Explicit calculations in rings of differential operators

S´eminai es & Cong `es 8, 2004, p. 89–128 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS by F ancisco J. Cas o-Jim´enez & Michel G ange Abs ac . — We use he no ion o a s anda d basis o s udy algeb as o linea di - e en ial ope a o s and ini e ype modules o e hese algeb as. We conside he polynomial and he holomo phic cases as well as he o mal case. Ou aim is o demons a e how o calcula e classical in a ian s o ge ms o cohe en (le ) modules o e he shea Do linea di e en ial ope a o s o e Cn. The main in a ian s we deal wi h a e: he cha ac e is ic a ie y, i s dimension and he mul i- plici y o his a ie y a a poin o he co angen space. In he inal chap e we shall s udy mo e e ined in a ian s o D-modules linked o he ques ion o i egula i y: The slopes o a D-module along a smoo h hype su ace o he base space. Résumé (Calculs explici es dans l’anneau des opé a eu s di é en iels). — Dans ce cou s on d´e eloppe la no ion de base s anda d, en ue d’´e udie les alg`eb es d’op´e a eu s di ´e en iels lin´eai es e les modules de ype ini su ces alg`eb es. On consid`e e le cas des coe icien s polynomiaux, des coe icien s holomo phes ainsi que le cas des alg`eb es d’op´e a eu s `a coe icien s o mels. No e bu es de mon e commen les bases s anda ds pe me en de calcule ce ains in a ian s classiques des ge mes de modules (`a gauche) coh´e en s su le aisceaux Ddes op´e a eu s di ´e en iels lin´eai es su Cn. Les p incipaux in a ian s que nous examinons son : la a i´e ´e ca ac ´e is ique, sa dimension e sa mul iplici ´e en un poin du ib ´e co angen . Dans le de nie chapi e nous ´e udions des in a ian s plus ins des D-modules qui son eli´es aux ques ions d’i ´egula i ´e : les pen es d’un D-module, le long d’une hype su ace lisse. 2000 Ma hema ics Subjec Classi ica ion. — 13N10, 13P10, 16S32. Key wo ds and ph ases. — D-modules, G ¨ obne basis, slopes. F.C.: Pa ially suppo ed by DGESIC-PB97-0723; BFM2001-3164 and FQM-218. Bo h au ho s pa ially suppo ed by Picasso-HF2000-0044. c S´eminai es e Cong `es 8, SMF 2004 90 F.J. CASTRO-JIM´ ENEZ & M. GRANGER In oduc ion The pu pose o hese no es is o make an accoun o explici me hods, using he no- ion o a s anda d basis, which could be used in s udying algeb as o linea di e en ial ope a o s and ini e ype modules o e hese algeb as. We conside in pa allel each o he ollowing cases: coe icien s in a ing o polynomials k[x1, . . . , xn] o he Weyl algeb a An(k), in he ing o ge ms o holomo phic unc ions a 0 ∈Cn o Dn, o in he ing o o mal powe se ies o c Dn. We deno e Rany o hese ings o ope a o s and B he co esponding commu a i e ing o coe icien s. Ou aim is o demons a e how o calcula e classical in a ian s o ge ms o cohe en (le ) modules o e he shea Do linea di e en ial ope a o s o e Cn. In p ac ice we shall look a ini e ype modules o e Dno c Dn. The main in a ian s we a e dealing wi h a e: he cha ac e is ic a ie y, and he mul iplici y o his a ie y a a poin o he co angen space. See [25] and [19] o an in oduc ion o he heo y o D-modules and o he de ini ion o he cha ac e is ic a ie y, o i s dimension and and o i s mul iplici y. In he las chap e we shall s udy mo e e ined in a ian s o R-modules linked o he ques ion o i egula i y: The slopes o a Dn-module o an An(k)-module along a smoo h hype su ace o he base space. In hese no es we deal mainly wi h he case o monogenic modules M=R/I wi h Ia (le ) ideal o R. We p o ide an algo i hm o build s anda d bases o Iand in he con ex o chap e II hese bases yield a special kind o sys em o gene a o s o which he module o ela ions is easy o desc ibe. The e is a s aigh o wa d gene alisa ion o he case M=Rp/Nin ol ing a submodule No Rp. Then con inuing he p ocess o building s anda d bases o submodules we can hus ob ain a (locally) ee esolu ion o M. The echniques used a e he no ion o p i ileged exponen s wi h espec o an o de ing and a heo em o di ision. They we e in oduced by H.Hi onaka (c . [26] o [1]). In he polynomial case he no ion o a s anda d basis was de eloped by Buchbe ge unde he name o a G ¨ obne basis in [13] whe e he also gi es an algo i hm o i s calcula ion. The commu a i e case is ea ed in chap e I, whe e we ecall he no ions o a p i ileged exponen o a polynomial o a powe se ies wi h espec o a con enien o de ing, he de ini ion o a s anda d basis and he algo i hm o calcula ing i , which is he Buchbe ge ’s algo i hm in he polynomial case. We also d aw a en ion o he elegan p oo in he con e gen case aken om Hause and Mulle (c . [20].) We inish by gi ing some applica ions in commu a i e algeb a such as calcula ing mul iplici ies, syzygies, and he in e sec ions o ideals. In chap e II, we conside di ision p ocesses in algeb as o ope a o s which a e compa ible wi h a il a ion which may ei he be he il a ion by he o de o ope a o s o in he pa icula case o An(k), he Be ns ein il a ion by he o al o de . A he same ime, o he sake o comple eness we ea a weigh ed homogeneous e sion o hese il a ions. Using a compa ible o de ing on monomials we again de elop a di ision algo i hm and an algo i hm o he cons uc ion o a s anda d basis. These S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 91 algo i hms a e e y simila o hose de eloped in chap e I, since in ac a di ision by a amily o ope a o s {P1, . . . , P }, o by a s anda d basis o an ideal Iinduces he same objec ia he p incipal symbols in he commu a i e associa ed g aded ings. The e e ences o hese esul s a e [11] and [14]. Le us also no ice ha i is only in he case o k[x1, . . . , x ] o An(k) ha he sui able o de ings used in chap e s I and II a e well o de ings and he e o e ha he algo i hms a e e ec i e. In he powe se ies case hey depend on o mal o con e gen p ocesses in he local ings o se ies. In chap e III we gi e an algo i hm o he calcula ion o he slopes o a cohe en R-module along a smoo h hype su ace Yo kno Cnin he neighbou hood o a poin o Y. The ma e ial is essen ially aken om ou wo k wi h A.Assi [2] whe e howe e only he case o An(k) is conside ed. The no ion o a slope o a cohe en D-module Mwas in oduced by Y. Lau en unde he name o a c i ical index. He conside s, in he mo e gene al con ex o mic odi e en ial ope a o s a amily o il a ions L =pF +qV (wi h a a ional numbe such ha 0 ⩽ =p/q ⩽+∞), which is an in e pola ion be ween he il a ion by he o de Fand he V- il a ion o Malg ange and Kashiwa a (c . [22]). The c i ical indices a e hose o which he L -cha ac e is ic a ie y o Mis no bihomogeneous wi h espec o Fand V. Lau en p o ed in loc. ci . he ini eness o he numbe o slopes and hen C.Sabbah and