Sums of residues on algebraic surfaces and application to coding theory

SUMS OF RESIDUES ON ALGEBRAIC SURF A CES AND APPLICA TION TO CODING THEOR Y ALAIN COUVREUR Abstra ct. In this pap er, w e study residues of diﬀerential 2-forms on a smooth algebraic surface ov er an arbitrary ﬁeld and give sev eral statemen ts abou t sums of residues. Afterw ards, using these results w e construct algebraic- geometric co des which are an extension to surfaces of the well -kn o wn diﬀeren- tial co des on curves. W e also study some prop erties of these co des and extend to them some known prop erties for cod es on curv es. AMS Classiﬁcation: 14J99, 14J20, 14G50 , 94B27 . Keyw ords: Algebraic surf ac es, diﬀerentia ls, residues, algebraic-geometric co des. Introduction The present pap er is divided in tw o parts. The ﬁrst one is a th eo retical study of residues of diﬀerentia l 2-forms on alg ebraic sur face s o v er an a rbitrary ﬁeld. The seco nd one u s es results o f the ﬁr st part to construct d iﬀeren tia l cod es on algebraic surfaces and to study some of their prop erties. The r eader esp ecially in terested in cod ing theory is encouraged to read brieﬂy the deﬁnitions and th e results of the ﬁrst part and then to jump to the second part. Ab out residues. If the notion of r esidue is we ll-kno wn for diﬀerentia l forms on curv es, there is no uniﬁed d eﬁnition in higher dimension. On complex v arieties, one can distinguish t w o ob jects called residues in the literature. The ﬁ rst one app ears for instance in Griﬃths an d Harris [12] c hapter V. In this b ook, giv en an n -dim en sional v ariet y X , the residue of a meromorph ic n -form ω at a p oint P is a complex n umber obtained by computing an integral on a r eal n -cycle. Th is ob ject dep end s on some n -uplet of divisors whose sum contai ns the p oles of ω in a neigh b orho od of P . Another deﬁn itio n is giv en in Comp act c omp lex surfac es b y Ba rth, Hulek, P eters a nd V an De V en ([2] I I.4). In th is b o ok, giv en an n - dimensional v ariet y X and a one-co dimensional su b v ariet y Y of X , the r esidue along Y of a q -form on X ha ving a simp le p ole along Y is a ( q − 1)-form on Y . The computation of th is residu e can b e obtained by a com b inatorial w ay , or b y computing an in tegral on a real sub v ariet y ([2] I I.4). In algebraic geometry o v er an arb itrary ﬁeld, sev eral references deal with r esidues , for instance Hartshorne [14] or Lipman [1 9]. Actually their main ob- jectiv e is to establish dualit y theorems generalizing S erre’s one. Th us, their ﬁrst in ten tion is n ot to deﬁne residu es of diﬀerential forms on higher-dimensional v arieties ov er an arbitrary ﬁ eld. The goal of the ﬁrst part of this pap er is to generalize to surfaces o ver an arb i- trary ﬁ eld, the deﬁnitions of residues give n for complex v arieties in [2] and in [12]. Then, w e will establish results of indep endence on the c hoice of lo cal coord in ate s, and fo cus on su mmation prop erties. Notice that Hartshorn e, in [14] I I I.9, in tro- duces a Gr othendie ck r esidue symb ol havi ng sligh tly the same pr operties as the 1 2 ALAIN COUVREUR residue deﬁned in Griﬃth s and Harris’s b ook. Moreov er, Lipman in [19] section 12 states a summ ati on residue formula whic h is cl osed to the theorem 6.8 in the present p ap er . Finally , most o f th e results of t his ﬁrst part can b e considered as consequences of sev eral statemen ts lyin g in [14] or [19]. Neve rtheless, b oth references are long and con tain an imp ortan t functorial mac h inery whic h is not necessary to d eﬁne residues on su rfaces, to study their pr operties and to obtain summation formulas. That is why we decided to presen t a self-con tained pap er for wh ic h r eferences [14] and [19] are not prerequisites. Notice that we c h ose to w ork only on surfaces. A t least t w o main r easo ns justify this choic e. Firs t, w orking on n -d im en sional v arieties w ould h a v e give n to o hea vy notations. Seco nd, the most diﬃcult s tep in algebraic-geometric co ding theory lies b et wee n curv es and higher-d imensional v arieties Ab out algebraic-geometric co des. In cod ing theo ry , t wo main p roblems are frequent ly studied. Th e ﬁrst one is: how to ﬁnd a lower b ound for its minimal distanc e of a given c o de? The second one is: h ow to ﬁnd algorithm s c orr e cting a suitable numb er of err ors in a r e asonable time? Giv en an arbitrary code deﬁned b y a generator or a parit y-c hec k matrix, b oth problems are very d iﬃcult. A go od w a y to solv e them, is to get a geometric (or arithmetic) r e alization of the cod e. Then, one or b oth problems ma y b e tran s lat ed into geometric (or arithmetic) problems. This is, for instance, su cce ssfull for the study of Reed-Muller co des. Consequent ly , geometric constructions of co d es are often in teresting. Co des on curves. In 1981, Goppa introd u ced in a construction of error-correcting co des usin g algebraic curves (see [11]). Their stu dy has b een a fruitful topic of researc h during last thir t y y ears. Hundred s of pap ers are dev oted to this sub ject. One of the main reasons why these co des hav e b een so in tensiv ely studied is that some families of suc h co des h a v e excellen t asymptotic parameters. P articularly , T sfasman, Vl˘ adut ¸ and Zink pro v ed in [30] that some families of algebraic-g eometric cod es b eat the Gilb ert-V arshamov b ound. Most of the b asic results ab out co des on cur v es are summarized in [27], [28] c hapter I I and [29]. Co des on higher-dimensional varieties. In higher dimension, the topic has n ot b een as extensiv ely explored. The ﬁ rst general construction of algebraic-ge ometric co de from a v ariet y of arbitrary dimension has b een giv en b y Manin in the pap er with Vl˘ adut ¸ [31]. Afterwards, co des coming from some particular v arieties hav e b een studied. Among others, in [1], Aubry dealt with co des on quadric v arieties. His results ha v e b een impro v ed in dimension 2 and 3 b y Ed ouk ou in [8] and [6]. Co des on Grassmannians ha v e b een discu s sed b y Nog in in [22] then b y Ghor- pade and Lac haud in [10]. Cod es on He rmitian v arieties hav e b een treated by Chakra v arti in [4], then b y Hirschfeld, Tsfasman and Vl˘ adut ¸ in [16], afterwards b y Sørensen in his PhD thesis [26] and b y Edouko u in [7]. In [23], Ro dier pr e- sen ted a uniﬁed p oint of view f or all the ab o ve-c ited examples regarding these v arieties as ﬂag-v arieties an d gav e some more examp les of codes. Zarzar studied in [33] the parameters of co des on sur f ace s ha ving a small Picard num b er. Th e author prop osed also a decodin g alg orithm for su ch co des in a join w ork with V olo c h [32]. General b ounds on the parameters of codes on a lgebraic v arieties of arbitrary dimension hav e b een giv en by L ac haud in [ 18] and by S øren Ha v e Hansen in [13]. Finally , a surv ey p ap er [20] by Little summarizing m ost of the kno wn works on co des on h igher-dimensional v arieties app eared recently . RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 3 Notice that almost all the references cited b elo w, deal with th e question of b ounding or ev aluating the parameters of some error-correcting codes. This will not b e the purp ose of the presen t pap er whose ob jectiv e is to giv e general theoretical statemen ts extending some kno w n results for co des on curv es. Diﬀer ent c onstruction of c o des on curves. In t he theory o f algebraic-geomet ric co des on curv es, one can d istinguish t wo diﬀerent constructions. F unctional c o des are obtained b y ev aluating elemen ts of a Riemann-Ro c h s p ace at some set of ra- tional p oin ts on a curv e. Diﬀer ential c o des are obtained b y ev aluating residues of some rational diﬀeren tial forms at these p oin ts. F or higher-dimensional v arieties, only the functional constru cti on has b een extended and studied (see r eferen ces b elo w). The diﬀerentia l one d oes not seem to ha v e a natur al generaliza tion and this qu estio n h as nev er b een treated b efore. Motivations. There are at least three motiv ations for an extension to surfaces of the d iﬀeren tial constru cti on. The ﬁrst one is historical. Indeed, the ﬁ rst construc- tion of algebraic-geometric co des giv en b y Goppa in [11] used diﬀerent ials. T his construction generalized that of classical Goppa codes wh ic h can b e regarded as diﬀeren tial cod es on the pro jectiv e line. The second one is that the orthogonal of a functional cod e on a cur v e i s a diﬀeren tial one. Moreo ver, this statemen t is used in almost all known algebraic decodin g algorithms (see [17]). The th ird motiv ation is t hat, as said before, it is alw a ys in teresting to ha v e a geometric realizatio n of a co de. T o ﬁn ish w ith motiv atio ns, notice that the in tro duction of the ab o v e cited survey pap er [20] of Little conta ins the follo wing sen tences. “In a sense, the ﬁrst major diﬀer enc e b etwe en higher dimensional varieties and curves is that p oints on X o f dimension ≥ 2 ar e subvarieties of c o dimension ≥ 2 , not divisors. This me ans that many familiar to ols use d for Gopp a c o des (e.g. R iemann-R o ch the or ems, the the ory of diﬀer entials and r esidues etc.) do not apply exactly in the same way.” Th us, ﬁ nding another w a y of applying residues and diﬀeren tials for co des on surfaces m ust b e inte resting. This is the purp ose of the second part of this pap er, whic h st arts w ith the pr esentati on o f a co nstru ction of co des using residu es of diﬀeren tial 2-forms on surfaces. T hen, connections b et we en these cod es and the functional ones are studied. W e prov es that an y diﬀeren tial co de is includ ed in the orthogonal of a functional one but th at the rev erse inclusion is false, whic h is an imp ortan t diﬀerence with the theory of co des on curv es. Notice that V olo c h and Zarzar suggested the existence of such a diﬀerence in [32] section 3 without pro ving it. Fin ally , w e prov e that, as for co des on curves, a d iﬀeren tial cod e can alw a ys b e regarded as a functional one asso ciated with some parameters dep ending on a canonical divisor. Con ten ts. Th e ﬁ rst part conta ins sections 1 to 6. In sectio n 1, w e recall the deﬁnition of one-co dimensional residues along a curv e C of a diﬀeren tial 2-form ω h a ving C as a simple p ole. Then, w e deﬁne naturally th e t wo -co dimensional residue of ω along C at a smo oth p oin t P ∈ C to be the residu e at P of the one-cod imensional residue. In section 