Jounal of lgeba Numbe heo: dance and pplcaon Volume 5 Numbe 6 Page -9 alable a hp://cenfcadance.co.n DOI: hp://d.do.og/.86/janaa_7686 MRIX COMPUIONS ON PROJCIV MODULS USING NONCOMMUIV GRÖBNR BSS CLUDI GLLGO Depaameno de Maemáca Semnao de Álgeba Conuca-SC Unedad Naconal de Colomba Sede Bogoá Colomba e-mal: cmgallegoj@unal.edu.co bac Conuce poof of fac ha a abl fee lef S-module M wh ank( M ) ( S) fee whee (S) denoe he able ank of an aba ng S wee deeloped n [7] (ee alo [5] and [5]). ddonall n uch pape ae peened algohmc poof fo calculang pojece dmenon and o check whehe a lef S-module M abl fee. Gen a lef -module M wh a bjece kew PBW eenon we wll ue hee eul and Göbne bae heo o eablh algohm ha allow u o calculae effecel he pojece dmenon fo h module o check whehe abl fee o conuc mnmal peenaon and o oban bae fo fee module. Mahemac Subjec Clafcaon: Pma: 6Z5; Seconda: 6D 65. Kewod and phae: noncommuae Göbne bae kew PBW eenon abl fee module fee module compuaon of bae conuce algohm. Receed Jul 6 6 Scenfc dance Publhe
CLUDI GLLGO. Inoducon he Göbne bae heo pode u wh a emakable aa of ool fo he effece calculaon of dee algebac objec. We hae deeloped h heo fo kew PBW eenon (ee [5] and [9]) whch n un ha enabled u ca ou calculaon n a boad cla of noncommuae ng. In addon gen an aba lef -module M a bjece kew PBW eenon h Göbne bae heo along wh he eence of ma conuce eul ha allow o eablh algohm fo calculang he pojece dmenon of M o check whehe o no M abl fee and o oban effecel a ba when M a abl fee module wh ank( M ) ( S) wll enable u o peen houghou cuen acle effece algohm and compuaon of h knd fo module defned on kew PBW eenon. he ke ool fo hee algohm wll be he calculaon of lef and gh nee mace. Relaed o he compuaon of pojece dmenon we hae ha he followng noable geneal fac oe aba ng. S wll epeen an aba noncommuae ng. heoem.. Le M be an S-module and fm Pm fm fm f f f Pm Pm P P M (.) be a pojece eoluon of M. If m and hee e a homomophm gm : uch ha g f hen we hae he followng Pm Pm pojece eoluon of M: m m hm hm fm f f f Pm Pm Pm Pm P P M wh P m fm h : m hm : [ fm ]. gm (.)
MRIX COMPUIONS ON PROJCIV Poof. See [5] Popoon. Coolla.. Le M be an S-module and S fm f f f f f m S m m S m m S S M be a fne fee eoluon of M. Le bae m. hen (.) F be he ma of f n he canoncal m () If m and hee e a homomophm g : S S m uch ha g f hen we hae he followng fne fee eoluon of M: m m S m m h m hm fm S m S m m S m S M (.) f f wh h In a ma noaon f j fm : hm : [ fm ]. g m m G m he ma of g m and H j he ma of h n he canoncal bae j m m hen H m : m [ F G ] H :. m m m F () If m and hee e a homomophm : g S S uch ha g f hen we hae he followng fne peenaon of M: S S h h S M (.5) wh f h : h : [ f ]. g
CLUDI GLLGO In a ma noaon H : [ F G ] H :. f Poof. See [5] Coolla. Wh epec o abl feene he followng chaacezaon hold. heoem.. Le M be an S-module wh eac equence S f S f M ae equalen:. hen M ( M S) and he followng condon S () M abl fee. () M pojece. () M. () F ha a gh nee. () f ha a lef nee. Poof. See [5] heoem..5. Fnall egadng fee module we nclude below a ma conuce chaacezaon. Lemma.. Le S be a ng and M be a abl fee S-module gen b f f a mnmal peenaon S S M. Le g : S S uch ha g f. hen he followng condon ae equalen: S () M fee of dmenon. I () hee e a ma U GL ( S) uch ha UG whee G he ma of g n he canoncal bae. In uch cae he la column of U confom a ba fo M. Moeoe he f column of U confom he ma F of f n he canoncal bae.
MRIX COMPUIONS ON PROJCIV 5 () hee e a ma V GL ( S) uch ha he f column of V.e. V of GL ( S). Poof. See [5] Lemma..5. G concde wh G can be compleed o an neble ma Some defnon and elemena popee ae necea n wha follow. hee can alo be eewed n [7]. If S an aba ng S denoe he lef S-module of column of ze ; fo each f S-homomophm S S hee a ma aocaed o f n he canoncal bae of S and S denoed F : m( f ) and dpoed b column.e. F M ( S); moeoe f a S hen f ( a ) ( a F ). I aghfowad o how ha funcon m : HomS ( S S ) M ( S) bjece; and ha f g p S S a homomophm hen he ma of G gf n he canoncal bae m ( gf ) ( F ). hu f : S S an omophm f and onl f F GL ( S) and f C M ( S ) we hae ha column of C confom a ba of S f and onl f C GL ( S). When S commuae o when we conde gh module nead of lef module we hae ha f ( a) Fa and n uch cae he ma of a compoe homomophm gf gen b m ( gf ) m ( g) m( f ). Fuhe f S S : an omophm f and onl f F GL ( S); bede C GL ( S) f and onl f column confom a ba of S (ee Subecon. n [7]). Now le S be a ng; we a ha S afe he ank condon ( RC ) f fo an nege f S S an epmophm hen. Fuhemoe S an IBN ng (naan ba numbe) f fo an nege S S f and onl f. I well known ha RC mple IBN. Fom now on we wll aume ha all ng condeed n he peen pape ae RC. We hae he he followng elemena chaacezaon fo ( RC ) ng. f
6 CLUDI GLLGO Popoon.5. Le S be a ng. () S RC f and onl f gen an ma F M ( S) he followng condon hold: f F ha a gh nee hen. () S RC f and onl f gen an ma F M ( S) he followng condon hold: Poof. C.f. [7] Popoon. f F ha a lef nee hen.. Skew PBW enon In h econ we ecall he defnon of kew PBW (Poncaé- Bkhoff-W) eenon defned fl n [6] and we wll eew alo ome bac popee abou he polnomal nepeaon of h knd of noncommuae ng. wo pacula ubclae of hee eenon ae ecalled alo. Defnon.. Le R and be ng. We a ha a kew PBW eenon of R (alo called a σ -PBW eenon of R) f he followng condon hold: () R. () hee e fne elemen R-fee module wh ba uch ha a lef n α α α n n Mon( ) : { n n α ( α α ) N }. () Fo ee n and R { } hee e R {} c uch ha c R. (.)
MRIX COMPUIONS ON PROJCIV 7 () Fo ee j n hee e c j R { } uch ha j c j j R R Rn. (.) Unde hee condon we wll we σ( R). : n pacula cae of kew PBW eenon when all deaon δ ae zeo. nohe neeng cae when all σ ae bjece and he conan c j ae neble. We ecall he followng defnon (cf. [6]). Defnon.. Le be a kew PBW eenon. (a) qua-commuae f he condon () and () n Defnon. ae eplaced b ( ) Fo ee uch ha n and R { } hee e R {} c c. (.) ( ) Fo ee j n hee e c j R { } uch ha j c j j. (.) (b) bjece f σ bjece fo ee neble fo an < j n. n and c j emakable pope afe b kew PBW eenon peened below whch a he ame me aue u ha he algohm ued fnh. Popoon. (Hlbe Ba heoem). Le be a bjece kew PBW eenon of R. If R a lef (gh) Noehean ng hen alo a lef (gh) Noehean ng. Poof. See [] Coolla.. Snce he objec uded n he peen pape ae kew PBW eenon necea o guaanee he IBN and RC popee fo hee ng. Fo h we hae he followng mpoan fac:
8 CLUDI GLLGO heoem.. Le be a kew PBW eenon of a ng R. hen RC ( IBN ) f and onl f R RC ( IBN ). Poof. See [8] heoem.9. Remak.5. We deeloped he Göbne bae heo fo an bjece kew PBW eenon. Specfcall we eablhed a Buchbege algohm fo hee ng he compuaon of zge module a well a ome applcaon a membehp poblem calculaon of neecon quoen peenaon of a module compung fee eoluon he kenel and mage of an homomophm (ee Chape 5 and Chape 6 n [5] o [9]).. Compung he Inee of a Ma In h econ we wll peen an algohm ha deemne f a gen ecangula ma oe a bjece kew PBW eenon lef neble and n uch cae he algohm compue one of lef nee. mla algohm wll be conuced fo he gh de cae. We a wh he followng elemena fac abou lef neble mace. Popoon.. Le F be a ecangula ma of ze wh ene n a ng S. If F ha lef nee hen. Moeoe F ha a lef nee f and onl f he module geneaed b he ow of F concde wh S. Poof. he f aemen follow fom he fac ha we ae aumng he S RC (ee Popoon.5). Now uppoe ha F ha a lef nee L M ( S).e. LF. Defne he followng S-homomophm: I f : S S l : S S a ( a F ) b ( b L )
MRIX COMPUIONS ON PROJCIV 9 hen m ( f ) F and m ( l ) L. Whence m ( f l ) ( LF ) I I.e. f an epmophm. Hence Im( f ) S.e. he lef ubmodule geneaed b he ow of F concde wh he fee module S. Coneel uppoe ha he module geneaed b he ow of F concde wh S hen fo f defned a aboe hee e S a a uch ha f ( a ) e fo each and whee e e denoe he canoncal eco of S. hu f a [ a a a ] we hae [ a a a ] F a F() a F() e a F whee F ( j) denoe he j-h ow of F j. heefoe f L he ma whoe ow ae he eco a hen LF.e. F ha a lef nee. Coolla.. Le be a bjece kew PBW eenon and le F M ( ) be a ecangula ma oe. he algohm below deemne f F lef neble and n he poe cae compue a lef nee of F: I
CLUDI GLLGO lgohm fo he lef nee of a ma INPU: ecangula ma F M ( ). OUPU: ma L M ( ) afng LF I f e and n ohe cae. INIILIZION: IF < RURN IF le G : { g g } be a Göbne ba fo he lef ubmodule geneaed b ow of F and le { } e be he canoncal ba of. Ue he don algohm o ef f e G fo each. IF hee e ome e uch ha e G. RURN IF G le H M ( ) wh he pope G H F and conde K : [ k j ] M whee he k j ae uch ha e k g k g k g fo. hu I K G. RURN L : K H. ample.. Le σ( Q) defned hough he elaon. Gen he ma F
MRIX COMPUIONS ON PROJCIV we appl he aboe algohm n ode o ef f F ha a lef nee. Fo h we compue a Göbne ba of he lef module geneaed b he ow of F. Condeng he degle ode on Mon() wh and he OPRV ode on Mon( ) wh e > e a Göbne ba fo F { e e}. In conequence F ha a lef nee and fom calculaon obaned dung he poce of Buchbege algohm we hae ha L a lef nee fo F. Coolla.. Le F be a quae ma of ze wh ene n a ng S. hen F neble f and onl f he ow of F confom a ba of S. Poof. Le L M ( ) uch ha LF I FL. Fom LF I follow ha he ow of F geneae S. Le f and l be lke n he poof of Popoon.; nce FL I hen l f and heefoe S f a monomophm.e. Sz ( F ). hu he ow of F ae lneal ndependen and h complee he f mplcaon. Coneel nce he ow of F geneae S b Popoon. F ha a lef nee. Le L be a uch nee hen LF I. We hae FLF F h mple ha ( FL I ) F bu Sz ( F ) hen FL.e. F L. I
