Integrable systems and local fields

INTEGRA BLE SYSTEMS AN D LOCAL FIELDS A. N. P arshin T o the memory of A lexey Ivanovich Kostrikin In 70’s there w as disco v ered a construction ho w to attach to some algebraic- geometric data an infinite-dimensional subspace in the space k (( z )) of the Lauren t p o w er se ries. The construction was succe ssfully us ed in the theory of in tegrable sy stems, particularly , for the KP and KdV equations [12, 26]. There w ere also found some applications to the mo duli of algebraic curv es [2, 4 ]. No w it is kno wn as the Kric hev er corresp ondence or the Kric hev er map [2, 17, 1, 24, 5]. The original w ork by I. M. Kric hev er has also included comm utative ring s of differen tial o p erators as a third part of the corresp on- dence. The map we w a n t to study here was first describ ed in an explicit w a y b y G. Segal and G. Wilson [26]. They hav e used a n analytical v ersion of the infinite dimensional Gr a ssmanian in tro duced b y M. Sato [25, 23]. In the sequel we consider a purely algebraic approach as dev elop ed in [17]. Let us just note that the core of the construction is an em b edding of the affine co or dina t e r ing on an algebraic curv e in to the field k (( z )) corresp onding to the p o w er dec omp ositions in a p oin t at infinity (the details see b elow in section 2). In num b er theory this corresp onds to an embedding of the ring of algebraic integers to the fields C or R . The latt er one is well known starting fr o m the XIX-th cen tury . The idea introduced b y Kriche v er w as to insert the lo cal parameter z . This trick lo oking so simple enormously extends the area of the corresp ondence. It allow s to consider a ll algebraic curv es sim ultaneously . But there still remained a hard restriction by the case of curv es, so by dimension 1. Rec en tly , it was p ointed out by the author [21, 2 2] tha t there 1 are some connections betw een the theory of the KP-equations and the theory of n -dimensional local fields [19], [7]. F ro m this p oin t of view it b ecomes clear that the Kric hev er construction should hav e a generalization to the case of higher dimensions. This generalization is suggested in the pap er f or the case of algebraic surfaces (see theorem 4 in section 4). A further generalization to the case of a r bitrary dimension was recenly prop osed b y D . V. Osip o v [18]. W e start with description of a connection b et we en the KP hierarch y in the Lax f o rm and the v ector fields on infinite G rassmanian manifolds (section 1). These results are kno wn but w e prov e them here in more transparen t and simple wa y (w e hav e used basically [17] and [15]). In app endix 1 we remind ho w to get the standard KP a nd KdV equations f r o m the Lax op erator form. In app endix 2 w e ouline a construction of the semi-infinite mono mes for the field k (( z )) whic h is an imp ortant part of the theory of Sato Grassmanian. An imp ortan t feature o f all t hese considerations is their purely algebraic c haracter. Ev erything can b e done o v er an a r bitrary field k of c haracteristic zero 1 . Let us also note that the construction of the restricted adelic complex in section 3 is of an indep enden t interes t, also in arithmetics. It has a lready app eared in a description of vector bundles on algebraic surfaces [20]. This work was partly done during my visit to the Institut f ¨ ur Algebra und Zahlen t heorie der T ec hnisc he Univ ersit¨ at Braunsc hw eig in 1999. I am v ery m uc h gr a teful to Professor Hans Op olk a for the ho spitality 2 . 1 Sato corresp onde nce fo r dimension 1 This cor r esp o ndence connects t wo seemingly distant ob j ects: infinite G rass- manian manifolds and rings of pseudo-differen tial op erators. Let K = k (( z )) b e the field o f Lauren t pow er series with filtr a tion K ( n ) = z n k [[ z ]]. If V ∼ = k (( z )) is a v ector space of dimension 1 o ver K then w e can c ho ose a filtratio n V ( n ) suc h that K ( n ) V ( m ) ⊂ V ( n + m ). Let V 1 := V (0 ). 1 F urther developement of differen t aspec ts o f the Krichev er co rresp ondence for dimen- sion t wo see in [27, 13, 1 4] 2 The text was published in Commun ic ations in A lgebr a , 29(200 1), No.9, 4157 -4181 . This version includes a cor rected pro of o f the prop o s ition 2. I’m grateful to Alexa nder Zheglov for the co rrection of a mistake in the orig inal pro of. Also, we include s o me additional remark s on the de ductio n of concrete equations from the Lax hierar ch y and app endix 2. 2 Denote b y Gr( V ) the set of the subspaces W in V suc h that the complex W ⊕ V 1 → V (1) is a F redholm one. It is a (infinite-dimensional) pro jective v ariet y with connected comp onen ts marked by the Euler characteristic of the complex (1) [1, 11](see also, app endix 2). Let us no w in tro duce t he ring E = k [[ x ]](( ∂ − 1 )) of for ma l pseudo-differen tial op erators with co efficien t s from the ring k [[ x ]] of regular formal p o w er series as the left k [[ x ]]-mo dule o f all express ions L = n X i> −∞ a i ∂ i , a i ∈ k [[ x ]] . Then a m ultiplication can b e defined according to the Leibnitz rule: ( X i a i ∂ i )( X j b j ∂ j ) X i,j ; k ≥ 0  i k  a i d k ( b j ) ∂ i + j − k . Here w e put  i k  = i ( i − 1) . . . ( i − k + 1) k ( k − 1) . . . 1 , if k > 0 ,  i 0  = 1 and d is the deriv ation b y x . P articularly , for f ∈ k [[ x ]]: [ ∂ , f ] = ∂ f − f ∂ = d ( f ) , (Heisen b erg commutativ e relation), [ ∂ − 1 , f ] = ∂ − 1 f − f ∂ − 1 − d ( f ) ∂ − 2 + d 2 ( f ) ∂ − 3 − . . . . It can be c hec k ed tha t k [[ x ]](( ∂ − 1 )) will be an asso ciative ring ( the details see in [22]). There is a decompo sition E = E + + E − , where E − = { L ∈ E : L = P n< 0 a n ∂ n } and E + consists of the operato rs con taining only ≥ 0 p o we rs of ∂ . The elemen ts from E + =: D are the differen tial o p erators and the elemen ts from E − are the V olterra op erators. 