2, w e study Lauren t series expansions in t w o v ariables, in order to ha v e a combinatoria l d eﬁnition for residues, whic h will b e more conv enien t for computations. I n section 3, w e in tro duce n ew deﬁnitions of one- and t w o-cod imensional residu es holding for any rational 2-form. Then, w e pro v e that the tw o-co dimensional residue at a p oin t P a long a curve C ∋ P o f 4 ALAIN COUVREUR a r atio nal 2-form d oes not d epen d on the c hoice of lo cal co ordinates. In sectio n 4, we study some prop erties of one- and t wo-c o dimensional residues. In section 5, w e deﬁne tw o-co dimensional residues along a curv e at a singular p oin t of it. Finally , sectio n 6 conta ins three statemen ts ab out summations of resid u es. The s econd part con tains s ect ions 7 to 10. Section 7 is a quic k review on the theory of co des on curves. In section 8 , after a brief o verview on fu nctional co des on higher-dimensional v arieties, w e deﬁne diﬀerentia l co des on surfaces. Then , prop erties of these co des and their relatio ns w ith fu nctional ones are studied in section 9. Particularly , w e p r o ve that a d iﬀerential co de is conta ined in the or- thogonal of a functional one. Finally , s ect ion 10 pro v es that the rev erse inclusion ma y b e false b y treating the elemen tary example of the surface P 1 × P 1 . P a rt 1. Residues of a rational 2 -form on a smo oth surface Not a tions F or an y irreducible v ariet y X o v er a ﬁeld k , we denote b y k ( X ) its fu nction ﬁeld. If Y is a closed irredu cible su b v ariet y of X , then the local ring (resp. its maximal ideal) of regular functions in a neighb orho o d of Y , that is functions whic h are regular in at least one p oint of Y , is denoted b y O X,Y (resp. m X,Y ). The m X,Y -adic completion of the ring O X,Y is denoted b y b O X,Y and its maximal ideal m X,Y b O X,Y b y b m X,Y . F or any fun ction u ∈ O X,Y , we denote b y ¯ u its restriction to Y . Recal l that, if Y has co dimension one in X and is n ot conta ined in the singular lo cus of X , t hen O X,Y is a discrete v aluation rin g with residue ﬁeld k ( Y ). In th is situation, the v aluation along Y is denoted b y v al Y . Finally , w e den ote b y Ω i k ( X ) /k the space of k -rati onal diﬀeren tial i -forms on X . 1. One and two-codimensional residues Con text. In this sec tion, k d enotes an arbitrary ﬁ eld of arbitrary c haracteristic and S a smo oth geometrica lly in tegral quasi-pro j ect iv e su rface ov er k . Moreo ver, C d en ote s an irreducible geometrically reduced curve em b edd ed in S and P a smo oth rational p oint of C . 1.1. First deﬁnitions for residues. Giv en a 2-form ω ∈ Ω 2 k ( S ) /k , on e can construct t wo ob jects ca lled r esidues in the literature. Th e ﬁrs t one is a ratio nal 1-form on a curve em b edd ed in S and the second one is an elemen t of k (or of some ﬁn ite extension of it). Their deﬁn itio ns will b e the r espective p urp oses of deﬁn itions 1.3 and 1.4. W e ﬁ rst need next prop osition, asserting the w ell- deﬁnition of one-co ndim en sional residues (deﬁnition 1.3). Prop osition 1.1. L et v b e a uniform izing p ar ameter of O S,C and ω b e a r ationa l 2 -form on S having m S,C -valuation gr e ater than or e qual to − 1 . Then, ther e exists η 1 ∈ Ω 1 k ( S ) /k and η 2 ∈ Ω 2 k ( S ) /k , b oth r e gu lar in a neig hb orh o o d of C and such that (1) ω = η 1 ∧ dv v + η 2 . Mor e over, the diﬀer ential form η 1 | C ∈ Ω 1 k ( C ) /k is unique and dep ends neither on the choic e of v nor on that of the de c omp osition (1). Pr o of. W e ﬁrst p r o ve the existence of a d eco mp osition (1). Recall that (2) dim k ( S ) Ω 1 k ( S ) /k = 2 and dim k ( S ) Ω 2 k ( S ) /k = 1 , RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 5 (see [25] thm I I I.5.4 .3). C onsequen tly , there exists a rational 1-form µ , whic h is non- k ( S )-colinear with dv v . T h u s, µ ∧ dv v 6 = 0. F rom (2), ther e exists also a unique function f ∈ k ( S ) satisfying ω = f µ ∧ dv v . Since v al C ( ω ) ≥ − 1, the 1-form f µ has no p ole alo ng C . W e obtain a decomp o- sition (1) b y setting η 1 := f µ and η 2 := 0. Ob viously , this decomp osition is far from b eing un iqu e. Only η 1 | C is unique. T o pro v e uniqu eness and indep endence of η 1 | C under the c hoice of v , see [2] I I.4. Ev en if th is b o ok only deals with complex surfaces, the v ery s ame pr oof holds for su rfaces o ve r an arbitrary ﬁ eld.  Remark 1.2. Another pr o of of pr op osition 1.1 wil l b e given in se ction 3 in a mor e gener al c ontext (se e lemma 3.9). Deﬁnition 1.3. Under the assumptions of pr op osition 1 .1 and given a de c om- p osition of the f orm (1) for ω , the 1 -form η 1 | C ∈ Ω 1 k ( C ) /k is c al le d the one- c o dimensional r esidue (or the 1 -r esidue) of ω along C and denote d by r es 1 C ( ω ) := η 1 | C . Deﬁnition 1.4. Under the assumptions of pr op osition 1.1, let P b e a k -r ational p oint of C . The two-c o dimensional r e si due (or the 2 -r esidue) of ω at P along C is the r esidue at P of the 1 -r esidue of ω along C . That is r es 2 C,P ( ω ) := r es P ( r es 1 C,P ( ω )) . Notice that to deﬁne residues in this w a y , ω needs to h av e v aluation greater than or equal to − 1 along C . Ho we ve r, t w o-cod imensional r esidues can actually b e deﬁned for any rational diﬀeren tial f orm even if it has a m u ltiple p ole along C . This will b e the purp ose of sections 2 to 4. Remark 1.5. It wo uld have b e en natur al to deﬁne 2 -r esidues at a close d p oint P of C . Ne vertheless, we de cide d to ke ep a mor e g e ometric p oint of v i ew, even if the b ase ﬁeld is not supp ose d to b e algebr aic al ly close d. Notic e that any ge ometric p oint of S (i.e. a close d p oint of S × k ¯ k ) is a r ational p oint of S × k L for a sui table ﬁnite sc alar extension L/k . Conse qu ently, if we deﬁne r esidues at r ational p oints of S , i t is e asy t o extend this deﬁnition to ge ometric p oints using such a sc alar extension. The only arithm etic statement we wil l ne e d in the se c ond p art of the pr esent p ap er is that, if C is deﬁne d over k and P ∈ C ( k ) , then the 2 -r e sidue along C at P of a k -r ational 2 -form is in k . That is why we ke ep c onsidering non-algebr aic al ly close d ﬁelds in se ctions 1 to 3 and 5. However, in se ctions 4 and 6, when we de al with pr op erties of r esidues and p articularly with summations of them, we work over an algebr aic al ly close d ﬁe ld. 2. Laurent series in two v ar iab l es As is w ell-kno wn , the residue at a p oint P on a curve C of a 1-form can b e computed using Laur en t series exp ansions. The residue of a diﬀeren tial form at a p oin t P is the co eﬃcien t of degree − 1 of its Laurent series expansion. W e lo ok for a sim ilar deﬁnition in the t w o-dimensional c ase. F or this pu rp ose, w e in tro duce Laurent series in tw o v ariables. Con text. The conte xt of this section is exactly that of section 1 (see page 4). 6 ALAIN COUVREUR 2.1. Lauren t series expansion, the ﬁrst construction. Recall that, C is assumed to b e a geometricall y reduced irreducible cur ve o ver k e mbedd ed in S and P a smooth rational p oin t of C . Deﬁnition 2.1. A p air ( u, v ) ∈ O 2 S,P is said to b e a str ong ( P , C ) -p air if the fol lowing c onditions ar e satisﬁe d. (1) ( u, v ) is a system of lo c al p ar ameters at P . (2) v is a uniformizing p ar ameter of O S,C . Lemma 2.2. L et ( u, v ) b e a str ong ( P, C ) - p air, then ther e exists a morphism φ : k ( S ) ֒ → k (( u ))(( v )) sending O S,P into k [[ u, v ]] and O S,C into k (( u )) [[ v ]] . Pr o of. W e will pro ve the existence o f φ 0 : O S,C ֒ → k (( u ) )[[ v ]] en tailing that of φ , thanks to the un iv ersal pr op erty of f racti on ﬁelds. F r om [25] I I.2, an y elemen t of O S,P has a un ique T aylo r series expansion in the v ariables u, v . Then , notice that O S,C and O S,P ( v ) are isomorphic and consider the f oll o wing d iagram. O S,P lo c O S,C comp ∃ ! b O S,C ∃ ! k [[ u, v ]] lo c k [[ u, v ]] ( v ) comp \ k [[ u, v ]] ( v ) . The horizonta l arrows in the left hand square corresp ond to lo calizat ions, the ones in the right hand square corresp ond to ( v )-adic completions. V ertical arr o ws are obtained by applying resp ectiv ely u niv ers al p rop erties of lo calizatio n and completion. W e no w ha v e to p ro ve that \ k [[ u, v ]] ( v ) is isomorphic to k (( u ))[[ v ]], whic h is a consequence of C ohen’s structure theorem (see [9] thm 7.7 or [5] thm 9 for an historical reference).  2.2. Lauren t series, the second construction. Let ( u, v ) b e a strong ( P , C )- pair. Cohen’s s tr ucture theorem asserts that b O S,C is isomorphic to k ( C )[[ v ]]. Unfortunately , this isomorph ism is not alw a ys u nique. Indeed, [5] thm 10(c) asserts that, if Char( k ) > 0, then there are in ﬁnitely man y subﬁ elds of b O S,C whic h are isomorp hic to th e residue ﬁ eld k ( C ). T herefore, to u se this isomorp hism for Lauren t series expansions, w e h a ve to c ho ose a representan t of k ( C ) which is, in some sense, related to u . Prop osition 2.3 (Th e ﬁeld K u ) . L et u ∈ O S,C whose r estriction ¯ u to C is a sep ar ating element (se e [28] p. 127 for a deﬁnition) of k ( C ) /k . Then, ther e exists a u ni q ue subﬁeld K u ⊂ b O S,C c ontaining k ( u ) and isomorphic to k ( C ) under the morphism b O S,C → b O S,C / b m S,C . F urthermor e, this ﬁeld is gener ate d over k ( u ) by an element y ∈ b O S,C . Pr o of. The extension k ( C ) /k ( ¯ u ) is ﬁnite and separable. Th us, from the prim i- tiv e elemen t theorem, there exists a fu nction ¯ y ∈ k ( C ) generating k ( C ) o ver k ( ¯ u ). F rom Hensel’s lemma, ¯ y lifts to an elemen t y ∈ b O S,C and the subrin g K u := k ( u )[ y ] ⊂ b O S,C is the exp ecte d cop y of k ( C ). The uniqueness of K u is a consequence of the uniquen ess of the Hensel Lift y of ¯ y .  Corollary 2.4. Under the assumptions of pr op osition 2.3, any element f ∈ b O S,C has a unique exp ansion in K u [[ v ]] . RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 7 Pr o of. Existence. Let f b e an elemen t of b O S,C and f 0 b e the Hensel-lift in K u of f mo d b m S,C . The b m S,C -adic v aluation of f − f 0 is greater than or equal to one. By induction, using the same reasoning on v − 1 ( f − f 0 ), w e obtain an expansion f = f 0 + f 1 v + · · · for f . Uniqueness. Assume that f has t w o d istinct expansions P j f j v j and P j e f j v j in K u [[ v ]]. Let j 0 b e th e sm alle st intege r suc h that f j 0 6 = e f j 0 . F r om p rop ositio n 2.3, a nonzero elemen t of K u ⊂ b O S,C has b m S,C -adic v aluation zero. Consequen tly , 0 has b m S,C -adic v aluation j 0 whic h is absurd.  