F. Cas o p o ed he same esul in [30] by using a local la ene . In [28] Z. Mebkhou in oduced he no ion o a anscenden al slope o a holonomic D-module M, as being a jump in he Ge ey il a ion I ( ) Y(M) o he i egula i y shea I Y(M). The i egula i y shea is he complex o solu ions o M wi h alues in he quo ien o he o mal comple ion along Yo he s uc u al shea O, by Oi sel . By he main esul o [28], i is a pe e se shea , and I ( ) Y(M) is he sub- pe e se shea o solu ions in o mal se ies o Ge ey ype along Y. In [23] Lau en and Z.Mebkhou p o ed ha he anscenden al slopes o an holonomic D-module a e equal o he slopes in he sense o Lau en called algeb aic slopes. The analogue in dimension one is Malg ange’s pape [27] o he pe e si y o he i egula i y shea and Ramis’s pape [29] o he heo em o he compa ison o slopes. In chap e III, we ecall he p inciple o he algo i hm o calcula ion o he algeb aic slopes o an R-module ha we de eloped in [2] and we gi e some supplemen a y in o ma ion. He e he addi ional di icul y is ha he linea o m L which yields he simila ly called il a ion now possesses a nega i e coe icien in he a iable x1. Al hough we can s ill speak o p i ileged exponen s and s anda d bases, he s anda d bases a e no longe sys ems o gene a o s o he ideal Iwhich we conside bu only induce a s anda d basis o he g aded associa ed ideal. A mo e se ious consequence o non-posi i i y, is ha he s aigh o wa d di ision algo i hm does no wo k inside ini e o de ope a o s. The way o sol e his p oblem is o homogenize he ope a o s in R[ ] wi h espec o he o de il a ion o , in he case o An(k), wi h espec o he Be ns ein il a ion. We no ice in chap e III, ollowing a ema k made by L.Na ´aez SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 92 F.J. CASTRO-JIM´ ENEZ & M. GRANGER [16] ha we can simpli y he o iginal p oo in [2] by conside ing on An[ ] a di e en s uc u e as a Rees ing. Ano he imp o emen o [2] lies in he dis inc ion be ween he slopes in he sense o Lau en and he alues o o which he ideal Igi es a non-bihomogeneous g aded ideal g L (I). We call hose , he idealis ic slopes o I. In [2] we conside ed only his se o slopes and p o ed i s ini eness; his pape howe e al eady con ains he ha d pa o he algo i hm o he calcula ion o algeb aic slopes. Le us end his in oduc ion by poin ing ou wo o he ex ensions o he o iginal ma e ial o ou pape [2]. Fi s we make he same algo i hm wo k o he ings o ope a o s Dn, o c Dn. Secondly we gi e some signi ican examples o he calcula ions o slopes: he slopes o he di ec image o DCe1/xkby an imme sion in C2, wi h espec o a smoo h cu e Y angen o he suppo . This example con ains idealis ic slopes which end up no being algeb aic slopes. Finally, we calcula e he slopes o DC2e1/(yp−xq)along any line h ough he o igin. Added on Ma ch 21, 2003. — This pape was w i en in Sep embe 1996, as ma e- ial o a six hou cou se gi en in he CIMPA summe school “Di e en ial Sys ems” (Se illa, Sep embe 1996). Consequen ly, he bibliog aphy is ou da ed. Since hen, many pape s ha e been published abou he compu a ional aspec s in D-modules heo y. We ha e he e o e decided o add, a e he e e ences, a complemen a y lis o ecen publica ions on he subjec . 1. Di ision heo ems in polynomial ings and in powe se ies ings 1.1. Le kbe a ield, wi h an a bi a y cha ac e is ic unless o he wise s a ed. Le n be a posi i e in ege . We deno e by: •k[X] = k[X1, . . . , Xn] he ing o polynomials wi h coe icien s in kand a iables X1, . . . , Xn. •k[[X]] = k[[X1, . . . , Xn]] he ing o o mal powe se ies wi h coe icien s in k and a iables X1, . . . , Xn. •k{X}=k{X1, . . . , Xn} he ing o con e gen powe se ies wi h coe icien s in kand a iables X1, . . . , Xn, i k=Ro C.(1) I ∈k[[X]], 6= 0, we w i e =Pα∈Nn αXαwhe e α∈k. I ∈k[X] 6= 0, hen his sum is ini e. The se N( ) = {α∈Nn| α6= 0}is called he New on diag am o he powe se ies o o he polynomial . 1.2. L-deg ee and L- alua ion. — Le L:Qn→Qbe a linea o m wi h non nega i e coe icien s. De ini ion 1.2.1. — Le 0 6= ∈k[X]. We de ine he L-deg ee o (and we deno e i by degL( )) as being max{L(α)| α6= 0}. We se degL(0) = −∞. (1)O , mo e gene ally, a comple e alued ield. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 93 De ini ion 1.2.2. — Le 0 6= ∈k[[X]]. We de ine he L- alua ion o (which we deno e by alL( )) as being min{L(α)| α6= 0}. We se alL(0) = +∞. We ha e degL( g) = degL( ) + degL(g) i , g ∈k[X] and alL( g) = alL( ) + alL(g) i , g ∈k[[X]]. De ini ion 1.2.3. — Le 0 6= ∈k[[X]]. We call he sum inL( ) = PL(α)= alL( ) αXα he L- ini ial o m o he powe se ies (2). Le Ibe an ideal o k[[X]]. We call he ideal o k[[X]] gene a ed by {inL( )| ∈I}, he ini ial ideal o I. We deno e i by InL(I) (o simply In(I)) No a ion. — The ollowing no a ion will be use ul. I =Pα αXαis a powe se ies, we se inL,ν( ) = PL(α)=ν αXα. When no con usion can occu , we w i e inν( ) ins ead o inL,ν( ). We ha e: =Pνinν( ). De ini ion 1.2.4. — Le 0 6= ∈k[X]. We call he sum inL( ) = PL(α)=degL( ) αXα he L- inal o m o he polynomial . Le Ibe an ideal o k[X]. We call he ideal o k[X] gene a ed by { inL( )| ∈I} he inal ideal o I. We deno e i by FinL(I) (o simply by Fin(I)). 1.3. O de ings in Nn. — Le <be a o al well o de ing on Nncompa ible wi h sums (i.e. i α, β ∈Nnand α < β hen we ha e α+γ < β +γ o any γ∈Nn). Le L:Qn→Qbe a linea o m wi h non nega i e coe icien s . The ela ion <L, de ined by: α <Lβi and only i L(α)< L(β) o L(α) = L(β) and α < β is a o al well o de ing on Nncompa ible wi h sums. 1.4. The p i ileged exponen o a polynomial o o a powe se ies. — The no ion o he p i ileged exponen o a powe se ies is due o H.Hi onaka. I was in oduced in [26] (see also [1], [10]). We ix, once and o all, a o al well o de ing <, compa ible wi h sums, in Nn. Le L:Qn→Qbe a linea o m as abo e. De ini ion 1.4.1. — Le =Pα αXα∈k[X], 6= 0. We call: •The n-uple expL( ) = max<L{α| α6= 0}, he L-p i ileged exponen o •The monomial mpL= expL( )XexpL( ), he L-p i ileged monomial o Le =Pα αXα∈k[[X]], 6= 0. We call: •The n-uple expL( ) = min<L{α| α6= 0}, he L-p i ileged exponen o . •The monomial mpL= expL( )XexpL( ), he L-p i ileged monomial o . (2)I all he coe icien s o La e posi i e, hen he ini ial o m o a powe se ies is a polynomial. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 94 F.J. CASTRO-JIM´ ENEZ & M. GRANGER When i becomes necessa y, we shall use he mo e p ecise no a ion, exp<L( ) = expL( ) and mp<L( ) = mpL( ). In all he cases, when no con usion can esul , we shall w i e exp( ) ins ead o expL( ) and mp( ) ins ead o mpL( ). No e 1.4.2. — When ∈k[X], 6= 0, we shall ake ca e no o con use he p i ileged exponen o he polynomial wi h he p i ileged exponen o he powe se ies , in spi e o he no a ion. I necessa y, we shall use he no a ion expp( ) o he p i ileged exponen o he polynomial and exps( ) o he p i ileged exponen o he powe se ies . P oposi ion 1.4.3. — Le , g ∈k[X]( esp. , g ∈k[[X]]) be non ze o elemen s. We ha e: (1) exp( g) = exp( ) + exp(g). (2) mp( g) = mp( ) mp(g). (3) I exp( )6= exp(g) hen exp( +g) = max <L{exp( ),exp(g)}( esp. exp( +g) = min <L{exp( ),exp(g)}). Le Ibe a non ze o ideal o k[X] ( esp. k[[X]]). We deno e E<L(I) = {expL( )| ∈I {0}}. When no con usion can esul , we w i e E(I) ins ead o E<L(I). Because o 1.4.3, we ha e E(I) + Nn= E(I). We deno e by mp(I), he ideal o k[X] gene a ed by he amily o monomials {mp( )| ∈I}(3). P oposi ion 1.4.4. — Le Ibe a non ze o ideal o k[X]( esp. k[[X]]). Then we ha e: E(I) = E(mp(I)) = E(Fin(I)) ( esp. E(I) = E(mp(I)) = E(In(I))). P oo . — By de ini ion, o e e y non ze o polynomial , we ha e exp( ) = exp( in( )) and exp( ) = exp(mp( )) (see 1.4.1). I is a non ze o powe se ies, hen we ha e: exp( ) = exp(in( )) and exp( ) = exp(mp( )) (see 1.4.1). No e 1.4.5. — Wi h he no a ions o 1.4.2, i is a powe se ies such ha in( ) is a polynomial, ( his condi ion is e i ied i e e y coe icien in he linea o m Lis posi i e) hen we ha e, in gene al, exp( )6= expp(in( )). Assume ha e e y coe icien in he linea o m Lis posi i e (we hen jus say ha Lis a posi i e linea o m). Conside he o de ing CLde ined on Nnby he o mula: αCLβi and only i L(α)< L(β) o L(α) = L(β) and β < α (3)This is a monomial ideal, which means ha a polynomial is an elemen o he ideal i and only i e e y monomial o is in he ideal. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 95 This is a o al well o de ing (4) on Nncompa ible wi h he sum. I is a powe se ies, hen we ha e: exp<L( ) = exps<L(inL( )) = exppCL(inL( )). P oposi ion 1.4.6. — Le E⊂Nnsuch ha E+Nn=E. Then Econ ains a ini e amily o gene a o s; In o he wo ds, he e exis s a ini e amily F⊂Esuch ha E=∪α∈F(α+Nn). P oo . — This is a e sion o Dickson’s lemma. The p oo is by induc ion on n. Fo n= 1 a ( ini e) amily o gene a o s is gi en by he smalles elemen o E( o he usual o de ing in N). Assume ha n > 1 and ha he esul is ue o n−1. Le E⊂Nnbe such ha E+Nn=Nn. We can assume ha Eis non emp y. Le α∈E. Fo any i= 1, . . . , n and j= 0, . . . , αiwe conside he bijec i e mapping φi,j :Ni−1×{j}×Nn−i−→ Nn−1 (β1, . . . , βi−1, j, γi+1, . . . , γn)7−→ (β1, . . . , βi−1, γi+1, . . . , γn) and we deno e Ei,j =φi,j(E∩(Ni−1×{j}×Nn−i)). I is clea ha Ei,j +Nn−1=Ei,j and by he induc ion hypo hesis he e is a ini e subse Fi,j ⊂Ei,j gene a ing Ei,j. The amily F={α}∪∪i,j(φi,j)−1(Fi,j)gene a es E. The p oo abo e is aken om [18]. Rema k. — The p e ious p oposi ion can be eph ased as ollows: Any monomial ideal in k[X]is ini ely gene a ed. This is a pa icula case o he Hilbe basis heo em. In he same way we can see ha any inc easing sequence Eko subse s o Nn, s able unde he ac ion o Nn, is s a iona y. We shall o en use his p ope y called he Noe he ian p ope y o Nn. We can adap he p oo abo e o show ha , gi en E⊂Nnas in he p oposi ion, we can ind in any se o gene a o s, a ini e subse o gene a o s o E. This p o es in pa icula ha in any sys em o gene a o s made o monomials o a monomial ideal o k[X], we can ind a ini e subse o gene a o s. This is Dickson’s lemma. De ini ion 1.4.7. — Le Ibe a non ze o ideal o k[X] ( esp. k[[X]]). A s anda d basis(5) o I, ela i e o L(o L-s anda d basis o I) is any amily 1, . . . , mo elemen s in Isuch ha E(I) = ∪m i=1(expL( i) + Nn). Rema k. — The e always exis a s anda d basis o I, because o he de ini ion o E(I) and 1.4.6. (4)I he o m Lhas a leas one non posi i e coe icien he p e ious o mula de ines a o al o de ing o e Nn, bu no a well o de ing. (5)The no ion o a s anda d basis, in oduced by H. Hi onaka in [21], is simila o he no ion o a G ¨ obne basis, in oduced by Buchbe ge in [13]. We shall come back o his analogy la e . SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 96 F.J. CASTRO-JIM´ ENEZ & M. GRANGER 1.5. He e a e he di isions. — We shall p o e he e ha a s anda d basis o an ideal Iis a sys em o gene a o s o his ideal. Wi h any m-uple (α1, . . . , αm) o elemen s o Nnwe shall associa e a pa i ion(6) ∆1, . . . , ∆m,∆ o Nnin he ollowing way. We se : ∆1=α1+Nn,∆i+1 = (αi+1 +Nn) (∆1∪···∪∆i) i i⩾1, ∆ = Nn (∪m i=1∆i) Theo em 1.5.1. — Le ( 1, . . . , m)be an m-uple o non ze o elemen s o k[[X]] ( esp. o k[X]). We deno e by ∆1, . . . , ∆m,∆ he pa i ion o Nnassocia ed wi h (exp( 1), . . . , exp( m)). Then, o any in k[[X]] ( esp. in k[X]) he e exis s a unique (m+ 1)-uple (q1, . . . , qm, )o elemen s o k[[X]] ( esp. o k[X]) such ha : 1) =q1 1+···+qm m+ , 2) exp( i) + N(qi)⊂∆i, i = 1, . . . , m, 3) N( )⊂∆. I kis ei he Ro Cand i he ia e con e gen powe se ies, hen o any con e gen powe se ies he se ies qiand a e con e gen . Rema k. — The elemen qiin he heo em is called he i- h quo ien and is called he emainde o he di ision o by ( 1, . . . , m). We shall deno e he emainde by ( ; 1, . . . , m). O cou se, he quo ien s as well as he emainde depend on he well o de ing <L. P oo o heo em 1.5.1. — Assume ha wo (m+ 1)-uples, (q1, . . . , qm, ) and (q0 1, . . . , q0 m, 0), sa is y he condi ions o he heo em. We ha e: (1) m X i=1 (qi−q0 i) i+ − 0= 0 I qi6=q0 i hen exp((qi−q0 i) i)∈∆i. I 6= 0 hen exp( − 0)∈∆. Since ∆1, . . . , ∆m,∆ is a pa i ion o Nn, he equali y (1) is only possible i qi=q0 i o any iand i = 0. This p o es he uniqueness in he heo em. We shall now p o e he exis ence. Le us i s conside he polynomial case. Since he se Nnis well o de ed wi h espec o <L, we use an induc ion on uni a y monomials o k[X]. I Xα= 1 (i.e. i α= (0, . . . , 0)), hen ei he exp( i)6= (0, . . . , 0) o any iand in his case i is enough o w i e 1 = Pm i=1 0 i+ 1, o he e exis s an in ege jsuch ha exp( j) = (0, . . . , 0). In his case jis a non ze o cons an .(7) Assume ha jis minimal. We w i e 1 = Pi6=j0· i+ (1/ j) j+ 0. This p o es he esul a he i s s ep o he induc ion. Assume ha he esul is p o ed o any βsuch ha β <Lα. Le jbe such ha α∈∆j. I he e is no such jwe w i e Xα=Pm i=1 0 i+Xα. I (6)We use he wo d pa i ion in a b oad sense, which means ha an elemen o he amily may be emp y. (7)We use he e he ac ha o he well o de ing <L, (0,...,0) is he i s elemen o Nn. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 97 jexis s, le γ∈Nnbe such ha α= exp( j) + γ. We can w i e, Xα=1 cjXγ j+gj whe e cjis he coe icien o he p i ileged monomial o jand all he monomials in gja e smalle (wi h espec o <L) han α. By he induc ion hypo hesis he e exis s (q0 1, . . . , q0 m, 0) sa is ying he condi ions o he heo em o =gj. In pa icula we ha e: Xα=X i6=j q0 i i+1 cj Xγ+q0 j j+ 0. This p o es he esul o α. Thus, exis ence is p o ed o he polynomials. We say ha a polynomial gis L-homogeneous i all i s monomials ha e he same L-deg ee. I is clea in he p oo abo e ha i is L-homogeneous o L-deg ee d∈Qand i iis L-homogeneous o L-deg ee di∈Q( o any i) hen he quo ien qi, i i is non ze o is L-homogeneous o L-deg ee d−di, and he emainde , i i is non ze o is L-homogeneous o L-deg ee d. Assume now ha is a powe se ies. Le us now see he exis ence in ha case, i s assuming ha Lis a posi i e linea o m (see 1.4.5). Any non ze o powe se ies =Pα αXαcan be ep esen ed, in a unique way, as a sum =Pν∈L(N2) νwhe e ν=PL(α)=ν αXαis a L-homogeneous polynomial. By de ini ion (see 1.2.2) we ha e: alL( ) = min{ν| ν6= 0}. Because o 1.4.5 we ha e, o any i: exp( i) = exppCL(in( i)) and we can apply he di ision, in he polynomial case, o in( ) by (in( 1), . . . , in( m)). The e exis s a (unique) (m+ 1)-uple (σ1, . . . , σm, ρ) such ha in( ) = m X i=1 σiin( i) + ρ and sa is ying he condi ions simila o 2) and 3) in he heo em. The ollowing no a ions will be use ul: σi( ) = σi,ρ( ) = ρand o any powe se ies g,bg=g−in(g). We ha e: = in( ) + b = m X i=1 σi( ) i+ρ( ) + b − m X i=1 σi( )b i We in oduce he ollowing no a ion: s0( ) = , s( ) = s1( ) = b − m X i=1 σi( )b i, sj( ) = s(sj−1( )). We ha e: • alL(sj+1( )) > alL(sj( )) o any j. •degL(σi(sj+1( ))) >degL(σi(sj( ))) o any iand any j. •degL(ρ(sj+1( ))) >degL(ρ(sj( ))) o any iand any j. •Fo any i, he se ies X j⩾0 σi(sj( )) SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 104 F.J. CASTRO-JIM´ ENEZ & M. GRANGER we ha e: FHSA(k) = dimkA Amk+1 = #{α∈(Nn E(I)) | |α|⩽k}. P oo . — Le us conside he o mal powe se ies case, he con e gen case being sim- ila . We ha e a na u al isomo phism o ec o spaces A/Amk+1 ≃k[[X]]/(I+mk+1). Fo he o de ing <Lwe ha e he equali y E(I+mk+1) = E(I)∪E(mk+1). Indeed, i is enough o p o e he inclusion E(I+mk+1)⊂E(I)∪E(mk+1), he o he being ob ious. Le ∈Iand g∈mk+1. I al( )< al(g) hen in( +g) = in( ) and hus exp( +g) = exp( )∈E(I). I al( )⩾ al(g) hen al( +g)⩾min{ al( ), al(g)}⩾ al(g)⩾k+ 1. Whence +g∈mk+1. We end he p oo o he p oposi ion by applying 1.5.3. Le us deno e by ℘ he se o he subse s {1, . . . , n}. We in oduce he ollowing no a ions: •Fo each σ∈℘we w i e: –S(σ) = {α∈Nn|αi= 0 i i∈σ} –T(σ) = S({1, . . . , n} σ) – #σ= ca dinal o σ •Fo each non-emp y subse E⊂Nnsuch ha E+Nn=E: –cd(E) = min{#σ|S(σ)∩E=∅} –d(E) = n−cd(E) P oposi ion 1.9.4. — Le ∅6=E⊂Nnbe such ha E+Nn=E. Le σ∈℘be such ha #σ=cd(E). Then he se {α∈T(σ)|(α+S(σ)) ∩E=∅} is ini e. P oo . — We ema k ha he se de ined in he p oposi ion is he complemen o p(E) in T(σ), pbeing he na u al p ojec ion o Nnon o T(σ). Since p(E) is s able by addi ion in T(σ), his complemen could only be in ini e i i con ained a coo dina e axis in T(σ), which would con adic he minimali y o he ca dinal o σ. Le us deno e by eσ(E) he ca dinal o he se de ined in he p e ious p oposi ion and by e(E) he sum e(E) = X #σ=cd(E) eσ(E) Theo em 1.9.5. — Wi h he no a ions abo e we ha e: (1) d(E(I)) = dim(A) (2) e(E(I)) = e(A). P oo . — See [15], [7]. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 105 2. Di ision heo ems in he ings o di e en ial ope a o s 2.1. The aim o his sec ion is o adap he di ision heo ems p o ed in chap e I o he case o he ings o di e en ial ope a o s and o gi e some applica ions: The calcula ion o ee esolu ions, o cha ac e is ic a ie ies and o mul iplici ies. The e e ences a e [11] and [14]. Le kbe a ield o cha ac e is ic ze o. We deno e: •An(k) = k[X,∂] = k[X1, . . . , Xn;∂1, . . . , ∂n] he Weyl algeb a, i.e. he ing o linea di e en ial ope a o s wi h polynomial coe icien s in n a iables. •b Dn(k) = k[[X]][∂] = k[[X1, . . . , Xn]][∂1, . . . , ∂n] he ing o linea di e en ial ope a o s wi h o mal powe se ies in n a iables as coe icien s. •Dn(k) = k{X}[∂] = k{X1, . . . , Xn}[∂1, . . . , ∂n] he ing o linea di e en ial ope a o s wi h con e gen powe se ies in n a iables as coe icien s, i k=Ro Co , mo e gene ally, a comple e alued ield o cha ac e is ic ze o. Fo he sake o b e i y we shall w i e when no con usion is possible: An,c Dn,Dn. We deno e by Rany o hese h ee ings. I Pis an ope a o we de elop i in he ollowing way: P=X (α,β)∈N2n a(α,β)Xα∂β=X β∈Nn β∂β whe e a(α,β)∈k, β∈k[X],k[[X]] o k{X}. We call he ollowing subse o N2n, deno ed by N(P), he New on’s diag am o P: N(P) = {(α, β)∈N2n|a(α,β)6= 0} 2.2. The o de o an ope a o . — We ix a linea o m Lon Q2nwi h non nega i e coe icien s, whose es ic ion L2 o {0}×Qnhas s ic ly posi i e coe icien s. This condi ion is only necessa y in he case o powe se ies coe icien s. De ini ion 2.2.1. — Le 0 6=P∈R=An,c Dno Dn. We de ine he L2-o de o P(and we deno e i by o dL2(P)) as being max{L2(β)| β6= 0}. We se o dL2(0) = −∞. We ha e o dL2(PQ) = o dL2(P) + o dL2(Q) o any ope a o s Pand Q. Fo each k∈L2(Qn), we w i e FL2 k(R) = {P∈R|o dL2(P)⩽k}. The amily FL2 •(R) is an inc easing il a ion o he ing R. Le g L2 k(R) (o , mo e b ie ly, g k(R)) deno e he quo ien FL2 k(R)/FL2 <k(R). We call he mapping σL2 k: Fk(R)→g k(R) he symbol unc ion o o de k. De ini ion 2.2.2. — Le P∈Fk(R) F<k(R). We call σL2 k(P) he L2-p incipal symbol o P. We deno e he L2-p incipal symbol o ∂iby ξi. Thus, σL2 k(P) = PL2(β)=k βξβ. We shall w i e i simply σL2(P). SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 106 F.J. CASTRO-JIM´ ENEZ & M. GRANGER The ing g L2(R) = ⊕ kg L2 k(R) is commu a i e and isomo phic o he ing B[ξ1, . . . , ξn] whe e as he case may be B=k[X],k[[X]],o k{X}. De ini ion 2.2.3. — Le Ibe an ideal (9) o R. We call he ideal o g L2(R), deno ed by g L2(I), gene a ed by {σL2(P)|P∈I} he L2-g aded ideal associa ed wi h I. De ini ion 2.2.4. — Le Ibe an ideal o R. We call he se {(x,ξ)∈k2n|σL2(P)(x,ξ) = 0 o all P∈I}, deno ed by Cha L2(R/I), he L2-cha ac e is ic a ie y o he R-module R/I. When R=Anwe also ha e he possibili y o mixing he a iables Xand ∂: De ini ion 2.2.5 (The L-Be ns ein il a ion). — Le P∈An(k). We call he in ege max{L(α, β)|a(α,β)6= 0} he L-o de o P(and we deno e i by o dL(P)). The L-p incipal symbol o Pis he sum σL(P) = PL((α,β))=o dL(P)a(α,β)Xαξβ. We ha e once again he no ion o g aded ideal associa ed wi h an ideal Io An and he no ion o L-cha ac e is ic a ie y o An/I, o he L-Be ns ein il a ion. On he o he hand when L2(β) = β1+···+βn, he il a ion induced by L2is he usual il a ion by he o de o ope a o s wi h espec o de i a ion a iables. 