3 F rom the comm uta t io n relations w e see that E = k (( ∂ − 1 )) ⊕ E x a nd th us the map E → E /E x = V (we iden tify the image of ∂ − 1 with z ) defines a linear action of the ring E on V a nd also on Gr( V ). The subspace V 1 is trans- formed by the action of op erato r s from E in to a subspace V ′ commesurable with V 1 (it means that the quotient V ′ + V 1 /V ′ ∩ V 1 is of finite dimension). Th us the F red holm condition from the definition of Grassmanian manifold will b e preserv ed. W e will call the map E → V by the Sato map. Prop osition 1 ( Lemma on a st abilizer ). L et P ∈ E and W 0 = k [ z − 1 ] ∈ Gr ( V ) if V = k (( z )) . Then P W 0 ⊂ W 0 if an d only if P ∈ D . Proo f . Since W 0 is equal to the imag e of E + under the Sato map (by lemma 1 b elow) w e can replace the first condition b y the f o llo wing one P · E + ⊂ E + + E x (2) and work in the ring E . Since E + is a ring we ha v e P · E + ⊂ E + for P ∈ E + = D . No w assume ( 1 ) and decompose P as P = P + + P − . W e will a lso use the notation L ∼ M for L, M ∈ E suc h that L − M ∈ E x . Lemma 1 . In the ring E , x n ∂ n ∼ c n wher e c n = ( − 1) n n ! . Proo f . By comm utation relations x n ∂ n = ∂ n x n +(some co efficien t ) ∂ n − 1 x n − 1 + . . . + c n and applying this op erato r to x − 1 w e get the v alue of c n . Returning to our prop osition w e see t ha t (2) implies P − E + ∈ E + + E − x. (3) T o prov e that P − = 0 it is enough to sho w P − ∈ E − x n for all n ≥ 1. First, 1 ∈ E + and consequen tly P − ∈ E + + E − xE + ⊕ E − x . W e see that P − ∈ E − x . Pro ceeding by induction w e assume P − = Q − x n with Q − ∈ E − . W e ha v e Q − x n ∂ n ∈ E + + E − x, b y (3) Q − x n ∂ n ∈ Q − E x + Q − ⊂ E x + Q − , b y lemma . T aking a ll together w e get Q − ∈ E + + E x ⊂ E + ⊕ E − x . It means Q − ∈ E − x and P − ∈ E − x n +1 . 4 Prop osition 2 ( Tra nsitivity theorem ). L et W ∈ Gr ( V ) and W ⊕ V 1 = V . Then ther e exists an unique op er ator S ∈ 1 + E − such that W = S − 1 W 0 . Proo f . If W satisfies the conditions o f the theorem then W is a union of subspaces W ∩ V − n and one can ch o ose basis w n in W suc h that w n = z − n + v n , v n ∈ V 1 . W e wan t to construct an op erat or S = 1 + P ∈ 1 + E − suc h that S z − n = w n + c 0 w 0 + . . . + c n − 1 w n − 1 for some c 0 , . . . c n − 1 ∈ k , or P z − n = v n + c 0 w 0 + . . . + c n − 1 w n − 1 for all n ≥ 0. The Sato map E → V transforms ∂ n in to z − n and w e can find Q n ∈ E − suc h that Q n go es to v n b y the map. Th us in o r der to construct our S from the conclusion o f the theorem w e ha v e to find a n op erat or P ∈ E − suc h that P ∂ n = Q n + c 0 (1 + Q 0 ) + . . . + c n − 1 ( ∂ n − 1 + Q n − 1 ) + E x, (4) where Q n is a giv en sequence of o p erators from E − . F rom now on w e w ork in the ring E . Put P 0 = Q 0 . CLAIM . F or n ≥ 1 there exist o p erators P n ∈ E − suc h that i) P n ∂ n + P 0 ∂ n + . . . + P n − 1 ∂ n ∈ Q n + c 0 (1 + Q 0 ) + . . . + c n − 1 ( ∂ n − 1 + Q n − 1 ) + E x, fo r some c 0 , . . . , c n − 1 ∈ k ii) P n , P n ∂ , . . . , P n ∂ n − 1 ∈ E x iii) P n ∈ E x n First, we show that t he claim implies the existence of the P with the prop erty (4). W e put P = P 0 + P 1 + . . . . Then P 0 ∂ n + P 1 ∂ n + . . . + P n ∂ n + P n +1 ∂ n + . . . ∼ (by ii) P 0 ∂ n + P 1 ∂ n + . . . + P n ∂ n ∼ (b y i) Q n + c 0 (1 + Q 0 ) + . . . + c n − 1 ( ∂ n − 1 + Q n − 1 ) . Here w e use the no t ation ∼ from the lemma 1. The pro p ert y iii) implies the con ve rgence of the series for P by the follow ing Lemma 2 . In the ring E , the series P n ≥ 0 P n , P n ∈ E − wil l c onver ge if P n ∈ E − x n . No w w e pro ve the claim a nd then the lemma. W e can define P n b y induction. Obv iously , by induction w e ha v e P 0 ∂ n + . . . + P n − 1 ∂ n ∈ c 0 (1 + Q 0 ) + . . . + c n − 1 ( ∂ n − 1 + Q n − 1 ) + E x + E − 5 for some c 0 , . . . , c n − 1 ∈ k . So, the image of the op erator P 0 ∂ n + . . . + P n − 1 ∂ n − c 0 (1 + Q 0 ) + . . . + c n − 1 ( ∂ n − 1 + Q n − 1 ) under the Sat o map lies in V 1 and we can find an op erator Q ′ n ∈ E − suc h that Q ′ n go es to this image b y the map. Then we can tak e P n = ( Q n − Q ′ n ) x n ∈ E − . It giv es iii) a nd P n ∂ n = ( Q n − Q ′ n ) x n ∂ n ∼ Q n − Q ′ n ∼ Q n + c 0 (1 + Q 0 ) + . . . + c n − 1 ( ∂ n − 1 + Q n − 1 ) − P 0 ∂ n − . . . − P n − 1 ∂ n b y lemma 1 a nd P n ∂ k = Q ′ n x n ∂ k = Q ′ n x n − k x k ∂ k ∼ Q ′ n x n − k ∼ 0 , k = 0 , 1 , . . . , n − 1 again by lemma 1 and w e a r e done. The uniquene ss of the op erator S ∈ 1 + E − suc h that W = S − 1 W 0 follo ws from prop osition 1. Indeed, let S W 0 = W 0 . Then S m ust b elong to E + and th us S = 1. Proo f of Lemma 2. Let P n = P ′ n x n where P ′ n = X m ≥ 0 a ( n ) m ∂ − m and P n = X m ≥ 0 b ( n ) m ∂ − m . W e see b ( n ) m = a ( n ) m x n ± a ( n ) m − 1 d ( x n ) ± a ( n ) m − 2 d 2 ( x n ) ± . . . and for ev ery m the series P n ≥ 0 b ( n ) m will conv erge in k [[ x ]]. If W ∈ G r( V ) then w e hav e T W = Hom( W, V /W ) for the tangent space in the p oin t W and there is a natural map Hom( V , V ) → T W . F or n ∈ Z w e can define a ve ctor field T n on Gr( V ). It is equal to the image of the m ultiplication op erator b y z − n in the space V . The KP hierarc hy is a dynamical system defined on an affine space E ′ := ∂ + E − . The tangent space to any p oint L ∈ E ′ is canonically equal to E − . Definition 1 . The n -th v ector field of the KP hierarc h y on E ′ is defin ed as K P n = [( L n ) + , L ]. Since ( L n ) + = L n − ( L n ) − the field K P n b elongs to E − . The set G = 1 + E − is a group carrying the vec tor fields − ( S ∂ n S − 1 ) − S . The tangent space to any p oint from G is again E − . A t last w e denote b y Gr + the big cell from the Grassmanian manifiold, Gr + ( V ) = { W ∈ Gr ( V ) : W ⊕ V (1) = V } . (5) All these spaces, E ′ , G, Gr + ha ve the distinguished p oin ts: ∂ , 1 , W 0 = k [ z − 1 ](if V = k (( z ))). 