The second Lauren t series expansion using Cohen’s structure theorem needs w eak er conditions on the pair ( u, v ). Thus, b efore w e deﬁ n e it, w e give a new deﬁnition. Deﬁnition 2.5. A p air ( u, v ) ∈ O 2 S,C is said to b e a we ak ( P , C ) -p air if ¯ u is a uniformizing p ar ameter of O C,P and v is a uniformizing p ar ameter of O S,C . Remark 2.6. Obviously, a str ong ( P , C ) -p air is we ak, bu t the c onverse statement is false (se e next example). Example 2.7. Assume that S is the aﬃne plane over C , the curve C is the line of e quation y = 0 and P is the origin. Set u := ( x + y )( x − y ) x and v := x y . Then, ( u, v ) is a we ak ( P , C ) -p air which is not str ong. No w , we can deﬁne the second w a y of Lauren t series expan s ion. Lemma 2.8. Given a we ak ( P, C ) -p air ( u, v ) , ther e is an inje ction ϕ : k ( S ) ֒ → k (( u ))( ( v )) sending O S,C in k (( u ) )[[ v ]] . Pr o of. As in the pro of o f lemma 2.2 , w e just ha ve to prov e the existence of a morphism ϕ 0 : O S,C ֒ → k (( u ))[[ v ]]. The curv e C is assumed to b e geomet rically reduced, th u s from [21] prop I I.4.4 (i), the extension k ( C ) /k is separable, hence has a separable tran s cendence basis. Moreo v er, the function ¯ u is a unif orm izing parameter of O C,P , thus its diﬀeren tial d ¯ u is n onzero and, from [3 ] thm V.16.7.5, it is a separat ing elemen t of k ( C ) /k . F rom pr oposition 2.3, there is an injec- tion O S,C ֒ → K u [[ v ]]. F u rthermore, there is a natural extension K u ֒ → k (( u )), coming fr om the ( ¯ u )-adic completion of k ( C ) ∼ = K u . Applying this extension co eﬃcie nt wise on K u [[ v ]], we obtain the morph ism ϕ 0 .  Next p rop ositi on links b oth L auren t series expansions. Prop osition 2.9. If ( u, v ) i s a str ong ( P , C ) -p air, then L aur ent series exp ansions of lemmas 2.2 and 2.8 ar e the same. That is φ = ϕ . Pr o of. Consider aga in the diagram in 2.1 including the new expansion O S,P O S,C ϕ 0 φ 0 b O S,C ∼ K u [[ v ]] γ δ k (( u ))[ [ v ]] id k [[ u, v ]] k [[ u, v ]] ( v ) \ k [[ u, v ]] ( v ) ∼ k (( u ))[ [ v ]] . Maps γ and δ corresp ond resp ecti ve ly to the ﬁ rst and the seco nd expansion. W e ha v e to pro v e that φ 0 = ϕ 0 , wh ich is equiv alen t with γ = δ . 8 ALAIN COUVREUR Recall that, from prop ositio n 2. 3, the ﬁeld K u is generated o v er k ( u ) by an elemen t y ∈ b O S,C . Thus, a local morph ism K u [[ v ]] → k (( u ))[[ v ]] is en tirely determined by the images of u, v and y . O b viously , δ sends u and v resp ectiv ely on t hemselve s and from the c omm utativit y of the left part of the diagram, so do es γ . The only nonobvious part is to pro ve that γ sends y on ψ ( u ), where ψ ( ¯ u ) is the ( ¯ u )-adic expansion of ¯ y . Let F ∈ k ( ¯ u )[ T ] b e the minimal p olynomial of ¯ y ov er k ( ¯ u ). T he formal function y is the unique ro ot of F in b O S,C whose class in the r esidue ﬁeld k ( C ) is ¯ y . Therefore, the morphism γ m ust sen d y o n t he unique ro ot of F in k (( u ))[[ v ]] whic h is congruen t to ψ ( u ) mo dulo ( v ). Moreo v er, ψ ( ¯ u ) = ¯ y , then F ( ¯ u, ψ ( ¯ u )) = 0, th us the formal series F ( X , ψ ( X )) ∈ k [[ X ]] is zero. Consequently , F ( u, ψ ( u )) is zero in k (( u )), hence is zero in k (( u ))[[ v ]]. Then , ψ ( u ) is a root of F ( u, T ) ∈ k (( u ))[ [ v ]][ T ] whose class in the residue ﬁeld k (( u )) equals ψ ( u ), such a ro ot is unique. Thus, γ ( y ) = ψ ( u ).  2.3. Change of coordinat es. In this subsection, w e d eﬁne 1- and 2-residues of an y diﬀeren tial 2-form ω using w eak ( P , C )-pairs. These deﬁnitions hold ev en if C is a m u ltiple pole of ω . Afterw ards, w e pro v e that the new deﬁnition of 2-residue do es not dep end on the c hoice of a w eak ( P, C )-pair. F or that, w e m ust describ e c hanges of w eak ( P , C )-pairs. Lemma 2.10. L et ( u, v ) and ( x, y ) b e tw o we ak ( P, C ) -p airs, then the L aur ent series exp ansions of u and v in k (( x ))[[ y ]] ar e of the form (CV)  u = f ( x, y ) with f ( x, 0) ∈ xk [[ x ]] r x 2 k [[ x ]] v = g ( x, y ) with g ∈ y k (( x ))[[ y ]] r y 2 k (( x ))[[ y ]] . Pr o of. F un ctio ns ¯ u and ¯ x are b oth un iformizing parameters in O C,P , th us ¯ u = f ( ¯ x, 0) ∈ ¯ xk [[ ¯ x ]] r ¯ x 2 k [[ ¯ x ]]. Both fu nctions v and y are un iformizing parameters of O S,C , then v /y is in v ertible in O S,C , that is v /y ∈ k (( x ))[[ y ]] × .  3. Genera l definition of tw o-codimens ional residues Lauren t series hav e b een in tro duced in section 2 b ecause they are useful for computations. Using them, one can deﬁn e 1- and 2-residues in a m ore general con text. Con text. The conte xt of this section is exactly that of section 1 (see page 4). Deﬁnition 3.1. L et ω ∈ Ω 2 k ( S ) /k and ( u, v ) b e a we ak ( P , C ) -p air. Then, th er e exists an unique function h ∈ k ( S ) , such that ω = hdu ∧ dv and h has a L aur ent series exp ansion h = P j h j ( u ) v j . (1) The ( u, v ) - 1 -r esidue of ω along C in a ne i ghb orh o o d of P is deﬁne d by ( u, v ) r es 1 C,P ( ω ) := h − 1 ( ¯ u ) d ¯ u ∈ Ω 1 k ( C ) /k . (2) The ( u, v ) - 2 -r esidue of ω at P along C is deﬁne d by ( u, v ) r es 2 C,P ( ω ) := r es P (( u, v ) r es 1 C,P ( ω )) = h − 1 , − 1 ∈ k . Remark 3.2. Pr op osition 2.9 asserts that ( u, v ) r es 1 C,P ( ω ) is a r ational diﬀer en- tial form and not a formal one. This is the r e ason why we intr o duc e d this se c ond way of L aur ent serie s exp ansion. Remark 3.3. Obviously, if v al C ( ω ) ≥ − 1 , deﬁnition 3.1 c oincides with deﬁni- tions 1.3 and 1.4. RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 9 Remark 3.4. In this deﬁnition of one-c o dimensional r esidues, we sp e cify the p oint P . This 1 -form is supp ose d to give us informat ion ab out ω only in a neigh- b orho o d of P . However, we wil l se e in se c tion 4 .2 that this one-c o dimensional r esidue is actual ly a glob al obje ct on C , henc e indep endent on P . No w , we will pro v e the follo wing statemen ts. (1) One-co dimensional resid ues do not dep end on the c hoice of v . (2) Two-c o dimensional residues do not dep end on the c hoice of u and v . Caution. In wh at follo ws, we sometimes deal with formal d iﬀeren tial form s, that is ob jects of th e form f du ∧ dv , where f ∈ k (( u ) )(( v )). Using suc h a general p oin t of view is necessary in some parts of next pro ofs (for instance that of theorem 3.6 and prop osition 4.6). Deﬁnitions of one- and t wo- co dimensional residues extend naturally to formal forms. Lemma 3.5. The morp hism k (( u ))(( v )) → k (( x ))(( y )) g iven by a change of variables (CV) in lemma 2.10 is wel l-deﬁne d and sends series (r esp. formal forms) with ( v ) -adic valuation n ∈ Z o n series (r e sp. formal form s) with ( y ) - adic valuation n . Pr o of. See app endix A.  Theorem 3.6 (Inv ariance of 2-residues und er (CV)) . L et ω = h ( u, v ) du ∧ dv b e a form al 2 -form an d ( x, y ) ∈ k (( u ))(( v )) 2 c onne cte d with ( u, v ) by a change of variables of the form (CV). Then, ( u, v ) r es 2 C,P ( ω ) = ( x, y ) r es 2 C,P ( ω ) . The pro of of this prop osition will use forthcoming lemmas 3 .7 and 3.9. Fi rst, notice the c hange of co ord inates (CV) in lemma 2.10 can b e applied in t wo steps. First, from ( u, v ) to ( u, y ), then from ( u, y ) to ( x, y ). That is, ﬁrst (CV1)  u = u v = γ ( u, y ) , then (CV2)  u = f ( x, y ) y = y , where γ is a series in y k (( u ) )[[ y ]] r y 2 k (( u ))[ [ y ]] satisfying g ( x, y ) = γ ( f ( x, y ) , y ). W e will p r o ve successiv ely that 2-residues are inv arian t u nder (CV1) and (CV2). Lemma 3.7 (In v ariance of 1-residues under (CV1)) . L et ω b e a formal 2 -form. F or al l y linke d to ( u, v ) by a change of variables (CV1): v = γ ( u, y ) , we have ( u, v ) r es 1 C,P ( ω ) = ( u, y ) r es 1 C,P ( ω ) . Pr o of. The 2-form ω is of the form ω = hdu ∧ dv for some h ∈ k (( u ))(( v )). After applying (CV1), w e get ω = h ( u, γ ( u, y )) ∂ γ ∂ y du ∧ dy . The ﬁeld k (( u ))(( v )) is the ( v )-adic completion of the k (( u ))( v ) regarded a s a fu nction ﬁeld o v er k (( u )) . F rom [28] IV.2.9, the c o eﬃcien t of v − 1 in h ( u, v ) equals that of y − 1 in h ( u, γ ( u, y )) ∂ γ /∂ y .  Remark 3.8. Notic e th at in the whole chap ter IV of [ 28] , the b ase ﬁeld is as- sume d to b e p erfe ct, which is no t true for k (( u )) if Cha r ( k ) > 0 . However,the pr o of of IV . 2.9 is pur ely f ormal and holds for non-p erfe ct b ase ﬁelds. Op eration (CV2) might c h ange 1-residues. Nev ertheless, we will see that it pr e- serv es 2-residues. 10 ALAIN COUVREUR Lemma 3.9. L et ω b e a forma l 2 -form, ω = h ( u, v ) du ∧ dv with h ∈ k (( u ))(( v )) such that v al ( y ) ( h ) ≥ − 1 . Then, for any p air ( x, y ) ∈ k (( u ))(( v )) 2 r elate d to ( u, v ) by a change of variables (CV) of lemma 2.10, we have ( u, v ) r es 1 C,P ( ω ) = ( x, y ) r es 1 C,P ( ω ) . Remark 3.10. Notic e that the pr o of of pr op osition 1.1 is a dir e ct c onse quenc e of lemma 3.9. Pr o of. F rom lemma 3.7, ( u, v )res 1 C,P ( ω ) = ( u, y )res 1 C,P ( ω ). Thus, w e only study the b ehavi or of residues un der (C V2). Decomp ose ω b y isolating its degree − 1 term, ω = h − 1 ( u ) y du ∧ dy +   X j ≥ 0 h j ( u ) y j   du ∧ dy = ω − 1 + ω + . The formal form ω + has p ositiv e ( y )-adic v aluation. F rom lemma 3.5, th e c hange of v ariables (CV2) do es not c hange this v aluation. Cons equen tly , the ( x, y )-1- residue of ω is that of ω − 1 and after applying (CV2), we ha ve ω − 1 = h − 1 ( f ( x, y )) y ∂ f ∂ x dx ∧ dy . F rom lemma 3.5, h − 1 ( f ( x, y )) h as ( y )-a dic v aluation zero. Th us, ( x, y )res 1 C,P ( ω ) = h − 1 ( f 0 ( ¯ x )) f ′ 0 ( ¯ x ) d ¯ x = h − 1 ( f 0 ( ¯ x )) d ( f 0 ( ¯ x )) , where f 0 ( x ) := f ( x, 0). This formal 1-form equals ( u, y )res 1 C,P ( ω ) = h − 1 ( ¯ u ) d ¯ u , using the c hange of v ariables ¯ u = f ( ¯ x, 0).  F or the pro of of theorem 3.6, w e need also the follo wing lemma. Lemma 3.11. L et A, B ∈ k (( u ))(( v )) , then for al l p air of series ( x, y ) asso ciate d with ( u, v ) by a change of variables (CV), we have ( x, y ) r es 2 C,P ( dA ∧ dB ) = 0 . Pr o of. See app endix B.  