The Horrocks correspondence for coherent sheaves on projective spaces

THE HORROCKS CORRESPONDE NCE FOR COHERENT SHE A VES ON PR OJECTIVE SP ACES IUSTIN COAND ˘ A Abstract. W e establish an equiv alence bet ween the stable category of coherent sheav es (satisfying a mild restriction) on a pro jective space and the homoto p y catego ry of a certain class of minimal complexes of free mo dules ov er the exterio r algebra Koszul dual to the homogene o us coor dinate algebra of the pro j ective space. W e also relate these complexes to the T ate r esolutions of the resp ective sheav es. In this wa y , we extend from vector bundles to coherent sheav es the re sults of G. T ra utmann and the author [9], which interpret in terms of the B GG cor resp ondence the r esults o f T rautmann [23] ab out the corres p ondence of Horro cks [15], [1 6]. W e als o give direct pro ofs of the BGG corres p ondence s for gr a ded mo dules and for co herent sheav es and of the theor em o f Eisenbud, Fløysta d and Schreyer [12] desc r ibing the linear part of the T ate resolution asso ciated to a coherent sheaf. Moreov er, we pr ovide a n ex plicit des c r iption of the quotient of the T ate reso lution b y its linear strand co rresp onding to the mo dule of global sections o f the v a rious t wists of the sheaf. Intr oduction Tw o lo cally free shea v es E and E ′ on the pr o jectiv e space P n o v er a field k a re stabl y e quival e nt if there exist finite direct sums of in vertible shea ves O P ( a ), a ∈ Z , L and L ′ suc h that E ⊕ L ≃ E ′ ⊕ L ′ . Let S = k [ X 0 , . . . , X n ] b e the homogeneous co ordinate ring of P n . P n b eing a quotien t of V \ { 0 } , where V = k n +1 , S can b e identifie d with the symmetric algebra S ( V ∗ ) of the dual v ector space V ∗ . Let Λ := V ( V ) b e the exterior algebra of V . F or 0 < i < n , the g raded S - mo dule H i ∗ E := L d ∈ Z H i ( E ( d )) is an inv arian t fo r stable equiv alence. How eve r, these coho mo lo gy S -mo dules alone do not determine uniquely the stable equiv alence class of E . G. Horro c ks [1 5] sho w ed that the stable equiv alence class is determined b y these mo dules an b y a seq uence of extension classes. Unfortunately , the arguments of the gro up Ext 1 in whic h any one of these extension classes liv es dep end on the previous extension classes. This incon v enience was remo v ed by G. T rautmann [23] who sho w ed that the stable equiv alence class is determined b y a system of matrices whose en tries are (essen tially) elemen ts o f the exterior algebra Λ. T rautmann’s approac h is related to the approa c h from Horro c ks’ pap er [16]. 2000 Mathematics Subje ct Classific ation. Pr imary: 14F05; Seconda ry: 1 3A02, 13D25 , 18E30 . Key wor ds and phr ases. Coherent sheaf, pro jective space, stable c a tegory , derived categ ory . Partially suppor ted by CNCSIS grant ID-PCE no.5 1/28.0 9 .2007 (co de 30 4). 1 2 I. CO AND ˘ A The meaning of the matrices considered b y T rautmann w as clarified, following a sug- gestion of W. D ec k er, by T rautmann and the author in [9] using the Bernstein-Gel’fand- Gel’fand functors. The se functors originate in the following easy observ a t io n: giving a linear complex of graded free S -mo dules: · · · − → S ( p ) ⊗ k N p − → S ( p + 1) ⊗ k N p +1 − → · · · is equ iv alent to giving a (left) Λ-mo dule structure on the g raded k - v ector space N := L p ∈ Z N p . One denotes t he ab o v e complex by F( N ). Similarly , to a graded S - mo dule M one can asso ciate a linear complex G( M ) of g raded free Λ -mo dules: · · · − → M p ⊗ k V ( V ∗ )( p ) − → M p +1 ⊗ k V ( V ∗ )( p + 1) − → · · · where one considers on the exterior alg ebra V ( V ∗ ), graded suc h tha t V ∗ has degree − 1, the structure o f left Λ-mo dule defined b y contraction. The tec hnical reason f or whic h one uses V ( V ∗ )( p ) instead of Λ( p ) is that F( V ( V ∗ )) is the Koszul resolution of S/ ( X 0 , . . . , X n ) 0 → S ( − n − 1 ) ⊗ k n +1 ∧ V ∗ → · · · → S ( − 1) ⊗ k V ∗ → S → 0 . The idea o f I.N. Bernste in, I.M. Gel’fand a nd S.I. Gel’fand [4] w as to extend thes e functors to complexes of mo dules by the form ula F( N • ) := tot ( X •• ), w here N • is a comple x o f graded Λ-mo dules and X •• is the double complex defined b y X p, • := F( N p ), and similarly for G( M • ). No w, consider the linear complex L n − 1 i =1 T − i G(H i ∗ E ) ( T the tr a nslation functor for com- plexes) with terms G p = L n − 1 i =1 H i ( E ( p − i )) ⊗ k V ( V ∗ )( p − i ) a nd let λ b e its differen tial. The first main result of the pap er [9 ] asserts that the stable equiv alence class of E is determined b y a p erturbation d = λ + δ of λ obtained by addition of terms of degree ≥ 2. Here “p erturbation” means that d ◦ d = 0, i.e., G • := (( G p ) p ∈ Z , d ) is a complex, and “obtained by addition of terms of degree ≥ 2” means t ha t : δ p (H i ( E ( p − i )) ⊗ k V ( V ∗ )( p − i )) ⊆ L j < i H j ( E ( p + 1 − j )) ⊗ k V ( V ∗ )( p + 1 − j ) . The second main result of [9] relates the Horro c ks corr esp ondence to the BGG corr espo n- dence via the results o f Eisen bud, Fløystad and Sc hrey er [12] ab out T ate resolutions ov er the exterior alg ebra. Let F b e a coherent sheaf on P n . Eisen bud et al. [12 ] sho w that there is a unique (up to isomorphism) p erturbation of the differential of the linear com- plex L n i =0 T − i G(H i ∗ F ) obtained b y addition of terms of degree ≥ 2 suc h tha t the resulting complex I • is a cyclic . It is sho wn in [9] that the complex G • whic h determines t he stable equiv alence class of a lo cally f ree sheaf E can b e obtained from its T ate r esolution I • b y remo ving the linear strands G(H 0 ∗ E ) and T − n G(H n ∗ E ). In this pap er, we g eneralize the results from [9] to the case of coheren t sheav es using differen t, more natural, arguments : while in [9] one av oids the use of the BG G corr espo n- dence, the pro ofs in the presen t pap er dep end on it. In the first section w e sho w that, using ar gumen ts close to the argumen ts of Horr o c ks [15], one can extend from v ector bundles to coheren t sheav es the splitting criterion of Hor ro c ks and his criterion of stable THE HORROCKS CORRESPONDENCE 3 equiv alence. The extens ion t o coheren t shea v es o f the first criterion is a result obtained recen tly by Ab e and Y oshinaga [1] (see, also, Bertone and Roggero [5]). In the second section w e introduce and pro v e the prop erties o f the BGG functors needed in the pro of of the Hor ro c ks corresp ondence. These are : (1) t he BGG equiv alence b et w een the b o unded deriv ed cat ego ry of finitely generated graded S - mo dules and the corresp onding category of Λ-mo dules, for whic h w e prov ide a direct pro of, av oiding the use of Koszul dualit y; (2) the easy half of the Koszul dualit y phenomenon whic h says that if N is a graded Λ-mo dule then GF( N ) is a righ t resolution of N with graded free Λ- mo dules; (3) using, additionally , the functors Hom S ( − , S ) for S - mo dules and Hom k ( − , k ) for Λ-mo dules, one deduc es, mo dulo some unpleasant sign problems , a functorial (left) free r esolution fo r ev ery Λ-mo dule N ; (4) a ke y tec hnical p oint of the pap er of Eisen bud et a l. [12] describing the linear part of the minimal complex asso ciated to a complex of free mo dules of t he form F( N • ) or G( M • ). The last result is a consequenc e of a general lemma abo ut double complexes, see [12], (3.5). W e explain, in App endix A, that this lemma is a particular case of a general lemma w ell-kno wn in homotop y theory under the name of Basic Pe rturbation Lemma. In the third section w e establish t he Horro c ks corresp ondence for coheren t shea v es. It asserts that the stable categor y of cohe ren t shea v es F on P n with the property tha t H 0 F ( − t ) = 0 fo r t >> 0 is equiv alent to the homotopy category