6 Theorem 1 ( Sa to corresp ondence ). The maps E ′ ϕ ← G ψ → Gr + ( V ) , wher e ϕ ( S ) = S ∂ S − 1 , ψ ( S ) = S − 1 ( W 0 ) hav e the fol lowing pr op erties i) for any S ∈ G , L = ϕ ( S ) ∈ E ′ and W = ψ ( S ) ∈ Gr + the diagr am ‘ T L dϕ S ← T S dψ S → T W k k k H om ( W , V /W ) ↑ E − ϕ ′ ← E − ψ ′ → H om ( V , V ) c ommutes. Her e dϕ S and dψ S ar e jac obian maps of the maps ϕ, ψ on the tangent s p ac e T S , and ϕ ′ ( A ) = [ AS − 1 , L ] , ψ ′ ( A ) = − S − 1 A acting on V by the Sato m ap with A ∈ E − . ii) fo r any S ∈ G , L = ϕ ( S ) ∈ E ′ and W = ψ ( S ) ∈ Gr + and any n ≥ 1 dϕ S ( − ( S ∂ n S − 1 ) − S ) = K P n , dψ S ( − ( S ∂ n S − 1 ) − S ) = T n Proo f . First w e consider t he left hand side of the diagr a m. If A ∈ E − = T S and R = S + A is an infinitely small deformation o f S then up to the higher p o w ers of A R = S (1 + S − 1 A ) = (1 + AS − 1 ) S, R − 1 = (1 − S − 1 A ) S − 1 = S − 1 (1 − AS − 1 ) , R∂ R − 1 = (1 + AS − 1 ) S ∂ S − 1 (1 − AS − 1 ) ( S ∂ S − 1 + A∂ S − 1 )(1 − AS − 1 ) S ∂ S − 1 + A∂ S − 1 − S ∂ S − 1 AS − 1 L + [ AS − 1 , L ] = L + d ϕ S ( A ) . 7 Using this result w e can c hec k up the statemen t on the ve ctor fields: dϕ S ( − ( S ∂ n S − 1 ) − S ) = [ − ( S ∂ n S − 1 ) − S S − 1 , L ] = [( L n ) + , L ] . It remains to consider the righ t hand side of the diag r a m. Let W = S − 1 W 0 as in the diagram and let R = S + A b e as ab ov e. Then W ′ : R − 1 W 0 and w e get W ′ = R − 1 S S − 1 W 0 = R − 1 S W = (1 − S − 1 A ) S − 1 S W = (1 − S − 1 A ) W . Since W, W ′ are t w o spaces fro m the big cell W ⊕ V (1) = V , W ′ ⊕ V (1) = V and the space W ′ defines a linear map W → V → V / W ′ ∼ ← V (1) ∼ → V /W whic h is an elemen t of the tangent space T W corresp onding to the deformation W ′ of W (see, for example, [11 ]) . It is easy to see that for our space W ′ the linear map will coincide with the action of op erator − S − 1 A through t he Sato map. The last step of the pro of is t o c heck that dψ S tak es − ( S ∂ n S − 1 ) − S in t o T n . But w e ha ve − S − 1 ( − ( S ∂ n S − 1 ) − S ) = S − 1 L n S − S − 1 ( L n ) + S ∂ n − S − 1 ( L n ) + S and w e ha ve to sho w that the second term is trivial in T W . This can b e seen from the commutativ e diagram W 0 → V → V → V /W 0 S − 1 ↓ S − 1 ↓ S − 1 ↓ S − 1 ↓ W → V → V → V /W The b ot t o m map from V to V is equal to the Sato image of S − 1 ( L n ) + S and the corresp o nding top horison ta l map is t he Sato action of the op erator ( L n ) + . By the prop osition 1 the last one is trivial in T W 0 = H om ( W 0 , V /W 0 ). The theorem is pro v ed. Remark 1 . The maps ϕ and ψ can b e deduced f rom the corresp onding actions of the group G on the manifolds E ′ and Gr ( V ). One has to consider the actions on the orbits going through ∂ and W 0 , respective ly . The first orbit is one o f the co-adjoint orbits f o r the (infinite dimens ional) Lie gr o up G . 8 Corollary 1 . The Sato c orr esp ondenc e induc es the diag r am of bije ctions E ′ ϕ ← G/G 0 ψ → Gr + ( V ) /k [[ z ]] ∗ , wher e G 0 = G ∩ k (( ∂ − 1 )) and the ac tion of k [[ z ]] ∗ on Gr ( V ) is define d by the mo dule structur e on V over K . Proo f for the map ψ easily follows from the definitions, theorem 1 and prop osition 2. T o c hec k up the bijectivit y of the map ϕ one has to apply theorem 1 from [22]. 2 Kric hev er corre sp onden ce for dimen sion 1 W e first disc uss the adelic complexes for the case o f dimension 1 . Concerning a definition of the adelic no t io ns w e refer to [7],[10]. W e a lso note t ha t the sign Q denotes the adelic pro duct. Let C b e an pro j ectiv e algebraic curv e o ver a field k , P b e a smo oth p oin t and η a general p o in t on C . F or ev ery po in t (in Grothendiec k’s sense) α ∈ C w e ha ve a field K α . K α is a quotient ring of the completed lo cal ring ˆ O α of the p oin t α . It is a 1- dimensional lo cal field. Let F b e a torsion free coheren t sheaf on C . W e denote b y ˆ F α = F ⊗ ˆ O α the completed fiber at the po in t α ∈ C . The fiber F η at a general po int is also the space of all ratio na l sections of the sheaf F . Prop osition 3 . The fol lowi n g c omplexes ar e quasi -isomorphic: i) ad e lic c omplex F η ⊕ Y x ∈ C ˆ F x − → Y x ∈ C ( ˆ F x ⊗ ˆ O x K x ) ii) the c omplex W ⊕ ˆ F P − → ˆ F P ⊗ ˆ O P K P wher e W = Γ( C − P , F ) ⊂ ˆ F η . Proo f will b e done in t wo steps. First, the adelic complex con tains a tr ivial exact sub complex Y x ∈ U ˆ F x − → Y x ∈ U ˆ F x , 9 where U = C − P . The quotient-complex is equal to F η ⊕ ˆ F P − → Y x ∈ U ( ˆ F x ⊗ K x ) / ˆ F x ⊕ ˆ F P . It has a surjectiv e homomorphism to the exact complex F η /W − → Y x ∈ U ( ˆ F x ⊗ K x ) / ˆ F x . The exac tness of the complex is the strong approx imation theorem for the curv e C [6][c h.I I, § 3, corollar y of prop. 9; ch. VI I, § , prop. 2]. The kerne l of this surjection will b e the second complex from prop osition. Let us now explain t he Kriche v er corresp ondence for dimension 1. Definition 2 . M 1 := { C , P , z , F , e P } C pro jectiv e irr educible curv e /k P ∈ C a smo o t h p oin t z formal lo cal para meter at P F torsion free rank r sheaf o n C e P a trivialization of F at P Indep enden tly , we ha v e the field K = k (( z )) o f Laurent p o w er series with filtration K ( n ) = z n k [[ z ]]. Let K 1 := K (0). If V = k (( z )) ⊕ r then V ( n ) = K ( n ) ⊕ r and V 1 := V (0 ). Theorem 2 [17]. The r e exists a c anonic al m a p Φ 1 : M 1 − → { ve ctor subsp ac es A ⊂ K , W ⊂ V } such that i) the c ohomolo gy of c omplexes A ⊕ K 1 − → K , W ⊕ V 1 − → V ar e isomorphic to H · ( C , O C ) and H · ( C , F ) , r esp e ctively ii) if ( A, W ) ∈ Im Φ 1 then A · A ⊂ A, A · W ⊂ W , iii) if m, m ′ ∈ M 1 and Φ 1 ( m )Φ 1 ( m ′ ) then m is isomorphic to m ′ 10 Proo f . If m = ( C , P , z , F , e P ) ∈ M 1 then we put A := Γ( C − P , O C ) , W := Γ( C − P , F ) . Also we ha ve b y the choice of z and e P ˆ O P = k [[ z ]] , K P = k (( z )) , F P = O P e P = O ⊕ r P , ˆ F P = ˆ O ⊕ r P . This defines the po in t Φ 1 ( m ) ∈ M 1 . Indeed, f o r the subspace W we ha v e the fo llo wing canonical iden tificatio ns Γ( C − P , F ) ⊂ F η ⊗ O P K P = ˆ F P ⊗ K P = ˆ O ⊕ r P ⊗ K P = k (( z )) ⊕ r . The same w o rks for the subspace A . The pro p ert y ii) is ob vious, the property i) fo llo ws from the propo sition 3. T o get iii) let us start with a p oin t Φ 1 ( m ) = ( A, W ). The standard v aluation on K gives us increasing filtrations A ( n ) = A ∩ K ( n ) and W ( n ) = W ∩ V ( n ) on the spaces A and W . Then w e ha v e C − P = Spec( A ) , C = Pro j( ⊕ n A ( n )) , F = Pro j( ⊕ n W ( n )) , b y lemma 9. Th us we can reconstruct the quin tiple m from the p o int Φ 1 ( m ). Remark 2 . It is p ossible to r eplace the gr ound field k in the Krich ev er construction b y an arbitrary sche me S , see [2 4 ]. Using the Kric hev er corresp ondence Φ 1 one can construct the integral v arieties in X = Gr( V ) /k [[ z ]] ∗ for the v ector fields T n (see section 1). Let us fix C, P , z . Then the image Φ 1 ( C , P , z , F , e P ) does not depend on e P in X and will run thr o ugh the generalized jacobian Jac( C ) of the curv e C when w e v ary the in v ertible sheaf F . T o sho w this fact we consider the comm utativ e diagram k [[ z ]] = k [[ z ]] ↓ ↓ 0 → A W → K α → H om ( W, V /W ) = T Gr,W k ↓ ↓ 0 → A W → K / k [[ z ]] → T X,W ↓ ↓ 0 0 11 Here α is the action of K o n V by mu ltiplications, W ∈ Gr ( V ) is the second space from the image Φ 1 ( C , P , z , F , e P ) and A W = { f ∈ K : f W ⊂ W } . The diagram explains what happ ens with tangen t space s when w e go fro m the Grassmanian manifold to it’s quotien t X by the group k [[ z ]] ∗ . The space k [[ z ]] is the Lie algebra of the last group. F or the case of inv ertible sheaf F w e know that A W = A (see [26][n 6]). Th us w e get the exact sequence A + k [[ z ]] → k (( z )) → T X,W , where k (( z )) / A + k [[ z ]] = H 1 ( C , O C ) is the tangen t space to the generalized jacobian of the curv e C . F rom the seq uence w e conclude that all v ector fields T n b elong to the image of H 1 ( C , O C ) in T X,W and consequen tly t hey are tangen t to the image of g eneralized jacobian Jac( C ) of the curve C under the map Φ 1 (see details in [17]). This resu lt w orks for the component of Gr ( V ) ( V = k (( z ))) con taining the subspace W suc h that W ⊕ k [[ z ]] = V . The image of the Sato corresp on- dence b elongs to another comp onent con taining t he space W 0 and the big cell Gr + ( V ) (see (5), section 1). The mu ltiplication b y z transforms one comp onen t on to another preserv- ing the v ector field s T n . Th us w e get the in tegr a l v arieties for the KP flo w whic h are ab elian v arieties for smo oth curv es C . If k = C t hey are top ological toruses with the mo v emen t along the straigh t lines. This explain wh y the KP system can b e considered to some exten t as an in tegrable one. But we m ust ha v e in mind that there is a lot of p oin ts W in Gr ( V ) wh ere t he dynamical b eha vior is quite differen t. T ak e, for example, the p oin ts with A W = k . If C is a pro jectiv e line with a double p oin t then Jac( C ) = k ∗ and we get a 1- solito n solution of the KP equation. W e see that lo cal fields en ters in to this picture in essen tial w ay . The ring E is a subring o f 2-dimensional lo cal sk ew-field P = k ( ( x ))(( ∂ − 1 )) (see [22]). The fields k (( z )) a nd K P are 1- dimensional lo cal fields. The complex (1)(section 1) from definition of the Grassmanian manifold intimately con- nected with adelic complex o n an algebraic curv e (prop ert y i) f r om Theorem 2). W e su ggest that analogous constructions should exist fo r higher dimen- sions as w ell (concerning the rings of PDO see [11 ]). Here w e consider a generalization of the Kric heve r map Φ 1 to the case of a lg ebraic surfaces. 12 3 Adelic complexes in di mension 2 Let X b e a pro jectiv e irreducible alg ebraic surface ov er a field k , C ⊂ X b e an irreducible pro jectiv e curv e, and P ∈ C b e a smo oth p oint on bo t h C and X . Let F b e a torsioh free coherent sheaf on X . W e remind the definition of the standard rings attached to a pair x ∈ D ⊂ X on the surface X . Here the D is a n irreducible divisor. It correspond to an ideal ℘ ⊂ O x in the lo cal ring of the po in t x . First w e apply lo calization b y ℘ to ˆ O x and then take a completion b y t he ideal ℘ . W e get a ring O x,D whic h is a complete discrete v aluation ring if the p oint x is smo oth on b oth D and X . The lo cal field K x,D is a quotien t ring o f the O x,D . It is really a field in the smo oth case. In this case we ha ve O x,D = k (( u ))[[ t ]] , K x,D = k (( u ))(( t )) if ˆ O x = k [[ u, t ]] a nd ℘ = ( t ). The field K x,D is an example of 2-dimensional lo cal field. There a re some rings attached to the p oin t x a nd the divisor D . Denote b y K the field of rational functions on the X . Let K x K ˆ O x ⊂ a quotien t ring of the lo cal ring ˆ O x . If ˆ O D is a lo cal ring of t he divisor D then let K D b e it’s quotient ring. In the smo o th case ˆ O D = k ( D )[[ t ]] , K D = k ( D )(( t )). Definition 3 . Let x ∈ C . W e let B x ( F ) = \ D 6 = C (( ˆ F x ⊗ K x ) ∩ ( ˆ F x ⊗ ˆ O x,D )) , where t he inte rsection is done inside the g roup ˆ F x ⊗ K x , B C ( F ) = ( ˆ F C ⊗ K C ) ∩ ( \ x 6 = P B x ) , where t he inte rsection is done inside ˆ F x ⊗ K x,C , A C ( F ) = B C ( F ) ∩ ˆ F C , A ( F ) = ˆ F η ∩ ( \ x ∈ X − C ˆ F x ) . W e will freely use the following shortcuts: K ˆ F x = ˆ F x ⊗ ˆ O x K x , K ˆ F D = ˆ F D ⊗ ˆ O D K D , ˆ F x,D = ˆ F x ⊗ ˆ O x O x,D , K ˆ F x,D = ˆ F x ⊗ ˆ O x K x,D . 