EXTENDED BÉZOUT IDENTITIES

Transcription

1 _ XTNDD ÉZOUT IDNTITIS A Quadra Schl f Mahemaics Uiversiy f Leeds Leeds LS2 9T Uied Kigdm fax: quadraamsaleedsacuk Keywrds: Mulidimesial sysems primeess exeded ézu ideiies -flaess algebraic aalysis Absrac We sudy primeess f mulidimesial crl sysems defied i erms f algebraic prperies f - mdules ad shw hw pass frm e aher by iversi f a cerai We use hese resuls deermie effecively exeded ézu ideiies f mulidimesial crl sysems ad he miimal umber f caied i 1 Irduci I [13] i has bee shw ha primeess f mulidimesial sysems defied by a full rw rak marix wih eries i are liked wih exeded ézu idei- such ha ies ie he exisece f a marix ad " $# where # is he ideiy marix Hwever ly w differe ypes f primeess ZLP ad MLP have bee defied i [13] which crrespd he case caiig &( variables ad a plymial T my kwledge hig has bee de fr he her cases uil he wrk f Obers [] surely because he cmplexiy f he marices icreases wih he umber & The mai cribui f [] has bee he irduci f algebraic aalysis cceps [2] i he hery f mulidimesial sysems Fllwig a idea f Malgrage i is shw i [] hw assciae wih ay mulidimesial sysem a fiiely preseed - mdule The he auhr shws ha ZLP (resp MLP crrespds a prjecive (resp rsi-free -mdule ad he defies a ew ype f primeess WZLP which crrespds caiig e variable I [12] i is shw ha fr a mulidimesial crl sysem defied by a full rw rak marices here exis a e--e crrespdece bewee he umber f i ad a chai f & differe ype f primeess defied by he dimesi f he algebraic variey frmed by he zers f all he maximal rder mirs f (his chai icludes ZLP WZLP ad MLP $>=? Fially usig he classificai f Palamdv-Kashiwara - [2] i is shw i [8] ha fr he rig f differeial perars wih cefficies i a differeial field / ( if is a csa field here are e--e equivaleces bewee he chai f algebraic prperies f (rsi-free 5 reflexive 5 5 pr- jecive he umber f successive paramerizais f he mulidimesial crl sysem he idex f he firs -zer 78:9 A fr C where is he raspsed mdule f ad ex is he exesi fucr ad i he case f a full rw rak marix he dimesi f he characerisic variey GFIH ad he ype f primeess baied i [12] if 4 i addii I his paper we shw hw pass frm a eleme f he previus equivale chais aher e by meas f he iversi f a cerai plymial i K caiig mre r less L I paricular i is shw hw pass frm a ype f primeess aher e by lcalizai f a plymial ad hw bai effecively exeded ézu ideiies fr each ype f primeess 2 Mdules ad xesi fucr I he curse f his paper we shall e MN O where P4Q r R Defiii 1 [11] Le The: =D be a fiiely geeraed -mdule S 0UT is a free -mdule if fr a cerai V?XWZY S 0UT is a prjecive -mdule if [] fr a cerai -mdule ad VP(W^Y S is a reflexive -mdule if he -mrphism defied by _Z` bacfedgf Fdhf >i kj j>i Ll j A l i mfdgf m is a ismrphism S is a rsi-free -mdule if i prqsuwv N S is a rsi -mdule if X i Therem 1 [11] We have he fllwig asseris: Szy 77 5 {Ad} 7 9 ~ g e78~ƒh7 5 9 dha ~ dg P y 77 S G If he ay rsi-free -mdule is free S Ay prjecive -mdule is free (Quille-Susli [14] x