2.3. O de ings in N2nand he p i ileged exponen o an ope a o . — Le < be a o al well o de ing on N2ncompa ible wi h sums. We de ine an o de ing deno ed by <L, on N2n, in a di e en way acco ding o whe he we a e in Ano wi h powe se ies coe icien s. •In An: (α, β)<L(α0, β0) i and only i        L2(β)< L2(β0) o L2(β) = L2(β0) and L(α, β)< L(α0, β0) o L2(β) = L2(β0), L(α, β) = L(α0, β0) and (α, β)<(α0, β0) This is a o al well o de ing compa ible wi h sums. •In c Dno Dn: (α, β)<L(α0, β0) i and only i        L2(β)< L2(β0) o L2(β) = L2(β0) and L(α, β)> L(α0, β0) o L2(β) = L2(β0), L(α, β) = L(α0, β0) and (α, β)>(α0, β0) De ini ion 2.3.1. — Le P∈An,c Dno Dn. We call he 2n-uple expL(P) = max<L{(α, β)|a(α,β)6= 0}, he L-p i ileged exponen o P. (9)All he ideals unde conside a ion a e le ideals. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 107 Rema k. — We ha e in e e y case he o mula expL(P) = expL(σL2(P)) wi h σL2(P)∈k[X,ξ], k[[X]][ξ] o k{X}[ξ] espec i ely, he wo las ings being seen as sub ings o k[[X,ξ]] o o k{X,ξ}and he p i ileged exponen s being aken in he sense o he i s chap e . Then we can s a e he ollowing p oposi ions which can be p o ed exac ly as in he i s chap e : P oposi ion 2.3.2. — Le P, Q ∈R. We ha e: 1) exp(PQ) = exp(P) + exp(Q). 2) I exp(P)6= exp(Q) hen exp(P+Q) = max<L{exp(P),exp(Q)}. Fo each non ze o ideal Io Rle E<L(I) deno e he se {expL(P)|P∈I {0}}. I no con usion is possible we w i e E(I) ins ead o E<L(I). We ha e, by 2.3.2, E(I) + N2n= E(I) and as we p o e in 1.4.6 we ha e: P oposi ion 2.3.3. — Le E⊂N2nbe such ha E+N2n=E. Then he e is a ini e subse F⊂Esuch ha E=∪(α,β)∈F((α, β) + N2n). De ini ion 2.3.4. — Le Ibe a non ze o ideal o R. We call any amily P1, . . . , Pmo elemen s in Isuch ha E(I) = ∪m i=1(expL(Pi) + N2n), a s anda d basis o I, ela i e o L(o an L-s anda d basis o I) Rema ks 1) The e always exis s a s anda d basis o Iby de ini ion o E(I) and 2.3.3. 2) In he case o Anwe can also conside he L-Be ns ein il a ion, and he ol- lowing o de ing simila o he one gi en in he p eceding chap e up o he change o nin o 2n: (α, β)<L(α0, β0) i and only i L(α, β)< L(α0, β0) o L(α, β) = L(α0, β0) and (α, β)<(α0, β0) 2.4. Mo e di isions. — The s a emen s below na owly ollow hose in he p e- ceding chap e and we shall only gi e he p oo s o he poin s speci ic o he case o he ope a o s. Wi h each m-uple ((α1, β1), . . . , (αm, βm)) o elemen s o N2n, we associa e a pa - i ion ∆1, . . . , ∆m,∆ o N2nin he same way as in chap e I. We se : ∆1= (α1, β1) + N2n,∆i+1 = ((αi+1, βi+1) + N2n) (∆1∪···∪∆i) i i⩾1, ∆ = N2n (∪m i=1∆i). Theo em 2.4.1. — Le (P1, . . . , Pm)be an m-uple o non ze o elemen s o Rand le ∆1, . . . , ∆m,∆be he pa i ion o N2nassocia ed wi h (exp(P1), . . . , exp(Pm)). Then, o any Pin R, he e is a unique (m+ 1)-uple (Q1, . . . , Qm, R)o elemen s in R, such ha : (1) P=Q1P1+···+QmPm+R. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 108 F.J. CASTRO-JIM´ ENEZ & M. GRANGER (2) exp(Pi) + N(Qi)⊂∆i, i = 1, . . . , m. (3) N(R)⊂∆. P oo . — Uniqueness can be p o ed as in he commu a i e case. Fo exis ence, we conside σL2(P) and σL2(Pi) as elemen s o k[X,ξ] ( esp. k[[X,ξ]],o k{X,ξ}), which a e L2-homogeneous wi h espec o he a iable ξ. Le us w i e he di ision in he sense o chap e I, in any o he h ee cases: σL2(P) = m X i=1 qiσL2(Pi) + , he qi(X,ξ) and (X,ξ) being polynomials and L2-homogeneous wi h espec o a iables ξ(since he coe icien s o L2a e s ic ly posi i e). Suppose ha d= o dL2(P) and ha di= o dL2(Pi). Then he deg ees o he quo ien s and o he emainde a e gi en by he ela ions: o dL2(qi) = d−di,o qi= 0,o dL2( ) = do = 0. Le hen Qiand Rbe he ob ious ope a o s such ha qi=σL2(Qi) and =σL2(R) ( o example i qi=PL2(β)=d−dia(α,β)Xαξβ, Qi=PL2(β)=d−dia(α,β)Xα∂β). Then he ope a o P0=P−Pm i=1 QiPi−Ris o L2-o de s ic ly smalle han d. We ema k ha he Qiand Rha e he p ope ies 2) and 3) abo e since qiand ha e he co esponding p ope ies and exp(Pi) = exp(σL2(Pi)). We end he p oo by an induc ion ( ini e since he coe icien s o L2a e >0) on he L2-o de . Rema k. — The elemen Qiin he heo em is called he i- h quo ien and Ris called he emainde o he di ision o Pby (P1, . . . , Pm). The emainde will be deno ed by R(P;P1, . . . , Pm). Rema k. — I ollows om he p oo ha o any di ision P=Q1P1+···+QmPm+R as in he heo em we ha e max{maxi{expL(QiPi)},expL(R)}= expL(P) and as a consequence max{maxi{o dL2(QiPi)},o dL2(R)}= o dL2(P). Rema k. — We ha e a simila (and simple o p o e) di ision heo em in he ing g L2(R) = B[ξ]. We le he eade s a e (and p o e) a di ision heo em in An, ela i e o he L-Be ns ein il a ion. See [14]. Co olla y 2.4.2. — Le Ibe a non ze o ideal o R(o g L2(R)) and le P1, . . . , Pmbe a amily o elemen s o I. The ollowing condi ions a e equi alen s: 1) P1, . . . , Pmis a s anda d basis o I. 2) Fo any Pin R, we ha e: P∈Ii and only i R(P;P1, . . . , Pm) = 0. Co olla y 2.4.3. — Le Ibe a non ze o ideal o R(o g L2(R)) and le P1, . . . , Pmbe a s anda d basis o I. Then P1, . . . , Pmis a sys em o gene a o s o I. These wo s a emen s can be p o ed exac ly as in he commu a i e case. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 109 Rema k. — Le Ibe an ideal o R. Then {P1, . . . , Pm}is a s anda d basis o Ii and only i {σ(P1), . . . , σ(Pm)}is a s anda d basis o g L2(I). 2.5. The calcula ion o a s anda d basis and i s applica ions. — Le P1, P2 be wo ope a o s wi h p i ileged exponen s (α1, β1),(α2, β2). As in chap e I, we call he semisyzygy o P1, P2 he ope a o M1P1−M2P2=S(P1, P2) whe e M1, M2a e wo monomials whose exponen s ν1, ν2a e such ha ν1+ (α1, β1) = ν2+ (α2, β2) and minimal o his p ope y and u he mo e such ha he leading coe icien s sa is y c(M1)c(P1) = c(M2)c(P2) so ha we ge expL(S(P1, P2)) <LexpL(M1P1) = expL(M2P2). We ha e again: P oposi ion 2.5.1. — Le P1, . . . , P be a sys em o gene a o s o he ideal Io Rsuch ha o any (i, j) he emainde o he di ision o S(Pi, Pj)by (P1, . . . , P )is ze o. Then, {P1, . . . , P }is a s anda d basis o he ideal I. P oo . — We deduce he p oo om he esul in he commu a i e case by conside ing he σ(Pi)∈B[ξ1, . . . , ξn], and by using he ac ha Piand σ(Pi) ha e he same p i ileged exponen . I MiPi−MjPj=S(Pi, Pj) = A1P1+···+A P is a di ision, we ha e o dL2(AkPk)⩽o dL2(MiPi) = o dL2(MjPj). We se mi=σ(Mi), ak=σνk(Ak) whe e νk= o dL2(MiPi)−o dL2(Ak), and hen we ge he ela ion: miσ(Pi)−mjσ(Pj) = a1σ(P1) + ···+a σ(P ). This is a di ision in k[X,ξ], k[[X,ξ]] o k{X,ξ}as he case may be. Fu he mo e, i is L2-homogeneous, hence in B[ξ]. Thus, {σ(P1), . . . , σ(P )}gi es a s anda d basis o he ideal which hey gene a e in he abo e ings hence also in B[ξ]. I emains o p o