CLUDI GLLGO Coolla.5. Le be a bjece kew PBW eenon and F M ( ) a quae ma oe. he algohm below deemne whehe F neble and n he poe cae compue he nee of F: INPU: quae ma F M ( ). lgohm fo he nee of a quae ma OUPU: ma L M ( ) afng LF I FL f e and n ohe cae. INIILIZION: Ue he algohm n Coolla. o deemne f F lef neble. IF F no lef neble RURN LS Compue Sz ( F ) IF Sz ( F ) RURN LS Compue he mace H and K n he algohm of Coolla.. RURN L : K H. ample.6. Fo h eample we conde he adde analogue of he Wel algeba. Recall ha f k a feld hen k -algeba n ( q q n ) geneaed b n n and ubjec o he elaon: j j n j j j j j j q n whee q k {}. I no dffcul o how ha n ( q q n ) a bjece kew PBW eenon. We ake n q and ; q on Mon() we conde he degle ode and oe Mon( ) he OPRV ode wh e > e. Le F be he followng ma:
MRIX COMPUIONS ON PROJCIV F. We wan o check f column of F confom a ba fo and we know ha h ue f and onl f F neble. Ung he aboe algohm we a efng f F ha a lef nee; fo h pupoe we compue a Göbne ba of he lef -module geneaed b he ow of F.e. of he lef -module Im(F). Ung he Buchbege algohm fo module poble o how ha G { f f f } a Göbne ba e fo h module whee f e e f e and f e e e e. pplng he don 9 algohm we can check ha e G heefoe G. hu F ha no a lef nee and hence he column of F ae no a ba fo. Remak.7. If S a lef (o gh) Noehean ng hen ee epmophm α : S S an auomophm (ee Popoon. n []). h mple ha ee lef (o gh) Noehean ng WF (ee []). heefoe o e f F M ( S) neble enough o how ha F ha a gh o a lef nee. So n he aboe algohm when a bjece PBW eenon of a LGS ng no necea he compuaon of Sz S ( F ) o e whehe he ma neble would be uffcen o appl he algohm fo he lef nee gen n Coolla.. Now we wll conde he gh nee of a ecangula ma. We a wh he followng heoecal eul: Popoon.8. Le F be a ecangula ma of ze wh ene n he ng S. If F ha gh nee hen and he module of zge of he ubmodule geneaed b he ow of F zeo.e. Sz ( F ). In he ohe wod f F ha a gh nee hen he ow of F ae lneal ndependen.
CLUDI GLLGO Poof. o begn nce we ae aumng ha S RC (Popoon.5). Le L M ( S) uch ha FL. Conde he homomophm f and I l a n Popoon. hen f a monomophm. Hence ke ( f ).e. Sz ( F ). Popoon.9. Le F be a ecangula ma of ze wh ene n he ng S. If F ha gh nee hen. Moeoe F ha a gh nee f and onl f Sz ( F ) and Im ( F ) a ummand dec of S whee Im ( F ) denoe he module geneaed b he column of F.e. he module geneaed b he ow of F. Poof. o begn nce we ae aumng ha S RC (Popoon.5). Now le L M ( S) uch ha FL. Conde he homomophm f and l a n Popoon. hen l f.e. f a pl monomophm. hu S Im( f ) ke( l ) and Im ( f ) a dec ummand of S. Coneel le M be a ubmodule of ha S Im( f ) M. So gen I S S uch f S hee e unque elemen f Im( f ) and f M uch ha f f f. Defne he homomophm l S S : a l ( f ) h whee h S uch ha : f f ( h f ). B hpohe Sz ( F ) o f njece and we ge ha f l well defned. I no dffcul o how ha l an S-homomophm. Conequenl FL I.e. F ha a gh nee. l f and f : m( l ) S f L hen Remak.. If we had a compuaonal ool fo o check f a ubmodule of a fee module a ummand dec hen Popoon.9 would eablh an algohm o check he eence of a gh nee.