of mi n imal complexes G • of graded free Λ-mo dules whose linear part is of the f o rm L n − 1 i =1 T − i G( H i ), where H i is the k -v ector space graded dual of a finitely generated graded S - mo dule of Krull dimension ≤ i + 1. This is equiv a len t to the fact that G • is minimal and satisfies the follo wing three conditions: (i) G • is right b ounde d and H p ( G • ) = 0 for p << 0, (ii) ∀ p ∈ Z , G p is of the form L n − 1 i =1 V ( V ∗ )( p − i ) c pi , (iii) lim p →∞ ( c − p,i /p i +1 ) = 0, i = 1 , . . . , n − 1. In the fourth section we relate the Horro cks corresp ondence a nd the BGG corresp on- dence. W e first giv e a direct pro of of the BGG description of the b ounded deriv ed category of coheren t shea v es on P n . This pro of is based on an elemen tary comparison lemma whic h is discussed in App endix B. Using the comparison lemma w e also g et a quick pro o f of the theorem of Eisen bud, Fløystad a nd Sc hrey er [12], Theorem 4.1., ab out T ate resolutions of coheren t shea v es on P n . Moreo v er, we pro vide a concrete description of the quotien t I • / G(H 0 ∗ F ), where I • is the T ate resolution of a coheren t sheaf F with H 0 ( F ( − t )) = 0 for t >> 0. Using this concrete description w e derive that the complex G • asso ciated to F b y the Horr o c ks correspondence can b e obtained from its T ate resolution I • b y remo ving the linear strands G(H 0 ∗ F ) and T − n G(H n ∗ F ). Notation. Throughout this pap er, V will denote an ( n + 1)-dimensional v ec tor space o v er a field k , e 0 , . . . , e n a fixed basis of V and X 0 , . . . , X n the dual basis of V ∗ := Hom k ( V , k ). 4 I. CO AND ˘ A (i) Let S = S ( V ∗ ) = L i ≥ 0 S i ( V ∗ ) ≃ k [ X 0 , . . . , X n ] b e t he symmetric algebra of V ∗ and S + = L i ≥ 1 S i ( V ∗ ) its irr elev an t homogeneous ideal. W e denote b y S -Mo d the category of graded S -mo dules with al l the h o mo ge n e ous c omp o n ents finite di m ensional ve ctor sp ac es , and with morphisms of degree 0. S - mo d denotes the full sub categor y of S -Mo d consisting of finitely generated gra ded S -mo dules, and P denotes the full sub category of S -mo d consisting of its free ob jects. (ii) If M is an ob ject o f S -mo d, the finitely generated S -mo dule M ∨ := Hom S ( M , S ) has a natura l grading giv en by ( M ∨ ) d = Hom S - m od ( M , S ( d )). If M is an o b j ect of S -Mo d, the graded dual v ector space M ∗ := L d ∈ Z Hom k ( M − d , k ) has a natura l structure of graded S -mo dule. (iii) L et Λ = V ( V ) = L n +1 i =0 i ∧ V b e the (p ositiv ely graded) exterior algebra of V . Λ + := L n +1 i =1 i ∧ V is an ideal of Λ. Let k denote the quotien t Λ / Λ + . W e denote by Λ- mo d the category of finitely generated graded left Λ-mo dules with mo r phisms of degree 0, and b y I its full sub category consisting of free ob jects. If N ∈ Ob(Λ-mo d), so c( N ) denotes the submo dule o f N consisting of the elemen ts annihilated by Λ + . It can b e iden tified with Hom Λ ( k , N ). Remark that so c (Λ) = n +1 ∧ V . (iv) Let P = P n = P ( V ) denote t he (classical) pro jective space parametrizing the 1-dimensional v ector subspaces of V . T he homogeneous co ordinate ring of P is S . W e denote by Coh P the category of coheren t shea v es o n P and b y ( − ) ∼ : S -mo d → Coh P the functor of Serre [21 ] asso ciating to a graded S - mo dule its sheafification. If F is a coheren t sheaf on P and 0 ≤ i ≤ n we shall denote b y H i ∗ F the graded S -mo dule L d ∈ Z H i ( F ( d )). (v) W e denote by C ( A ), C b ( A ), C ± ( A ) the categories of complexes in an ab elian cat- egory A , by K ( A ), K b ( A ), K ± ( A ) the corresp onding homotop y categories, and b y D( A ), D b ( A ), D ± ( A ) the corresp onding deriv ed categories. “T” will denote the tr anslation func- tor and “Con” the mapping c one . When A is S - mo d or Λ-mo d or Coh P we shall use the shorter notatio n C( S ), C(Λ), C( P ) etc. Our main reference for category theory will b e Chapter I of the b o ok of Kashiw ara and Sc hapira [18]. One may also use the b o oks o f Kashiw ara and Sc hapira [19] or G el’fand and Manin [13]. 1. Tw o crite ria of Horrocks In t his section w e include pro ofs o f t w o in tro ductory results whic h extend to coherent shea v es the splitting criterion o f Horro c ks for v ector bundles on pro jectiv e spaces and his criterion c haracterizing stable equiv alence s in the same con text. There are (at least) t w o recen tly published pro of s of the first result in Abe and Y oshinaga [1] and Bertone and Roggero [5]. W e follow, ho w ev er, Horro c ks’ original approac h. It is based on the next theorem, whic h is usually prov ed using lo cal cohomology and lo cal duality (see, for THE HORROCKS CORRESPONDENCE 5 example, [11], ( A.4.1.) and (A.4 .2 )). W e shall giv e, for the reader’s con v enience, a direct pro of av o iding the use of lo cal cohomology . 1.1. Theorem. (Gra ded Serre Dualit y) If M is a fini tely gener ate d g r ade d S - mo dule then ther e exist an e xact se quenc e : 0 → Ext n +1 S ( M , ω S ) ∗ → M → H 0 ∗ f M → Ext n S ( M , ω S ) ∗ → 0 and isomorph i s ms : H i ∗ f M ≃ Ext n − i S ( M , ω S ) ∗ , 1 ≤ i ≤ n , wher e ω S := S ( − n − 1) ⊗ k n +1 ∧ V ∗ . Pr o of. e ω S = O P ( − n − 1 ) ⊗ k n +1 ∧ V ∗ ≃ ω P . One know s that H n ( ω P ) ≃ k and that, ∀ a ∈ Z : Hom O P ( O P ( − a ) , ω P ) ∼ − → Hom k (H n ( O P ( − a )) , H n ( ω P )) . It follows that if L is a free graded S -mo dule of finite ra nk then there exists a functorial isomorphism: Hom S ( L, ω S ) ∼ − → (H n ∗ e L ) ∗ . No w, let 0 → L − n − 1 → · · · → L 0 → M → 0 b e a free resolution of M in S -mo d. Let C − i b e the cok ernel of L − i − 1 → L − i . One has short exact sequences : 0 → C − i − 1 → L − i → C − i → 0 . (1) W e consider the complex H n ∗ e L • ≃ Hom S ( L • , ω S ) ∗ . Since H n ∗ is right exact, w e ha v e exact sequence s: H n ∗ e L − i − 1 → H n ∗ e L − i → H n ∗ e C − i → 0 hence H 0 (H n ∗ e L • ) ≃ H n ∗ f M and H − i (H n ∗ e L • ) ≃ Ker(H n ∗ e C − i → H n ∗ e L − i +1 ) for i ≥ 1. Since H p ∗ e L − j = 0 for 0 < p < n , ∀ j , one deduces easily , using the sheafifications of the exact sequence s ( 1 ), that: H − i (H n ∗ e L • ) ≃ H n − i ∗ f M , 0 ≤ i ≤ n − 1 and that one has exact sequences : L 0 = H 0 ∗ e L 0 → H 0 ∗ f M → H − n (H n ∗ e L • ) → 0 L − 1 = H 0 ∗ e L − 1 → H 0 ∗ e C − 1 → H − n − 1 (H n ∗ e L • ) → 0 , hence: H − n (H n ∗ e L • ) ≃ Cok er( M → H 0 ∗ f M ) H − n − 1 (H n ∗ e L • ) ≃ Cok er( C − 1 → H 0 ∗ e C − 1 ) . Finally , applying the Snak e Lemma to the dia g ram: 0 − − − → C − 1 − − − → L 0 − − − → M − − − → 0   y   y ≀   y 0 − − − → H 0 ∗ e C − 1 − − − → H 0 ∗ e L 0 − − − → H 0 ∗ f M one gets t hat: Cok er( C − 1 → H 0 ∗ e C − 1 ) ≃ Ker( M → H 0 ∗ f M ).  6 I. CO AND ˘ A 1.2. Lemma. If M 6 = 0 is a fini tely gen er ate d gr ade d S - mo dule of pr oj e ctive dimen- sion m then Ext m S ( M , S ) 6 = 0. Pr o of. Let 0 → L − m → · · · → L 0 → M → 0 b e a free resolution of M in S -mo d. If Ext m S ( M , S ) = 0 then L − m +1 ∨ → L − m ∨ is surjectiv e, hence its k ernel L ′− m +1 is free and the sequence: 0 → L ′− m +1 → L − m +1 ∨ → L − m ∨ → 0 is split exact. It follows that the dua l sequence: 0 → L − m → L − m +1 → ( L ′− m +1 ) ∨ → 0 is exact. One gets an exact sequence: 0 → ( L ′− m +1 ) ∨ → L − m +2 → · · · → L 0 → M → 0 from whic h w e deduce that the pro jective dimension of M is ≤ m − 1, a contradiction.  1.3. Theorem. (Horro c ks’ splitting criterion) L et F b e a c oher ent she af o n P n with the pr op erty that H 0 ( F ( − t )) = 0 for t >> 0. If H i ∗ F = 0 for 0 < i < n then F is a dir e ct sum of inv e rtible she a v es O P ( a ), a ∈ Z . Pr o of. By h yp othesis, M := H 0 ∗ F is a finitely gener ate d graded S -mo dule. It follo ws that M → H 0 ∗ f M is an isomorphism . One deduces, now, from the h ypot hesis and fr o m The orem 1.1., that Ext i S ( M , ω S ) = 0, ∀ i > 0. It f ollo ws, from Lemma 1.2., that M is a graded free S -mo dule, hence a direct sum of graded S -mo dules of the f o rm S ( a ), a ∈ Z .  