n ) → k ( q ) → 0 to o btain an exact sequence O ( n − 1) ⊗ k ( q ) β n ⊗ k ( q ) → O ( n ) ⊗ k ( q ) ∼ = → k ( q ) → 0. It follows that β n ⊗ k ( q ) = 0 as desir e d.  Recall that by Lemma 3.1(1), there exist in tegers n 0 and m such that the di- mension o f ker Γ F ( O ( n )) eq uals m for all n ≥ n 0 . This notation is employ ed in the following re sult, which is used in the pro of of P rop osition 3 .6. Lemma 3.5. Su pp ose m 6 = 0 . Then t her e exists a close d p oint r in the supp ort of W ( F ) c orr esp onding to γ 0 : O ( − 1 ) → O such t hat cok(ker Γ F ( γ n )) 6 = 0 for al l n > n 0 . Pr o of. If not, then for an arbira ry point q in the suppo r t of W ( F ), there ex- ists an integer n q > n 0 such that cok(ker Γ F ( β n q )) = 0 and s o ker Γ F ( β n q ) is an isomorphism. It follows from Lemma 3.1(2) that for any q in the supp ort of 8 A. NYMAN W ( F ), ker Γ F ( β n 0 +1 ) is a n isomorphism. On the other hand, by Lemma 3.1(1), ker Γ F ( α n 0 +1 ) is an isomorphism. Th us, ker Γ F ( δ ) is a n isomor phism for all nonzero δ ∈ Hom O P 1 ( O ( n 0 ) , O ( n 0 + 1)). Howev er , if x 0 , x 1 are indeterminates, det( x 0 ker Γ F ( α n 0 +1 ) + x 1 ker Γ F ( β n 0 +1 )) is a homogeneo us po lynomial of degr ee m > 0 in x 0 , x 1 . Therefor e, it has a nontriv- ial zero . Since ker Γ F is k - line a r, this pr ovides a nonzero δ ∈ Hom O P 1 ( O ( n 0 ) , O ( n 0 + 1)) suc h that ker Γ F ( δ ) is no t an iso morphism, and the contradiction e stablishes the res ult.  Prop osition 3.6. The c onstant m in L emma 3.1(1) is 0 . Ther efor e, ther e exist nonne gative inte gers l i and l such that ther e is an exact se quenc e (3-4) 0 → ⊕ ∞ i = − l H 1 ( P 1 , ( − )( i )) ⊕ l i → F Γ F → H 0 ( P 1 , − ⊗ W ( F )) → 0 Pr o of. The second part o f the pro po s ition follows from the first pa r t in lig h t o f Corollar y 2.2 and Coro llary 3.2. T o prov e the first part of the prop osition, we pro ceed b y c on tradiction. Supp ose m 6 = 0 . W e let r denote the closed p oint in the supp ort o f W ( F ) shown to exist in Lemma 3.5. W e s ho w that there is a mo nic natural trans fo rmation ∆ : R r → ker Γ F . This contradicts the fa ct that if u : U → P 1 is inclusion of a n a ffine op en subscheme co n taining r then R r ( u ∗ u ∗ O P 1 ) 6 = 0 while, since ker Γ F is tota lly globa l, ker Γ F ( u ∗ u ∗ O P 1 ) = 0 . The functor R r is k - linear and co mm utes with dir ect limits b y Lemma 3.4(1) and (2), and ker Γ F is k - linear and commutes with direct limits s inc e it is a kernel of a na tural transfo rmation be tw een k -linear functors which commute with dire ct limits b y Corolla ry 2.2. F urthermore, if T is co he r en t torsion, then R r ( T ) = 0 b y Lemma 3 .4(3). Therefore, to define a monomorphis m ∆ : R r → ker Γ F , it s uffices, by Lemma 3.3, to define, for a ll n ∈ Z , a monomor phism ∆ O ( n ) : R r ( O ( n )) → ker Γ F ( O ( n )) s uc h that (3-5) R r ( O ( i )) R r ( ψ ) → R r ( O ( i + 1)) ∆ O ( i )   y   y ∆ O ( i +1) ker Γ F ( O ( i )) → ker Γ F ( ψ ) ker Γ F ( O ( i + 1)) commutes for all i ∈ Z and all ψ ∈ Hom O P 1 ( O ( i ) , O ( i + 1)). In our co nstruction of ∆ , we will reta in the notation