Pr o of of the or em 3.6 if Char ( k ) = 0 . F r om lemma 3.7, w e allready kno w that 1- residues are inv ariant u n der (CV1). Th us, w e will only stu dy their b eha vior under (CV2). Consider an y f orm al 2-form ω = − 2 X j = − l h j ( u ) y j du ∧ dy + X j ≥− 1 h j ( u ) y j du ∧ dy = ω − + ω inv . F rom lemma 3.9, the formal form ω inv has a n in v ariant 1-residue under (CV ), th us so is its 2-residue. W e now h a ve to stu d y ω − . Since extractio n of ( x, y )-1- and -2-residues are k -linear operations, w e may only consid er 2-forms of the form ω = φ ( u ) du ∧ dy y n with φ ∈ k (( u )) and n ≥ 2 . The formal 2-form ω has a zero ( u, y )-2-residue b ecause its ( u, y )-1-residue is also z ero. Th en, w e ha ve to pro ve that its ( x, y )-2-residue is zero to o. Before applying (CV2), w e will w ork a little bit m ore on ω . First, isol ate the term in u − 1 of the Laur en t series φ . φ ( u ) = e φ ( u ) + φ − 1 u , where e φ i =  φ i if i 6 = − 1 0 if i = − 1 . RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 11 The ser ies e φ has a formal p rimitiv e e Φ. Set s := 1 (1 − n ) y n − 1 , whic h is a p rimitiv e of 1 /y n (it mak es sen se b ecause Char( k ) is assumed to b e zero). Then, we ha v e ω = d e Φ ∧ ds + φ − 1 du u ∧ ds = ω r + φ − 1 ω − 1 . F rom lemma 3.11 , the form ω r has a zero 2-residue for all pair ( x, y ) ∈ k (( u ))(( v )) 2 connected to ( u, v ) b y a c hange of v ariables (C V ). Now consid er ω − 1 = du u ∧ dy y n and apply (C V2), ω − 1 = d f ( x, y ) f ( x, y ) ∧ dy y n . Recall that f is of the form P j ≥ 0 f j ( x ) y j with f 0 ( x ) = f 1 , 0 x + f 2 , 0 x 2 + · · · and f 1 , 0 6 = 0 . Th us, one can factorize f 0 in f 0 ( x ) = f 1 , 0 x  1 + f 2 , 0 f 1 , 0 x + · · ·  . Set r ( x ) := f 2 , 0 f 1 , 0 x + f 3 , 0 f 1 , 0 x 2 + · · · ∈ k [[ x ]] and µ ( x, y ) := f 1 ( x ) f 0 ( x ) y + f 2 ( x ) f 0 ( x ) y 2 + · · · ∈ k (( x )) [[ y ]] . The series f has the follo wing factoriza tion (3) f ( x, y ) = f 1 , 0 x (1 + r ( x ))(1 + µ ( x, y )) . Moreo ver, for eve ry series S in xk [[ x ]] (resp. in y k (( x ))[[ y ]]) w e deﬁne the formal logarithm of 1 + S to b e log(1 + S ) := + ∞ X k =0 ( − 1) k +1 S k k . This m ak es sense b ecause Char( k ) = 0 and this series con v erges for the ( x )- adic (r esp. ( y )-a dic) v aluation. F urthermore, d (1+ S ) (1+ S ) = d log (1 + S ). Using factorizat ion (3), w e obtain ω − 1 = dx x ∧ dy y n + d log (1 + r ) ∧ ds + d log (1 + µ ) ∧ ds. F rom lemma 3.11, second and third term of the sum ha v e zero ( x, y )-2-residues, and the ﬁrst one has zero ( x, y )-1-residue, hence a zero ( x, y )-2-residue.  Pr o of of the or em 3.6 in p ositive char acteristic. Th e idea is basica lly the same as in the p roof of in v ariance of residues of 1-forms on cur v es (c.f. [28] IV.2.9 or [24] prop II.7.5). One p ro ves that the ( x, y )-2-residue of ω is a p olynomial expression in a ﬁn ite family of co eﬃcien ts of f . This p olynomial has in teger coeﬃcien ts and dep ends neither o n f nor on the base ﬁeld k . Thus, usin g the result of the p r oof in c haracteristic zero and the p rinciple of pr olo ngation of algebraic iden tities ([3] prop IV.3.9), w e conclude th at th is p olynomial is zero. F or more d etails see app endix C.  Consequent ly , from now on, when we deal with 2-residues at P along C , we w on’t hav e to precise th e ( P, C )-pair. 12 ALAIN COUVREUR 4. Pr oper ties of res idues Con text. In this s ection, k is an algebraically closed ﬁeld and S a smo oth geometrica lly in tegral quasi-pro jectiv e surface o ve r k . Moreo v er, C denotes an irreducible absolutely reduced curve em b edd ed in S and P a p oint of C . 4.1. Ab out 2 -residues. Next lemma giv es a n ece ssary condition on ω to hav e nonzero 2-residues at P along C . Lemma 4.1. L et ω ∈ Ω 2 k ( S ) /k having the cu rve C as a p ole. L et P ∈ C such that C is the only one p ole of ω in a neighb orho o d of P . Then, r es 2 C,P ( ω ) = 0 . Pr o of. Let ( u, v ) b e a strong ( P , C )-pair and n := − v al C ( ω ). There exists a function h ∈ O S,C suc h that ω = hdu ∧ dv v n . F urtherm ore, since ω h as no p ole but C in a neigh b orho o d of P , the fu n ctio n h is in O S,P . Consequent ly , h has a T a ylor expansion P i ≥ 0 h i ( u ) v i , where h i ∈ k [[ u ]] for all i . T hen, ( u, v ) r es 1 C ( ω ) = h n − 1 ( ¯ u ) d ¯ u , w hic h is regular at P , hence has zero residue at th is p oin t.  4.2. Ab out 1 -residues. W e will give a n ew deﬁnition for one-co dimensional residues generalizing the pr evious one. The goal is, as said in remark 3.4, to deﬁne 1-residues as global ob jects on the curv e C . Prop osition 4.2 . L et u, v b e e lements of O S,C such that ¯ u is a sep ar ating ele- ment 1 of k ( C ) /k and v is a uniformizing p ar ameter of O S,C . Then, any 2 -form ω ∈ Ω 2 k ( S ) /k c an b e exp ande d as (4) ω = X j ≥− l f j v j du ∧ dv , wher e f j ’s ar e elements of the Hensel lift K u of k ( C ) over k ( u ) in b O S,C (se e pr op osition 2.3). F urthermo r e, the 1 -f orm ¯ f − 1 d ¯ u is r ational on C and do es not dep end on the choic e of the unif ormizing p ar ameter v of O S,C . Pr o of. Recall that, fr om [25] thm I I I.5.4.3 , the s p ace Ω 2 k ( S ) /k has d imension one o ver k ( S ). Thus, there exists a unique function f ∈ k ( S ) suc h that ω = f du ∧ dv . F rom corollary 2.4, one can exp and f in K u (( v )), whic h giv es expansion (4 ). F rom the constru ctio n of K u (see prop ositi on 2.3), ¯ f − 1 ma y b e identi ﬁed to a rational function on C . Th us, the 1-form ¯ f − 1 d ¯ u is rational on C . T o pro v e its indep endence on the choice of v , the reasoning is exactly the same as in the pro of of lemma 3.7.  Deﬁnition 4.3. Under the assumpt ions of pr op osition 4.2, we c al l ( u ) - 1 -r esidue of ω along C and denote b y ( u ) r es 1 C ( ω ) the r ational 1 -form ( u ) r es 1 C ( ω ) := ¯ f − 1 d ¯ u ∈ Ω 1 k ( C ) /k . Remark 4.4. Using lemma 3.9, one c an pr ove that if v al C ( ω ) ≥ − 1 , then this 1 -form is also indep endent on the choic e of u . 1 See [28] p. 127 for a deﬁn ition. RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 13 Remark 4.5. L et ω ∈ Ω 2 k ( S ) /k and u ∈ b O S,C such that ¯ u is a sep ar ating element of k ( C ) /k . Set µ := ( u ) r es 1 C ( ω ) ∈ Ω 1 k ( C ) /k . Then, at e ach p oint P of C wher e ¯ u is a lo c al p ar ameter, we have ( ♠ ) ( u, v ) r es 1 C,P ( ω ) = µ and r es 2 C,P ( ω ) = r es P ( µ ) . This remark asserts that deﬁnition 4.3 generalizes th e notion one-co dimensional residue (d eﬁnition 3.1). Next prop ositio n extends ( ♠ ) to an y smo oth p oin t of C . Prop osition 4.6. L et ω ∈ Ω 2 k ( S ) /k and u ∈ O S,C such that ¯ u is a sep ar ating element of k ( C ) /k , then at e ach smo oth p oint Q of C , we have r es 2 C,Q ( ω ) = r es Q  ( u ) r es 1 C ( ω )  . Remark 4.7. In se ction 5, we gener alize the deﬁnition of 2 -r esidue at P along C when C may b e singular at P . Using this deﬁnition, the assumption “ C is smo oth at Q ” in pr op osition 4.6 c an b e c anc el le d (se e r emark 5.3). Pr o of of pr op osition 4.6. Set µ := ( u )res 1 C ( ω ) = ¯ f − 1 d ¯ u . Step 1. Let Q ∈ C at wh ic h ¯ u is regular and ( u − ¯ u ( Q ) , v ) is a w eak ( Q, C )-pair. Set u 0 := u − ¯ u ( Q ). The function ¯ u is a local parameter of O C,Q . Moreo v er, K u = K u 0 and du = du 0 . Consequen tly , ( u 0 , v )res 1 C,Q ( ω ) = ¯ f − 1 d ¯ u 0 = µ and res 2 C,Q ( ω ) = res Q ( µ ). Step 2. Let Q ∈ C at wh ic h ¯ u is regular but ¯ u − ¯ u ( Q ) is not a lo cal parameter of O C,Q . Set u 0 := u − ¯ u ( Q ). W e ha ve ω = P j f j v j du 0 ∧ dv , bu t ( u 0 , v ) is not a we ak ( Q, C )-pair. Let ( x, v ) b e a we ak ( Q, C )-pair. T h e function ¯ x is a local parameter of O C,Q and for some φ ∈ k [[ T ]], w e hav e ¯ u 0 = φ ( ¯ x ) in k ( C ) . Let σ b e the Hensel-lift of ¯ x in K u , last r ela tion lifts in K u and giv es u 0 = φ ( σ ). Consequent ly , we get a new formal expr ession for ω , ( ♣ ) ω = X j ≥− l f j v j φ ′ ( σ ) dσ ∧ dv . Notice that σ ∈ b O S,C and is congruen t to x mo d ulo ( v ). Therefore, σ expands in k (( x ))[[ v ]] as σ = x + σ 1 ( x ) v + σ 2 ( x ) v 2 + · · · Th us, the pair ( σ, v ) is asso ciate d with ( x, v ) b y a change of v ariables (CV). Using ( ♣ ) and theorem 3.6, we conclude that res 2 C,Q ( ω ) = res P  ¯ f − 1 φ ′ ( σ ) dσ  = res P  ¯ f − 1 φ ′ ( ¯ x ) d ¯ x  = res P ( µ ) . Step 3. Let Q ∈ C at wh ic h ¯ u is n ot regular. Set t := 1 /u and n otic e that u = 1 / t ⇒ k ( u ) = k ( t ) ⇒ K u = K t . Th us, exp an s ion of ω is of the form ω = X j ≥− l f j v j  − dt t 2  ∧ dv , for some v and ( t )res 1 C ( ω ) = − ¯ f − 1 d ¯ t ¯ t 2 = µ. 14 ALAIN COUVREUR Applying the argumen ts of the previous steps, we conclude the p r oof.  Summary . (1) A 2-residue depend s only o n a c urve and a p oint. Consequently , from no w on, w e will deal w ith res 2 C,P and not ( u, v )res 2 C,P (deﬁnition 3.1). (2) A 1-residue dep ends only on the cur v e and the c hoice of s ome ele ment u of b O S,C , whose r estricti on to C is a separating elemen t of k ( C ) /k . Mo reo v er this ob ject giv es a global information on C and in a neighborh o o d of a p oin t. F rom no w on, we will deal with ( u )res 1 C (deﬁnition 4.3) and not with ( u, v )res 1 C,P (deﬁnition 3.1). W e will also kee p usin g map res 1 C for 2-forms having m S,C -adic v aluation greater than or equal to − 1. Corollary 4.8. L et u b e a function in O S,C whose r estriction ¯ u to C is a sep a- r ating element of k ( C ) /k . L et π : e S → S b e the blowup of S at P and e C b e the strict tr ansform of C by π . Then, ( π ∗ u ) r es 1 e C ( π ∗ ω ) = π ∗ | e C  ( u ) r es 1 C ( ω )  . Pr o of. Sur faces e S r E and S r { P } are isomorphic under π . F urthermore, recall that P is assu med to b e a smo oth p oin t of C , th us π ind uces an isomorph ism b et ween e C and C . The 1-forms ( π ∗ u )res 1 e C ( π ∗ ω ) and ( u )res 1 C ( ω ) are pullback of eac h other b y π | e C and its inv erse.  Corollary 4.9. L et ( u, v ) b e a we ak ( P , C ) - p air and π : e S → S b e the blowup of S at P . Denote by e C the strict tr ansform of C by π and by Q the interse ction p oint b etwe en e C and the exc e ptio nal divisor. Then, r es 2 e C ,Q ( π ∗ ω ) = r e s 2 C,P ( ω ) . 