1.4. Theorem. (Horro c ks’ criterion o f stable equiv alence) L et φ : F → G b e a mor- phism o f c oher ent she aves on P n , n ≥ 2, with the pr o p erty that H 0 φ ( − t ) is an isomorphism for t >> 0. If H i ∗ φ is a n isomorphism for 0 < i < n then φ factorizes as : F ֒ → F ⊕ A ∼ − → G ⊕ B ։ G wher e the first morphism is the c anonic al inclusion , A and B ar e finite dir e ct sums of invertible she aves O P ( a ), a ∈ Z , and the last morphis m is the c anonic al pr oje ction . Pr o of. Cho ose m ∈ Z suc h that H 0 φ ( − t ) is an isomorphism fo r t > m and let M := L j ≥ − m H 0 ( F ( j )), N := L j ≥ − m H 0 ( G ( j )) . Cho ose a n epimorphism g : A → N , with A a finitely generated gra ded fr ee S -mo dule. Let π : F ⊕ e A → G b e the epimorphism defined b y φ and e g and let B b e the ke rnel of π . Using the exact sequence: 0 → B → F ⊕ e A → G → 0 (*) one sees that B satisfies the h yp othesis of Theorem 1.3., hence B is a direct sum of inv ert- ible sheav es O P ( b ), b ∈ Z , and, consequen tly , B := H 0 ∗ B is a graded free S -mo dule. Ap- plying H 0 ∗ to the exact seque nce (*) and cancellating the isomorphism L j < − m H 0 F ( j ) ∼ → THE HORROCKS CORRESPONDENCE 7 L j < − m H 0 G ( j ) one gets a short exact sequence: 0 → B → M ⊕ A → N → 0 . (**) Since H n − 1 ∗ φ is an isomorphism, it follows from Theorem 1.1 . that: Ext 1 S ( N , B ) − → Ext 1 S ( M ⊕ A, B ) is an isomorphism, hence Hom S ( M ⊕ A, B ) → Hom S ( B , B ) is surjectiv e, hence the exact sequence (**) splits.  2. The BGG functors 2.1. Defi nition. When dealing with the catego ry Λ-mo d one encounters sign prob- lems. In order to av oid any complication we shall observ e strictly the Koszul sign c on- vention (when t w o homogeneous sym bo ls ξ and η are p erm uted the result is m ultiplied b y ( − 1) deg ξ · deg η ). (i) If K , N ∈ Ob(Λ-mo d) the g raded k -v ector space K ⊗ k N has a structure of left Λ-mo dule give n b y: v · ( x ⊗ y ) := ( v · x ) ⊗ y + ( − 1) deg x x ⊗ ( v · y ) , for v ∈ V . In pa rticular, w e put, for a ∈ Z , N ( a ) := k ( a ) ⊗ k N . The g r ading of N ( a ) is give n by N ( a ) p = N p + a and the Λ-mo dule structure by: ( v · y ) N ( a ) = ( − 1) a ( v · y ) N , for v ∈ V , y ∈ N . With this definition, if v ∈ V then the morphism of k -v ector spaces ( v · − ) N : N → N is a morphism in Λ-mo d: N ( a ) → N ( a + 1), ∀ a ∈ Z . If φ : K → N is a morphism in Λ-mo d, φ ( a ) : K ( a ) → N ( a ) is just φ if one fo rgets t he g r adings. Ho w ev er, if N • ∈ Ob C(Λ- mo d) then N • ( a ) is, b y definition, the complex with terms ( N p ( a )) p ∈ Z but with d N ( a ) := ( − 1) a d N (the differen tial of a complex is a sym b ol o f degree 1 !). (ii) If K , N ∈ Ob(Λ-mo d) the g raded k - v ector space Hom k ( N , K ) has a structure of left Λ- mo dule giv en by: ( v · φ )( y ) := v · φ ( y ) − ( − 1) deg φ φ ( v · y ) , for v ∈ V . In pa rticular, for K = k , one puts N ∗ := Hom k ( N , k ). One ha s ( N ∗ ) p = ( N − p ) ∗ and, for v ∈ V , ( v · − ) N ∗ : ( N ∗ ) p → ( N ∗ ) p +1 is ( − 1) p +1 · the dual o f ( v · − ) N : N − p − 1 → N − p . The map µ : N → N ∗∗ , µ ( y )( φ ) := ( − 1) deg y · deg φ φ ( y ) (i.e., with µ p := ( − 1) p can : N p → ( N p ) ∗∗ , p ∈ Z ) is a n isomorphism in Λ-mo d. (iii) The ma p α : K ∗ ⊗ k N ∗ → ( K ⊗ k N ) ∗ giv en b y: α ( f ⊗ g )( x ⊗ y ) := ( − 1) deg g · deg x f ( x ) g ( y ) is a n isomorphism in Λ-mo d. In part icular, for a ∈ Z , one gets an isomorphism in Λ-mo d α : N ∗ ( − a ) ∼ → ( N ( a )) ∗ with α p = ( − 1) ( p − a ) a id ( N a − p ) ∗ , p ∈ Z . Under these iden tifications, if v ∈ V and a ∈ Z , the dual of the morphism ( v · − ) N : N ( − a − 1) → N ( − a ) is iden tified with the morphism ( − 1) a ( v · − ) N ∗ : N ∗ ( a ) → N ∗ ( a + 1). 8 I. CO AND ˘ A (iv) W e endo w V ( V ∗ ) = L n +1 i =0 i ∧ V ∗ , graded suc h that i ∧ V ∗ = V ( V ∗ ) − i , with the structure of graded left Λ -mo dule given b y: v · f 1 ∧ . . . ∧ f p := p X i =1 ( − 1) i − 1 f i ( v ) f 1 ∧ . . . ∧ b f i ∧ . . . ∧ f p , v ∈ V , f 1 , . . . , f p ∈ V ∗ . The unique mor phism of left Λ-mo dules Λ → V ( V ∗ ) ∗ (resp., Λ → V ( V ∗ )( − n − 1) ⊗ k n +1 ∧ V ) mapping 1 ∈ Λ 0 to 1 ∈ ( V ( V ∗ ) ∗ ) 0 (resp., to id n +1 ∧ V ∈ ( V ( V ∗ )( − n − 1) ⊗ k n +1 ∧ V ) 0 ) is an isomorphism in Λ-mo d. The follow ing lemma, whose standard pro of can b e f ound, f or example, in [8], (4) (i), sho ws, in pa rticular, that, ∀ a ∈ Z , V ( V ∗ )( a ) is an injectiv e ob ject of Λ-mo d. 2.2. Lemma. ∀ N ∈ O b(Λ-mo d), ∀ a ∈ Z , the ma p : Hom Λ-mo d ( N , V ( V ∗ )( a )) − → ( N − a ) ∗ , φ 7→ φ − a is bije ctive . 2.3. Remark. φ − a − 1 : N − a − 1 → V ( V ∗ )( a ) − a − 1 = V ∗ c an b e describ e d by : φ − a − 1 ( y )( v ) = ( − 1) a φ − a ( v · y ) , ∀ y ∈ N − a − 1 , ∀ v ∈ V , or, e quivalently, by : φ − a − 1 = ( − 1) a P n i =0 ( φ − a ◦ ( e i · − ) N ) ⊗ X i . Pr o of. R ecalling the D efinition 2.1.(i), (ii), one has, for v ∈ V , λ ∈ V ∗ : ( v · λ ) V ( V ∗ )( a ) = ( − 1) a ( v · λ ) V ( V ∗ ) = ( − 1) a λ ( v ) . In particular, for λ = φ − a − 1 ( y ), since φ is a mor phism in Λ- mo d: ( − 1) a φ − a − 1 ( y )( v ) = ( v · φ − a − 1 ( y )) V ( V ∗ )( a ) = φ − a ( v · y ) .  2.4. Definition. (The BG G functors) (i) One defines a functor F : Λ-mo d → C b ( P ) by: F( N ) p := S ( p ) ⊗ k N p , d F( N ) := P n i =0 ( X i · − ) S ⊗ ( e i · − ) N . F can b e extended to a functor F : C b (Λ-mo d) → C b ( P ) b y putting F( N • ) := tot( X •• ), where X •• is the double complex with X p, • := F( N p ) and with d ′ p X : X p, • → X p +1 , • equal to F( d p N ). (ii) One defines a functor G : S -Mo d → C( I ) by : G( M ) p := M p ⊗ k V ( V ∗ )( p ) , d G( M ) := P n i =0 ( X i · − ) M ⊗ ( e i · − ) V ( V ∗ ) . G can b e extended, in a similar w a y , to a functor G : C b ( S -Mo d) → C( I ). The (extended) functor G maps C b ( S -mo d) to C + ( I ) . (iii) F and G c ommute with the tr anslation functors and with m apping c one s . THE HORROCKS CORRESPONDENCE 9 2.5. Definition. Let Φ : A op → B b e an additiv e con tra v ar ia n t functor b etw een tw o additiv e categories A a nd B . If X • ∈ Ob C( A ), one defines a complex Φ( X • ) ∈ Ob C( B ) b y: Φ( X • ) p := Φ( X − p ) , d p Φ( X ) := ( − 1) p +1 Φ( d − p − 1 X ) : Φ( X − p ) → Φ( X − p − 1 ) . F or example , if M • ∈ Ob C( S - mo d) one can define the complex M •∨ ∈ Ob C( S - mo d) and if M • ∈ Ob C( S - Mo d) (res p., N • ∈ Ob C(Λ-mo d)) one can define the comple x M •∗ ∈ Ob C( S -Mo d) (resp., N •∗ ∈ Ob C(Λ- mo d)). F urthermore, if X •• is a double complex in A one defines a double complex Φ( X •• ) in B by: Φ( X •• ) pq := Φ( X − p, − q ) , d ′ pq Φ( X ) := ( − 1) p +1 Φ( d ′− p − 1 , − q X ) , d ′′ pq Φ( X ) := ( − 1) q +1 Φ( d ′′− p, − q − 1 X ) . If w e denote Φ( X •• ) by Y •• then Y p, • = Φ( X − p, • ) and d ′ p, • Y = ( − 1) p +1 Φ( d ′− p − 1 , • X ). 2.6. Lemma. (a) I f X •• and Y •• ar e two double c omplexes with X pq = Y pq , ∀ p, q , but with d ′ Y = ( − 1) a d ′ X and d ′′ Y = ( − 1) b d ′′ X , for som e a, b ∈ Z , then X •• ≃ Y •• . (b) Using the notations fr om the last p art of Definition 2 .5 ., assume that , ∀ m ∈ Z , the set { ( p, q ) | p + q = m, X pq 6 = 0 } is finite. T hen tot(Φ( X •• )) ≃ Φ(tot( X •• )). Pr o of. (a ) (( − 1) ap + bq id X pq ) p,q ∈ Z is an isomorphism of double complexes X •• ∼ → Y •• . (b) One can easily che c k that Φ(tot( X •• )) = t ot( Z •• ), where the double complex Z •• is defined by: Z pq := Φ( X − p − q ) , d ′ pq Z := ( − 1) p + q +1 Φ( d ′− p − 1 , − q X ) , d ′′ pq Z := ( − 1) p + q +1 Φ( d ′′− p, − q − 1 X ) . But (( − 1) pq id Φ( X − p, − q ) ) p,q ∈ Z is an isomorphism of double complexes Φ( X •• ) ∼ → Z •• .  