13 Next, w e need tw o lemmas connecting the adelic complexes o n X and C . They ar e the ve rsions of the relativ e exact sequences, see [19], [7]. The curve C defines the fo llowing ideals: K x,C ⊃ ˆ O x,C . . . ⊃ ℘ n x,C ⊃ . . . , K C ⊃ ˆ O C . . . ⊃ ℘ n C ⊃ ℘ n +1 C ⊃ . . . , K x ⊃ ˆ O x . . . ⊃ ℘ n x . . . , and ℘ x = ˆ O x ∩ ℘ x,C . Lemma 3 . We assume that the curve C is a lo c al ly c omplete interse ction. L et N X/C b e the normal she af for the curve C in X . F or al l n ∈ Z the map s Y x ∈ C ℘ n x,C ˆ F x,C /℘ n +1 x,C ˆ F x,C − → A C, 01 ( F ⊗ ˇ N ⊗ n X/C ) , Y x,C ℘ n x ˆ F x /℘ n +1 x ˆ F x − → A C, 1 ( F ⊗ ˇ N ⊗ n X/C ) , ℘ n C ˆ F C /℘ n +1 C ˆ F C − → A C, 0 ( F ⊗ ˇ N ⊗ n X/C ) , ar e bije ctive. In general, w e hav e an exact sequence 0 − → J n +1 − → J n − → J n | C − → 0 where J ⊂ O X is an ideal defining the curve C . In our case J = O X ( − C ) and N X/C = O X ( C ) | C . Th us the isomorphisms f r om the lemma a r e coming from the exact relativ e sequence 0 − → A X ( F ( − ( n + 1) C )) − → A X ( F ( − nC )) − → A C ( F ( − nC ) | C ) − → 0 . Lemma 4 . L et P ∈ C . F or al l n ∈ Z the c omplex ℘ n C ˆ F C /℘ n +1 C ˆ F C ⊕ Y x ∈ C ℘ n x ˆ F x /℘ n +1 x ˆ F x − → Y x ∈ C ℘ n x,C ˆ F x,C /℘ n +1 x,C ˆ F x,C is quasi-isomo rp hic to the c omplex ( A C ( F ) ∩ ℘ n C ˆ F C ) / ( A C ( F ) ∩ ℘ n +1 C ˆ F C ) ⊕ ℘ n P ˆ F P /℘ n +1 P ˆ F P − → ℘ n P ,C ˆ F P ,C /℘ n +1 P ,C ˆ F P ,C . 14 This lemma is an extension of the prop osition 1 ab ov e. The prov es of the b oth lemmas ar e straightforw ard and we will skip them. Theorem 3 . L et X b e a pr oje ctive irr e ducible algebr aic surfac e over a field k , C ⊂ X b e an irr e ducible pr oje ctive curve, and P ∈ C b e a smo oth p oint on b oth C and X . L et F b e a torsioh fr e e c ohe r ent she af on X . Assume that the the surfac e X − C is affine. T h en the fol lowing c omplexes ar e quasi-isomorph i c : i) the adelic c omplex ˆ F η ⊕ Y D ˆ F D ⊕ Y x ˆ F x − → Y D ( ˆ F D ⊗ K D ) ⊕ Y x ( ˆ F x ⊗ K x ) ⊕ Y x ∈ D ( ˆ F x ⊗ ˆ O x,D ) − → − → Y x ∈ D ( ˆ F x ⊗ K x,D ) for the she af F and ii) the c omplex A ( F ) ⊕ A C ( F ) ⊕ ˆ F P − → B C ( F ) ⊕ B P ( F ) ⊕ ( ˆ F P ⊗ ˆ O P ,C ) − → ˆ F P ⊗ K P ,C Proo f will b e divided in to sev eral steps. W e will subsequen tly trans- form the adelic complex c heck ing that ev ery time w e get a quasi-isomorphic complex. Step I . Consider the diagram Q D 6 = C ˆ F D ⊕ Q x ∈ U ˆ F x − → Q D 6 = C ˆ F D ⊕ Q x ∈ U ˆ F x ⊕ ↓ ↓ ↓ ↓ ˆ F η ⊕ Q D ˆ F D ⊕ Q x ˆ F x − → Q D K ˆ F D ⊕ Q x K ˆ F x ⊕ k ↓ ↓ ↓ ↓ ˆ F η ⊕ ˆ F C ⊕ Q x ∈ C ˆ F x − → ( Q D 6 = C K ˆ F D / ˆ F D ⊕ K ˆ F C ) ⊕ ( Q x ∈ U K ˆ F x / ˆ F x ⊕ Q x ∈ C K ˆ F x ) ⊕ ⊕ Q x ∈ D 6 = C ˆ F x,D − → Q x ∈ D 6 = C ˆ F x,D ↓ ↓ ⊕ Q x ∈ D ˆ F x,D − → Q x ∈ D K ˆ F x,D ↓ ↓ ⊕ Q x ∈ C ˆ F x,C − → Q x ∈ D 6 = C K ˆ F x,D / ˆ F x,D ⊕ Q x,C K ˆ F x,C where U = X − C . The middle ro w is the full adelic complex and the first row is an exact sub complex. The comm utativity of the upp er squares is ob vious. The exactness fo llows from the trivial 15 Lemma 5 . L et f 1 , 2 : A 1 , 2 − → B b e ho m omorphisms o f ab elian gr oups. The c omplex 0 − → A 1 ⊕ A 2 − → A 1 ⊕ A 2 ⊕ B − → B − → 0 , wher e ( a 1 ⊕ a 2 ) 7→ ( a 1 ⊕ − a 2 ⊕ − f ( a 1 ) + f ( a 2 )) , ( a 1 ⊕ a 2 ⊕ b ) 7→ ( f ( a 1 ) + f ( a 2 ) + b ) , is ex a ct. The third row in the diagra m is a quotien t- complex by the sub complex and w e conclude that it is quasi-isomorphic to the adelic complex. Step I I . W e can mak e the same step with the adelic complex for t he sheaf F on t he surface U . By assumption the surface U is affine and w e get an exact complex ˆ F η / A − → Y D 6 = C ( ˆ F D ⊗ K D ) / ˆ F D ⊕ Y x ∈ U ( ˆ F x ⊗ K x ) / ˆ F x − → Y x ∈ U x ∈ D 6 = C ( ˆ F x ⊗ K x,D ) / ( ˆ F x ⊗ ˆ O x,D ) , where A = Γ( U, F ). Lemma 6 . The c omplex 0 − → Y x ∈ C ( ˆ F x ⊗ K x ) /B x ( F ) − → Y x ∈ C x ∈ D 6 = C ( ˆ F x ⊗ K x,D ) / ( ˆ F x ⊗ ˆ O x,D ) − → 0 is exact. Proo f . The injectivit y follo ws directly fro m the definition of the ring B x . The surjec tivit y is the lo cal strong appro ximation around the p oin t x ∈ C (see [19][ § 1],[7][ch.4]). Step I I I . T ake the sum of the t wo complexes from step I I. Then we ha ve a map of the complex w e got in the step I to this complex ˆ F η ⊕ ˆ F C ⊕ Q x ˆ F x − → ( Q D 6 = C K ˆ F D / ˆ F D ⊕ K ˆ F C ) ⊕ ( Q x ∈ U K ˆ F x / ˆ F x ⊕ Q x ∈ C K ˆ F x ) ⊕ ↓ ↓ ↓ ↓ ↓ ˆ F η / A ⊕ (0) ⊕ (0) − → Q D 6 = C K ˆ F D / ˆ F D ⊕ ( Q x ∈ U K ˆ F x / ˆ F x ⊕ Q x ∈ C K ˆ F x /B x ) ⊕ 16 ⊕ Q x ∈ C ˆ F x,C − → Q x ∈ D 6 = C K ˆ F x,D / ˆ F x,D ⊕ Q x ∈ C K ˆ F x,C ↓ ↓ ↓ ⊕ (0) − → Q x ∈ U x ∈ D D 6 = C K ˆ F x,D / ˆ F x,D ⊕ Q x ∈ C x ∈ D D 6 = C K ˆ F x,D / ˆ F x,D ⊕ (0) F or this map a ll the comp onen ts whic h do not ha v e arro ws are mapp ed to zero. The diagram is comm utative and the k ernel of the map is equal to A ⊕ ˆ F C ⊕ Y x ∈ C ˆ F x − → K ˆ F C ⊕ Y x ∈ C B x ( F ) ⊕ Y x ∈ C K ˆ F x − → Y x ∈ C K ˆ F x,C . W e conclude that this complex is quasi-isomorphic to the a delic complex. Step IV . Using the em b edding ˆ F x − → B x ( F ) and lemma 5 we hav e an exact complex and it’s em b edding into the complex of the step I I I: Q x ∈ C − P ˆ F x − → Q x ∈ C − P B x ( F ) ⊕ Q x ∈ C − P ˆ F x − → ↓ ↓ ↓ A ⊕ ˆ F C ⊕ Q x ∈ C ˆ F x − → K ˆ F C ⊕ Q x ∈ C B x ( F ) ⊕ Q x ∈ C ˆ F x,C − → − → Q x ∈ C − P B x ( F ) ↓ − → Q x ∈ C K ˆ F x,C As a result w e get the fa ctor-complex A ⊕ ˆ F C ⊕ ˆ F P − → K ˆ F C ⊕ B P ( F ) ⊕ Y x ∈ C − P ˆ F x,C / ˆ F x ⊕ ˆ F P ,C − → − → Y x ∈ C − P K ˆ F x,C /B x ( F ) ⊕ K ˆ F P ,C . Step V . No w w e need Lemma 7 . The c omplex 0 − → ( ˆ F C ⊗ K C ) /B C ( F ) − → Y x ∈ C − P ( ˆ F x ⊗ K x,C ) /B x ( F ) − → 0 is exact. 