MRIX COMPUIONS ON PROJCIV 5 Followng [] and [5] conde a ma F : [ f ] M ( ) wh j whee a bjece kew PBW eenon endowed wh an noluon θ.e. a funcon θ : S S uch ha θ ( a b) θ( a) θ( b) θ ( ab) θ( b) θ( a) and θ fo all a b S. Noe ha θ ( ) and S hence θ an an-omophm of S. We defne ( F ) [ θ( f )]. ha f K M ( ) hen θ Obee : j θ ( FK ) θ( K ) θ( F ). (.) Popoon.. Le be a bjece kew PBW eenon endowed wh an noluon θ and le F : [ f ] M ( ) j wh. hen F ha a gh nee f and onl f fo each j e j whee G a Göbne ba of he lef -module geneaed b he column of θ ( F ) and { e j } j he canoncal ba of. Poof. G : [ g ] M ( ) a gh nee of F f and onl f j FG I and h equalen o a ha G e j f f f g j f f f g j j ; applng θ we oban e j θ( f ) θ( f ) ( ) ( ) θ f θ f θ( g j ) θ( gj ). θ( f ) θ( f )
6 CLUDI GLLGO hu G a gh nee of F f and onl f he canoncal eco of belong o he lef -module geneaed b he column of θ ( F ).e. e e θ( ). Le G be a Göbne ba of θ ( F ) hen G a F gh nee of F f and onl f fo each j e. G j Coolla.. Le be a bjece kew PBW eenon and F M ( ) be a ecangula ma oe. he algohm below deemne f F gh neble and n he poe cae compue he gh nee of F: lgohm fo he gh nee of a ma INPU: n noluon θ of ; a ecangula ma F M ( ). OUPU: ma H M ( ) afng FH I f e and n ohe cae. INIILIZION: IF < RURN IF le G : { g g } be a Göbne ba fo he lef ubmodule geneaed b column of θ ( F ) and le { } e j j be he canoncal ba of. Ue he don algohm o ef f e G fo each j. j IF hee e ome e j uch ha e j G. RURN IF G le J M ( ) wh he pope G J θ( F ) and conde K : [ k j ] M whee he k j ae uch ha e j k j g k j g kj g fo j. hu I K G. RURN H : θ( J ) θ( K ). Poof. pplng (.) we ge I K G K J θ( F ) θ( θ( K )) θ( θ( J )) θ( F ) θ( θ( J ) θ( K )) θ ( F ) θ( Fθ( J ) θ( K )) o I θ( Fθ( J ) θ( K )) θ( I ) and fom h we ge Fθ( J ) θ( K ). I
MRIX COMPUIONS ON PROJCIV 7 ample.. Le u conde he ng ( ) Q σ wh. Ung he aboe algohm we wll compue a gh nee fo F poded ha e. Fo h we conde he noluon θ on gen b ( ) θ and ( ). θ Wh h noluon we hae ha ( ). θ hu ( ). θ F We a compung a Göbne ba fo he lef module geneaed b he column of ( ). F θ We ge ha { } e e G a Göbne ba fo ( ). F θ In h cae F ha a gh nee and J uch ha ( ). F J G θ Snce I G hen I K and ( ) J L θ : a gh nee fo F whee ( ). θ J o fnd noluon of kew PBW eenon a dffcul ak o he aboe algohm no paccal. econd algohm fo eng he eence and compung a gh nee of a ma ue he heo of
8 CLUDI GLLGO Göbne bae fo gh module deeloped n [5]. Fo h we wll made a mple adapaon of Popoon. and Coolla. fo gh ubmodule ung he gh noaon. Popoon.. Le F be a ecangula ma of ze wh ene n a ng S. If F ha gh nee hen. Moeoe F ha a gh nee f and onl f he gh module geneaed b he column of F concde wh S. Poof. he f aemen follow fom Popoon.5. Now uppoe ha F ha a gh nee and le L be a ma uch ha FL I. Defne he followng homomophm of gh fee S-module: f : S S l : S S a Fa b Lb hen m ( f ) F and m () l L. Whence m ( f l) FL.e. f an epmophm. heefoe Im( f ) S.e. he gh ubmodule geneae b column of F concde wh he fee module S. Coneel f Im( F ) S hen fo f defned a aboe hee e a a S uch ha f ( a ) e fo each and whee e e denoe he canoncal eco of. S hu f [ a a a ] j j j j I a we hae [ ] ( ) ( a a a F a F ) Fa j F j j j j aj e j whee ( j F ) denoe he j-h column of F j. So f L he ma whoe column ae he eco a j hen FL I.e. F ha a gh nee. hu condeng he eul abou Göbne bae fo gh module (ee [5]) we hae he followng alenae algohm fo eng he eence of a gh nee.