2.7. Lemma. L et M • ∈ Ob C b ( S -Mo d), N • ∈ Ob C b (Λ-mo d) a n d a ∈ Z . Then : (a) F( N • ( a )) = T a F( N • )( − a ) and G( M • ( a )) = T a G( M • )( − a ) ( one has e quality, not only an iso morphism !), (b) F( N •∗ ) ≃ F( N • ) ∨ , (c) G( M • ) ∗ ≃ G( M •∗ )( − n − 1) ⊗ k n +1 ∧ V ≃ T − n − 1 G(( M • ⊗ S ω S ) ∗ ). Pr o of. (a ) One c hec ks , firstly , that if M ∈ Ob( S - Mo d) and N ∈ Ob(Λ -mo d) then F( N ) = T a F( N )( − a ) and G ( M ) = T a G( M )( − a ). F o r the general case , one ta kes into accoun t the sign conv ention at the end of Definition 2.1.(i). (b) If N ∈ Ob(Λ-mo d) then one c hec ks easily tha t F( N ∗ ) = F( N ) ∨ . Now, if N • ∈ Ob C b (Λ-mo d) then, by defin tion, F( N • ) = to t( X •• ) with X p, • = F( N p ), ∀ p ∈ Z . One deduces that, using the last part of Definition 2.5., F( N •∗ ) = tot(( X •• ) ∨ ). But, b y Lemma 2.6.(b), tot(( X •• ) ∨ ) ≃ tot( X •• ) ∨ . (c) If N ∈ Ob(Λ-mo d), o ne can define a functor G N : S -Mo d → C(Λ- mo d) b y G N ( M ) p := M p ⊗ k N ( p ), d G N ( M ) := P n i =0 ( X i · − ) M ⊗ ( e i · − ) N . As in Definition 2.4., G N can b e extended to a functor G N : C b ( S -Mo d) → C(Λ-mo d). 10 I. CO AND ˘ A Firstly , if M ∈ Ob( S -Mo d) then, by Definition 2.1.(iii), G N ∗ ( M ∗ ) p ∼ → (G N ( M ) − p ) ∗ , ∀ p ∈ Z , and under these iden tifications, d p G N ∗ ( M ∗ ) is iden tified with ( − 1) p ( d − p − 1 G N ( M ) ) ∗ . R e- calling t he Definition 2 .5 ., it f o llo ws that G N ∗ ( M ∗ ) is isomorphic to a complex wh ose terms coincide with the terms of G N ( M ) ∗ but whose differential equals “ − the differential of G N ( M ) ∗ ”. Using Lemma 2.6., one deduces now, for ev ery complex M • ∈ Ob C b ( S -Mo d), an isomorphism G N ∗ ( M •∗ ) ∼ → G N ( M • ) ∗ . Secondly , if M ∈ Ob( S -Mo d) a nd a ∈ Z then, taking into accoun t the sign conv ention at the end of Definition 2.1.(i), G N ( a ) ( M ) = G N ( M )( a ). Using Lemma 2.6.(a) one deduces, for ev ery complex M • ∈ Ob C b ( S -Mo d), an isomorphism G N ( a ) ( M • ) ≃ G N ( M • )( a ). Since G := G V ( V ∗ ) one gets, recalling the isomorphisms at the end of D efinition 2.1.(iv), the first isomorphism from the statemen t. The second isomorphism follows f rom (a).  2.8. Definition. (The linear part of a minimal complex) (i) Let L • ∈ Ob C( P ) . One may write L i = L j ∈ Z S ( i − j ) b ij . F or m ∈ Z one puts: F m L i := L j ≤ m S ( i − j ) b ij . Alternativ ely , F m L i is the S -submo dule o f L i generated b y the homogeneous elemen ts of degree ≤ m − i . The complex L • is called minimal if Im d L ⊆ S + · L • . This is equiv alen t to the fact, ∀ m ∈ Z , F m L • := ( F m L i ) i ∈ Z is a sub c omplex of L • . In this case, gr F ( L • ) is called the lin e ar p art of L • . (ii) Similarly , let I • ∈ Ob C( I ). One ma y write (by the last part of Definition 2 .1.(iv)) I i = L j ∈ Z V ( V ∗ )( i − j ) c ij . F or m ∈ Z , one puts: F m I i := L j ≤ m V ( V ∗ )( i − j ) c ij . Alternativ ely , F m I i is the Λ- submo dule of I i generated b y t he homogeneous elemen ts of degree ≤ m − i − n − 1. The complex I • is called minimal if Im d I ⊆ Λ + · I • or, equiv alen tly , if F m I • := ( F m I i ) i ∈ Z is a sub complex of I • , ∀ m ∈ Z . In this case, g r F ( I • ) is called the line ar p art of I • . (iii) If two minimal c omplexes fr om C( I ) ar e isom orphic in K( I ) ( i.e. , ar e homotopic al ly e quival e nt ) then they a r e isom orphic in C( I ) ( see, for example, [9], (4.2.)). The follow ing result, which is a direct consequence of Lemma A.7 . from App endix A, is one o f the k ey p oin ts of the pap er of Eisen bud, Fløystad, a nd Sc hrey er [12]. 2.9. Lemma. (a) If N • ∈ Ob C b (Λ-mo d) then F( N • ) ∈ Ob C b ( P ) c an b e c ontr acte d to a minimal c omplex L • whose line ar p art is F(H • ( N • )), whe r e H • ( N • ) is the c omplex with terms H p ( N • ), p ∈ Z , and with the diff e r ential e qual to 0. Mor e over, this c ontr action induc e s , ∀ m ∈ Z , a c ontr action of F( τ ≤ m N • ) onto F m L • and of F( τ >m N • ) onto L • /F m L • . THE HORROCKS CORRESPONDENCE 11 (b) If M • ∈ Ob C b ( S -Mo d) then G( M • ) ∈ O b C( I ) c an b e c ontr acte d to a minimal c om plex I • whose line ar p art is G(H • ( M • )). Mor e over, this c on tr action induc es , ∀ m ∈ Z , a c ontr action of G( τ ≤ m M • ) onto F m I • and of G( τ >m M • ) onto I • /F m I • . The next theorem is the Bernstein-Gel’fand- Gel’fand correspo ndence for graded mo d- ules. W e include here a direct pro of, whic h do es not use Koszul duality . W e use, instead, the Comparison L emma B.1. f rom App endix B. 2.10. Theorem. ([4], Theorem 3.) The functor F : C b (Λ-mo d) → C b ( P ) ex tends to an e quivalenc e of triangulate d c ate- gories F : D b (Λ-mo d) → K b ( P ). Pr o of. If φ : N ′• → N • is a quasi-isomorphism in C b (Λ-mo d) then Con( φ ) is a cyclic. By Lemma 2.9.(a), the complex Con(F ( φ )) = F (Con( φ )) is homotopically equiv alen t to 0, whence F( φ ) is a ho motopic equiv alence. O ne deduces that F e xtends to a functor F : D b (Λ-mo d) → K b ( P ). W e sho w, firstly , that this functor is ful ly faithful , i.e., that if N • , N ′• ∈ Ob C b (Λ-mo d) then: Hom D b (Λ) ( N ′• , N • ) ∼ − → Hom K b ( P ) (F( N ′• ) , F( N • )) . (*) W e en do w N • and N ′• with the filtrations F i N • = σ ≥ i N • , F i N ′• = σ ≥ i N ′• with suc cessiv e quotien ts T − i N i and T − i N ′ i , resp ectiv ely . If, ∀ i, j ∈ Z , the map: Hom D b (Λ) ( N ′ i , T p N j ) − → Hom K b ( P ) (F( N ′ i ) , F(T p N j )) w ould b e bijectiv e for i − j − 1 ≤ p ≤ i − j + 1 then Lemma B.1., applied to t he functor F : D b (Λ-mo d) → K b ( P ), would imply that (* ) is bijectiv e. Now, if K and K ′ are tw o ob jects of Λ-mo d t hen: Hom D b (Λ) ( K ′ , K ) ∼ − → Hom K b ( P ) (F( K ′ ) , F( K )) as one can easily see using the fact that Hom D b (Λ) ( K ′ , K ) ≃ Hom Λ-mo d ( K ′ , K ) . Moreo v er, Hom D b (Λ) ( K ′ , T p K ) = 0 for p < 0 (see (B.3.)) and, for p > 0, Hom D b (Λ) ( K ′ , T p K ) = 0 if K is a direct sum of Λ-mo dules of the form V ( V ∗ )( a ) b ecause, in t his case , K is an injectiv e ob ject of Λ- mo d. On the other hand, Hom K b ( P ) (F( K ′ ) , F(T p K ) ) = 0 for p < 0 b ecause F( K ′ ) i = S ( i ) ⊗ k K ′ i and F(T p K ) i = F( K ) i + p = S ( i + p ) ⊗ k K i + p , ∀ i ∈ Z . Moreo v er, if L • ∈ Ob C b ( P ) and H i ( L • ) − i = 0, ∀ i ∈ Z , t hen Hom K b ( P ) (F( K ′ ) , L • ) = 0 b ecause Hom K b ( P ) (T − i F( K ′ ) i , L • ) ≃ H i ( L • ) − i ⊗ k ( K ′ i ) ∗ = 0, ∀ i ∈ Z , and F( K ′ ) can b e endo w ed with the filtra tion σ ≥ i F( K ′ ), i ∈ Z , with successiv e quotien ts T − i F( K ′ ) i . F or L • = F(T p K ) , the condition H i ( L • ) − i = 0, ∀ i ∈ Z , is fulfilled if p > 0 and K is a direct sum of Λ-mo dules of the f o rm V ( V ∗ )( a ), b ecause F(T p V ( V ∗ )( a )) = T p + a F( V ( V ∗ ))( − a ) and F( V ( V ∗ )) is the Koszul resolution o f S/S + . 12 I. CO AND ˘ A Summing up, if N j is a direct sum of Λ - mo dules of the form V ( V ∗ )( a ), ∀ j ≤ sup { i ∈ Z | N ′ i 6 = 0 } , then Lemma B.1. implies that (*) is bijectiv e. If N • is arbitrary , one constructs, using Lemma 2.2 ., a quasi-isomorphism N • → I • with I • ∈ Ob C + ( I ) . F or m > sup { i ∈ Z | N ′ i 6 = 0 } large enough, one gets a quasi-isomorphism N • → τ ≤ m I • . By what hav e b een prov ed, (*) is bijectiv e for the pair ( N ′• , τ ≤ m I • ), hence also for the pair ( N ′• , N • ). Finally , the e ssential surje ctivity can b e c hec k ed as fo llo ws. By what ha v e b een pr ov ed, the image of F : D b (Λ-mo d) → K b ( P ) is a ful l sub categor y of K b ( P ), closed under the functors T and T − 1 and under mapping cones. Moreov er, F(T a k ( − a )) = S ( a ), ∀ a ∈ Z . If L • ∈ Ob K b ( P ) one deduces easily , b y induction o n P i ∈ Z rk L i , that L • is isomor phic in K b ( P ) to a complex in the image of F.  