e ha he σ(Pi)’s gene a e g (I). We conside P∈Iand we w i e: P=A1P1+···+A P (∗) I o dL2(P)< δ = max(o dL2(AkPk)), we ha e a1σ(P1) + ··· +a σ(P ) = 0, whe e ak=σδ−o dL2(Pk)(Ak). We deduce om 1.6.4 he ac ha in B[ξ], L2-homogeneous ela ions be ween he σ(Pk) a e gene a ed by hose which come om he di isions o semisyzygies. This allows us o change he ela ion (∗) in o de o lowe δ. We inally ob ain a decomposi ion (∗) o which δ= o dL2(P) in which case we ha e σ(P) = a1P1+···+a P ∈g (I). Le I⊂Rbe an ideal gi en by a sys em o gene a o s P1, . . . , Ps. The p ocess ha we a e going o desc ibe enables us o build a s anda d basis (P1, . . . , Ps, Ps+1, . . . , Ps+ ) by a ini e sequence o di isions. This algo i hm is he analogue o algeb aic di e en ial ope a o s o Buchbe ge ’s [13] (see 1.6.3). SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 110 F.J. CASTRO-JIM´ ENEZ & M. GRANGER •Assume ha (P1, . . . , Ps, Ps+1, . . . , Ps+q) a e al eady buil and de ine Eq= Ss+q k=1(exp(Pk) + N2n). •I he e is (i, j) such ha he emainde o he di ision o S(Pi, Pj) by (P1, . . . , Ps+q) is non ze o, le us choose he i s o hese (i, j) ( o he lexico- g aphic o de ing) and deno e by Ps+q+1 he emainde hus ob ained. Thus we ha e Eq⊂Eq+1 and Eq6=Eq+1 ⊂E(I). •By a Noe he ian a gumen , his p ocess s ops and he e exis s an in ege such ha Es+ = E(I). This can be de ec ed by he lack o a non ze o emainde since hen (P1, . . . , Ps, . . . , Ps+ ) is a s anda d basis. •We can elimina e (one by one) he Pkwhose p i ileged exponen s a e con ained in he N2n-subse gene a ed by he emaining exponen s. Applica ion 1. The calcula ion o he cha ac e is ic a ie y o a R-module o ype R/I P oposi ion 2.5.2. — Le (P1, . . . , P )be a L-s anda d basis o he ideal Io R. Then he equa ions o he L2-cha ac e is ic a ie y o R/I a e: σ(P1)(X,ξ) = ··· =σ(P )(X,ξ) = 0 Indeed he equa ions σ(P)(X,ξ) = 0 o all P∈Ia e linea combina ions o hese equa ions. Applica ion 2. F ee esolu ions o an R-module o ype R/I. — Le (P1, . . . , P ) be a s anda d basis o he ideal Io R. Le Sbe he module o ela ions be ween he ope a o s Pk. This module is he se o -uples R= (A1, . . . , A )∈R such ha A1P1+···+A P = 0. We say ha Ris o o de ki k= max(o dL2(AiPi)) and we se : σk(R) = (σk−d1(A1), . . . , σk−d (A )). Le us deno e he ela ions ollowing om he di ision o semisyzygies by Ri,j and i,j =σ(Ri,j). P oposi ion 2.5.3. — We ha e an exac sequence: D ( +1)/2ϕ −→ D ψ −→ D→D/I wi h: ψ(Q1, . . . , Q ) = Q1P1+···+Q P , ϕ((Ai,j)) = XAi,jRi,j. P oo . — This is equi alen o s a ing ha he ela ions be ween he P`a e gene a ed by he ela ions Ri,j. I Ris such a ela ion, σ(R) = is a homogeneous ela ion be ween he σ(P`), o deg ee k= o dL2(R). By he commu a i e analogue (see 1.6.4), we can w i e =Pλi,j i,j wi h o d(λi,j) + ki,j ⩽kwhe e ki,j = o d(Ri,j). We choose Λi,j ∈Rsuch ha σ(Λi,j) = λi,j. Then, R0=R−PΛi,jRi,j is a ela ion be ween he ope a o s P`o L2-o de < k. We conclude by an induc ion on he L2-o de . Applica ion 3. Elimina ion o a iables in Anand in e sec ion o ideals. — Rename he ec o (x1, . . . , xn, ∂1, . . . , ∂n) as (y1, . . . , yn, yn+1, . . . , y2n) and conside new a i- ables z1, . . . , zn, zn+1, . . . , z2n. Le τbe a pe mu a ion o 2nsymbols and deno e S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 111 zi=yτ(i). Deno e by ρ he in e se o τ. Then Anis isomo phic o he k-algeb a gene a ed by z1, . . . , zn, zn+1, . . . , z2nwi h ela ions [zρ(i), zρ(j)] = 0 (i⩽j) excep o j=i+nin which case [zρ(i), zρ(j)] = −1. Le Ibe a le ideal o Anand kbe an in ege 0 ⩽k⩽2n−1. We deno e by An,k he subalgeb a o Angene a ed by zk+1, . . . , z2n. We de ine Ik=I∩An,k. The (le ) ideal Iko An,k is he se o ope a o s in Iwhich depend only on zk+1, . . . , z2n. We w i e I2n=k∩I. The ideal Ikis called he k- h elimina ion ideal o I. We shall e u n la e o his de ini ion. Using he lexicog aphic o de ing on N2n(as in 1.7.1) we can p o e he ollowing esul s which a e simila as well as he p oo s o hose in 1.7 and 1.8.1. Lemma 2.5.4. — Le Pbe an elemen o An. Then mp<lex (P)is in An,k i and only i Pis in An,k. Theo em 2.5.5. — Le Ibe a le ideal o Anand kan in ege such ha 0⩽k⩽2n. Le Gbe a s anda d basis o he ideal I ela i e o he lexicog aphic o de ing. Le Gk=G∩An,k. Then we ha e: (1) I Gk=∅ hen Ik= (0). (2) I Gk6=∅ hen Gkis a s anda d basis o he ideal Ik ela i e o he lexicog aphic o de ing. Le I, J be wo le ideals o An. Le θbe a new inde e mina e. We deno e by Ie ( esp. Je) he ex ension o he ideal I( esp. J) o he ing An[θ] (he e θis a cen al elemen ). I his an elemen o k[θ] we deno e by hIe( esp. hJe) he p oduc o he ideals(10) (h) and Ie( esp. (h) and Je). Wi h hese no a ions we ha e: Theo em 2.5.6. — Le I, J be wo le ideals o An. Then I∩J= (θIe+(1−θ)Je)∩An. Rema k. — The heo y o s anda d bases can be easily gene alized o he case o sub–modules o RN, see [14]. Fo ha pu pose we only ha e o adap he no ions o o de ing and o p i ileged exponen s o exponen s in N2n×{1, . . . , N}. By apply- ing his o he calcula ion o a s anda d basis o ke (ϕ) and hen o he successi e ke nels, we build a ee esolu ion o any R-module Mo ini e p esen a ion, whence o example a ealiza ion o he complex o solu ions and o he De Rham complex RHomR(M,O) and ΩnL ⊗M. This is algo i hmic in he algeb aic case. 2.6. An example: The cha ac e is ic cycle o O[1/ ] o a quasihomo- geneous in wo a iables.— In his example we a e dealing wi h he o m L2(i, j) = i+j. In his case and mo e gene ally in he case o he diagonal o m L2on Qn, we e e o [25, 19] o he de ini ion o he mul iplici y a a poin o he co angen space. The cha ac e is ic cycle o a cohe en D-module is he linea (10) hese a e ideals o he ing An[θ] SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 112 F.J. CASTRO-JIM´ ENEZ & M. GRANGER combina ion o he i educible componen s o he cha ac e is ic a ie y, each coun ed wi h i s mul iplici y a a gene ic poin . Le ∈C[x, y] be a quasi-homogeneous polynomial. We deno e by w1and w2 he weigh s o a iables and by χ he Eule ec o ield: χ=w1x∂x+w2y∂y We ha e χ( ) = . We e i y ha O[1/ ] = D·1 , because he Be ns ein polynomial o has no ⩽−2 in ege oo (see [31]). I is easie o deal wi h he quo ien O[1/ ]/O and we ind ha he annihila o ideal o i s gene a o c`(1 ) is he ideal gene a ed by he ollowing h ee ope a o s: •P1= 0 y∂x− 0 x∂y •P2=w1x∂x+w2y∂y+ 1(= χ+ 1) •P3= Le us i s conside he case =yp+c1xq1yp−p1+···+ckxkq1yp−kp1+··· wi h q1> p1⩾1, p= 0 o 1(mod p1) and w2= 1/p,q1w1=p1w2. In his si ua ion we e i y by compu ing he semisyzygies ha {P1, P2, P3}is a s anda d basis o he o de ing (o se ies ype) associa ed wi h L(j, i, β, α) = j+i+ α+β he monomial wi h he same L-o de being u he o de ed by y > x > ∂y> ∂x. The p i ileged exponen s a e espec i ely: (p−1,0,0,1),(0,1,0,1),(p, 0,0,0). By applying 1.9 we can compu e he mul iplici y a he o igin o O[1/ ]/Owhich is he e o e (p−1)+0+p+0+0+0 = 2p−1. The cha ac e is ic cycle has he ollowing o m: sT∗ 0(C2) + 1.T∗ −1(0)(C2) o some in ege s. The mul iplici y o −1(0) a he o igin being pwe ge om his 2p−1 = s+ 1.p, o : s=p−1. Fo he case =x·g, whe e gis a polynomial as in he p e ious case we e e o [9]. 