17 Proo f . The injectivit y is aga in the definition of the B C and the surjec- tivit y follows from the strong approximation on the curv e C (see pro of o f prop osition 3) and lemma 4 ab ov e. As a corollary w e hav e an isomorphism ˆ F C / A C ( F ) ∼ = − → Y x ∈ C − P ( ˆ F x ⊗ ˆ O x,C ) / ˆ F x , where A C ( F ) := B C ( F ) ∩ ˆ F C . Com bining the isomorphisms from the lemma and its corolla ry into a single complex of length 2, we get the diagram A ⊕ ˆ F C ⊕ ˆ F P − → K ˆ F C ⊕ B P ( F ) ⊕ ( Q x ∈ C − P ˆ F x,C / ˆ F x ⊕ ˆ F P ,C ) − → ↓ ↓ ↓ ↓ ↓ ↓ (0) ⊕ ˆ F C / A C ⊕ (0) − → K ˆ F C /B C ⊕ (0) ⊕ Q x ∈ C − P ˆ F x,C / ˆ F x − → − → Q x ∈ C − P K ˆ F x,C /B x ( F ) ⊕ K ˆ F P ,C ↓ ↓ − → Q x ∈ C − P K ˆ F x,C /B x ( F ) ⊕ (0) The k ernel of the map of the complexes is ob viously equal to A ( F ) ⊕ A C ( F ) ⊕ ˆ F P − → B C ( F ) ⊕ B P ( F ) ⊕ ( ˆ F P ⊗ ˆ O P ,C ) − → ( ˆ F P ⊗ K P ,C ) and w e arriv e to the conclusion of the theorem. Remark 3 . Sometimes we will call the complex from the theorem as the r estricte d adelic complex. Lemma 8 .L et X b e a pr oje ctive irr e ducible variety ove r a field k and O (1 ) b e a very amp l e she af on X . Then 1. The fol lowing c onditions ar e e quivale nt i) X is a Cohen-Mac aulay variety ii) for any lo c al ly fr e e she af F on X and i < dim ( X ) H i ( X , F ( n )) = (0) for n << 0 2. If X is normal of dim ension > 1 then for any lo c al ly fr e e she af F on X H 1 ( X , F ( n )) = (0) for n << 0 18 Proo f see in [9][ch. I I I, Thm. 7.6, Cor. 7.8]. The la st statemen t is kno wn a s the lemma of Enriques-Sev eri-Zar iski. F or dimension 2 ev ery normal v ariet y is Cohen-Macaulay [1 6] and thus the second claim follo ws from the fir st one. Prop osition 4 . L et F b e a lo c al ly fr e e c oher ent she af on the pr oje ctive irr e ducible surfac e X . Assume that the lo c al ri n gs of the X ar e C o hen-Mac aulay and the c urve C is a lo c al ly c ompl e te interse ction. Then, i n side the field K P ,C , we have B C ( F ) ∩ B P ( F ) = A ( F ) . Proo f will b e done in sev eral steps. Step 1 . If w e kno w the prop osition for a sheaf F then it is true for the sheaf F ( nC ) for an y n ∈ Z . Th us ta king a t wist b y O ( n ) we can assume that deg C ( F ) < 0. Step 2 . No w w e show that A C ( F ) ∩ ˆ F P = (0). The filtrations from lemma 3 giv es the corresp onding filtratio n of the group A C ( F ). Lemma 4 implies tha t ( A C ( F ) ∩ ˆ F P ) ∩ ℘ n ˆ F P ( A C ( F ) ∩ ˆ F P ) ∩ ℘ n +1 ˆ F P ∼ = Γ( C , F ⊗ ˇ N ⊗ n X/C ) . Since deg C ( F ) < 0, N X/C = O X ( C ) | C and deg C ( N X/C ) > 0 w e get that the last g roup is trivial. Step 3 . The next step is to prov e the equality: B C ( F ( − D )) ∩ B P ( F ( − D )) = A ( F ( − D )) , where D is an sufficien tly a mple divisor on X distinct from the curv e C . By theorem 3 the cohomology o f F X ( − D ) can b e computed from the complex A ( F ( − D )) ⊕ A C ( F ( − D )) ⊕ ˆ F P ( − D ) − → B C ( F ( − D )) ⊕ B P ( F ( − D )) ⊕ ˆ F P ,C ( − D ) − → K F P ,C . No w tak e a 01 ∈ B C ( F ( − D )) , a 02 ∈ B P ( F ( − D )) suc h that a 01 + a 02 = 0. They define an elemen t ( a 01 ⊕ a 02 ⊕ 0) in the middle comp onen t of the complex. By our conditio n for D and lemma 8 we ha ve H 1 ( X , F X ( − D )) = ( 0 ) and th us there exist a 0 ∈ A ( F ( − D )) , a 1 ∈ A C F ( − D ) , a 2 ∈ ˆ F P ( − D ) suc h that a 01 = a 0 − a 1 , a 02 = a 2 − a 0 , 0 a 1 − a 2 . 19 By the second step a 1 = a 2 ∈ ( A C ( F ( − D )) ∩ ˆ F P ( − D )) ⊂ A C ( F ) ∩ ˆ F P = (0) and, consequen tly , w e hav e a 01 (= − a 02 ) ∈ A ( F ( − D )). Step 4 . The last step is to tak e tw o distinct divisors D , D ′ suc h that D ∩ D ′ ⊂ C . Since C is a hyperplane section we can c ho ose for D , D ′ t wo h yp erplane sec tions whose in tersection b elongs to C . Therefore their ideals in the ring A ( F ) are relativ ely prime and A ( F ) = A ( F ( − D ))+ A ( F ( − D ′ )) ∋ 1 = a + a ′ , a ∈ A ( F ( − D ) ) , a ′ ∈ A ( F ( − D ′ )) . If no w b ∈ B C ( F ) ∩ B P ( F ), then b = ba + ba ′ , where ba ∈ B C ( F ( − D )) ∩ B P ( F ( − D )) , ba ′ ∈ B C ( F ( − D ′ )) ∩ B P ( F ( − D ′ )). W e see that b ∈ A ( F ) b y the previous step. Remark 4 . The metho d we hav e used cannot b e applied if our v ariety is not Cohen-Macaula y (b y lemma 8 ab ov e). It w ould b e in t eresting to know ho w to extend the result to the a r bit r a ry surfaces X and the shea v es F suc h that F are lo cally free outside C . The last condition is really necessary . W e also note that an y norma l surface is Cohen-Macaulay [16 ][ § 1 7 ]. Remark 5 . This pro p osition is a v ersion for the reduced adelic complex of the corresp onding result for the full complex. Namely , A X, 01 ∩ A X, 02 = A X, 0 , see [7, c h.IV]. This should be gene ralized to a rbitrary dimension n in the fo llo wing w a y . Let I , J ⊂ [0 , 1 , . . . , n ] a nd A X,I ( F ) = ( Y { codimη 0 ,codimη 1 ,... }∈ I K η 0 ,η 1 ,... ) ⊗ F η 0 ) \ A X ( F ) . Then w e ha ve A X,I ( F ) ∩ A X,J ( F ) = A X,I ∩ J ( F ) for a lo cally free F and a Cohen-Macaulay X . Example . Let X = P 2 ⊃ C = P 1 ∋ P . W e in tro duce ho mogenous co ordinates ( x 0 : x 1 : x 2 ) suc h that C = ( x 0 = 0); P ( x 0 = x 1 = 0 and U = X − C = S peck [ x, y ] with x = x 1 /x 0 , y = x 2 /x 0 . Then k ( C ) = k ( y / x ) , x − 1 is t he last para meter fo r a ny tw o-dimensional lo cal field K Q,C with Q 6 = P . F or lo cal field K P ,C w e hav e K P ,C = k (( u ))(( t )) , u = xy − 1 , t = y − 1 . 20 Then w e can easily compute all the rings fr om the complex o f theorem 1 for the sheaf O X . B P = k [[ u ]](( t )) B C = k [ u − 1 ](( u − 1 t )) ˆ O P ,C = k (( u ))[[ t ]] A = Γ( U, O X ) = k [ u t − 1 , t − 1 ] A C = k [ u − 1 ][[ u − 1 t ]] ˆ O P = k [[ u, t ]] W e can draw the subspaces as some subsets o f the plane according to the supp orts of the elemen ts of the subspaces (on the plane with co ordinates ( i, j ) for elemen ts u i t j ∈ K P ,C . Then the first three subspaces B P , B C , ˆ O P ,C will correspond to some halfpla nes and the subspaces A, A C , ˆ O P to the in- tersections of them. 4 Kric hev er corre sp onden ce for dimen sion 2 W e need the follo wing we ll kno wn r esult. Lemma 9 . L et X b e an pr oje ctive variety, F b e a c oher ent she af on and C b e an ample div i s o r on X . If S = ⊕ n ≥ 0 Γ( X , O X ( nC )) , F = ⊕ n ≥ 0 Γ( X , F ( nC )) , then X ∼ = P r oj ( S ) , F ∼ = P r oj ( F ) . Proo f . Let mC b e a very ample divis or, S = ⊕ n ≥ 0 S n and S ′ := ⊕ n ≥ 0 S nm . Then