Actually , the author s of [4 ] pro v e something more, namely that one can get a quasi- inverse t o F by applying G and then ta king con v enien t truncations (see Beilinson et al. [3], (2.12.) for a detailed pro of ). W e shall only need the easy half of this fact, whic h is the con ten t of the f ollo wing: 2.11. Prop osition. Ther e exists a functorial quasi-isom orphism N • → GF( N • ), ∀ N • ∈ Ob C b (Λ-mo d). Pr o of. W e conside r, firstly , the case of an ob j ect N of Λ-mo d. In this case it turns out that GF( N ) is an injectiv e resolution of N in Λ-mo d. Indeed, by definition, GF( N ) = tot( Y •• ) with Y p, • = G(F( N ) p ) = G ( S ( p ) ⊗ k N p ), i.e., with Y pq = S p + q ( V ∗ ) ⊗ k N p ⊗ k V ( V ∗ )( q ). In particular, GF( N ) m = 0 fo r m < 0 and GF( N ) 0 = L p ∈ Z N p ⊗ k V ( V ∗ )( − p ). Let β p : N → N p ⊗ k V ( V ∗ )( − p ) b e the morphism corresp onding, according to Lemma 2.2., to ( − 1) p id N p , and let β : N → GF( N ) 0 b e the morphism defined b y β p , p ∈ Z . W e w an t to chec k that d 0 GF( N ) ◦ β = 0. This is equiv alen t to the fact that, ∀ p ∈ Z , the diagram: N p ⊗ k V ( V ∗ )( − p ) P X i ⊗ ( e i ·− ) N ⊗ id − − − − − − − − − − → V ∗ ⊗ k N p +1 ⊗ k V ( V ∗ )( − p ) β p x   x   ( − 1) p +1 P X i ⊗ id ⊗ ( e i ·− ) V ( V ∗ ) N β p +1 − − − − − − − − − − → N p +1 ⊗ k V ( V ∗ )( − p − 1) an ticomm utes. According to Lemma 2.2., this is equiv alen t to the fa ct that the diagram: N p P X i ⊗ ( e i ·− ) N − − − − − − − − → V ∗ ⊗ N p +1 ( − 1) p id N p x   x   ( − 1) p +1 P X i ⊗ id N p +1 ⊗ ( e i ·− ) V ( V ∗ ) N p β p +1 p − − − − − − − → N p +1 ⊗ V ∗ an ticomm utes. But, according to Remark 2.3 ., β p +1 p = ( − 1) p +1 · ( − 1) p +1 P ( e i · − ) N ⊗ X i = P ( e i · − ) N ⊗ X i . THE HORROCKS CORRESPONDENCE 13 W e hav e th us defined a morphism of complexes β : N → G F( N ). In or der to show that it is a quasi-isomorphism, one may assume, by induction on dim k N , tha t N = k ( a ) for some a ∈ Z , and then that a = 0, i.e., that N = k . In this case F( k ) = S and the complex G( S ): · · · → 0 → V ( V ∗ ) → V ∗ ⊗ k V ( V ∗ )(1) → · · · → S p ( V ∗ ) ⊗ k V ( V ∗ )( p ) → · · · is an injectiv e resolution of k in Λ-mo d, as one can easily che c k using the fact that t he Koszul complex 0 → S ( − n − 1 ) ⊗ k n +1 ∧ V ∗ → · · · → S ( − 1) ⊗ k V ∗ → S → 0 is a (free) resolution of S/S + in S -mo d. The general case N • ∈ Ob C b (Λ-mo d) can b e no w deduced from the follo wing easy ob- serv ation : GF( N • ) = tot( Z •• ), where Z •• is the double complex with Z p, • = GF( N p ) and with d ′ p Z : Z p, • → Z p +1 , • equal to GF( d p N ). Indeed, b y definition, F( N • ) = tot( X •• ) with X p, • = F( N p ) and G F( N • ) = tot( Y •• ) with Y m, • = G(to t( X •• ) m ) = L p + q = m G( X pq ). Consider the triple c omplex W ••• defined b y W pq , • = G( X pq ). W e ha v e: Z p, • = GF( N p ) = G( X p, • ) = tot( W p, •• ), hence: tot( Z •• ) = tot( W ••• ) = tot( Y •• ) = GF( N • ).  2.12. Corollary . ∀ N • ∈ Ob C b (Λ-mo d), ther e exis ts a functorial quasi-isomorphi sm : T − n − 1 G((F( N • ) ∨ ⊗ S ω S ) ∗ ) − → N • . Pr o of. By Prop osition 2.11., there exists a functorial quasi-isomorphism N •∗ → GF( N •∗ ) and, b y Lemma 2.7.(b), F( N •∗ ) ≃ F( N • ) ∨ . One gets a quasi-isomorphism G(F( N • ) ∨ ) ∗ → N • and, by Lemma 2.7.(c), G(F( N • ) ∨ ) ∗ ≃ T − n − 1 G((F( N • ) ∨ ⊗ S ω S ) ∗ ).  3. The Horrocks corresp ondence 3.1. Definition. (i) If M , M ′ ∈ Ob( S -mo d) let I P ( M ′ , M ) denote the subgroup of Hom S - m od ( M ′ , M ) consisting of the morphisms factorizing through an ob ject of P . The stable c ate gory S - mo d has, b y definition, the same ob jects as S -mo d, but the groups Hom are given b y: Hom S - m od ( M ′ , M ) := Hom S - m od ( M ′ , M ) /I P ( M ′ , M ) . (ii) Similarly , using t he full sub category e P of Coh P consisting of finite direct sums o f in v ertible sheav es O P ( a ), a ∈ Z , one defines the stable c ate gory Coh P . 3.2. Definition. A complex K • ∈ Ob C b ( P ) is called a Horr o cks c o mplex if it satisfies the following equiv alen t conditions: (1) H i ( K • ) = 0 for i ≤ − 2 and H i ( K •∨ ) = 0 for i ≤ 1 (2) H i ( K •∨ ) = 0 for i ≤ 1 a n d dim H i ( K •∨ ) ≤ n + 2 − i , for i > 1 (3) H i ( K • ) = 0 for i ≤ − 2 and dim H i ( K • ) ≤ n − 1 − i , fo r i ≥ − 1. 14 I. CO AND ˘ A Here “dim” stands for “ Krull dimension”. The equiv alence of these conditions fo llows from Lemma 3.3. b elow. Condition ( 1) implies that if K ′• ∈ Ob C b ( P ) is homotopically equiv alen t to a Horro ck s complex then it is a Hor r o c ks complex. Let M ∈ Ob( S -mo d) and let L • (resp., L ′• ) b e a free r esolution of M (resp., M ∨ ) in S - mo d. One can concatenate the complexes L ′• and T − 1 ( L •∨ ) using the comp osite morphism L ′ 0 ։ M ∨ ֒ → L 0 ∨ . The dual K • of the resulting complex is a Horro ck s complex. W e call it a Horr o cks r esolution of M . 3.3. Lemm a. L et A b e a No etherian ( c ommutative ) rin g and let P • b e a left b ounde d c om plex of finitely gener ate d pr oje ctive A - mo dules . Then the fol lo wing c onditions ar e e quival e nt : (i) H i ( P • ) = 0, ∀ i < 0 (ii) ∀ i > 0, ∀ p ∈ Supp H i ( P •∨ ) ⊆ Sp ec A , depth A p ≥ i . Pr o of. (i) ⇒ (ii) Let i > 0 and let p ∈ Sp ec A with depth A p < i . Let M := C 0 ( P • ) := Cok er( P − 1 → P 0 ). Condition (i) implies t hat M has finite pro jectiv e dimension. No w, the Auslander-Buc hsbaum form ula implies that the pro jectiv e dimension of t he A p -mo dule M p is ≤ depth A p < i , hence H i ( P •∨ ) p ≃ Ext i A p ( M p , A p ) = 0, whence p / ∈ Supp H i ( P •∨ ). (ii) ⇒ (i) W e use induction on m := sup { i ∈ Z | H i ( P •∨ ) 6 = 0 } . The case m ≤ 0 is clear. F or the pro of o f the induction step ( m − 1) → m , consider the A -mo dule N := C − 1 ( P • ) := Cok er( P − 2 → P − 1 ). Applying the induction hy p othesis to T − 1 P • , one gets that H i ( P • ) = 0, ∀ i < − 1, hence the sequence: 0 → P − r → · · · → P − 2 → P − 1 → N → 0 is exact. W e assert that Ass( N ) ⊆ Ass( A ). Indeed, let p ∈ Ass( N ) and let d := depth A p . It follo ws f r o m the Auslander-Buc hsbaum f orm ula that the pro jective dimension o f the A p -mo dule N p is d , whic h implies tha t Ext d A p ( N p , A p ) 6 = 0. If d > 0 then Ext d A p ( N p , A p ) ≃ H d +1 ( P •∨ ) p hence, b y (ii), depth A p ≥ d + 1, a c ontr adiction . It remains that d = 0, i.e., p ∈ Ass( A ). No w, H − 1 ( P • ) ≃ Ker( N → P 0 ), hence Ass(H − 1 ( P • )) ⊆ Ass( N ) ⊆ As s( A ). If p ∈ Ass( A ) then, b y (ii), the sequence: P 0 ∨ p → P − 1 ∨ p → · · · → P − r ∨ p → 0 is exact. Since it consists of free A p -mo dules, its dual is also exact. In particular, it follows that H − 1 ( P • ) p = 0. One deduces t ha t Ass(H − 1 ( P • )) = ∅ , i.e., H − 1 ( P • ) = 0.  3.4. Theorem. The functor C − 1 : C b ( P ) → S -mo d asso ciating to a c omplex L • the c oke rnel of the differ ential d − 2 L : L − 2 → L − 1 induc e s a functor C − 1 : K b ( P ) → S - mo d which , r estricte d to the ful l sub c ate gory H of K b ( P ) c onsisting of Horr o cks c omple x es , is an e quivalenc e of c ate gories . THE HORROCKS CORRESPONDENCE 15 Pr o of. If a morphism f : L ′• → L • in C b ( P ) is homotopic to 0 then C − 1 ( L ′• ) → C − 1 ( L • ) factorizes thro ugh L − 1 ։ C − 1 ( L • ) (and through C − 1 ( L ′• ) → L ′ 0 ) hence C − 1 induces a functor C − 1 as in the statemen t. The fact that C − 1 | H is ful l y faithful follo ws from the more general Lemma 3.5. b elo w. The fact tha t C − 1 | H → S -mo d is es s ential ly surje ctive was already observ ed in the last part of D efinition 3.2..  