MRIX COMPUIONS ON PROJCIV 9 Coolla.5. Le be a bjece kew PBW eenon and F M ( ) be a ecangula ma oe. he algohm below deemne f F gh neble and n he poe cae compue a gh nee of F: lgohm fo he gh nee of a ma INPU: ecangula ma F M ( ). OUPU: ma L M ( ) afng FL I f e and n ohe cae. INIILIZION: IF < RURN IF le G : { g g } be a gh Göbne ba fo he gh ubmodule geneaed b column of F and le { } e j j be he canoncal ba of. Ue gh eon of don algohm o ef f e G fo each. IF hee e ome e j uch ha e j G. RURN IF G le H M ( ) wh he pope G FH and conde K : [ k j ] M whee he k j ae uch ha e j g k j g k j gkj fo. hu I GK. RURN L : HK. ample.6. Conde he ng σ( Q) wh and le F be he ma gen b F. pplng he gh eon of Buchbege algohm we hae ha a Göbne ba fo he gh module geneaed b he column of F G { e e }. Fom Coolla.5 we can how ha F ha a gh nee; moeoe one gh nee fo F gen b
CLUDI GLLGO L.. Compung he Pojece Dmenon Gen M an S-module and f f f f f f P P P P M (.) a pojece eoluon of M no dffcul o how ha f he malle nege uch Im ( f ) pojece hen doe no depend on he eoluon and pd(m) (c.f. [5] heoem..). heefoe we can conde a fee eoluon { f } whch we can calculae ung he ome of he applcaon of Göbne bae heo. Hence b heoem. we oban he followng algohm whch compue he pojece dmenon m of a module M gen b a fne e of geneao whee a bjece kew PBW eenon of a LGS ng R (lef Göbne oluble ee [5] and [9]) wh fne lef global dmenon. Noe ha lef Noehean (heoem.) and lgld ( ) < (ee []). Pojece dmenon of a module oe a bjece kew PBW eenon lgohm INPU: lgld( ) < M f f wh f k. OUPU: pd(m). INIILIZION: Compue a fee eoluon { f } of M :. WHIL lgld( ) DO: IF Im ( f ) pojece HN pd(m). LS. m k
MRIX COMPUIONS ON PROJCIV Obee ha n he peou algohm we no need o compue fne fee eoluon of M; an fee eoluon compued ung zge enough. Ne we peen anohe algohm fo compung he lef pojece m dmenon of a module M gen b a fne fee eoluon: f fm fm fm f f m m m M. (.) h algohm uppoed b Coolla.. Pojece dmenon of a module oe a bjece kew PBW eenon lgohm INPU: n -module M defned b a fne fee eoluon (.). OUPU: pd(m). INIILIZION: Se canoncal bae. j : m and H j : Fm wh F m he ma of f m n he WHIL j m DO: Sep. Check whehe o no H j adm a gh nee G j : (a) If no gh nee of (b) If hee e a gh nee Sep. j : j. () If j hen pd(m). () If j hen compue (.5). H e hen pd(m) j. j G of j () If j hen compue (.). H and j ample.. Le be he ng σ ( Q) whee. We wll calculae he pojece dmenon of he lef module M ( ) ( ) ( ) ( ). Fo h we ue he degle ode on Mon() wh
CLUDI GLLGO and he OP ode oe Mon( ) wh e > e. Ung Göbne bae poble o how ha a fee eoluon fo M gen b: F F F M whee F F F. In ode o appl he aboe algohm we a checkng whehe F [ ] ha a gh nee. aghfowad calculaon how ha a gh nee fo (.5): F G [ ] o we compue H 5 H M (.) whee H : and H :. o ef f H ha a gh nee we mu calculae a Göbne ba fo he gh module geneaed b he column of H. Snce he ng condeed a bjece kew PBW eenon we can ue he gh eon of Buchbege algohm. Fo h we conde he degle ode on Mon() wh and he OP ode oe Mon( ) wh e < e < e. pplng h algohm we oban he followng Göbne ba fo
MRIX COMPUIONS ON PROJCIV H G {( ) ( ) ( ) ( ) ( )}. Noe ha e no educble b G hu G and hence H doe no hae a gh nee. heefoe pd(m). Remak.. he aboe algohm can be ued fo eng f a gen module M pojece: We can compue pojece dmenon and hence M pojece f and onl f pd(m). 5. e fo Sabl-Feene heoem. ge a pocedue fo eng abl-feene fo a m module M gen b an eac equence f f M whee a bjece kew PBW eenon. e fo abl-feene lgohm INPU: M an -module wh eac equence f f M. OUPU: RU f M abl fee FLS ohewe. INIILIZION: Compue he ma F of f. IF F ha gh nee HN RURN RU LS RURN FLS ample 5.. Le σ( Q) wh. We wan o know f he lef -module M gen b M e e e e e e e e e e abl fee o no. o anwe h queon we a compung a fne peenaon of M. Condeng he degle ode on Mon() wh
CLUDI GLLGO he OP ode on Mon( ) wh e > e > e > e and ung he mehod eablhed n he peou econ we hae ha a em of geneao fo Sz(M) gen b S {( ) ( ) ( )}. heefoe we ge a followng fne peenaon fo M: F 6 F M (5.) whee F : and F :. pplng he mehod fo compung he zg module we hae ha Sz ( F ) o he peenaon obaned n (5.) become Fnall we mu o e f F 6 F M. F ha a gh nee. Fo h we calculae a Göbne ba fo he gh module geneaed b he column of F. Ung he OP ode on Mon( ) wh e > e > e a Göbne ba fo F gen b G { f } whee f he -h column of 7 F fo 6 and f 7 e e. Noe ha fo eample e G 6 o ha G. hu F ha no gh nee and hence M no abl fee.