by [8][prop. 2.4.7] P r oj ( S ′ ) ∼ = P r oj ( S ) . The divisor mC defines an em b edding i : X − → P to a pro jective space suc h that i ∗ O P (1) = O X ( mC ). Let J X ⊂ O P b e an ideal defined b y X . If I := ⊕ n ≥ 0 Γ( P , J X ( n )) , A := ⊕ n ≥ 0 Γ( P , O P ( n )) , 21 then I ⊂ A and b y [8][prop. 2.9.2] P r oj ( A/I ) ∼ = X . W e hav e an exact sequence of shea v es 0 − → J X ( n ) − → O P − → O X ( n ) − → 0 , whic h implies the sequenc e 0 − → M n ≥ 0 Γ( J X ( n )) − → M n ≥ 0 Γ( O P ( n )) − → M n ≥ 0 Γ( P , O X ( n )) − → M n ≥ 0 H 1 ( P , J X ( n )) . Here the last term is trivial for sufficien tly large n . The first three terms are equal corresp ondingly to I , A and S ′ . It means that the homog enous comp onen ts o f A/I and S ′ ⊃ A/I are equal for sufficien tly big degrees. By [8][pro p. 2.9 .1] P r oj ( A/I ) ∼ = P r oj ( S ′ ) , and com bining ev erything together w e get the statemen t of the lemma. The statemen t concerning the sheaf F can b e prov ed along the same line. No w we mo ve to the case of algebraic surfaces. The corresp onding data has the following Definition 4 . M 2 := { X , C , P , ( z 1 , z 2 ) , F , e P } X pro jectiv e irr educible surface /k C ⊂ X pro jective irreducible curv e /k P ∈ C a smo oth p oint on X and C z 1 , z 2 formal lo cal para meter at P such that ( z 2 = 0) = C near P F torsion free rank r sheaf on X e P a trivialization of F at P Then w e ha ve ˆ O X,P = k [[ z 1 , z 2 ]] , K P ,C = k (( z 1 ))(( z 2 )) , ˆ F P = ˆ O P e P = ˆ O ⊕ r P . 22 F or the field K = k (( z 1 ))(( z 2 )) w e hav e the following filtrations and sub- spaces: K 02 = k [[ z 1 ]](( z 2 )) , K 12 = k (( z 1 ))[[ z 2 ]] , K ( n ) = z n 2 K 12 . T aking the direct sums w e in tro duce the subspaces V 02 , V 12 , V ( n ) of the space V = K ⊕ r . Theorem 4 . L et C b e a hyp erplane se ction on the surfac e X . T h en ther e exists a c anonic al map Φ 2 : M 2 − → { ve ctor subsp ac es B ⊂ K , W ⊂ V } such that i) fo r al l n the c omplex e s B ∩ K ( n ) B ∩ K ( n + 1) ⊕ K 02 ∩ K ( n ) K 02 ∩ K ( n + 1) − → K ( n ) K ( n + 1) W ∩ V ( n ) W ∩ V ( n + 1) ⊕ V 02 ∩ V ( n ) V 02 ∩ V ( n + 1) − → V ( n ) V ( n + 1) ar e F r e dholm of index χ ( C , O C ) + nC.C and χ ( C , F | C ) + nC.C , r esp e ctively ii) the c ohomolo gy of c omplexes ( B ∩ K 02 ) ⊕ ( B ∩ K 12 ) ⊕ ( K 02 ) ∩ K 12 ) − → B ⊕ K 02 ⊕ K 12 − → K ( W ∩ V 02 ) ⊕ ( W ∩ V 12 ) ⊕ ( V 02 ) ∩ V 12 ) − → W ⊕ V 02 ⊕ V 12 − → V ar e isomorphic to H · ( X , O X ) and H · ( X , F ) , r esp e ctively iii)if ( B , W ) ∈ Im Φ 2 then B · B ⊂ B , B · W ⊂ W iv) fo r al l n the ma p ( C , P , z 1 | C , F ( nC ) | C , e P ( n ) | C ) 7→ 7→ ( B ∩ K ( n ) B ∩ K ( n + 1) ⊂ K ( n ) K ( n + 1) k (( z 1 )) , W ∩ V ( n ) W ∩ V ( n + 1) ⊂ V ( n ) V ( n + 1) k (( z 1 )) ⊕ r ) 23 c oincides with the map Φ 1 . v) let the sh e af F b e l o c al ly fr e e and the s urfac e X b e Cohen- Mac aulay. If m, m ′ ∈ M 1 and Φ 2 ( m ) = Φ 2 ( m ′ ) then m is iso- morphic to m ′ Proo f . If m = ( X, C , P , ( z 1 , z 2 ) , F , e P ) ∈ M 2 then to define the map Φ 2 w e put B = B C ( O X ) , W = B C ( F ) , Φ 2 ( m ) = ( B , W ) . Since w e ha ve the lo cal co o r dina t es z 1 , 2 and the trivialization e P the subspaces B and W will b elong to the space k (( z 1 ))(( z 2 )) exactly as in the case of dimension 1 considered ab o ve . W e note tha t our condition on the curv e C implies that C ia a Cartier divisor and the surface X − C is affine. The prop ert y i) follows from lemma 4, the prop ert y ii) fo llo ws from theo- rem 3 and the general theorem of adelic t heory: the cohomologies of the adelic complex of a sh eaf F ar e equal to the cohomologies H · ( X , F ) of the sheaf F [19, 3, 10, 7]. The prop ert y iii) is trivial a g ain, to get iv) one needs ag ain to apply lemma 4 and to get v) it is enough to use prop osition 4 and lemma 9. They show that giv en a p oint ( B , W ) ∈ M 2 suc h that ( B , W ) = Φ 2 ( m ) w e can reconstruct the data m up to an isomorphism. Remark 6 . The pro p ert y v) of the theorem cannot b e extended to the arbitrary torsion free shea v es on X . W e certainly cannot reconstruct suc h sheaf if it is not lo cally free outside C . Indeed, if F , F ′ are t w o shea ves and there is a monomorphism F ′ − → F suc h that F / F ′ has supp ort in X − C then the restricted adelic complexes for the shea v es F , F ′ are isomorphic. Remark 7 . A definition of the map Φ n for all n w a s suggested in [18]. It has the prop erties that corresp ond to the prop erties i) - v) of the theorem. Also the pro ofs in [18] has the adv an tag e: they are direct and don’t use the general a delic machin ery . App endix 1 Here we show how to deduce from the L a x form of the KP hierarc hy f o r pseudo-differen tial op erators L the classical KP and KdV equations. Let L = ∂ + u 1 ∂ − 1 + u 2 ∂ − 2 + . . . , where u m = u m ( x, t 1 , t 2 , . . . ) ∈ k [[ x, t 1 , t 2 , . . . ]] for m ≥ 1. If w e denote ∂ /∂ t n as ∂ n then the KP hierarch y ha s the following 24 form ∂ n L = [( L n ) + , L ] . This gives us for any n an infinite sequence o f differential equations f or t he functions u m . Denote by u ′ , u ′′ , . . . the subsequen t deriv ativ es by x . First for n = 1, w e get ∂ 1 u m = u ′ m for all m ≥ 1 This means that w e can tak e t 1 = x . No w w e write down the first tw o equations for n = 2 and the first equation for n = 3. ∂ 2 u 1 = u ′′ 1 + 2 u ′ 2 (1) ∂ 2 u 2 = u ′′ 2 + 2 u ′ 3 + 2 u 1 u ′ 1 (2) ∂ 3 u 1 = u ′′′ 1 + 3 u ′′ 2 + 3 u ′ 3 + 6 u 1 u ′ 1 (3) Let us in tr o duce the new notations: u = u 1 ( x, y , t ) with y = t 2 , t = t 3 . Also w e use the standard notatio ns u t , u y , u y y , . . . for deriv ativ es. W e can eliminate u ′ 3 from equations (2) and (3) and then w e get 2 u t − 2 u ′′′ − 6 uu ′ = 3( u ′′ 2 + u 2 y ) . (4) Differen tiat ing this equation b y x w e hav e (2 u t − 2 u ′′′ − 6 uu ′ ) ′ = 3( u ′′′ 2 + u ′ 2 y ) (5) . ¿F rom ( 1 ) w e find u ′′′ 2 = 1 / 