3.5. Lemma. L et L • , L ′• ∈ Ob C( P ). If H i ( L • ) = 0 for i ≤ − 2 and if H i ( L ′•∨ ) = 0 for i ≤ 1, then the m orphism : Hom C( P ) ( L ′• , L • ) − → Hom S - m od ( C − 1 ( L ′• ) , C − 1 ( L • )) is surje ctive an d induc es an iso m orphism : Hom K( P ) ( L ′• , L • ) ∼ − → Hom S - m od ( C − 1 ( L ′• ) , C − 1 ( L • )) . Pr o of. The complex · · · → L − 2 → L − 1 → 0 is a free resolution of C − 1 ( L • ) in S -mo d, and the complex · · · → L ′ 1 ∨ → L ′ 0 ∨ → 0 is a fr ee resolution of C − 1 ( L ′• ) ∨ . No w, one uses the f ollo wing t w o elemen tary facts: (1 ) if P • ∈ Ob C ≤ 0 ( P ), M • ∈ Ob C ≤ 0 ( S -mo d) and if H i ( M • ) = 0 for i < 0, then an y morphism C 0 ( P • ) → C 0 ( M • ) can b e lifted to a morphism of complexes P • → M • ; (2) if, moreov er, C 0 ( P • ) → C 0 ( M • ) fa cto r izes through an ob j ect of P (hence through M 0 ։ C 0 ( M • )) then the morphism of augmented complexes: · · · − − − → P − 1 − − − → P 0 − − − → C 0 ( P • ) − − − → 0   y   y   y · · · − − − → M − 1 − − − → M 0 − − − → C 0 ( M • ) − − − → 0 is homotopic to 0.  3.6. Th eorem. F or N • ∈ Ob C b (Λ-mo d) the c omplex F( N • ) is a Horr o cks c omplex if and only if the l i n e ar p art of a minimal f r e e r esolution of N • in Λ- mo d is of the form L n − 1 i = − 1 T − i G( H i ), wher e H i is the k - v e ctor sp ac e gr ade d dual of a finitely gener ate d gr a d e d S - mo dule of Krul l dim ension ≤ i + 1, i = − 1 , . . . , n − 1. Pr o of. Let us denote F( N • ) by K • . By Corollary 2.12. and b y Lemma 2.9.(b), t he linear part of a minimal free resolution of N • is isomorphic to L i ∈ Z T − i G( H i ), where H i = H i (T − n − 1 (( K •∨ ⊗ S ω S ) ∗ )) ≃ (H n +1 − i ( K •∨ ⊗ S ω S )) ∗ . One can no w conclude, using condition (2) from D efinition 3.2..  3.7. Definition. A minima l complex G • ∈ Ob C − ( I ) with H p ( G • ) = 0 for p << 0 is called a Horr o cks-T r autmann c om plex if it satisfies the following conditio ns (compare with [9], (1.6.)): 16 I. CO AND ˘ A (1) F n − 1 G • = G • and F 0 G • = 0, i.e., G p ≃ L n − 1 i =1 V ( V ∗ )( p − i ) c pi , ∀ p ∈ Z , (2) lim p →∞ ( c − p,i /p i +1 ) = 0 , i = 1 , . . . , n − 1 . 3.8. Lemm a. A minimal c omple x G • ∈ Ob C − ( I ) is a Horr o cks-T r autmann c o mplex if and on l y if its line ar p art is of the form L n − 1 i =1 T − i G( H i ), wher e H i is the k - ve ctor sp ac e gr ade d dual of a finitely gener ate d gr ade d S - m o dule of Krul l dimension ≤ i + 1, i = 1 , . . . , n − 1 . Pr o of. The equiv alence can b e prov ed b y applying to G •∗ the following: Assertion. F or a minimal c omplex I • ∈ Ob C + ( I ) , the fol lowin g c onditions ar e e quiv- alent : (i) H p ( I • ) = 0, for p >> 0 (ii) The line ar p art of I • is of the form L i ∈ Z T − i G( M i ), wher e M i is a finitely gene r ate d gr ade d S - mo dule , ∀ i ∈ Z . (i) ⇒ (ii) H p ( I • ) = 0, ∀ p > m , for some m ∈ Z . Let Z m := Ker( I m → I m +1 ). σ ≥ m I • is a minimal right free resolution of T − m Z m in Λ-mo d. By Prop osition 2.11., GF(T − m Z m ) is a right free resolution of T − m Z m in Λ-mo d. F r o m Lemma 2.9.(b), it can b e con tracted to a minimal complex J • in C + ( I ) , whose linear part is L i ∈ Z T − i G(H i (F(T − m Z m ))) = L i ∈ Z T − i G(H i − m (F( Z m ))). σ ≥ m I • and J • are isomorphic in D + (Λ-mo d) hence, since ev ery free ob ject of Λ -mo d is an injective ob ject of this category , they are isomorphic in K + ( I ) and consequen tly , by (2.8.)(iii) , isomorphic in C + ( I ) . One deduces that the linear part of σ ≥ m I • is isomorphic to L i ∈ Z T − i G(H i − m (F( Z m ))). (ii) ⇒ (i) It suffices to pro v e that if M ∈ Ob( S -mo d) then H p (G( M )) = 0 for p >> 0 . Let L • b e a finite free resolution of M in S -mo d. G( M ) and G( L • ) are quasi-isomorphic (ev en homotopically equiv alen t). Since G( S ) is a r ig h t free resolution o f k in Λ-mo d (see the pro o f of Prop osition 2.11.) it follow s that H p (G( L • )) = 0 for p >> 0.  3.9. Theorem. Ther e exists an e quivalenc e of c a te gories b etwe en the ful l sub c ate gory HT of K − ( I ) c onsisting of Horr o cks-T r autmann c omplexes and the ful l sub c ate g o ry of Coh P ( V ) c onsisting of the c oher ent she aves F with the pr op erty that H 0 ( F ( − t )) = 0, for t >> 0. Pr o of. The equiv alence from the statemen t will app ear as a comp osition of previously established equiv alenc es. (1) Let B b e the full sub category of Coh P ( V ) consisting of the coheren t shea v es with the pro p ert y fro m the statemen t. Let A b e the full sub category of S -mo d consisting of the mo dules o f pro jectiv e dimension ≤ n − 1. Using Theorem 1.1. (Graded Serre Dualit y) one sees that the functor ( − ) ∼ : S -mo d → Coh P ( V ) induces an equiv alences of categories b et w een A and B . Moreov er, this equiv alence induces an equiv alence b etw een the correp onding full sub categories A and B of S -mo d and Coh P ( V ), resp ectiv ely . THE HORROCKS CORRESPONDENCE 17 (2) By Lemma 1.2 ., t he equiv alence C − 1 : H → S -mo d from Theorem 3.4. induces an equiv alence b et w een the full sub category H ′ of H consisting of the Horro cks complexes K • with the a dditional prop ert y that H i ( K •∨ ) = 0 for i ≥ n + 1 and A . (3) Finally , there is a w ell-kno wn equiv alence Φ b et w een the full su b category K of K − ( I ) consisting of t he complexes I • with H p ( I • ) = 0 f or p << 0 and D b (Λ-mo d). Φ asso ciates to I • a con v enien t truncation τ ≥ m I • with m << 0 (it suffices that H p ( I • ) = 0 for p < m ) and its quasi-in v erse asso ciates to a complex in D b (Λ-mo d) a free resolution of it. Now , by Theorem 3.6. and Lemma 3 .8., t he comp osition o f the BGG equiv alence (Theorem 2.10.) F : D b (Λ-mo d) → K b ( P ) and Φ induces an equiv alence b et w een HT and H ′ .  3.10. Example. (Eilen b erg-MacLane sheav es) Let 0 < i < n and let E b e a finitely generated graded S - mo dule o f K rull dimension ≤ i + 1. Consider a minimal free resolution of E in S -mo d: 0 → Q − n − 1 → · · · → Q 0 → E → 0 . Applying Lemma 3.3. to P • := T n − i ( Q •∨ ) one deriv es that H j ( Q •∨ ) = 0 for j ≤ n − i − 1 . Using condition (1) from Definition 3.2. one deduces that K • := T n − i +1 ( Q •∨ ) is a Horro cks complex. It is a Horro c ks resolution of M := Cok er(( Q − n + i +1 ) ∨ → ( Q − n + i ) ∨ ). M has a minimal free r esolution: 0 → ( Q 0 ) ∨ → · · · → ( Q − n + i +1 ) ∨ → ( Q − n + i ) ∨ → M → 0 . Let F := f M . Since H j ( K •∨ ) = 0 for j 6 = n − i + 1 and H n − i +1 ( K •∨ ) ≃ E , it follo ws from the pro of of Theorem 3.6. that the Horro cks -T rautma nn complex asso ciated to F is T − i G( H ) where H = ( E ⊗ S ω S ) ∗ . Moreov er, by Graded Serre Dua lit y (Theorem 1 .1.), H j ∗ F = 0 for 0 < j < n , j 6 = i , a nd H i ∗ F ≃ H . When E is o f finite length F is a lo cally free sheaf. The lo cally free shea v es of this kind w ere called Eile nb er g-MacL ane b und les in Horro cks [16]. 4. The Horrocks corresp ondence and the BGG correspondence 4.1. Definition. The ge om etric B GG functor is the functor L : Λ-mo d → C b (Coh P ) defined by L( N ) := F( N ) ∼ . W e denote b y Λ -mo d the stable c ate gory of Λ-mo d with resp ect to its full sub category I consisting of free o b jects (see Definition 3.1.). 4.2. Lemma. If N , N ′ ∈ Ob(Λ -mo d) then , ∀ p ≥ 1: Hom K b ( P ) (L( N ′ ) , T p L( N )) ∼ − → Hom D b ( P ) (L( N ′ ) , T p L( N )) . Pr o of. The lemma is an immediate application o f Lemma B.4., taking in to accoun t t ha t H i ( O P ( a )) = 0 for i > 0, i 6 = n , ∀ a ∈ Z , a nd H n ( O P ( a )) = 0, ∀ a ≥ − n .  18 I. CO AND ˘ A 4.3. Corollary . If N , N ′ ∈ Ob(Λ -mo d) then , ∀ p ≥ 1: Hom D b (Λ) ( N ′ , T p N ) ∼ − → Hom D b ( P ) (L( N ′ ) , T p L( N )) . Pr o of. By the BGG corresp ondence for graded mo dules Theorem 2.10.: Hom D b (Λ) ( N ′ , T p N ) ∼ − → Hom K b ( P ) (F( N ′ ) , T p F( N )) and, on the other hand, it is obvious that: Hom K b ( P ) (F( N ′ ) , T p F( N )) ∼ − → Hom K b ( P ) (L( N ′ ) , T p L( N )) . It only remains, now , to apply L emma 4.2..  The following theorem is the Bernstein-Gel’fand-Gel’fand corresp ondence for coheren t shea v es on pro jectiv e spaces. W e include here a direct pro of of this result. 