2 Y 0 Y 0 0 & x Defiii 2 We have he fllwig defiiis (see eg [11] fr mre deails: S A prjecive reslui (resp free reslui f a - mdule is a exac sequece f he frm 1a where -mrphism S If a a a Ia a (1 is a prjecive (resp free -mdule ad : is a is defied by a prjecive reslui (1 he he defecs f exacess f is de- ly deped Therefre: where Fdgf ad ` j j fied by l j ad (1 They are called 7 8:9 A 7 8:9 7 Fdgf 7 8:9 7 ~ f a (2 Remark 1 If is a fiiely geeraed -mdule he has a fiie free reslui $# a a " Ia a A Ia a & where is a marix wih eries i ad ` a is defied by leig perae a rw vecr f legh he lef f defied by ( bai a rw vecr f legh ` a is defied by leig perae a cl- where um vecr f legh he righ f vecr f legh The we have: 789 A (3 The (2 is # 7 ~ f bai a clum l Defiii 3 If is a -mdule defied by he fllwig fiie preseai a a a (4 =D is he -mdule de- 7 ie is he -mdule defied by he he is raspsed mdule fied by d fllwig fiie preseai: - We easily verify ha fr ay fiiely geeraed -mdule we have: =D>=? 3 h (5 Prpsii 1 Le be a fiiely preseed -mdule ad a muliplicaive se f he we have: =D / =D 0 Prf Takig he esr prduc he exac sequece f -mdules [11]: 21 X f (4 we bai : Ia a a =D The d 7 ~>= is defied by he fiie preseai defied i Figure 1 We have Fedgf =? X h ad ~A= X: 4 ~A= because ad are w fiiely geeraed -mdules [11] Mrever if we ake he esr we bai he fllwig exac se- prduc f (5 by quece: =D 84 7 Fially we have he cmmuaive exac diagram defied i Figure 2 which prves he prpsii Therem =? 2 If is a fiiely geeraed -mdule ad he we have: $ :9 2 is a rsi-free -mdule iff is a reflexive -mdule iff 7 8:9 4 is a prjecive -mdule iff 7 8:9 $ g & A gc Prf See [9] fr he prves f 1 ad 2 We have he fllwig exac sequece?1a 7 8:9 u a a D a 78:9F u1a (see [8] ad is refereces fr a prf which prves 3 A algebraic prf f 4 ca be easily adaped frm he prf f Crllary 4 i [8] Defiii 4 S The grade f a -mdule is defied by: G f IH ~ LK q ex v ux$ S We call dimesi f a -mdule f Hg (wih he cvei ha h [11] Therem 3 [1 2] Le he we have: G ` & X ( he Krull dimesi be a fiiely geeraed -mdule ~ y v LK ~ y h

3 x # ( G 0 DC G =D / Fdhf = / 4 7 8I Fdgf = / =D / =D Fdgf Figure 1: xac sequece / Fdhf 84 7 X 3 Mai resuls 31 Geeral case Defiii 5 Le be a fiiely geeraed -mdule ad he we defie: =D f IH ~ G $ q 789 A v ux Figure 2: Cmmuaive exac diagram ƒ ƒ &3 glk Remark 2 The ai is jusified by he fac ha ly depeds up a prjecive equivalece ad hus 7 8:9 $ ly depeds [10] Mrever by Therem 2 v C 0LK reflexive We shall dee by prjecive he grup f permuais f & elemes Therem 4 Le be a fiiely geeraed -mdule ad fr all m : 4 &w 4 1 :LK x is he muli- The fr all ieger here exiss such ha ( " hh where ( plicaive se frmed by #" " ad ( ($ I paricular #" fr all p here exiss such ha ($ #" is a prjecive ( -mdule LK Prf Firs f all le us ice ha if r he he resul is rivial (ake g f he prf we suppse h &m X g &m 4-2 ha is say ad fr &O 2 Y Therefre we have [11]: 7 8:9( & ~ f (7 (8 I he fllwig ad e: Xl G g : g is a fla -mdule ad is fiiely preseed he we l have G [11]: 7 8:9( 7 8:9( A Mrever Hece we bai 7 8:9( Ll G ie is a prjecive -mdule Fially he righ member f he ismrphism i (9 fr G cmbied wih he fac ha 7 8:9( is a rsi -mdule [9] fr G l implies ha we have G : ( ` Hg 7 8:9( v Fr G le us ake ( XH 7 8:9( A ad defie: We have -0/ Y 1 1 Y32 54 ($ 7 8 :9 ( ad: 78:9( we have: = 78:9( ( ($ ( 0 ($ 7 8:9( =? " ($ Therefre fr y Therem 2 ad Prpsii 1 (ie #" ($ we bai: ( w #" he ( If we ake #" & ( -mdule xample 1 Le us csider he 4 defied by he marix?> A > (9 is a prjecive?> -mdule