b oth pr e c eding the s tatemen t of Lemma 3.5 and defined in the statement of Le mma 3.5. W e begin by defining ∆ O ( n 0 ) . T o this end w e let a n 0 be a gener ator of R r ( O ( n 0 )) and w e let 0 6 = b n 0 ∈ ker Γ F ( O ( n 0 )) be in the k ernel of k er Γ F ( γ n 0 +1 ). Suc h an element exists by Lemma 3.5. W e define ∆ O ( n 0 ) ( a n 0 ) = b n 0 and extend linearly . W e define ∆ O ( n 0 +1) as follows: we let a n 0 +1 ∈ R r ( O ( n 0 + 1 )) b e R r ( α n 0 +1 )( a n 0 ). Since R r ( α n 0 +1 ) is a n isomo rphism by Lemma 3.4(4), a n 0 +1 6 = 0 so is a basis for R r ( O ( n 0 + 1)). W e define ∆ O ( n 0 +1) ( a n 0 +1 ) = ker Γ F ( α n 0 +1 )( b n 0 ) =: b n 0 +1 and extend linea r ly . The fact that ∆ O ( n 0 +1) is monic follows immediately from the fact that, by choice of n 0 , ker Γ F ( α n 0 +1 ) is an isomo rphism b y Lemma 3.1(1). W e no w chec k that (3-5) commu tes in ca se i = n 0 . Since, by construction, the diag ram commutes when ψ = α n 0 +1 , w e need o nly chec k that it commut es when ψ = γ n 0 +1 . By Lemma 3.4(5), R r ( γ n 0 +1 ) = 0 a nd b n 0 is chosen so that A STR UCTURE THEOREM FOR P 1 − Sp ec k -BIMODULES 9 ker Γ F ( γ n 0 +1 )( b n 0 ) = 0 , so the diagr a m commutes for all ψ ∈ Hom O P 1 ( O ( n 0 ) , O ( n 0 + 1)) as des ired. Now supp ose n > n 0 and for all j suc h that n 0 < j ≤ n , we ha ve defined a monic ∆ O ( j ) such that (3-5) co mm utes when i = j − 1. W e define ∆ O ( n +1) as follows. Let a n +1 ∈ R r ( O ( n + 1)) b e R r ( α n +1 )( a n ) where a n is some nonzero element of R r ( O ( n )). Since R r ( α n +1 ) is an is omorphism b y Lemma 3.4(4), a n +1 6 = 0 so is a bas is for R r ( O ( n + 1)). If b n = ∆ O ( n ) ( a n ), then b y the induction h y p othesis, b n 6 = 0 a nd b n = k er Γ F ( α n )( b n − 1 ) for some b n − 1 ∈ ker( ker Γ F ( γ n )). W e define ∆ O ( n +1) ( a n +1 ) = ker Γ F ( α n +1 )( b n ) and extend linearly . Since ker Γ F ( α n +1 ) is an isomorphism b y Lemma 3.1(1), ∆ O ( n +1) is monic. W e c heck that (3-5) comm utes when i = n and ψ = γ n +1 , from whic h it will follow that the diagra m co mm utes for all ψ ∈ Ho m O P 1 ( O ( n ) , O ( n + 1)). By Lemma 3.4(5), R r ( γ n +1 ) = 0. Thus, we must show that k er Γ F ( γ n +1 )( b n ) = 0. T o s how this, we note that the exa c tnes s of (3-2) implies the sequence (3-6) ker Γ F ( O ( n − 1)) (ker Γ F ( α n ) , − k er Γ F ( γ n )) − → ker Γ F ( O ( n )) ⊕ 2 ker Γ F ( γ n +1 )+ker Γ F ( α n +1 ) − → ker Γ F ( O ( n + 1)) → 0 is exact. Thus, since ( b n , 0) = (k er Γ F ( α n ) , − ker Γ F ( γ n ))( b n − 1 ) ∈ ker Γ F ( O ( n )) ⊕ 2 , we must hav e ker Γ F ( γ n +1 )( b n ) + ker Γ F ( α n +1 )(0) = 0 ∈ ker Γ F ( O ( n + 1)) so that ker Γ F ( γ n +1 )( b n ) = 0 as desir ed. W e now define ∆ O ( n ) for all n < n 0 . W e b egin by defining ∆ O ( n 0 − 1) . T o this end, we let a n 0 − 1 = R r ( α n 0 ) − 1 ( a n 0 ), which makes sense by Lemma 3.4(4). W e claim there exists an elemen t b n 0 − 1 ∈ ker Γ F ( α n 0 ) − 1 ( b n 0 ) ∩ ker(k e r Γ F ( γ n 0 )) and we define ∆ O ( n 0 − 1) ( a n 0 − 1 ) = b n 0 − 1 and ex tend linearly . W e no te that the claim will imply that ∆ O ( n 0 − 1) is