5. Residues alo n g a singula r cur ve Con text. The cont ext of th is sectio n is that of sections 1, 2 and 3 with only one diﬀerence, the curv e C ma y b e singular at P . Prop osition 5.1. L et π : e S → S b e a morph ism obtaine d by a ﬁnite se quenc e of blowups of S and such that the strict tr ansform e C of C by π is a desingularization of C at P . Then, the sum X Q ∈ π − 1 ( { P } ) r es 2 e C ,Q ( π ∗ ω ) do es not dep end on the choic e of the desingu lar ization π : e S → S . Pr o of. Let π 1 : e S 1 → S and π 2 : e S 2 → S b e t wo morp h isms as in the w ording of the prop osition. Denote by e C 1 and e C 2 the resp ectiv e strict transform s of C b y th ese t wo morp hisms. Since b oth maps π 1 and π 2 induce desingularizations of C at P , th e p oin t P h as the same n umber of preimages by π 1 and by π 2 . These pr eimag es resp ectiv ely d enote d b y P 1 , 1 , . . . , P 1 ,n and P 2 , 1 , . . . , P 2 ,n . By construction of π 1 and π 2 , there exists an op en set U 1 ⊆ e C 1 (resp. U 2 ⊆ e C 2 ) con taining P 1 , 1 , . . . , P 1 ,n (resp. P 2 , 1 , . . . , P 2 ,n ) and an isomorphism ϕ : U 1 → U 2 suc h that π 1 | U 1 = π 2 | U 2 ◦ ϕ . Moreo v er, for a suitable ordering of indexes, ϕ sends P 1 ,i on P 2 ,i for all i . RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 15 Let u b e an elemen t of O S,C whose restriction to C is a separating elemen t of k ( C ) /k . F r om corollary 4.8, the 1-forms ( π ∗ 1 u )res 1 e C 1 ( π ∗ 1 ω ) and ( π ∗ 2 u )res 1 e C 2 ( π ∗ 2 ω ) are pu llb ac k of eac h other by ϕ and ϕ − 1 . Consequently , ∀ i ∈ { 1 , . . . , n } , res 2 e C 1 ,P 1 ,i ( π ∗ 1 ω ) = res 2 e C 2 ,P 2 ,i ( π ∗ 2 ω ) . W e conclude by adding last equalities for all i .  Deﬁnition 5.2. U nder the assumptions of pr op osition 5.1 the 2 -r esidue of a 2 -form ω ∈ Ω 2 k ( S ) /k at P along C is deﬁne d by r es 2 C,P ( ω ) = X Q ∈ π − 1 ( { P } ) r es 2 e C ,Q ( π ∗ ω ) . Remark 5.3. As said in r emark 4.7, using deﬁnition 5.2 the statement of pr op o- sition 4.6 ho lds f or singular p oints of C . T o pr ove this, apply the same ar guments as in the pr o of of pr op osition 4.6 on a surfac e e S such that ther e exists a map π : e S → S inducing a normalization of C . 6. Residue formulas W e lo ok for an analogous deﬁnition of the residue form u la on curv es ([24] lem I I.12.3 or [28 ] IV 3.3) in the t wo -dimensional case. W e will giv e three statemen ts ab out summations of 2-residues. Con text. In this s ection, k is an algebraically closed ﬁeld and S a smo oth geometrica lly integral pro jectiv e surface o v er k . Theorem 6.1 (First Residue form ula) . L et C b e a r e duc e d irr e ducible pr oje ctive curve emb e dde d in S . Then, ∀ ω ∈ Ω 2 k ( S ) /k , X P ∈ C r es 2 C,P ( ω ) = 0 . Pr o of. Let u b e an elemen t of O S,C whose restriction ¯ u to C is a separating elemen t of k ( C ). If C is smooth, then apply p rop ositio n 4 .6 and th e classical residue form u la on curves to ( u )res 1 C ( ω ). Else, use d eﬁnition 5.2 and apply the same arguments to a morp hism φ : e S → S inducing a n ormaliza tion of C .  Remark 6.2. If v al C ( ω ) ≥ − 1 , then fr om pr op osition 1.1 and deﬁnition 1.3, the 2 -form ω has a 1 -r e sidue along C denote d by r es 1 C ( ω ) . Thus, in th is p articular situation, last the or em is an e asy c onse quenc e of the classic al r esidue formula on curves applie d to the 1 -form r es 1 C ( ω ) . The nonobvious p art of this pr op osition is that the statement holds even if v al C ( ω ) < − 1 . Theorem 6.3 (Second residue F orm ula) . L et C S,P b e the set of germs of irr e- ducible r e duc e d curves emb e dde d in S and c ontaining P . Then, ∀ ω ∈ Ω 2 k ( S ) /k , X C ∈C S,P r es 2 C,P ( ω ) = 0 . Remark 6.4. Notic e that this sum i s actual ly ﬁnite b e c ause almost al l C ∈ C S,P is not a p ole of ω thus the 2 -r esidue at P along this curve is zer o. Pr o of. Let ω ∈ Ω 2 k ( S ) /k and C 1 , . . . , C n ∈ C S,P b e th e set of its p oles in a n eig h- b orho o d of P . W e will prov e the theorem by induction on n . 16 ALAIN COUVREUR Step 1. Assume that, for eac h pair of curv es C i , C j with i 6 = j , their intersecti on m ultiplicit y at P is one. If n = 1. F rom lemma 4.1 the su m is ob viously zero. If n = 2 . Let u 1 , u 2 b e resp ectiv ely local equations of C 1 and C 2 . Then, ( u 1 , u 2 ) is a strong ( P , C 2 )-pair and ( u 2 , u 1 ) a strong ( P , C 1 )-pair, b eca use C 1 and C 2 are assumed t o ha v e a normal crossing at P . Thus, f or some h ∈ O S,P and some p ositiv e in tegers n 1 and n 2 , we ha v e ω = h du 1 u n 1 1 ∧ du 2 u n 2 2 . Expand h in T a ylor series h = P h ij u i 1 u j 2 . Usin g the an ticomm utativit y of the ex- ternal pro duct, a br ief computation giv es res 2 C 2 ,P ( ω ) = − res 2 C 1 ,P ( ω ) = h n 1 − 1 ,n 2 − 1 . If n ≥ 2. Consider π : e S → S the blo wup of S at P . Denote b y E the exceptional divisor, by e C i the strict transform of C i and b y Q i the in tersection p oin t b et wee n E and e C i . P oin ts Q i ’s are all distinct and curv es E and e C i ha v e normal crossing at Q i . Th e curve E is pro jective and the e C i ’s are the only p oles of π ∗ ω wh ic h cross E . F urtherm ore, the p r evious case en tails res 2 e C i ,Q i ( π ∗ ω ) = − res 2 E ,Q i ( π ∗ ω ) for all i . Consequen tly , from corollary 4.9, w e ha v e n X i =1 res 2 C i ,P ( ω ) = n X i =1 res 2 f C i ,Q i ( π ∗ ω ) = − n X i =1 res 2 E ,Q i ( π ∗ ω ) and last su m is zero from theorem 6.1. Step 2. In the general case, a curve C i migh t b e sin gu lar at P or inte rsect the other C j ’s with higher multiplicit y . After a ﬁnite num b er of blo wups, using deﬁnition 5.2 and app lyin g same argumen ts to t he resolution tree, w e get t he exp ected result.  Remark 6.5. Notic e that the valuation of π ∗ ω al ong the exc e ptio nal divisor E is not always g r e ater than or e qual to − 1 . This valuation is given by the formula v al E ( π ∗ ω ) = 1 + X C ∈C S,P v al C ( ω ) , wher e the set C S,P is that of the or em 6.3. F or a pr o of of this form ula se e [15] pr op V .3.3 and V.3.6. Ther efor e, the or em 6.1 is ne c essary to c onclude in the ﬁrst step of last pr o of. T o state the third residue form u la, we need to extend the deﬁnition of 2-residues at a p oint along a curve to 2-residues at a p oin t along a divisor. Deﬁnition 6.6. L et D = n 1 C 1 + · · · + n p C p b e a divisor on S and ω ∈ Ω 2 k ( S ) /k . We deﬁne the 2 -r esidue of ω at P to b e r es 2 D ,P ( ω ) := n X i =1 r es 2 C i ,P ( ω ) . Remark 6.7. N otic e that c o eﬃcients n i ’s of D ar e not involve d in this deﬁnition. A ctual ly, r es 2 D ,P dep ends only on the supp ort of the D . The most lo gic notation would have b e en “ r e s 2 Supp ( D ) ” which is to o he avy. RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 17 Theorem 6.8 (Third residue form ula) . L et D a , D b b e two divisors such that the set Supp ( D a ) ∩ Supp ( D b ) is ﬁnite. L et ∆ b e the zer o-cycle given by the scheme- the or etic interse ction ∆ := D a ∩ D b . Set D := D a + D b . Then, ∀ ω ∈ Ω 2 ( − D ) , X P ∈ Supp ∆ r es 2 D a ,P ( ω ) = X P ∈ S r es 2 D a ,P ( ω ) = 0 . Pr o of. F rom theorem 6.1 , fo r e ac h irreducible comp onent C i of the supp ort of D a , w e h a v e P P ∈ C i res 2 C i ,P ( ω ) = 0. F rom theorem 6.3, if a p oin t P is out of the supp ort of ∆, then res 2 D a ,P ( ω ) = 0. A combinatio n of b oth claims conclud es the pro of.  Remark 6.9. F r om the or em 6.3 and under the assumptio ns of the or em 6.8, for al l P in S , we have r es 2 D a ,P ( ω ) = − r es 2 D b ,P ( ω ) . Conse quently, the statement in the or em 6.8 hold s r eplacing D a by D b . P a rt 2. Application to co ding theory Not a tions Let X b e a v ariety deﬁned ov er ﬁ nite a ﬁ eld k , we denote by Div k ( X ) the group of rational W eil divisors on X . That is, the free ab elian group spanned by irreducible one-codimensional closed subv arietie s of X . If G ∈ Div k ( X ), then we use the follo wing notations. (1) G + denotes the eﬀectiv e part of G . (2) L ( G ) denotes the Riemann-Ro c h s p ace of rational fun ctio ns L ( G ) := { f ∈ k ( X ) , ( f ) + G ≥ 0 } ∪ { 0 } . (3) Ω 2 ( G ) den otes the Riemann-Ro c h space of rational 2-forms Ω 2 ( G ) := n ω ∈ Ω 2 k ( X ) /k , ( ω ) − G ≥ 0 o ∪ { 0 } . (4) If G ′ ∈ Div k ( X ) such that t he supp orts of G and G ′ ha v e n o common comp onen t in a neigh b orh oo d of P , we denote b y m P ( G, G ′ ) ∈ Z the in tersection multiplicit y of these divisors at P . 7. About cod es from cur ve s, cl assical const r uctions In this sectio n C is a smo oth pro jectiv e absolutely irreducible curv e o ver a ﬁnite ﬁeld F q . Let G b e a rational divisor on C and P 1 , . . . , P n b e a family of rational p oin ts of C a v oiding the supp ort of G . Set D := P 1 + · · · + P n ∈ Div F q ( C ) and ev D :  L ( G ) → F n q f 7→ ( f ( P i )) i =1 ...n , res D :  Ω 1 ( G − D ) → F n q ω 7→ (res P i ( ω )) i =1 ...n . W e deﬁne the co des C L ( D , G ) := Im(ev D ) and C Ω ( D , G ) := Im(res D ) called resp ectiv ely functional co de and diﬀeren tial co de. Both constructions are link ed b y the follo wing pr operties. (OR): C Ω ( D , G ) = C L ( D , G ) ⊥ . (LΩ): F or some canonical divisor K , we ha ve C Ω ( D , G ) = C L ( D , K − G + D ). 18 ALAIN COUVREUR See [27], [28] or [29] for the pro ofs of th ese statemen ts. Relat ion (OR) is a consequence of the residu e formula for inclusion “ ⊆ ” and of Riemann -Roc h’s theorem for the rev erse one. This r ela tion is used in almost all algebraic deco ding algorithms (see [17]). Rel ation (LΩ) is a consequence of the w eak appro ximation theorem ([28] thm I.3 .1). It allo ws to restrict the stud y of alge braic-geometric co des to only one class, for example functional co des which seems to b e easier to study . The goal of this second part is to extend s ome of these statement s to surfaces. 8. Algebraic-geometr ic c odes on sur f aces F rom no w on, S denotes a smo oth geometrica lly in tegral pro jectiv e sur face ov er F q and S := S × F q F q . Moreo ve r, G den otes a rational divisor on S and P 1 , . . . , P n a s et of r atio nal p oin ts of S av oiding the s upp ort of G . S et ∆ := P 1 + · · · + P n . Notice that ∆ is not a divisor bu t a 0-c ycle. Most of the diﬃculties w e will meet come fr om this diﬀerence of dimens ion b et we en G and ∆. 8.1. F unctional codes. As said in the in tro duction, the functional constr u ctio n of co des extends to higher-dimensional v arieties (see [31] I.3.1). Deﬁn e the map ev ∆ :  L ( G ) → F n q f → ( f ( P 1 ) , . . . , f ( P n )) . The fun cti onal co de is C L (∆ , G ) := Im( ev ∆ ). The stud y of su c h co des is really more complicated th an that of co des on cur v es. P articularly , ﬁ nding a min oratio n of the minimal distance b ecomes a v ery diﬃcult problem. F or more details about this topic, see references cited in in tro duction. 