4 # > > v > 0 i & x crrespdig he curl perar i Q > fllwig free reslui f Thus we have he crrespds he diver- is defied by We easily check ha we have P a > 1a > a Ia a P D?> where he marix =D gece perar The he -mdule > > 78:9 78:9 4 > 78:9( pl G where U C w 0C Thus we bai e^ Mrever 78:9 A 4 > by he fllwig equais > l ad we verify ha ]m : Hg 78:9 2 u e we have: 1 > ad is defied ad hus -mdule Fi- is a pr- 78:9 7 8:9 Mrever we have 7 8:9( l G which implies ha is a prjecive ally if #" we e he jecive " " " -mdule where wih hh x y Therem 1 is a free mdule ad we easily verify ha a basis is give by where saisfies ad Cz we have C - because Remark 3 Le us ice ha Therem 4 des predic he miimal umber f idepede variables i he plymial Ideed i he previus example we ly eed iver 4 which cais jus e idepede variable whereas frm Therem 4 we ly kw ha we have iver a plymial i w variables The ex herem gives a mre precise saeme he miimal umber f i Lemma =D 1 Le be a fiiely geeraed -mdule ad The is a prjecive :LK -mdule 0LK iff is a prjecive -mdule ie Prf We have he fllwig exac sequece If is prjecive he his exac sequece splis [7 11] ad we bai ha is prjecive Thus is sill prjecive [11] Mrever we have 7 8:9 7 8:9 A because is prjecive hus usig he exac sequece ( we bai ha is prjecive Chagig i we bai he cverse resul which prves he lemma Therem 5 Le be a fiiely geeraed 2 32 =D -mdule ad: 1 LK The fr all ad r here exiss such ha we have (8 where i (7 #" I paricular here exiss ha $ #" is a prjecive -mdule Prf If is defied such is prjecive he he resul is rivial Le us suppse ha is a prjecive -mdule The by Lemma 1 we have # & The -mdule has a prjecive reslui f he frm: Ia Y a 1a f ~ H / Ia a Ia $ q 78:9 A v ux Usig he fac ha we bai by dualiy he fllwig exac sequece where d 7 Y 7 8:9 Y Y d 7 Y Le us e Frm (10 we deduce ha: Applyig Therem 4 such ha: Hece we deduce ha prves (8 Thus fr 7 8:9 Y #" 0 $ 7 8:9 Y 0 $ 7 8:9 07 8:9 $ $ $ $ 07 8:9 A l l he we have: (10 here exiss A#" $ which xample 2 If we ake agai xample 1 we easily shw ha h ad Thus here exiss # such ha is ak " prjecive - mdule We have see i xample 1 ha Therem 5 predics ha here exiss caiig jus e variable which gives a aswer Remark 3 xample 3 If 4 4 > is he 2 > -mdule defied by he gradie C perar hec we easily check ha ad Therefre here " exiss #" such ha is a prjecive -mdule We le he reader check by himself ha we ca chse ad

5 v > Remark 4 If & he fllwig he prf f Therem 4 we bai ha he ideal # defied by # H 7 8:9 A is pricipal fr every ad Thus up a csa f here exiss a uique lwer degree plymial such ha # ad / H " 8 :9 This is exacly he case fr xamples 2 ad 3 32 Paricular case Lemma 2 [5] If fiie preseai Ia is a -mdule defied by he fllwig a he is prjecive iff LK Ia a =? $ 078:9 A (11 ie Therem If is a -mdule defied by he exac sequece (11 he we have: G ^ h (12 ie: &w K h v 3 Prf If is prjecive he Lemma 1 shws ha LK LK $ If prjecive Lemma 2 shws ha 07 8:9 Xv This shws he firs equaliy f (12 =D Mrever is defied by a full rak marix he -mdule ad hus 7 8:9 Fially we bai: f H ~ L q 7 8:9 A G g v x y Therem 3 we have her equaliies f (12 ad hus is ie is a rsi which prves he xample 4 Le be he -mdule defied by he marix > We easily verify ha 7 8:9 >h 7 8:9 7 8:9 A A A # > where is defied i xample 1 Therefre G ad by Therem we bai ( ad he exisece f wih " such ha # # " is a prjecive -mdule wih " hh x (we ca ake Crllary 1 Le & be a full rak marix ( wih eries i =D ad he here exis ad ad XmW Y such ha we have he fllwig exeded ézu ideiies fr all p : 1 # C 2 ^ # # Prf Applyig Therem he here exiss such ha is a prjecive ad hus free $ -mdule by Therem 1 Therefre here exiss a ismrphism ` $ a $ Usig he fac ha is a fla -mdule [11] he we bai he fllwig cmmuaive exac diagram: a $ a " 4 8 " $ 1a a " 4 8 #" a " Le us call he marix crrespdig ~ i he " caical basis f " ad $ he we bai he fllwig spliig exac sequece [7 11]?1a where " Ia =$ $ Ia =$ " a ad " are marices wih eries i Chasig heir demiars we fially fid he ideiies 1 ad 2 Nice ha &m is give here by (12 Defiii A -mdule has pure dimesi if as ay f is -zer submdule have dimesi as well Therem 7 If is a fiiely MN geeraed - mdule which saisfies { he: 1 78:9 Y A &m L :9 Y 7 8:9 Y 3 if v he has pure dimesi & : Prf The fac ha { N meas ha here exiss a prjecive reslui f a Y y defiii $ Ia f IH ~ which meas ha 78:9 $ 78:9 Y v 7 8:9 Y a 2 A q 7 8:9 f he frm: a a a v x fr ie we have he exac sequece: A Y $ ad