4.4. Theorem. ([4 ], Theorem 2.) The functor L : Λ -mo d → C b (Coh P ) in duc es an e quival e nc e o f c ate gories L : Λ-mo d → D b (Coh P ). Pr o of. L( V ( V ∗ )) is the tautological Koszul complex on P ( V ): 0 → O P ( − n − 1) ⊗ k n +1 ∧ V ∗ → · · · → O P ( − 1) ⊗ k V ∗ → O P → 0 hence, if P is a free ob ject of Λ- mo d then L( P ) is an a cyclic complex. It follows tha t L : Λ-mo d → C b (Coh P ) induces a functor L : Λ-mo d → D b (Coh P ). W e firstly sho w that the induced functor is ful ly faithful . Let N , N ′ ∈ Ob(Λ -mo d). W e ha v e to show that the morphism: Hom Λ-mo d ( N ′ , N ) − → Hom D b ( P ) (L( N ′ ) , L( N )) (*) is surjectiv e and that its k ernel consists of the morphisms factorizing throug h a free o b j ect of Λ-mo d. Consider an exact sequence 0 → K → P → N → 0 with P a free ob j ect of Λ-mo d. F rom Lemma 4 .2.: Hom K b ( P ) (L( N ′ ) , TL( P )) ∼ − → Hom D b ( P ) (L( N ′ ) , TL( P )) and Ho m D b ( P ) (L( N ′ ) , TL( P )) = 0 since L( P ) is acyclic. Now, applying Hom K b ( P ) (L( N ′ ) , − ) and Hom D b ( P ) (L( N ′ ) , − ) to the complex in K b ( P ): L( P ) → L( N ) → TL( K ) → T L ( P ) deduced (see [8], (2) ( i) ,(ii)) from the semi-split short exact sequence: 0 → L( K ) → L( P ) → L( N ) → 0 , one gets a comm utativ e diagram with exact rows: Hom K( P ) (L( N ′ ) , L( P )) − − − → Hom K( P ) (L( N ′ ) , L( N )) − − − → Hom K( P ) (L( N ′ ) , TL( K ) ) → 0   y   y   y ≀ Hom D( P ) (L( N ′ ) , L( P )) − − − → Hom D( P ) (L( N ′ ) , L( N )) − − − → Hom D( P ) (L( N ′ ) , TL( K ) ) → 0 THE HORROCKS CORRESPONDENCE 19 By Lemma 4 .2 ., the v ertical arrow from the right hand side of the diagram is an isomor- phism. Moreov er, Hom D( P ) (L( N ′ ) , L( P )) = 0 b ecause L( P ) is acyclic. Since the v ertical arro ws in the comm utativ e diagram: Hom Λ-mo d ( N ′ , P ) − − − → Hom Λ-mo d ( N ′ , N )   y ≀   y ≀ Hom K( P ) (L( N ′ ) , L( P )) − − − → Hom K( P ) (L( N ′ ) , L( N )) are clearly isomorphism s, one deduces that the morphism (*) is surjectiv e and that its k ernel consists of the morphisms facto r izing through P → N . The essential surje ctivity o f L : Λ-mo d → D b ( P ) can no w b e c hec ked, in a we ll-kno wn manner, using the fo llo wing observ ations: (1) By what ha v e b een pro v ed, the image of L is a ful l sub category of D b ( P ). (2) If N ∈ Ob(Λ-mo d) and o ne considers a short exact sequen ce 0 → N → I → Q → 0 with I a free ob ject of Λ- mo d then the connecting morphism w : L( Q ) → TL( N ) deduced (see [8], (2 )(ii)) from the semi-split short exact sequence : 0 → L( N ) → L( I ) → L( Q ) → 0 is a quasi-isomorphism b ecause L( I ) is acyc lic. Similarly , considering a short exact se- quence 0 → K → P → N → 0 w ith P a free ob j ect of Λ-mo d one gets a quasi-isomorphism T − 1 L( N ) → L( K ). (3) Let u : N ′ → N b e a mor phism in Λ- mo d. Consider an em bedding v : N ′ → I ′ of N ′ in to a fr ee ob ject I ′ of Λ- mo d and define C ∈ Ob(Λ-mo d) by the short exact sequence : 0 → N ′ ( u,v ) − → N ⊕ I ′ − → C → 0 . By applying L to this short exact se quence one gets a semi-sp lit short exact se quence, hence L( C ) is homotopically equiv alen t to Con L(( u, v )). Moreov er, Con L(( u , v )) is quasi- isomorphic to Con L( u ) b ecause L( I ′ ) is acyclic , whe nce one gets a quasi-isomorphism L( C ) → Con L( u ). (4) L( k ( − a )) = T − a O P ( a ), ∀ a ∈ Z . Using these o bserv ations and the fact that ev ery coheren t sheaf on P ( V ) has a finite resolution with finite direct sums of in v ertible sheav es O P ( a ), one deduces immediately (as, for example, in the last part of the pro of of [8], Theorem 7) that L : Λ-mo d → D b ( P ) is essen tially surjectiv e.  4.5. Corollary . ([4], R emark 3 after Theorem 1 ) F or every F • ∈ Ob C b (Coh P ) ther e exists N ∈ Ob(Λ-mo d) annihilate d by so c(Λ) = n +1 ∧ V such that F • ≃ L( N ) in D b (Coh P ). Mor e over , N is unique up to iso m orphism . F or a pro of see, for example, [8 ],(8). 20 I. CO AND ˘ A 4.6. Corollary . If F • and N ar e as in Cor ol lary 4.5. then , ∀ i ∈ Z , H i (F( N )) ≃ L j ≥ − i H i ( F • ( j )) as S - mo dules ( wher e H de n otes the hyp er c ohomolo gy ). Pr o of. H i (F( N )) j = 0 fo r j < − i b ecause F( N ) i = S ( i ) ⊗ k N i . F or ev ery j one has: H i (F( N )) j ≃ Hom K b ( P ) (T − i O P , L( N )( j )) hence it remains to show that for j ≥ − i : Hom K b ( P ) (T − i O P , L( N )( j )) ∼ − → Hom D b ( P ) (T − i O P , L( N ) ( j )) or, equiv alen tly , that: Hom K b ( P ) (L( k ( − i )) , T i + j L( N ( − i − j ))) ∼ − → Hom D b ( P ) (L( k ( − i )) , T i + j L( N ( − i − j ))) . F or j > − i this follows fro m Lemma 4.2.. F or j = − i , the ab ov e morphism can b e iden tified with the morphism: Hom Λ-mo d ( k ( − i ) , N ) − → Hom D b ( P ) (L( k ( − i )) , L( N )) . By Theorem 4.4., the last morphism is surjectiv e and its k ernel consists of the comp o site morphisms k ( − i ) → P → N with P a free o b j ect of Λ-mo d. But the image o f k ( − i ) → P m ust lie in so c( P ) = so c(Λ) · P . Since N is annihilated b y so c( Λ ), any suc h composite morphism must b e 0.  4.7. Definition. Let N b e an ob j ect of Λ - mo d annihilated b y so c(Λ). Let P • (resp., P ′• ) b e a minimal free resolution of N (resp., N ∗ ) in Λ-mo d. By concatenating the complexes P ′• and T − 1 ( P •∗ ) using the comp osite morphism P ′ 0 ։ N ∗ ֒ → P 0 ∗ one gets an acyclic c om plex which is minimal (since N ∗ is annihilated b y so c(Λ), the image of the ab ov e comp o site morphism is con tained in Λ + · P 0 ∗ ). The k -vec tor space dual I • of this complex (see Definition 2 .5.) is called a T ate r esolution of N . The next theorem, whic h is one of the main results o f the pap er of Eisen bud, Fløystad and Sch rey er [12], is a direct consequence of Coro lla ry 4.6.. 4.8. Theorem. ([12], Theorem 4.1.) If F • and N ar e as in C or ol la ry 4.5. and if I • is a T ate r esolution of N then the l i n e ar p art of I • is isom o rphic to L i ∈ Z T − i G( H i ∗ F • ). Pr o of. Let Z − m := Ker( I − m → I − m +1 ), m > 0. As w e sho wn in the pro of o f Lemma 3.8., the linear part of σ ≥− m I • is isomorphic to L i ∈ Z T − i G(H i + m (F( Z − m ))). No w, b y definition, N ≃ Cok er ( I − 2 → I − 1 ) and, since I • is acyclic, N ≃ Ker( I 0 → I 1 ). It follo ws, from observ at ion (2) in the second part of the pro of of Theorem 4.4., that L( Z − m ) ≃ T − m L( N ) ≃ T − m F • in D b ( P ). Moreov er, since Z − m is contained in Λ + · I − m , it is annihilated b y so c( Λ ). Corollary 4.6 . implies, now, that H i + m (F( Z − m )) ≃ L j ≥ − i − m H i + m ((T − m F • )( j )) = L j ≥ − i − m H i ( F • ( j )). T aking in to accoun t what ha v e b een recalled in the first paragraph, one deduces that the linear par t of σ ≥− m I • is isomorphic to THE HORROCKS CORRESPONDENCE 21 L i ∈ Z T − i G( L j ≥ − i − m H i ( F • ( j ))). Finally , letting m → ∞ one gets the desired conclusion.  4.9. Theorem. L et F b e a c oher ent she af on P n with H 0 F ( − t ) = 0 for t >> 0, le t M := H 0 ∗ F and let 0 → L − n +1 → · · · → L 0 → M → 0 b e a minimal fr e e r esolution o f M in S -mo d. L et N ∈ Ob(Λ-mo d) b e as in Co r ol lary 4.5. and let I • b e a T ate r esolution of N . Then : (a) I • /F 0 I • is a c ontr action of T − n G(( L •∨ ⊗ S ω S ) ∗ ). (b) The Horr o cks-T r aut mann c omplex c orr esp ond ing to F via the e quivalenc e of c ate- gories fr om The or em 3 .9. is iso m orphic to F n − 1 I • /F 0 I • . Pr o of. (a ) Cho ose m ∈ Z suc h tha t H i ( F ( j )) = 0, ∀ i > 0, ∀ j ≥ m − i . Since, as a consequenc e of Theorem 4.8., I p ≃ L n i =0 H i ( F ( p − i )) ⊗ k V ( V ∗ )( p − i ), ∀ p ∈ Z , one sees that I p = F 0 I p for p ≥ m , hence I • /F 0 I • = ( σ − n (( F •∨ ⊗ S ω S ) ∗ ) → ( L •∨ ⊗ S ω S ) ∗ and then a quasi- isomorphism τ > 0 (T − n (( F •∨ ⊗ S ω S ) ∗ )) → T − n (( L •∨ ⊗ S ω S ) ∗ ) . It follows no w, from the last part of Lemma 2.9.