8.2. Diﬀeren tial co des. T o deﬁn e diﬀerential co des, w e need more than G and ∆. W e w an t to ev aluate 2-residues of some rational diﬀeren tial forms with prescrib ed p oles. Unfortunately , 2-residu es dep end not only on a p oin t bu t on a ﬂag P ∈ C ⊂ S . Th us, we ha ve to inp ut another divisor. Deﬁnition 8.1. L et D ∈ Di v F q ( S ) and assume that D is the sum of two divisors D a , D b whose supp orts ha ve no c ommon irr e ducible c omp onent. Then, one c an deﬁne the map r es 2 D a , ∆ :  Ω 2 ( G − D ) → F n q ω 7→ ( r es 2 D a ,P 1 ( ω ) , . . . , r es 2 D a ,P n ( ω )) . The diﬀer ential c o de is deﬁne d by C Ω (∆ , D a , D b , G ) := Im ( r es 2 D a , ∆ ) . Remark 8.2. We c an also deﬁne a map r es 2 D b , ∆ , but fr om the or em 6.3, we have r es 2 D b , ∆ = − r es 2 D a , ∆ . Thus, b oth maps have the same image and C Ω (∆ , D a , D b , G ) = C Ω (∆ , D b , D a , G ) . 8.3. ∆ -con v enience. Actually , if one c ho oses an arbitrary d ivisor D , last de- ﬁnition is not v ery con venien t. Recall that, from lemma 4 .1 and theorem 6.3, res 2 D a ,P i ( ω ) is nonzero o nly if the supp orts of D + a and D + b in tersect at P i . There- fore, if w e wa nt to hav e a co de which is linke d to the fu nctional cod e C L (∆ , G ), the divisor D m ust b e itself r elate d to the 0-cycle ∆. W e will ﬁ rst deﬁne the notion of ∆-con v enien t pair of divisors. Afterwards, w e will give a criterion o f ∆-con v enience. Although this one ma y lo ok ugly , it is actually easy to handle. RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 19 Let D a , D b b e a p air of F q -rational divisors on S whose sup p orts ha ve no common comp onen t and s et D := D a + D b . F r om now on, F denotes the sheaf on S deﬁn ed b y F ( U ) = n ω ∈ Ω 2 F q ( S ) / F q , ( ω | U ) ≥ − D | U o . Moreo ver, for all p oint P ∈ S , the stalk of F at P is denoted by F P (see [15] II.1 p. 62 f or deﬁnition of “stalk”). Notice that H 0 ( S , F ) = Ω 2 ( − D ) ⊗ F q F q . Deﬁnition 8.3. The p air ( D a , D b ) i s said to b e ∆ -c onvenient if it satisﬁes the fol lowing c onditions. ( i ) Supp orts of D a and D b have no c ommon irr e ducible c omp onents. ( ii ) F or al l p oint P ∈ S , the map r es 2 D a ,P : F P → F q is O S ,P -line ar. ( iii ) This map is su rje ctive for al l P ∈ Supp (∆) and zer o elsewher e. Remark 8.4. The structur e of O S ,P -mo dule of F q is induc e d by the morphism f → f ( P ) . Thus, if the map r es 2 D a ,P satisﬁes ( ii ) , then it vanishes on m S ,P F P . Remark 8.5. L et P ∈ S and ω ∈ F P . F r om r emark 6.9, we have r es 2 D a ,P ( ω ) = − r es 2 D b ,P ( ω ) . Conse quently, if at a p oint P ∈ S , the map r es 2 D b ,P is O S,P -line ar and satisﬁes ( ii ) and ( iii ) , so do es r es 2 D a ,P . Prop osition 8.6 (Cr iterio n for ∆-con v enience) . L e t ( D a , D b ) b e a p air of F q - r ational divisors having no c ommon irr e ducible c omp onent on S and set D := D a + D b . If ( D a , D b ) satisﬁes the fol lowing c onditions, then it is a ∆ -c onvenient p air. (1) F or e ach P ∈ Supp (∆) , ther e exists an irr e ducible curve C smo oth at P such that in a neighb orho o d U P of P , either ( D + a ) | U P or ( D + b ) | U P e quals C | U P and m P ( C, D − C ) = 1 . (2) F or e ach P ∈ S r Supp (∆) , either D ∗ = D a or D ∗ = D b satisﬁes the fol lowing c onditions. F or e ach F q -irr e ducible c omp onent C of D + ∗ c on- taining P , (a) the cu rve C is smo oth at P , (b) this cu rve C app e ars in D ∗ with c o eﬃcient one, (c) m P ( C , D − C ) ≤ 0 . Remark 8.7. In c ondition (2) of this criterion, the divisor D ∗ may b e zer o in a neighb orho o d of P (actual ly that is what happ e ns at almost al l p oint P ). In this situation, c onditions (2a), (2b ) and (2c) ar e obviously satisﬁe d. F or the pro of of this prop osition, w e need next lemma and its corollary . Lemma 8.8 . L et C b e an irr e ducible curve over F q emb e dde d in S and P b e a smo oth p oint of C . L et ω ∈ Ω 2 F q ( S ) / F q having a simple p ole along C . Then v al P ( r es 1 C ( ω )) = m P ( C , ( ω ) + C ) , wher e v al P denotes the valuation at the p oint P on Ω 1 F q ( C ) / F q . 20 ALAIN COUVREUR Pr o of. Let ϕ , ψ and v b e resp ectiv e local equations of  ( ω ) + C  + ,  ( ω ) + C  − and C in a neigh b orho o d of P . Let u ∈ F q ( S ) suc h that ( u, v ) is a strong ( P , C )-pair, th en for some h ∈ O × S ,P , we ha v e ω = h ϕ ψ du ∧ dv v . Th us, res 1 C ( ω ) = ¯ h ¯ ϕ ¯ ψ − 1 d ¯ u and since ¯ h ∈ O × C ,P , w e ha ve v al P ( ¯ hd ¯ u ) = 0. Conse- quen tly , v al P (res 1 C ( ω )) = v al P ( ¯ ϕ ) − v al P ( ¯ ψ ) . F urtherm ore, m P ( C , ( ω ) + C ) = m P ( C , (( ω ) + C ) + ) − m P ( C , (( ω ) + C ) − ) = dim F q O S ,P / ( ϕ, v ) − dim F q O S ,P / ( ψ , v ) = dim F q O C ,P / ( ¯ ϕ ) − dim F q O C ,P / ( ¯ ψ ) = v al P ( ¯ ϕ ) − v al P ( ¯ ψ ) .  Corollary 8.9. L et C b e an irr e ducible curve emb e dde d in S and P b e a smo oth p oint of C . L et ω ∈ Ω 2 F q ( S ) / F q such that v al C ( ω ) ≥ − 1 and m P ( C , ( ω ) + C ) ≥ − 1 . Then, ∀ f ∈ O S ,P , r es 2 C ,P ( f ω ) = f ( P ) r es 2 C ,P ( ω ) . Pr o of. Let ( u, v ) b e a strong ( P , C )-pair and f b e an elemen t of O S ,P . Since v al C ( ω ) ≥ − 1, there exists ψ ∈ O S ,C suc h that ω = ψ du ∧ dv v . Set µ := res 1 C ( ω ) = ¯ ψ d ¯ u . T he cond itio n v al C ( ω ) ≥ − 1 en tails also res 1 C ( f ω ) = ¯ f ¯ ψ d ¯ u = ¯ f µ. F rom lemma 8.8, we ha v e v al P ( µ ) = m P ( C , ( ω ) + C ) ≥ − 1. Thus, res 2 C ,P ( f ω ) = res P ( ¯ f µ ) = ¯ f ( P )res P ( µ ) = f ( P )res 2 C ,P ( ω ) .  Pr o of of pr op osition 8.6. Let ( D a , D b ) b e a pair of divisors s ati sfying conditions of prop osition 8.6. Cond itio n ( i ) of deﬁn itio n 8.3 is ob viously satisﬁed, b ecause supp orts of D a and D b are assumed to ha v e no common irreducible compon ent. No w , we pro ve th at ( ii ) and ( iii ) are satisﬁed. First, recall that F denotes the sheaf of r ati onal 2-forms ω on S satisfying lo cally ( ω ) ≥ − D = − D a − D b . Condition ( ii ) . Let P ∈ Supp(∆) and ω ∈ F P , where F P denotes th e stalk of the s h eaf F at P . F rom (1), there is an irredu cible curve C , s m ooth at P suc h th at either D + a or D + b equals C in a neigh b orho od of P . Using remark 8.5, we ma y assume that D + a = C without loss of generalit y . Thus, C is the only one irredu cible comp onen t of Sup p ( D + a ) in a neigh b orho o d of P . Therefore, res 2 D a ,P ( ω ) = res 2 C,P ( ω ). Consequently , v al C ( ω ) ≥ − 1 and corollary 8.9 asserts that res 2 C,P (hence res 2 D a ,P ) is O S,P -linear. RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 21 Condition ( iii ) . Let P ∈ S b e a p oin t out of the supp ort of ∆. F r om remark 8.5, we ma y assu me without loss of generalit y that condition (2) in prop osition 8.6 is satisﬁed b y D a (i.e. D ∗ = D a at P ). Let C b e an F q -irreducible comp onen t of Supp( D + a ) an d ω ∈ F P . F rom (2b), v al C ( ω ) ≥ − 1 and fr om lemma 8.8 and (2c), w e ha ve v al P (res 1 C ( ω )) ≥ 0. Consequent ly , r es 2 C ,P ( ω ) = 0, which concludes the pr oof.  Example 8.10. L et S = P 2 and let ∆ b e the sum of the r ational p oints of an aﬃne ch art U . L et x, y b e aﬃne c o or dinates on U . F or al l α, β ∈ F q , set D a,α the line { x = α } and D b,β := { y = β } . Now, set D a := P α ∈ F q D a,α and D b := P β D b,β . The p air ( D a , D b ) satisﬁes the criterion of pr op osition 8.6, henc e i s ∆ -c onvenient. N otic e that the c omp onents of D a (r esp. D b ) interse ct themselves at a p oint lying on the line at inﬁnity, which do es not r epr esent any c ontr adiction with the deﬁnition of ∆ -c onvenienc e. Notice that in the deﬁn itio n, neither D a nor D p are assumed to b e eﬀectiv e. In s ome situation it is necessary to use n oneﬀec tiv e divisors. This h app ens in next example. Example 8.11. Consider again S = P 2 and assume that the b ase ﬁeld is F q with q o dd. Set ∆ = P 1 + P 2 + P 3 with P 1 = (0 : 0 : 1) , P 2 = (1 : 0 : 1) a nd P 3 = (0 : 1 : 1) . The p air ( D a , D b ) deﬁne d by D a = { Y = 0 } + { Y = 1 } and D b = { X = 0 } + { X = 1 } − { X + Y − 2 = 0 } , is ∆ -c onvenient. However, in this situation, ther e d o es not exist any ∆ -c onvenient p air of eﬀe ctive divisors. The pr o of of last c laim is left to the r e ader. 9. Pr oper ties of diffe r ential codes 9.1. Orthogonalit y. Theorem 9.1. L et ( D a , D b ) b e a ∆ -c onvenient p air and set D := D a + D b , then C Ω (∆ , D a , D b , G ) ⊆ C L (∆ , G ) ⊥ . Pr o of. Let ω ∈ Ω 2 ( G − D ) and f ∈ L ( G ), then f ω ∈ Ω 2 ( − D ) and from the deﬁnition of ∆-con venien t pairs, ∀ P ∈ S , res 2 D a ,P ( f ω ) =  0 if P / ∈ S upp(∆) f ( P )r es 2 D a ,P ( ω ) if P ∈ Sup p (∆) . Th us, < ev ∆ ( f ) , res 2 D a , ∆ ( ω ) > = X P ∈ Supp(∆) f ( P )r es 2 D a ,P ( ω ) = X P ∈ Supp(∆) res 2 D a ,P ( f ω ) . And last s um is zero from th eorem 6.8.  In section 10, we pro v e that in some situatio n, the r ev erse inclusion is f alse for an y choic e of a ∆-con venien t pair of divisors. Thus, in general, we d o n ot hav e equalit y . An in terp retat ion of th is statement is that, ev en if a pair of ∆-con v enien t divisors is link ed to ∆, it is not in v olv ed in the functional constru ctio n. This lac k of canonicit y in the c hoice of D might b e the reason of this non-equalit y . In a forthcoming pap er, w e will study ho w to get the whole orthogonal of a fu nctional co de, using diﬀerentia ls. 22 ALAIN COUVREUR 9.2. A diﬀerential co de is functional. Recall that in section 7, w e d iscussed ab out t w o relations denoted b y (O R) and (LΩ) . W e just said that it is not p ossible to extend p erfectly the orth og onalit y relation (OR) . Nev er th ele ss, next prop osition asserts that relation (LΩ) holds on surfaces, a d iﬀeren tial co de is alw a ys a fun ctio nal one associated with some canonical d ivisor. Rec all that, the pro of of (LΩ) for curv es is a consequence of the w eak ap p ro ximation theorem. Here is the needed statemen t for surfaces. Prop osition 9.2. L et P 1 , . . . , P m and Q 1 , . . . , Q n b e two families of close d p oints of S and C b e an irr e ducible curve emb e dde d in S . Supp ose that the P i ’s ar e c ontaine d in C and th e Q i ’s ar e out of it. Then, ther e exists a function u ∈ F q ( S ) satisfying the fol lowing c onditions. ( i ) ∀ i ∈ { 1 , . . . , m } , u is a lo c al e quation of C in a neighb orho o d of P i . ( ii ) ∀ j ∈ { 1 , . . . , n } , Q j / ∈ Supp ( u ) , i.e. u ∈ O × S,Q i Pr o of. Cho ose u 0 , a uniformizing parameter of O S,C . Then, ( u 0 ) = C + D where D ∈ Div F q ( S ) whose supp ort do es not cont ain C . F rom the moving lemma ([25] thm I I I.1.3.1), there exists a divisor D ′ linearly equiv alent to D w h ose supp ort a voids P 1 , . . . , P m , Q 1 , . . . , Q n . Thus, for s ome fun ctio n f ∈ F q ( S ), we ha v e D ′ = D + ( f ) and u := f u 0 is a solution of the prob lem.  