3. Gene alized di ision heo ems. The calcula ion o slopes The e e ence o his chap e is [2] o he case o he Weyl algeb a. We deno e by Rany o he ings An,Dno c Dn. 3.1. O de s and il a ions wi h espec o a smoo h hype su ace. Le Ybe a hype su ace o Cnde ined by x1= 0. Gi en a linea o m L(a, b) = pa +qb on Q2(wi h non nega i e and ela i ely p ime in ege coe icien s p, q), we de ine he L-o de along Yo P=P(x, ∂) in Rdeno ed by o dL(P), as he maximum o L(|β|, β1−α1) o (α, β) in he New on diag am o P. To sho en we w i e he e x ins ead o Xand ∂ins ead o ∂. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 113 No ice ha he e Lis a linea o m on Q2whe eas in he p e ious chap e s his le e was used o deno e a linea o m on Q2nwhose pa is now aken by: e L(α, β) = L(|β|, β1−α1) = (p+q)|β|−q(β2+···+βn+α1). Le FL,•(R) be he il a ion induced by he L-o de on Ri.e. FL,k is he se o ope a o s Psuch ha o dL(P)⩽k. Le F( esp. V) deno e he il a ion associa ed wi h he linea o m L(a, b) = a( esp. L(a, b) = b). By ex ension we also w i e F ( esp. V) o he co esponding linea o ms. I L6=F, V he g aded ing associa ed wi h his il a ion g L(R) = L k∈Z FL,k(R)/FL,k−1(R) is isomo phic o one o he g aded commu a i e ings C[x, ξ] = C[x1, . . . , xn, ξ1, . . . , ξn] o C{x2, . . . , xn}[x1, ξ1, . . . , ξn] o C[[x2, . . . , xn]][x1, ξ1, . . . , ξn] whe e he deg ee o he monomial xαξβis L(|β|, β1−α1). I L=F, he il a ion FL,•is he il a ion by he o de o ope a o s. The g aded ing g V(R) is isomo phic o one o he ings An, C{x2, . . . , xn}[x1, ∂1, . . . , ∂n] o C[[x2, . . . , xn]][x1, ∂1, . . . , ∂n] whe e he deg ee o he monomial xα∂βis β1−α1. Gi en an ideal Io Rle g L(I) be he g aded ideal associa ed wi h he il a ion induced by FL,•on I. The ideal g L(I) is gene a ed by he se {σL(P)|P∈I}whe e σL(P) is he p incipal symbol o Pwi h espec o L. By de ini ion, i L6=V, σL(P) = X L(|β|,β1−α1)=o dL(P) pα,βxαξβ. I Lis he o m V, he symbol o Pwi h espec o Vis he di e en ial ope a o σV(P) = X β1−α1=o dV(P) pα,βxα∂β. No ice ha o L6=V, (α, β)→L(|β|, β1−α1) is a linea o m whose coe icien s on he βia e all s ic ly posi i e. Wha ollows wo ks in he same way o any amily o linea o ms o his ype o which he a iables αiha ing non-posi i e coe icien s a e ixed and o which o dL([P, Q]) <o dL(P)+o dL(Q) whence g L(R) is commu a i e. We shall no w i e his gene aliza ion. In he case o an ideal o An, he ollowing lemma shows how o deal wi h he ideal gene a ed by Iin Dn(o in c Dn) and con e sely: Lemma 3.1.1. — Le Ibe an ideal in An. Then g L(DnI) = g L(Dn) g L(I). Mo e p ecisely, i F={P1, . . . , P }is a sys em o gene a o s o Isuch ha G={σL(Pi)} i=1 gene a es g L(I), hen Ggene a es g L(DnI)o e g L(Dn). Rema k. — We shall see la e ha such a amily Fcan be calcula ed e ec i ely s a ing om a sys em o gene a o s o he ideal I. P oo . — See [2]. The same esul is alid in c Dn. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 120 F.J. CASTRO-JIM´ ENEZ & M. GRANGER Recall, o end his chap e , he p ocess which allows us o de e mine hese idealis ic slopes. O cou se his is an algo i hm only in he case o An. •We de e mine an L-s anda d basis {P1, . . . , P }o Iwhe e Lis Fo a p e iously de e mined slope. We make su e ha i is also a V-s anda d basis o g L(I). •We de e mine he o m L(1) wi h minimal slope >slope(L) such ha one o he σL(1) (Pi) is no bihomogeneous. P ecisely L(1) is he linea o m wi h smalles slope g ea e han slope(L) appea ing in he (F, V )-New on diag am o he ope a o s Pi. •By a ini e di ision p ocess we can decide whe he one o he bihomogeneous componen s o one o he σL(1) (Pi) is no an elemen o g L(1) (R). In his case L(1) is a new idealis ic slope. In he o he case we can modi y Piin o de o elimina e L(1), and ob ain a basis which is s anda d o Land o L(1). We p o e in [2] ha his ype o cancella ion can happen only a ini e numbe o imes be o e we come upon a new slope o upon V. 3.6. Examples o calcula ions o slopes Example 1. — In his example we conside he di ec image o he DC-module DCe1/ k, by an imme sion in C2and he slopes ela i e o a hype su ace angen o he suppo . The ad an age o his example is ha we can ca y ou all he calcula ions in many cases and ha i shows idealis ic slopes which a e no slopes. Fo k∈Nwe w i e: M=DCe1/ k≃DC DC( k+1∂ + 1),N=i+M≃DC2 DC2( k+1∂ + 1) + DC2u whe e iis he imme sion C→C2gi en by i( ) = (0, ). We wan o calcula e he slopes o Nalong he cu e m+u= 0. We ca y ou he change o a iables: u=x−ym, =y. We ha e: ∂u=∂x, ∂ = ∂y+mym−1∂x. We hen ind ha Nis he quo ien o DC2by he ideal Igene a ed by he ollowing ope a o s: •P0 1=yk+1∂y+myk+m∂x+ 1 •P2=ym−x We hen ha e o look a he slopes along x= 0. In wha ollows we say ha he slope is −p/q i L(a, b) = pa +qb. Subexample 1.1: m= 1. — We ind he slope −k. We a e in he same si ua ion as o he calcula ion o he slope o Nalong = 0. This is also a pa icula case o he ollowing. Subexample 1.2: k=mp. — We hen ind he slope −p. We ha e: P0 1=ymp+1∂y+mym(p+1)∂x+ 1 = (y∂y−mp)ymp +m∂xxp+1 + 1 = (y∂y−mp)xp+mxp+1∂x+m(p+ 1)xp+ 1 (mod DP2) This gi es he p esen a ion o N: S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 121 •P1=mxp+1∂x+xpy∂y+mxp+ 1 •P2=ym−x We choose he o de ing <F o which he a iables a e o de ed as ollows: ∂x> ∂y> y > x. We hen ind ha (P1, P2) is an F-s anda d basis. Indeed he emainde o he di ision o : S(P1, P2) = ymP1−mxp+1∂x by {P1, P2}is ze o. In his s anda d basis he p i ileged exponen o Piis also he p i ileged exponen o σV(σF(Pi)), so ha by looking a P1we can say ha he −p is an idealis ic slope. Fo ha we e i y ha σL(P1) = mxp+1ξ+xpyη +mxp+ 1 is no bihomogeneous (11). This las esul comes o example om he ac ha 1/∈g V(g F(I)) (N6= 0). The e is no o he slope because o any L0o slope >−p, g L0(I) = (1). Subexample 1.3: m= 2 and k= 2n−1. — By changing P0 1=y2n∂y+ 2y2n+1∂x+ 1 modulo P2=y2−x, as in he p eceding subexample, we ind I=DP1+DP2, wi h: P1= 2xny∂x+xn∂y+ 1. The i s semisyzygy S(P1, P2) = yP1−2xn∂xP2di ided by {P1, P2}gi es a e- mainde P3whence he ollowing gene a o s o I: •P1= 2xny∂x+xn∂y+ 1 •P2=y2−x •P3= 2xn+1∂x+xny∂y+y+ 2xn We ind