monic a nd that (3-5) co mm utes in case i = n 0 − 1. T o prov e the claim, we note that (3-6) is exact for an y v a lue of n . In particular, when n = n 0 , exa ctness of (3-6) implies that ( b n 0 , 0) ∈ ker Γ F ( O ( n 0 )) ⊕ 2 m ust be of the form (k er Γ F ( α n 0 ) , − ker Γ F ( γ n 0 ))( b n 0 − 1 ) for some b n 0 − 1 ∈ ker Γ F ( O ( n 0 − 1)) whence the claim. Finally , suppo se n < n 0 and for all j such that n ≤ j < n 0 , we hav e defined monic ∆ O ( j ) such that (3-5) co mm utes in case i = j . The definition of ∆ O ( n − 1) as w ell as the pro of tha t it is monic and ma k es (3- 5) commute in case i = n − 1 is identical to the pro of of these prop erties for ∆ O ( n 0 − 1) , and we omit it. The pr opo sition follows.  4. Proof of the structure theorem In this section, we assume F ∈ bimo d k ( P 1 − Spec k ). By P r op o sition 2.1, we know W ( F ) is coher en t torsio n. Theorem 4.1. Ther e exist nonn e gative inte gers l , l i such that F ∼ = ⊕ ∞ i = − l H 1 ( P 1 , ( − )( i )) ⊕ l i ⊕ H 0 ( P 1 , − ⊗ W ( F )) . 10 A. NYMAN Pr o of. In case W ( F ) = 0, Theorem 1 .2 implies that F ∼ = ker Γ F . Therefor e, the result follows from [4, Lemma 7.2] and [4, Theorem 7.1 2]. If W ( F ) 6 = 0 , Prop ositio n 3.6 applies. W e prove that the shor t exact se- quence (3-4) splits, i.e. we construct a natural transformatio n Λ : F → ker Γ F which splits the inclusion Θ : ker Γ F → F from Theorem 1.2 in the category F unct k ( Qcoh P 1 , Qcoh (Spec k )). The functors F and ker Γ F are k -linea r a nd com- m ute with direct limits a nd ker Γ F is totally global. Thus, to construct Λ it suffices, by Lemma 3.3, to define, fo r all n ∈ Z , a morphism Λ O ( n ) such that Λ O ( i ) Θ O ( i ) is the identit y for a ll i ∈ Z and suc h that the diag r am (4-1) F ( O ( i )) F ψ → F ( O ( i + 1)) Λ O ( i )   y   y Λ O ( i +1) ker Γ F ( O ( i )) → ker Γ F ( ψ ) ker Γ F ( O ( i + 1)) commutes for all i ∈ Z and all ψ ∈ Hom O P 1 ( O ( i ) , O ( i + 1 )). Once Λ is constructed, we will show that ΛΘ is the identit y . By P rop osition 3.6, ther e exists n 0 ∈ Z suc h that k er Γ F ( O ( n )) = 0 for all n ≥ n 0 . F or such n , w e define Λ O ( n ) = 0. W e now pro ceed to define Λ O ( n ) for n < n 0 inductively . W e b egin by defining Λ O ( n 0 − 1) . T o this end, we pick a subspace B n 0 − 1 ⊂ F ( O ( n 0 − 1)) co mplimen tary to the image of Θ O ( n 0 − 1) . If a ∈ ker Γ F ( O ( n 0 − 1)) and b ∈ B n 0 − 1 then we define Λ O ( n 0 − 1) (Θ O ( n 0 − 1) ( a ) + b ) = a . It fo llows immediately that Λ O ( n 0 − 1) Θ O ( n 0 − 1) is the identit y and that (4-1) commut es in case i = n 0 − 1. Next, supp ose there is a n n < n 0 such that for all j with n ≤ j ≤ n 0 we hav e defined Λ O ( j ) with the pro per t y that Λ O ( j ) Θ O ( j ) is the ident ity and makes (4- 1) commute when i = j . W e no w define Λ O ( n − 1) . W e do this b y constructing a subspa ce B n − 1 of F ( O ( n − 1)) co mplemen tary to Θ O ( n − 1) (ker Γ F ( O ( n − 1))) a nd defining Λ O ( n − 1) (Θ O ( n − 1) ( a )+ b ) = a , where b ∈ B n − 1 . W e will then see that, by choice of B n − 1 , Λ O ( n − 1) makes (4-1) commute when i = n − 1. Let B n ⊂ F ( O ( n )) denote