N.B. In the whole b o ok of Shafarevic h [25] the b ase ﬁeld is assumed to b e algebraical ly closed. Nev ertheless the very same pro of h olds o v er an arbitrary ﬁeld. Corollary 9.3. L et ( D a , D b ) b e a ∆ - c onvenient p air and set D := D a + D b , then ther e exists a diﬀer ential ω 0 ∈ Ω 2 k ( S ) /k satisfying the fol lowing c onditions. (1) F or some op en set U c ontaining Supp (∆) , we have ( ω 0 | U ) = − D | U . (2) ∀ P ∈ Supp (∆) , r es 2 D a ,P ( ω 0 ) = 1 . (3) ∀ P ∈ Supp (∆) , ∀ f ∈ O S,P , r es 2 D a ,P ( f ω 0 ) = f ( P ) r es 2 D a ,P ( ω 0 ) . Pr o of. Let X 1 , . . . , X r and Y 1 , . . . , Y s b e r esp ective ly the irreducible comp onen ts of Supp( D a ) and S upp( D b ). That is D a = m 1 X 1 + · · · + m r X r and D b = n 1 Y 1 + · · · + n s Y r for some in tegers m i ’s and n j ’s. F rom p rop ositio n 9. 2 there is an op en subset U of S con taining the supp ort of ∆ and functions u 1 , . . . , u r , v 1 , . . . , v s suc h that u i (resp v j ) is an equation of X i (resp. Y j ) in U . Set u := Q i u m i i and v := Q i v n i i . Let µ b e a r ati onal 2-form on S ha ving neither zeros nor p oles in a n eigh b orho o d of the supp ort of ∆ and set ω 0 := µ uv . Replacing U by a smaller op en set con taining Sup p(∆), we m ay assume that µ has n either zeros nor p oles in U . Thus, statemen ts (1) and (3) are satisﬁed by ω 0 . Moreo v er, from the deﬁn ition 8.3 of ∆-con v enience, w e hav e ∀ P ∈ Su pp (∆) , res 2 D a ,P ( ω 0 ) = a P 6 = 0 . Cho ose g ∈ ∩ P ∈ Supp(∆) O × S,P suc h that g ( P ) = a − 1 P for all P ∈ Su pp (∆). Th en, replacing U by a smaller op en set con taining Supp(∆) and ω 0 b y g ω 0 , the thr ee conditions are satisﬁed.  RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 23 Theorem 9.4. L et D = D a + D b such that ( D a , D b ) is ∆ -c onvenient. Ther e exists a c anonic al divisor K such that C Ω (∆ , D a , D b , G ) = C L (∆ , K − G + D ) . Pr o of. F rom corollary 9.3, there exists a 2-form ω 0 satisfying (1), (2) and (3). Set K := ( ω 0 ), this divisor is of the form K = − D + R where the supp ort of R av oids that of ∆. Let ω ∈ Ω 2 ( G − D ), th en for some function f ∈ L ( K − G + D ), w e h a v e ω = f ω 0 . Notice that K − G + D = G + R , then any function f ∈ L ( K − G + D ) is regular in a neigh b orho od of eac h P ∈ Sup p(∆). Consequentl y , from condition (3) in coroll ary 9.3, w e h a v e res 2 D a , ∆ ( ω ) = res 2 D a , ∆ ( f ω 0 ) = ev ∆ ( f ) .  An y diﬀeren tial code is actually a functional one. Notice th at, if the con ve rse statemen t is trivial for co des on curve s, it is less easy in our situation. Indeed, to pro v e th at a fun ctio nal cod e is diﬀeren tial, w e h a ve to build a ∆-con venien t pair of d ivisors. 9.3. Con v erse statement, a functional co de is diﬀerential. Lemma 9.5. L e t Q 1 , . . . , Q m b e r ational p oints of S and set Γ := Q 1 + · · · + Q m . Then, ther e exists a Γ -c onvenient p air ( D a , D b ) . Pr o of. Step 1: Construction of D a . Cho ose a curve C (whic h may b e re- ducible) con taining the whole supp ort of Γ and r egular at eac h p oin t of it and set D a := P k C k where C k ’s are the irredu cible comp onents of C . Finding su c h a curve is an interp ola tion p roblem with inﬁnitely many solutions. Step 2: C onstruction of D b . Cho ose another divisor D ′ in terp olating all the p oin ts of Sup p (Γ) and h a ving no c ommon comp onen t with D a . Let Λ b e the 0-cycle giv en by th e sc heme-theoretic i nte rsection D a ∩ D ′ . Unf ortunately , the supp ort of Λ migh t b e b igg er than that of Γ. Thus, w e ha v e Λ = Γ + Γ ′ where Γ ′ is an eﬀectiv e 0-cycle. No w c ho ose a divisor D ′′ suc h that D ′′ ∩ D a = Γ ′ + Γ ′′ where Γ ′′ and Γ hav e d isjoin t supp orts. S et D b := D ′ − D ′′ . T he pair ( D a , D b ) satisﬁes the criterion of p rop ositi on 8.6, wh ic h concludes the pr oof.  Theorem 9.6. L et G b e a r ational divisor on S , then for some c anonic al divisor K and some divisor D := D a + D b such that ( D a , D b ) is ∆ -c onvenient, we have C L (∆ , G ) = C Ω (∆ , D a , D b , K − G + D ) . Pr o of. Lemma 9.5 asserts the existence of a ∆-con venien t pair ( D a , D b ). Then, construct a 2-form ω 0 using corollary 9.3. Set K := ( ω 0 ). No w the result is an easy consequence of theorem 9.4.  10. The r everse inclusion is f a lse As said in sectio n 9.1, if a d iﬀeren tial co de is included in the orthogonal of a functional one, the rev er s e in clusion is in general false. The study of the follo wing example will pro ve this. In this section, the surface S is the p rod uct of t w o p ro jectiv e lin es S := P 1 × P 1 . Let U b e an aﬃne c hart of S w ith aﬃn e co ordinates x, y . The complement of U in S is a union of t wo lines E and F . The Picard group of S is generated b y the classes of E and F . T h u s , without loss of generalit y , one can c ho ose f or G the 24 ALAIN COUVREUR divisor G n,m := mE + n F , with m, n ∈ Z . F inally , ∆ is d eﬁned as the formal sum of all rational p oin ts of U . 10.1. F unctional co des on P 1 × P 1 . On U , the v ector sp ace L ( G m,n ) ma y b e iden tiﬁed with F q [ x ] ≤ m ⊗ F q F q [ y ] ≤ n , where F q [ t ] ≤ d denotes th e sp ac e of p oly- nomials in t with degree less than or equ al to d . F urthermore, the functional co de C L (∆ , G m,n ) ma y b e iden tiﬁed with a tensor p rod uct of t w o codes on the pro j ect iv e line, which are Reed-Solomon co des. Thus, (  ) C L (∆ , G m,n ) = R S q ( m + 1) ⊗ F q RS q ( n + 1) , where RS q ( k ) denotes the Reed-Solomon co de o v er F q of length q and dimension k . 10.2. Orthogonal of functional co des on P 1 × P 1 . In this subsection, we pro v e that the orthogonal of some functional co de on P 1 × P 1 cannot b e d iﬀer- en tial. Prop osition 10.1. L et m, n b e two inte gers such that 0 ≤ n, m < q − 2 , th en for al l ∆ - c onvenient p air of divisors ( D a , D b ) , we have C Ω (∆ , D a , D b , G m,n ) C L (∆ , G m,n ) ⊥ . Pr o of. F rom (  ) and lemma D.1 in app end ix D, w e hav e C L (∆ , G m,n ) ⊥ = RS q ( m + 1) ⊥ ⊗ F q q + F q q ⊗ RS q ( n + 1) ⊥ . Supp ose that for some ∆-con v enien t pair ( D a , D b ), we had C Ω (∆ , D a , D b , G m,n ) = C L (∆ , G m,n ) ⊥ . F rom theorem 9.4, the co de C Ω (∆ , D a , D b , G m,n ) is functional. Th us, from (  ), it is a tensor pro duct of tw o Reed-Solomon cod es. But C L (∆ , G m,n ) ⊥ is of the form A ⊗ F q q + F q q ⊗ B with A, B nonzero and strictly con tained in F q q . This con tradicts lemma D.2 in app en d ix D.  Remark 10.2. The c ondition 0 ≤ m, n < q − 2 asserts that in the tensor pr o duct r epr esentation C L (∆ , G m,n ) = R S q ( m + 1) ⊗ RS q ( n + 1) , none of the terms of the tensor pr o duct i s zer o or F q q . A solution to a v oid th is lac k of reverse inclusion is to try to constru ct the orthogonal as a sum of d iﬀeren tial cod es. Th e purp ose of next subsection is the realizatio n of C L (∆ , G m,n ) ⊥ as a su m of t w o diﬀeren tial co des a sso ciated with t w o d istinct ∆-con v enien t pairs. 10.3. A construction of t he orthogonal code. F or eac h α ∈ F q , consider the lines D 1 ,α := { x = α } , D 2 ,α := { y = α } D 3 ,α := { x − y − α } . No w set D 1 := X α ∈ F q D 1 ,α , D 2 := X α ∈ F q D 2 ,α and D 3 := X α ∈ F q D 3 ,α P airs ( D 1 , D 3 ) and ( D 2 , D 3 ) are ∆ -co nv enient . Using them, one can realize the orthogonal of C L (∆ , G m,n ) as a sum of t w o diﬀeren tial co des. Prop osition 10.3. The thr e e fol lowing r elations ar e satisﬁe d. ( i ) C Ω (∆ , D 1 , D 3 , G m,n ) = F q q ⊗ RS q ( q − 2 − n ) . ( ii ) C Ω (∆ , D 2 , D 3 , G m,n ) = R S q ( q − 2 − m ) ⊗ F q q . ( iii ) C L (∆ , G m,n ) ⊥ = C Ω (∆ , D 1 , D 3 , G m,n ) + C Ω (∆ , D 2 , D 3 , G m,n ) . RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 25 Pr o of. As said in last pr oof, relation (  ) and lemma D.1 en tail C L (∆ , G m,n ) ⊥ = RS q ( m + 1) ⊥ ⊗ F q q + F q q ⊗ RS q ( n + 1) ⊥ . Consequent ly , ( i ) + ( ii ) ⇒ ( iii ). F urth ermore, by symmetry ( i ) ⇔ ( ii ). Thus, we only hav e pro v e ( i ). S et ν := dy Q β ∈ F q ( x − β ) ∧ dx Q α ∈ F q ( x − y − α ) . This f orm satisﬁes conditions (1), (2) and (3) in corollary 9.3. Compute the divisor of ν . On U , we ha v e ( ν | U ) = − D 3 | U − D 1 | U , m oreov er D 1 ∼ q E and D 3 ∼ q ( E + F ). Since the canonical cl ass on P 1 × P 1 equals that of − 2( E + F ), w e hav e ( ν ) = (2 q − 2) E + ( q − 2) F − D 1 − D 3 and C Ω (∆ , D 1 , D 3 , mE + nF ) = C L (∆ , (2 q − 2 − m ) E + ( q − 2 − n ) F ) = RS q (2 q − 2 − m ) ⊗ RS q ( q − 2 − n ) . T o conclude, notice that if m ≤ q − 1, then 2 q − 2 − m ≥ q − 1 and RS q (2 q − 2 − m ) equals F q q .  11. Conclusion This new constru ctio n of co des generalizes the d iﬀeren tial constru ctio n of co des on curv es. The main d iﬀerence is that it is not al wa ys p ossible to r ealize the orthogonal of a functional co de as a diﬀeren tial (or equiv alen tly functional) one. A natural qu estio n comes from the study of last examp le. Question 11.1. Is the ortho g onal of a f unctional c o de a sum of diﬀer e ntial c o des? If yes, is ther e a b ound on th e numb er of diﬀer ential c o des involve d in this sum? Moreo ver, we n o w know t hat the orthogonal of a functional co de migh t b e non-functional. Consequ ently , th e study of suc h co des migh t b e in teresting. Appendix A. Proof of lemma 3.5 If w e pro ve the well- deﬁnition of the morphism k (( u ))[ [ v ]] → k (( x ))[[ y ]], then w e conclude a b out that of the morphism k (( u ))(( v )) → k (( x ))(( y )), using the unive rsal pr operty of fraction rings. First, we ha ve to d eﬁne a top ology on k (( u ))[ [ v ]] (resp. k (( x ))[[ y ]]). Recall that k (( u ))[ [ v ]] = lim ← − k (( u ))[ v ] / ( v n ) . Afterw ards, usin g the ( u )-adic top olog y of k (( u )), on e can d eﬁne a top ology of pro j ect iv e limit on k (( u ))[[ v ]]. F or this t op ology , a sequence ( s ( n ) ) n ∈ N deﬁned for all n b y s ( n ) = P j ∈ N s ( n ) j ( u ) v j con v erges to zero if and only if ∀ j ∈ N , lim n → + ∞ s ( n ) j ( u ) = 0 , for the ( u )-adic topology . Afterw ards, usin g a Cauch y criterion, one pro ves that, for this to p ology , a series of elemen ts of k (( u ) )[[ v ]] co nv erges if and only if its general term con v erges to zero. 26 ALAIN COUVREUR Remark A.1. Notic e that this top olo gy on k (( u ))[[ v ]] is we aker than the ( v ) - adic one (for which the subset k (( u )) is discr ete). Thus, if a se quenc e (r esp. a series) c onver ges for the ( v ) -adic top olo gy, henc e c onver ges for the pr oje ctive limit top olo g y. Pr o of of lemma 3.5. Step 1. Recall that f is of the f orm f = f 0 ( x ) + f 1 ( x ) v + · · · and su c h that f 0 has ( x )-adic v aluation one. W e will p ro ve that t he sequence ( f n ) n ∈ N con v erges to zero. Let i b e a nonnegativ e inte ger, for n large enough, the co eﬃcie nt of y i in f n is of the form f n 0 P i ( f 0 , . . . , f i ), where P i is a p olynomial whic h do es not dep end on n . Thus, for the ( x )-adic top ology this co eﬃcien t tends to zero. Consequentl y , for all L au r en t series φ ( u ) ∈ k (( u )), the series φ ( f ( x, y )) con v erges in k (( x ))[[ y ]]. Step 2. Th e series g has ( y )-adic v aluation one, th us the s equence ( g n ) n ∈ N con v erges to zero f or the ( y )-adic top olo gy , hence for the pro jectiv e limit top ol- ogy (see remark A.1). Using step 1, we conclude that for ev ery series ψ ( u, v ) ∈ k (( u ))[ [ v ]], the series ψ ( f , g ) con v erges in k (( x ))[[ y ]]. Moreo v er, its ( y )-adic v al- uation equals the ( v )-adic one of ψ . Step 3. If ω is a formal form ω = h ( u, v ) du ∧ dv with ( v )-adic v aluation n ∈ Z , then we hav e to pro v e that the ( y )-adic v aluation of h ( f , g ) d f ∧ dg is n too. If we pro v e that the ( y )-adic v aluation of d f ∧ dg is zero, then w e can conclude using step 2. F or that, consider the expression d f ∧ dg = ∂ f ∂ x ∂ g ∂ y | {z } val ( y ) =0 − ∂ f ∂ y ∂ g ∂ x | {z } val ( y ) ≥ 1 ! dx ∧ dy . This concludes the pro of.  