ha he emainde s o he di isions o S(P1, P3) = xP1−yP3and S(P2, P3) = −2xn+1∂xP2+y2P3by {P1, P2, P3}a e ze o. Thus his is an F-s anda d basis which is also as in he p eceding subexample a V-basis o g F(I). Le Lbe he linea o m co esponding o he i s e en ual slope, he slope −n. We ha e σL(P3) = 2xn+1ξ+xnyη+y. I is impossible ha y∈g V(g F(I) because Iwould con ain he wo elemen s o o de 0, y2−xand y+xφ(x, y) and Nwould be suppo ed by he o igin. Thus we ha e poin ed ou an idealis ic slope o he ideal I. I is no a slope o DC2/I because he equa ions o he L-cha ac e is ic a ie y a e: •σL(P1) = 2xnyξ = 0 •σL(P2) = y2= 0 •σL(P3) = 2xn+1ξ+xnyη +y= 0 and he associa ed educed a ie y is bihomogeneous wi h equa ions y=xξ = 0. Le us se Fi=σL(Pi) and look o a V-s anda d basis o g L(I). Le us ema k ha he p i ileged exponen o F3has changed and is now he monomial y. The semisyzygy S(F1, F3) = −F1+ 2xnξF3gi es he emainde : F4= 4x2n+1ξ2+ 2x2nyξη, and we ind ha all he o he emainde s a e ze o, so ha (11)He e Lis he linea o m o slope −p SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 122 F.J. CASTRO-JIM´ ENEZ & M. GRANGER {F1, F2, F3, F4}is a V-s anda d basis o g L(I). We can li he p eceding semisyzygy in D, as −P1+ 2xn∂xP3, and his gi es he ope a o : P4= 4x2n+1∂2 x+ 2x2ny∂x∂y+ 4(n+ 2)x2n∂x+ 2nx2n−1y∂y−xn∂y+ 4nx2n−1−1. The amily {P1, P2, P3, P4}is he e o e a sys em o gene a o s o Iwhich gi es a s anda d basis o g V(g L(I)) and by P4we poin ou he linea o m L0o slope −(n−1/2). By he algo i hm abo e, i is enough o e i y ha σL0(P4) = 4x2n+1ξ2+ 2x2nyξη −1 is no in g V(g L(I)), which amoun s o ind ou ha 1 /∈g V(g L(I)) = (2xnyξ, y2, y, 4x2n+1ξ2+ 2x2nyξη). Finally, he e is no o he slope because o any o m L00 o slope >−(n−1/2) we ha e 1 = −σL00 (P4)∈g L00 (I). Example 2: De1/(yp−xq)(wi h G. B e e ). — To apply ou algo i hm o he de e mi- na ion o he slopes o he D-module gene a ed by e1/(yp−xq), i is necessa y o know he annihila o in Do he unc ion e1/(yp−xq). The answe o his las ques ion was gi en by J. B ian¸con and Ph. Maisonobe in [12] in he mo e gene al case whe e is quasi-homogeneous wi h an isola ed singula i y. The annihila o is: D( χ + 1) + D(∂ ∂x ∂ ∂y −∂ ∂y ∂ ∂x) whe e χis a ec o ield such ha χ( ) = . Thus we ha e: I= AnnD(e1/(yp−xq)) = D(P1, P2) wi h P1=pyp−1∂x+qxq−1∂y, P2=qyp+1∂y−pxq+1∂x−2qxqy∂y+pq Fu he mo e, we ake as an L-o de ing (whe e Lis a linea o m on Q2wi h a ional posi i e coe icien s) he o de ing on N4de ined as ollows: (i, j, α, β)<L(i0, j0, α0, β0)⇐⇒                  L(α+β, α −i)< L(α0+β0, α0−i0) o L(α+β, α −i) = L(α0+β0, α0−i0) and i+j > i0+j0 o    L(α+β, α −i) = L(α0+β0, α0−i0), i+j=i0+j0 and (α, β, j, i)<lex(α0, β0, j0, i0) 1. The calcula ion o a s anda d basis o I o he o m L=F. — I q > p, we ha e: mpF(P1) = pyp−1∂xand mpF(P2) = qyp+1∂y. Le us se hen ∆1= (0, p−1,1,0)+N4 and ∆2= ((0, p + 1,0,1) + N4) ∆1. The syzygy ela i e o P1and P2is equal o: S(P1, P2) = qy2∂yP1−p∂xP2. The emainde o he di ision gi es a hi d ope a o P3=p2xq+1∂x2+ 2pqxqy∂x∂y+q2xq−1y2∂y2 +p2(q+ 1)xq∂x+q2(p+ 1)xq−1y∂y−p2q∂x. P oposi ion 3.6.1. — Fo 2⩽p < q,{P1, P2, P3}is an F-s anda d basis o I. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 123 P oo . — We ha e mpF(P3) = p2xq+1∂x2and ∆3= ((q+1,0,2,0)+N4) (∆1∪∆2). We hen ha e o p o e ha he emainde s o he di isions by (P1, P2, P3) o he semisyzygies S(P1, P3) and S(P2, P3) a e ze o. We ind i s : S(P1, P3) = pxq+1∂xP1−yp−1P3 = (p2xq+1yp−1∂x2+pqx2q∂x∂y+pq(q−1)x2q−1∂y) −p2xq+1yp−1∂x2+ 2pqxqyp∂x∂y+q2xq−1yp+1∂y2+p2(q+ 1)xqyp−1∂x +q2(p+ 1)xq−1yp∂y−p2qyp−1∂x =··· = ((pq −2q−p)xq−2qxqy∂y+pq)P1−qxq−1∂yP2. Le us deno e by Q1= (pq −2q−p)xq−2qxqy∂y+pq he quo ien ela i e o P1in his di ision. We mus now deal wi h he semisyzygy S(P2, P3) = p2xq+1∂x2P2−qyp+1∂yP3. Ins ead o di ec ly applying he di ision algo i hm we a e going o use he abo e equali ies: yp−1P3=Q1P1+qxq−1∂yP2 p∂xP2= (qy2∂y−(p−1)qy)P1−P3 and we deno e by Q0 1=qy2∂y−(p−1)qy he quo ien ela i e o P1. Thus we ha e on one hand: qy2(∂yyp−1−(p−1)yp−2)P3= (qy2∂y−(p−1)qy)yp−1P3 = (qy2∂y−(p−1)qy)(Q1P1+qxq−1∂yP2) and he ob ained quo ien s o P1and P2a e allowed o he di ision. We ha e, on he o he hand: p2xq+1∂2 xP2=pxq+1∂x(Q0 1P1−P3) and he ob ained quo ien s a e allowed o he di ision. This shows ha S(P2, P3) has by di ision a ze o emainde . 2. The calcula ion o he slopes. — Le us d aw i s he New on polygons associa ed wi h he ope a o s P1, P2, P3: 6 V - F (1,0) • (1,−q+1) • N(P1) 6 V - F (1,0) •• (1,−q) • N(P2) 6 V - F (1,1) BBBB • (2,−q+1) • • N(P3) SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 124 F.J. CASTRO-JIM´ ENEZ & M. GRANGER Since {P1, P2, P3}is an F-s anda d basis o Iand a sys em o gene a o s o I, he ideal g F(I) is gene a ed by he p incipal symbols σF(P1) = pyp−1ξ+qxq−1η σF(P2) = qyp+1η−pxq+1ξ−2qxqyη σF(P3) = p2xq+1ξ2+ 2pqxqyξη +q2xq−1y2η2 P oposi ion 3.6.2. — (σF(Pi))1⩽i⩽3is a V-s anda d basis o g F(I). P oo . — We ha e mpV(σF(P1))=pyp−1ξ, mpV(σF(P2))=qyp+1ηand mpV(σF(P3))= p2xq+1ξ2. The di isions by σF(P1), σF(P2), σF(P3) gi e: S(σF(P1), σF(P2)) = qy2ησF(P1)−pξσF(P2) = σF(P3)≡0, S(σF(P1), σF(P3)) = pxq+1ξσF(P1)−yp−1σF(P3) =−2q2xqyησF(P1)−qxq−1ησF(P2)≡0, S(σF(P2), σF(P3)) = p2xq+1ξ2σF(P2)−qyp+1ησF(P3)) =−2q2xqy3η2σF(P1)−q2xq−1y2η2σF(P2)−pxq+1ξσF(P3)≡0. This p o es he p oposi ion. We now ha e o conside he linea o m L(L < F) wi h he g ea es possible slope such ha one o he p incipal symbols o one o he Piis no bihomogeneous. We ha e: L(a, b) = qa +b(slope equal o −q). P oposi ion 3.6.3. — The D-module De1/(yp−xq)has only he slope −qalong he hy- pe su ace x= 0. P oo Fi s s ep: −qis a slope. We know (see [2]) ha i L < Λ< F , hen g Λ(I) = g V(g F(I)). I g L(I) was bihomogeneous, hen i would also be equal o g V(g F(I)). Bu by he p e ious p oposi ion, we ha e: g V(g F(I)) = (σV(σF(P1)), σV(σF(P2)), σV(σF(P3)))C{y}[x, ξ, η] = (pyp−1ξ, qyp+1η, p2xq+1ξ2+ 2pqxqyξη +q2xq−1y2η2)C{y}[x, ξ, η] Since σL(P3) = σV(σF(P3)) −p2qξ and σV(σL(P3)) = −p2qξ i is he e o e enough o p o e ha ξ6∈ g V(g F(I)): his can be seen by w i ing ξas a linea combina ion o σV(σF(Pi)) hen by e alua ing a x=y= 0 (we ind hen ξ≡0!). Second s ep: he e is no o he slope. Take V < L0< L. Le us show ha L0is no a slope. We ha e: (σL0(P1), σL0(P2), σL0(P3))C{y, x}[ξ, η]⊂g L0(I) S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 125 ha is (yp−1ξ, yp+1η, ξ)⊂g L0(I) and so Cha L0 (De1/(yp−xq))⊂ {y=ξ= 0}∪{η= ξ= 0}. Thus his cha ac e is ic a ie y is o dimension 2 and: qg L0(I) = (ξ, y) o (ξ, η) o (ξ, yη). pg L0(I) is he e o e bi-homogeneous and L0is no a slope. 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