the k ernel of Λ O ( n ) . W e define B n − 1 ⊂ F ( O ( n − 1)) as follows. W e note that the imag e under Θ O ( n − 1) of the k ernel o f (ker Γ F ( α n ) , − ker Γ F ( β n )) : k er Γ F ( O ( n − 1)) → ker Γ F ( O ( n )) ⊕ 2 , which we de no te by K , is c on tained in F α − 1 n ( B n ) ∩ F β − 1 n ( B n ). W e define B n − 1 ⊂ F α − 1 n ( B n ) ∩ F β − 1 n ( B n ) to b e a complimen tary subspace to K in F α − 1 n ( B n ) ∩ F β − 1 n ( B n ). W e claim tha t B n − 1 is complimentary to the imag e of Θ O ( n − 1) . T o prov e the claim, consider the exa ct sequence (4-2) 0 − → O ( n − 1) ( α n , − β n ) − → O ( n ) ⊕ 2 β n +1 + α n +1 − → O ( n + 1 ) − → 0 . The sequence (4-2) induces a diagr am (4-3) ker Γ F ( O ( n − 1)) → ker Γ F ( O ( n )) ⊕ 2 → ker Γ F ( O ( n + 1)) → 0   y   y   y F ( O ( n − 1)) → F ( O ( n )) ⊕ 2 → F ( O ( n + 1)) → 0 A STR UCTURE THEOREM FOR P 1 − Sp ec k -BIMODULES 11 whose verticals ar e inclusions and whose top row is exa c t b y the exactness o f (2-3) in case Υ = Γ F and (2- 1) is (4-2). W e b egin the pro of of the claim b y showing that B n − 1 int ersected with imag e of Θ O ( n − 1) is 0 . Suppos e x is in the intersection. By the commutativit y of (4- 3), F α n ( x ) ∈ B n ∩ Θ O ( n ) (ker Γ F ( O ( n ))). Thus, F α n ( x ) = 0 and similarly , F β n ( x ) = 0. Since the middle vertical o f (4-3) is an inclusion, and since x ∈ Θ O ( n − 1) (ker Γ F ( O ( n − 1))), say x = Θ O ( n − 1) ( a ), it follows ag ain from the comm utativity of (4-3) that ker Γ F ( α n )( a ) = 0 = ker Γ F ( β n )( a ). Th us, x is in the image, under Θ O ( n − 1) , of the k ernel of the top left horizontal. Therefor e, by choice of B n − 1 , x = 0. T o complete the pro of of the claim, it remains to show that Θ O ( n − 1) (ker Γ F ( O ( n − 1))) + B n − 1 = F O ( n − 1) . T o this end, it suffices to show (4-4) Θ O ( n − 1) (ker Γ F ( O ( n − 1))) + F α − 1 n ( B n ) ∩ F β − 1 n ( B n ) = F ( O ( n − 1)) . T o prov e (4- 4), supp ose x ∈ F ( O ( n − 1)). Then ( F α n , − F β n )( x ) has the form (Θ O ( n ) ( a 1 ) + b 1 , Θ O ( n ) ( a 2 ) + b 2 ) ∈ F ( O ( n )) ⊕ 2 where a 1 , a 2 ∈ k er Γ F ( O ( n )) and b 1 , b 2 ∈ B n . Since the b ottom row o f (4-3) is e x act, (Θ O ( n ) ( a 1 ) + b 1 , Θ O ( n ) ( a 2 ) + b 2 ) is in the kernel o f the second map in the b ottom ro w. By the induction hypo thesis, the diagram (4-1) comm utes in case i = n . It follows that ( a 1 , a 2 ) ∈ ker Γ F ( O ( n )) ⊕ 2 is in the kernel of the second top horizon tal of (4-3). Therefore, there ex is ts a c ∈ k er Γ F ( O ( n − 1)) which maps to ( a 1 , a 2 ) ∈ k er Γ F ( O ( n )) ⊕ 2 via the top left horizo n tal of (4-3) so that ( F α n , − F β n )( x − Θ O ( n − 1) ( c )) = ( b 1 , b 2 ) ∈ F ( O ( n )) ⊕ 2 . W e conclude that x − Θ O ( n − 1) ( c ) ∈ F α − 1 n ( B n ) ∩ F β − 1 n ( B n ), establis hing the claim that B n − 1 is co m- plement ary to the ima ge of Θ O ( n − 1) in F ( O ( n − 1)). W e define Λ O ( n − 1) (Θ O ( n − 1) ( a ) + b ) = a , where b ∈ B n − 1 . It follows immediately that Λ O ( n − 1) Θ O ( n − 1) is the identit y and that (4-1) c omm utes in the case t hat i = n − 1. T o show that ΛΘ is the identit y , we note that by construc tio n of Λ, (ΛΘ) M = id ker Γ F ( M ) for all coherent M . Th us, b y the uniqueness statemen t of Lemma 3 .3, the comp osition ΛΘ is the identit y functor. The result follows.  