Appendix B. Pr oof o f lemma 3.11 If ω = dA ∧ dB f or some series A, B ∈ k (( u ))(( v )), after a change of coordin ates (CV), ω = dF ∧ dG for s ome other series F , G ∈ k (( x ))(( y )). Th en , in ord er to pro v e the lemma we only h a v e to pr o ve th at the ( u, v )-2-residue of ω = dA ∧ dB is zero. W e ﬁrst in tro duce some notations. Let ρ and Jac b e the maps ρ :  k (( x ))( ( y )) → k (( x )) P i ≥− n h j ( u ) v j 7→ h − 1 ( u ) , Jac :  k (( x ))( ( y )) 2 → k (( x ))( ( y )) ( A, B ) 7→ ∂ A ∂ u ∂ B ∂ v − ∂ A ∂ v ∂ B ∂ u . Th us, ω = dA ∧ dB = Jac( A, B ) du ∧ dv . W e will prov e the follo wing lemma. Lemma B.1. F or al l A, B ∈ k (( u ))(( v )) , we have ρ ( Jac ( A , B )) = φ ′ ( u ) for some φ ∈ k (( u )) , wher e φ ′ denotes the formal derivative of φ . Pr o of. Maps Jac and ρ are resp ectiv ely k -bilinear an tisymmetric and k -linear. Then, w e can restrict the pr o of to the three follo win g situations and ext end it b y linearit y . (1) A, B ∈ k (( u ))[[ v ]]. (2) A ∈ k (( u )) [[ v ]] and B = b ( u ) v n with n ∈ N ∗ and b ∈ k (( u )). (3) A = a ( u ) v m and B = b ( u ) v n with m, n ∈ N ∗ and a, b ∈ k (( u )). RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 27 Let us consider these three situations. (1) The series A and B don’t hav e terms with negativ e p o we rs of v , th us s o are their p artial deriv ativ es, then ρ ◦ Jac ( A, B ) = 0. (2) Th e series A is of the form A = P j ≥ 0 a j ( u ) v j . Then, ρ (Jac( A, B )) = − n ( a ′ n ( u ) b ( u ) + a n ( u ) b ′ ( u )) = ( − na n ( u ) b ( u )) ′ . (3) Jac( A, B ) =  − n a ′ ( u ) b ( u ) v m + n +1 − ( − m ) a ( u ) b ′ ( u ) v m + n +1  . In tegers m and n are p ositiv es, there is no term in v − 1 , thus ρ (Jac( A, B )) = 0.  Conclusion. Usin g lemma B.1, we get ( u, v )res 1 C,P ( ω ) = φ ′ ( ¯ u ) d ¯ u and the coef- ﬁcien t of u − 1 in φ ′ is zero, b ecause it is a deriv ative . Appendix C . Proof of theorem 3.6 when k has a positive characteristic W e only hav e to w ork on the p oint s of the pr oof of prop osition 3.6 in whic h we used sp eciﬁc p rop erties of c haracteristic zero. Thus, we will stud y the b eha vior under (CV2) of diﬀeren tials of the form ω = φ ( u ) du ∧ dy y n +1 , where φ ∈ k (( u ) ) and n ≥ 1 . Let N b e a nonn ega tiv e in teger. In what follo ws, w e consider a c hange of v ariables of the form (CV2) u = f ( x, y ) , with f = X j ≥ 0 f j ( x ) y j and f 0 ∈ xk [[ x ]] r x 2 k [[ x ]] , suc h that min k =1 ...n  v al ( x ) ( f k )  = − N , where v al ( x ) denotes the ( x )-adic v alu- ation of an elemen t of k (( x )). Step 1. Assume that ω = u m du ∧ dy y n +1 with m ∈ N . Then, ω = ( f ′ 0 ( x ) + f ′ 1 ( x ) y + · · · )( f 0 ( x ) + f 1 ( x ) y + · · · ) m dx ∧ dy y n +1 . The ( x, y )-1-residue of ω is the coeﬃcien t in y n of the series f m ∂ f /∂ x . This residue is of the form ( x, y )res 1 C,P ( ω ) = P m,n ( f 0 , . . . , f n , f ′ 0 , . . . , f ′ n ) d ¯ x, where P m,n ∈ Z [ X 0 , . . . , X n , Y 0 , . . . , Y n ] dep end s n eit her on the ﬁeld k nor on f . Actually , P m,n dep ends only on m and n . By the same wa y , its coeﬃcient of x − 1 is a p olynomial expression Q in the f i,j ’s with 0 ≤ j ≤ n and − N ≤ i ≤ N + 1, suc h that Q h as co eﬃcien ts in Z an d dep end s n either on k nor on f . F u rthermore, if k has charact eristic zero, we kno w fr om section 3 that Q v anishes on the set { f 1 , 0 6 = 0 } , hence is the zero p olynomial. Step 2. Assume th at ω = φ ( u ) du ∧ dy y n +1 , with φ = P m ≥ 0 φ m u m ∈ k [[ u ]]. F rom step 1, w e ha v e (5) ( x, y )res 1 C,P ( ω ) = X m ≥ 0 φ m P m,n ( f 0 , . . . , f n , f ′ 0 , . . . , f ′ n ) d ¯ x, 28 ALAIN COUVREUR where P m,n ’s den ote the p olynomials inv olv ed in Step 1. The ( x, y )-1-residue of ω is well-deﬁned. Th us, the series in (5) con v erges in k (( x )) . Consequent ly , the ( x )-adic v aluation of it terms is p ositiv e for eac h m ≥ M and ( x, y )res 1 C,P ( ω ) = M X m =0 φ m P m,n d ¯ x + X m>M φ m P m,n d ¯ x. The r igh t term has p ositiv e ( x )-adic v aluation, th u s its residue is zero. The left one has zero residue zero b ecause of step 1 extended b y linearit y . Step 3. Assume that ω = du u m ∧ dy y n +1 , with m ∈ N . Then, ω = 1 f m ∂ f ∂ x dx ∧ dy y n +1 . W e hav e to study the fraction 1 f m ∂ f ∂ x . First, compute its co eﬃcient of y n corresp onding to the ( x, y )-1-residue. W e ha v e (6) 1 f m = 1 f m 0  1 + R m, 1 ( f 0 , f 1 ) f 0 y + · · · + R m,p ( f 0 , . . . , f p ) f p 0 + · · ·  , for some homogeneous p olynomials R m,i ∈ Z [ X 0 , . . . , X p ] of degree p and de- p ending only on m . Th us, the co eﬃcien t of y n in 1 f m ∂ f ∂ x is C ( x ) := 1 f m 0  f ′ n + f ′ n − 1 R m, 1 ( f 0 , f 1 ) f 0 + . . . + f ′ 0 R m,n ( f 0 , . . . , f n ) f n 0  . F or all k ∈ { 1 , . . . , n } , set S m,n,k ( f 0 , . . . , f k ) := f n − k 0 R m,k ( f 0 , . . . , f k ) and S m,n, 0 ( f 0 ) := f n 0 . P olynomials S m,n,k ’s are homogeneous of degree n and (7) C ( x ) := 1 f m + n 0 | {z } A ( x ) n X k =0 f ′ n − k S m,n,k ( f 0 , . . . , f k ) | {z } B ( x ) . Recall that f 0 ∈ x k [[ x ]], that is f 0 := f 1 , 0 x + f 2 , 0 x 2 + · · · , then A ( x ) = 1 ( f 1 , 0 x ) m + n 1 + R m, 1 ( f 1 , 0 , f 2 , 0 ) f 1 , 0 x + · · · · · · + R m,p − 1 ( f 1 , 0 , f 2 , 0 , . . . , f p, 0 ) f p 1 , 0 x p + · · · ! , for p olynomials R m,k as in (6). Finally , w e w an t to express th e co eﬃcien t C − 1 of x − 1 in C ( x ). Recall that, for all p , the degree of S m,n,p is n . T herefore, there exists an integ er M and a p olynomial V ∈ Z [ X i,j ] with − N ≤ i ≤ max ( m + n , ( n + 1) N + 1) and 0 ≤ j ≤ n , depen ding only o n m , n a nd N and such that C − 1 = 1 f M 1 , 0 V ( f i,j ) . Ov er a ﬁ eld of c h arac teristic zero, V v anishes on the set { f 1 , 0 6 = 0 } , hence is the zero p olynomial. Remark C .1. N otic e that, in the whole pr o of, we de al with the value N such that − N is the minimal valuation of the f i , for 0 ≤ i ≤ n . Then, we have pr ove d that the 2 -r esidue is invariant u nder a change of variables u = f ( x, y ) such that the f i have valuation minor e d by − N . But we pr ove d it for al l N , which c oncludes the pr o of. RESIDUES ON SURF ACES AND APPLICA TION TO CODING THEOR Y 29 Appendix D. About t ensor products Statemen ts of this app en dix are qu ite elemen tary results of linear algebra. W e pro v e them b ecause of a lack of references. Lemma D.1. L et ( E , <, > E ) and ( F , <, > F ) b e two ﬁnite-dimensional ve ctor sp ac es over an arbitr ary ﬁeld k with r esp e ctive non-de gener ate biline ar forms < , > E and < , > F . L et A and B b e r esp e c tiv e su bsp ac es of E and F , then for the biline ar form < , > E ⊗ F := < , > E ⊗ < , > F on E ⊗ k F , we have ( A ⊗ k B ) ⊥ = A ⊥ ⊗ k F + E ⊗ k B ⊥ . Pr o of. Inclusion “ ⊇ ” is obvious. F or t he rev erse one, w e will p r o ve that b oth spaces hav e the same dimension. First, w e hav e to prov e that (8) A ⊥ ⊗ k F ∩ E ⊗ k B ⊥ = A ⊥ ⊗ B ⊥ . Here aga in, inclusion “ ⊇ ” is ob vious. F or the rev erse one, consider bases ( e i ) i ∈ I 0 and ( f j ) j ∈ J 0 resp ectiv ely of A ⊥ and B ⊥ and complete them as b ases ( e i ) i ∈ I and ( f j ) j ∈ J of E and F . Then, for all s = P i,j s ij e i ⊗ f j ∈ E ⊗ F , we ha ve s ∈ A ⊥ ⊗ k F ∩ E ⊗ k B ⊥ = ⇒  ∀ ( i, j ) ∈ ( I r I 0 ) × ( J r J 0 ) , s ij = 0  . Th us, (8 ) is prov ed and en tails dim( A ⊥ ⊗ F + E ⊗ B ⊥ ) = d im( A ⊥ ⊗ F ) + d im( E ⊗ B ⊥ ) − dim( A ⊥ ⊗ B ⊥ ) . After an easy computatio n, w e p ro ve that s p ace s ( A ⊗ B ) ⊥ and ( A ⊥ ⊗ F + E ⊗ B ⊥ ) ha v e the same dimension, wh ic h concludes the p r oof.  Lemma D.2. L et E and F b e two ve ctor sp ac es over a n arbitr ary ﬁeld k . L e t A (r esp. B ) b e a strict nonzer o subsp ac e of E (r esp F ). Then, the subsp ac e A ⊗ k F + E ⊗ k B of E ⊗ k F c annot b e written as an elementary tensor pr o duct U ⊗ V . Pr o of. Assume that A ⊗ F + E ⊗ B = U ⊗ V for some subsp ace U (resp. V ) of E (resp. F ). Let ( e i ) i ∈ I 0 (resp. ( f j ) j ∈ J 0 ) b e a basis of A (resp. B ) completed in a basis ( e i ) i ∈ I of E (resp. ( f j ) j ∈ J of F ). Assume that U * A and c ho ose u ∈ U su c h that u / ∈ A . Then, for all v ∈ V , the v ector u ⊗ v is of the form u ⊗ v = X i,j u i v j e i ⊗ f j . F rom th e assu m ption A ⊗ F + E ⊗ B = U ⊗ V , the pr o du ct u i v j is zero for all couple ( i, j ) ∈ I r I 0 × J r J 0 . S ince u / ∈ A , there exists at least one index i 1 ∈ I r I 0 suc h that u i 1 6 = 0. Thus, for all j ∈ J r J 0 , w e ha v e u i 1 v j = 0 wh ic h en tails that v ∈ B . This statemen t works f or all v ∈ V , hence U ⊗ V ⊆ E ⊗ B . No w choose f ∈ F such that f / ∈ B and a ∈ A r { 0 } . Th en , a ⊗ f / ∈ E ⊗ B , th us a ⊗ f / ∈ U ⊗ V which con tradicts A ⊗ F + E ⊗ B = U ⊗ V . 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Institut de Ma th ´ ema tiques d e Toulouse, UMR 5219, Unive rsi t ´ e P a ul S aba tier, 118 r oute de Narbonne, 31062 Toulouse, France E-mail addr ess : cou vreur@math.un iv-toulouse.f r

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