6 & 0 ad by The- 1 We ca apply Therem 3 he -mdule 7 8:9 Y A bai: 78:9 Y G 7 8:9 Y A 7 8:9 Y A g h h 2 We have 7 8:9 Y 078:9 rem 2 we have $ 07 8:9 A which shws ha: 07 8:9 Y 789 Y 3 If v has pure dimesi &( xample 5 I xample 3 we have see ha > C saisfies ha U { ha >$ > 07 8:9 7 8:9 A A fac ha ca be prved direcly ce icig ha Crllary 2 If v he has pure dimesi &m he by Therem 710 f [1] we bai ha ad hus Therefre by Therem 7 we bai has pure dimesi 0 a is defied by he exac sequece (11 ad Le us ice ha Therem 7 ad Crllary 2 are als rue if /2Ah where is a differeial field [7 8] 4 Cclusi very resuls i his paper are effecive by meas f Gröber basis: we ca cmpue a fiie free reslui f a fiiely preseed -mdule ad by dualiy 7 8:9 A ad 789 A fr ad deermie Mrever he prves f Therem 4 ad 5 are ally csrucive: we firs cmpue 78:9 $ $ A fr ad heir aihilars Hg 7 8:9 A The by meas f echiques f elimiai we ca deermie expliciely # $ H 7 8:9 A l fially fid ]p Mrever exeded ézu ideiies as well as geeralized iverses baied i [7] ca be effecively baied fllwig he lie f [7] See als [12] fr cmpuaial aspecs y lack f space we jus give e applicai f he resuls baied i 2 his paper I he case f differeial delay sysems ie Therems 5 ad give a effecive mehd deermie he plymials irduced i [4] d mi plaig Hwever belgs he subgrup f permuais f he &D firs variables f This remark ad Therem 5 shw ha a sysem saisfyig ^ is -fla where 2 ]p ad & fr a sysem defied by a full rw rak marix T fiish le us ice ha Crllary 1 shws ha ca be cmpleed a square marix whse deermia divides a pwer f (if &] he is he greaes cmm divisr f he mirs f by Remark 4 See [3] fr relaed quesis Ackwledgemes This wrk was suppred by he gra HPMF-CT Refereces [1] jrk - (1979 Rigs f Differeial Operars Nrh-Hllad Mahemaical Library [2] rel A ad al (1987 Algebraic D-mdules Academic Press [3] Li Z ad se N K (2000 Sme cjecures mulivariae plymial marices 2 d Ieraial Wrkshp Mulidimesial Sysems (NDS pp [4] Fliess M ad Muier H (1998 Crllabiliy ad bservabiliy f liear delay sysems: a algebraic apprach SAIM COCV vl 1 pp [5] Kuz (1985 Irduci Cmmuaive Algebra ad Algebraic Gemery irkhaüser [] Obers U (1980 Mulidimesial Csa Liear Sysems Aca Applicadae Mahemaica vl 20 pp [7] Pmmare F ad Quadra A (1998 Geeralized ézu Ideiy Applicable Algebra i gieerig Cmmuicai ad Cmpuig vl 9 pp [8] Pmmare F ad Quadra A (1999 Algebraic aalysis f liear mulidimesial crl sysems IMA f Crl ad Ifrmai vl 1 pp [9] Pmmare F ad Quadra A (2000 A fucrial apprach he behaviur f mulidimesial crl sysems 2 d Ieraial Wrkshp Mulidimesial Sysems (NDS pp 91-9 [10] Pmmare F ad Quadra A (2000 quivaleces f liear crl sysems prceedigs f MTNS 2000 Perpiga Frace [11] Rma (1979 A Irduci Hmlgical Algebra Academic Press [12] Wd Rgers ad Owes D (1998 Frmal hery f marix primeess Mahemaics f Crl Sigal ad Sysems vl 11 pp [13] Yula D C ad Gavi G (1979 Nes & - dimesial sysem hery I Tras Circuis Sysems vl 2 pp [14] Yula D C ad Pickel P F (1984 The Quille-Susli herem ad he srucure f & -dimesial elemeary plymial marices I Tras Circuis Sys vl 31 pp