2( u ′′ y − u ′′′′ ) , u ′ 2 y = 1 / 2( u y y − u ′′ y ) and inserting these expressions in to (5) w e finally get the KP equation (4 u t − u ′′′ − 12 uu ′ ) ′ = 3 u y y . In the space E ′ there is an inv ar ia n t submanifold defined b y condition ( L 2 ) − = 0 [22][Lemma 2]. Let us lo ok what this conditio n means for the op erator L as a b o v e. W e ha ve L 2 = ∂ 2 + (2 u 2 + u ′ 1 ) ∂ − 1 + . . . and thus in our notat io ns 2 u 2 + u ′ = 0 . (6) 25 Com bining (1) a nd (6) w e se e that u 2 y = 0. T ogether with (4) this implies 3 u ′′ 2 = 2 u t − 2 u ′′′ − 6 u u ′ . Using once more (6) w e get at last the KdV equation 4 u t − 7 u ′′′ − 12 uu ′ = 0 . The n umerical co efficien ts here are not es sen tial since one can get a ll po ssible v alues b y rescaling of x, t and u . These computations and the choice of v ariables lo ok quite artificial com- paring with the previous constructions. It is an in teresting indep enden t problem ho w to deduce t he KP and KdV equations from the Lax op era- tor equations in a more conceptual w ay . No w, w e show ho w to do that for the classical K dV equation. Comparing with the previous deduction this one can b e done in a purely formal wa y , without a n y tric ks. Again, we order that our equation will satisfy the constraint condition ( L 2 ) − = 0. F or the o p erator L as ab ov e we ha ve L 2 = ∂ 2 + (2 u 2 + u ′ 1 ) ∂ − 1 + (2 u 3 + u 2 1 + u ′ 2 ) ∂ 2 + . . . and thus in our notat io ns 2 u 2 + u ′ = 0 , 2 u 3 + u 2 1 + u ′ 2 = 0 and so on. This means that w e can compute all u m for m > 1 starting from u 1 and its deriv ativ es. Under t his conditio n the first non- trivial equation of the KP hierarc h y is the follo wing o ne ∂ 3 L = [( L 3 ) + , L ] . and taking the co efficien t nearby ∂ − 1 w e hav e u ′′′ 1 + 3 u ′′ 2 + 3 u ′ 3 + 6 u 1 u ′ 1 = 0 . Substituting the giv en a b ov e expressions fo r u 2 and u 3 w e get at last the KdV equation 4 u t − 7 u ′′′ − 12 uu ′ = 0 . Problem. Let us go to the case of dimension tw o. Then we ha v e the constrain t conditions of the type ( L m M n ) − = 0 26 and the particular comp onen ts of hierarc h y . Can w e deduce some conc rete equations using the same w ay a s ab o v e ? Certainly , w e hav e more p ossibili- ties. The first question is how many initial functions u mn one has to use so to generate all other co efficien ts ? F or dimension 1 only u 1 w as enough ! Remark. F or higher dimensions it is crucial to transform the whole system to a void the asymme try of the v ariables. It means that w e ha v e to consider not t he ring P alone but at least the direc t sum P ⊕ P ′ where P ′ has in terchanged v ariables, namely y , x instead of x, y . As a g oal one can hop e to get in this w a y the equations of the plane h ydro dynamics. It is know n they hav e infinitely many conserv ation la ws. App endix 2 Here, we presen t a w ell-know n construction of se mi-infinite monomes in an infinite-dimensional vec tor space. Let V b e a v ector space o ver a fied k and V n , n ∈ Z b e an exhausted increasing filtr a tion in V . The example one has to ha ve in mind is when V = k (( z )) , V n = z − n k [[ z ]]. If V w ould b e a finite-dimensional space,the k -dimensional ve ctor subspaces W ⊂ V can b e described b y the elemen ts V k ( W ) ⊂ V k ( V ). In the infinite-dimensional case w e assume t hat dim W ∩ V n = k + n for large n. In the case of V = k (( z ) ) this k is exactly the index o f subspace W from the Sato G r a ssmanian. In general case fo r suc h n , we hav e the diagr a m W ∩ V n W ∩ V n +1 V n V n +1 / /     / / W e denote the o ne- dimensional space ( W ∩ V n +1 ) / ( W ∩ V n ) as 1 n . The diagram induces isomorphism 1 n → V n +1 /V n . F or an y exact sequence 0 → V → V ′ → V ′ /V → 0 with o ne-dimensional space V ′ /V , there is a canonical map V n +1 ( V ′ ) → V n ( V ) ⊗ ( V ′ /V ) . If v = v 1 ∧ v 2 ∧ . . . v n +1 ∈ V n +1 ( V ′ ) then the image of v is zero if either all v i b elong to V or at least tw o v i b elong to V ′ − V . If exactly one v i b elong to V ′ − V then the image is ( − 1) n − i v 1 . . . ∧ v i − 1 ∧ v i +1 ∧ . . . v n +1 ⊗ v i mo d V . 27 W e g et the new diag ram k + n +1 ^ ( W ∩ V n +1 ) k + n ^ ( W ∩ V n ) ⊗ 1 n k + n +1 ^ ( V n +1 ) k + n ^ ( V n ) ⊗ V n +1 /V n / /     / / F or a n y tw o subspaces A, B ⊂ V , we sa y they are commesurable iff the in tersection A ∩ B has finite co dimension in b oth o f them. In this case, we define ( A | B ) = det( A/ A ∩ B ) − 1 ⊗ det ( B / A ∩ B ) , where det( A ) = V dim ( A ) ( A ) and ( A ) − 1 is the space dual to A . Note that all the spaces ( A | B ) are one-dimensional. There are the canonical isomor- phisms ( A | B ) ∼ = ( B | A ) − 1 and ( A | C ) ∼ = ( A | B ) ⊗ ( B | C ) whic h w e will treat as equalities. These rules allows us to write V n +1 /V n = ( V n | V n +1 ) = ( V n | V 0 )( V n +1 | V 0 ) − 1 , 1 n = ( V n | V n +1 ) = ( V n | V 0 )( V n +1 | V 0 ) − 1 for larg e n. Th us, w e can rewrite the diagram as k + n +1 ^ ( W ∩ V n +1 )( V n +1 | V 0 )) k + n ^ ( W ∩ V n ) ⊗ ( V n | V 0 ) k + n +1 ^ ( V n +1 )( V n +1 | V 0 )) k + n ^ ( V n ) ⊗ ( V n | V 0 ) / /     / / A t last, the space of semi-infinite monomes (o f index k ) can b e defined as pro jectiv e limit resp ect these maps: k + ∞ 2 ^ ( V ) = lim n k + n ^ ( V n ) ⊗ ( V n | V 0 ) 28 and w e can a ttac h to the subspace W the line [ W ] = lim n k + n ^ ( W ∩ V n ) ⊗ ( V n | V 0 ) . Using these constructions w e can presen t the Sato Grassmanian as a disjoin t union of connected comp o nen ts Gr ( V ) = a k Gr k ( V ) and ev ery comp onent has a pro jectiv e em b edding Gr k ( V ) → P ( k + ∞ 2 ^ ( V )) where W 7→ [ W ]. Note that the ve ctor space ∧ k + ∞ 2 ( V ) do es dep end o n the c hoice of the subspace V 0 from the filtration but it’s pro jectivization do esn’t. References [1] ´ Alv arez V´ a squez A., Munoz Porras J. M., Plaza Mart ´ ın F. 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