(b), that ( σ m . One defines, similarly , a quotien t double complex τ >m I X •• of X •• . The followin g result, which is a particular case of (A.6.), is stated and pro v ed in Eisen- bud et al. [12], (3.5), and it is a k ey tec hnical p oint of that pap er. Lemma. Assume that the d o uble c omplex X •• satisfies the f o l lowing finiteness c on- dition : ∀ m ∈ Z , X p,m − p = 0 for p << 0. I f al l the r ows X • ,q := ( X pq , d ′ pq X ) p ∈ Z , q ∈ Z , of X •• split ( se e (A.3.)) then ther e exists a c ontr action of tot( X •• ) onto a c omplex Y • , endowe d with an incr e asing filtr ation ( F m Y • ) m ∈ Z by sub c omplexes, such that : Y n = M p + q = n H pq I ( X •• ) , ∀ n ∈ Z , (1) F m Y n = M p + q = n p ≤ m H pq I ( X •• ) , ∀ m, n ∈ Z , (2) gr F ( Y • ) = tot(H I ( X •• )) . (3) Mor e over, this c ontr action c an b e chosen in such a w a y that, for al l m ∈ Z , it induc es a c on tr action of tot( τ ≤ m I X •• ) onto F m Y • and of tot( τ >m I X •• ) onto Y • /F m Y • . Pr o of. R ecall that the differen tial of tot( X •• ) is d ′ X + δ ′′ X , where δ ′′ X | X pq := ( − 1) p d ′′ pq X . Let X •• I b e the double complex with the same terms as X •• , with d ′ X I = d ′ X , and with d ′′ X I = 0. (A.3.) pro vides a con traction ( π , u , h ) o f tot( X •• I ) onto a complex with terms Y n giv en by the form ula (1) from the statemen t and with the differen tial equal to 0. One ma y assume that the homo t op y op erator h maps X pq in to X p − 1 ,q , ∀ p, q ∈ Z . The differen tia l of tot( X •• ) is a p erturbation of d ′ X and the finiteness condition fro m the statemen t implies that hδ ′′ X is lo cally nilp ot ent. (A.6.) produces now a contraction ( b π , b u, b h ) o f tot( X •• ) on to 26 I. CO AND ˘ A a complex Y • with terms given b y fo rm ula (1) from the statemen t and with differen tial: d Y = π δ ′′ X u + X i ≥ 1 ( − 1) i π δ ′′ X ( hδ ′′ X ) i u. The explicit formulae from (A.6.) allo ws one no w to che c k easily the other a ssertions from the lemma.  Appendix B : A comp arison lemma B.1. Lemma. L et C , D b e triangulate d c ate gories and Φ : C → D an additive functor c omm uting with the tr anslation functors and sending distinguishe d triangles to distinguishe d triangle s . L et X , Y b e two obje cts of C endowe d with “ de c r e asin g filtr ations ”, i.e., with s e quenc es of morphisms : · · · → F i +1 X → F i X → · · · , · · · → F i +1 Y → F i Y → · · · such that F i X = X , F i Y = Y for i << 0 a n d F i X = 0, F i Y = 0 for i >> 0, and with the “ suc c essive quotients ” r eplac e d b y distinguishe d triangles : F i +1 X → F i X → X i → T F i +1 X , F i +1 Y → F i Y → Y i → T F i +1 Y . (a) If Hom C ( X i , Y j ) → Hom D (Φ( X i ) , Φ( Y j )) is surje c tive and Hom C ( X i , T Y j ) → Hom D (Φ( X i ) , Φ(T Y j )) is i n je ctive , ∀ i, j , then Hom C ( X , Y ) → Hom D (Φ( X ) , Φ( Y ) ) is sur- je ctive . (b) If Hom C ( X i , Y j ) → H om D (Φ( X i ) , Φ( Y j )) is inje ctive and Hom C (T X i , Y j ) → Hom D (Φ(T X i ) , Φ( Y j )) is surje ctive , ∀ i, j , then Hom C ( X , Y ) → Hom D (Φ( X ) , Φ( Y )) is inje ctive . Pr o of. F or p, i ∈ Z , w e endo w T p F i X with t he filtration whose j th term is T p F j X fo r j > i and T p F i X for j ≤ i , and similarly for T p F i Y . W e also endow T p X i with the filtration whose j th term is T p X i for j ≤ i and 0 for j > i , and similarly for T p Y i . W e pro v e (a) and (b) sim ultaneously , b y induction on N := card { i ∈ Z | X i 6 = 0 } + card { j ∈ Z | Y j 6 = 0 } . The case N ≤ 2 is ob vious. F or the induction step , assume, firstly , that card { j ∈ Z | Y j 6 = 0 } ≥ 2 and let n := inf { j ∈ Z | Y j 6 = 0 } . By applying Hom C ( X , − ) to the complex: T − 1 Y n → F n +1 Y → Y → Y n → T F n +1 Y (*) and Hom D (Φ( X ) , − ) to the complex Φ((*) ) , one gets a comm utativ e diag r am with exact ro ws and five v ertical a rro ws. If the pair ( X , Y ) v erifies the h yp othesis of (a) (resp., (b)) then ( X , F n +1 Y ) and ( X , Y n ) ve rify the hy p othesis of (a) (resp., (b)), and ( X , T F n +1 Y ) v erifies the hy p othesis of (b) (resp., ( X , T − 1 Y n ) v erifies the h ypothesis of (a)). Using, no w, the strong form o f t he “Five Lemma” (see [18], Chap. I, Ex 1 .8 . or [19], (8.3.1 3)) and taking into accoun t the induction h yp o thesis, o ne gets the desired conclusion. THE HORROCKS CORRESPONDENCE 27 Similarly , if card { i ∈ Z | X i 6 = 0 } ≥ 2 let m := inf { i ∈ Z | X i 6 = 0 } . One applies Hom C ( − , Y ) to the complex: T − 1 X m → F m +1 X → X → X m → T F m +1 X (**) and Hom D ( − , Φ( Y )) to the complex Φ((**)) a nd one uses again the “Fiv e Lemma”.  Before stating an useful consequence of (B.1.), namely Lemma B.4 . b elo w, w e recall the following w ell kno wn: B.2. Lemma. L et X • and Y • b e c o m plexes i n an ab elian c ate gory A and n ∈ Z . (a) I f X p = 0 ( r esp ., H p ( X • ) = 0) for p > n then : Hom K( A ) ( X • , τ ≤ n Y • ) ∼ − → Hom K( A ) ( X • , Y • ) ( r esp ., Hom D( A ) ( X • , τ ≤ n Y • ) ∼ − → Hom D( A ) ( X • , Y • )) . (b) If Y p = 0 ( r esp ., H p ( Y • ) = 0) for p < n then : Hom K( A ) ( τ ≥ n X • , Y • ) ∼ − → Hom K( A ) ( X • , Y • ) ( r esp ., Hom D( A ) ( τ ≥ n X • , Y • ) ∼ − → Hom D( A ) ( X • , Y • )) . Pr o of. The assertions ab out Hom K( A ) are easy . (a) The in v erse of Hom D( A ) ( X • , τ ≤ n Y • ) → Hom D( A ) ( X • , Y • ) asso ciates to a morphism X • qis ← − X ′• − → Y • in D( A ) the morphism X • qis ← − τ ≤ n X ′• − → τ ≤ n Y • . (b) The in v erse of Hom D( A ) ( τ ≥ n X • , Y • ) → Hom D( A ) ( X • , Y • ) asso ciates to a morphism X • − → Y ′• qis ← − Y • the morphism τ ≥ n X • − → τ ≥ n Y ′• qis ← − Y • .  B.3. Defin ition. If X , Y are ob jects of an ab elian category A and p ∈ Z then Ext p A ( X , Y ) := Hom D( A ) ( X , T p Y ). I t fo llo ws from (B.2.) that Ext p A ( X , Y ) = 0 f o r p < 0. Mor eov er, using the argumen ts from the pro of of (B.2.), one sees easily that Hom A ( X , Y ) ∼ → Ext 0 A ( X , Y ). The follo wing lemma app ears, in w eak er v arian ts, in sev eral pap ers lik e, fo r example, Kapranov [17] or Canonaco [7], ( A.5 .3.). In the more precise form (B.4.) b elo w, it w as pro v ed in [9], (3.3.), under the assumption that the ab elian category A con tains sufficien tly man y injectiv e ob jects. Here w e drop this assumption using an argumen t similar to t ha t used by Canonaco (this arg umen t actually app ears in the pro of of (B.1.)). B.4. Lemma. L et A b e an ab elian c ate gory , X • ∈ Ob C − ( A ) an d Y • ∈ Ob C + ( A ). Consider the c anon i c al morp h ism φ : Hom K( A ) ( X • , Y • ) → Hom D( A ) ( X • , Y • ) . (a) I f Ext p − q A ( X p , Y q ) = 0, ∀ p > q , then φ is surje ctive . (b) If Ext p − q − 1 A ( X p , Y q ) = 0, ∀ p > q + 1, then φ is inj e ctive . 28 I. CO AND ˘ A Pr o of. Let m := sup { p ∈ Z | X p 6 = 0 } and n := inf { q ∈ Z | Y q 6 = 0 } . T aking in to accoun t (B.2.), one ma y replace X • b y τ ≥ n X • and Y • b y τ ≤ m Y • , hence one may assume that X • and Y • are b ounde d complexes. In this case, one endow s X • with the filtration F i X • := σ ≥ i X • ( σ = “stupid trunca- tion”). T o the semi-split short exact sequence: 0 → σ ≥ i +1 X • → σ ≥ i X • → T − i X i → 0 one can asso ciate (see, for example, [8], (2)(ii)) a distinguished triang le in K b ( A ): σ ≥ i +1 X • → σ ≥ i X • → T − i X i → T σ ≥ i +1 X • . One also endows Y • with the similar filtration. The conclusion of the lemma follo ws no w from (B.1.) a pplied to the canonical functor K b ( A ) → D b ( A ). The h yp otheses of (B.1.) can b e easily chec ke d in this case b ecause most of the Hom groups inv olve d are zero a nd Hom K( A ) ( X , Y ) ∼ → Hom D( A ) ( X , Y ), ∀ X , Y ∈ Ob A (see (B.3.)).  Reference s [1] T. Ab e and M. 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