Corollary 4. 2. Th e fol lowing ar e e quivalent: (1) The natur al t ra nsformation Γ F is an isomorphism. (2) dim k ( F ( O ( i ))) is c onst ant. (3) F is exact on short exact se quenc es of ve ctor bund les. Pr o of. W e first prov e (1) if and only if (2): if Γ F is an isomorphism, then F ∼ = H 0 ( P 1 , − ⊗ W ( F )). T he r efore, since W ( F ) is co herent torsion, dim k ( F ( O ( i ))) is independent o f i . Conv ersely , if dim k ( F ( O ( i ))) is constant, Theorem 4 .1 implies ker Γ F = 0 . Next w e prove (1 ) if a nd only if (3): If Γ F is a n isomor phism then, since W ( F ) is coherent tors ion by Pro pos ition 2.1, F is exact on vector bundles. Co nversely , since Theorem 4.1 implies there exist nonnegative in tegers n , n i such that F ∼ = ⊕ ∞ i = − n H 1 ( P 1 , ( − )( i )) ⊕ n i ⊕ H 0 ( P 1 , − ⊗ W ( F )), F exact on vector bundles implies that n i = 0 for all i . It follows that Γ F is an isomorphism.  Corollary 4. 3. Th e fol lowing ar e e quivalent: 12 A. NYMAN (1) F ∼ = f ∗ wher e f : Spec k → P 1 is a morphism of k -schemes. (2) dim k ( F ( O ( i ))) = 1 for al l i . (3) F is ex act on short exact se quenc es of ve ctor bund les and dim k ( F ( O ( i ))) = 1 for some i . Pr o of. W e first prove (1) if and only if (2): if F ∼ = f ∗ where f : Spec k → P 1 is a morphism o f schemes o ver k , then f ∗ = H 0 ( P 1 , − ⊗ k ( r )) for some c lo sed po in t r in P 1 , whence the forward direction. Conv ersely , by Theorem 4 .1, ther e exist nonnegative in tegers n , n i such tha t F ∼ = ⊕ ∞ i = − n H 1 ( P 1 , ( − )( i )) ⊕ n i ⊕ H 0 ( P 1 , − ⊗ W ( F )). Therefore, if dim k ( F ( O ( i ))) = 1 for all i , n i = 0 for all i , and dim k ( O ( i ) ⊗ W ( F )) = 1 for all i . Since W ( F ) is co he r en t to rsion, W ( F ) ∼ = k ( r ) for s ome clos ed po in t r ∈ P 1 . Next we prove (2) if and only if (3): if dim k ( F ( O ( i ))) = 1 fo r a ll i , then Coro llary 4.2 implies that F is exact on v ector bundles and trivially implies the second part of (3). Conv er sely , if F is exact o n vector bund les, Corollar y 4.2 implies that dim k ( F ( O ( i ))) is cons ta n t. Since it equals 1 for some i , it equals 1 fo r all i , w he nce (2).  References [1] S. Eilenberg, Abstract description of some basic functors, J. Indian Math. So c., 24 (1960) 231-234. [2] R. Hartshorne, A lgebr aic Ge ometry , Spri nger- V erl ag, New Y ork 1977. [3] R. Hartshorne, L o c al Cohomolo g y , Lecture Notes i n Math. 41, Springer-V erlag, Heidelberg (1967). [4] A. N yman, The Eilenberg-W atts Theorem ov er s c hemes, J. Pur e Appl. Algebr a, 214 (2010), 1922-1954. [5] S. P . Smith, Subspaces of non-commutat ive spaces, T r ans. Amer. Math. So c., 35 4 (2002), 2131-2171. [6] C. E. W atts, In trinsic cha racterizations of s ome additive f unctors, Pr o c. Amer. Math. So c. , 11 (1960) 5-8. Dep ar tment of M a thema tics, 5 16 High St, Western W ashington University, Belling- ham, W A 98225 -9063 E-mail addr ess : adam.nym an@wwu.edu