MRIX COMPUIONS ON PROJCIV 5 Remak 5.. Fom heoem. f M a lef -module wh eac f f equence M S Im( f ) and M f : S S hen M ( M ) whee he homomophm of gh fee S-module nduced b he ma F. hu fo eng f M abl fee we can ue he eul of Subecon 5.6 n [5] and compung a Göbne ba fo he gh module geneaed b column of F. Ung he gh eon of he don algohm poble o check whehe S Im( F ). If h la equal hold hen M and M abl fee. Coolla. ge anohe pocedue fo eng abl-feene fo a m module M gen b a fne fee eoluon (.) wh S : Indeed f m and f m ha no lef nee hen M non abl fee; f f m ha a lef nee we compue hen he new fne fee eoluon (.) and we check f h m ha a lef nee. We can epea h pocedue unl (.5); f h ha no lef nee hen M no abl fee. If h ha a lef nee hen M abl fee. ample 5.. Le be he ng σ ( Q) whee and conde he lef module M ( ) ( ) ( ) ( ) gen n he ample.. we aw hee a fne peenaon fo M gen b: H 5 H M (5.) whee H : and H :.
6 CLUDI GLLGO In uch eample we howed ha M no a abl fee module. H ha no a gh nee hence 6. Compung Mnmal Peenaon m If M a abl fee module gen b he fne fee eoluon (.) wh S hen he Coolla. ge a pocedue fo compung a mnmal peenaon of M. In fac f m hen f m ha a lef nee (f no pd(m) m bu h mpoble nce M pojece). Hence we compue he new fne peenaon (.) and we wll epea he pocedue unl we ge a fne peenaon a n (.5) whch a mnmal peenaon of M. ample 6.. Le u conde agan he ng σ( Q) wh. Le M be he lef -module gen b peenaon Im( F ) whee F. Regadng he degle ode on Mon() wh and he OP ode oe Mon( ) wh e > e we hae ha Sz ( F ) geneaed b ( ). So he followng eac equence obaned: F F π M whee : [ ]. F Noe ha F ha a gh nee: G ; fom Coolla. we ge he followng fne peenaon fo M: hu h h M (6.)
MRIX COMPUIONS ON PROJCIV 7 wh H [ F G ] and [ f ]. ha h In ample.6 we howed H ha a gh nee; moeoe one gh nee fo H L. In conequence (6.) a mnmal peenaon fo M and M un ou o be a abl fee module. 7. Compung Fee Bae In [7] and [5] peened a ma conuce poof of a eul due Saffod abou abl fee module. heoem 7.. Le S be a ng. hen an abl fee S-module M wh ank( M ) ( S) fee wh dmenon equal o ank(m). Poof. See heoem n [7]. In he poof of uch affmaon he followng fac necea: Popoon 7.. Le S be a ng and : [ ] be an unmodula able column eco oe S hen hee e U ( S) uch ha U. e Poof. B compleene we nclude he poof of h fac (ee Popoon 8 n [5]). hee e elemen a a S uch ha : c ( ) Um ( S) wh : a. (7.)
CLUDI GLLGO 8 Conde he ma ( ); : S a a a (7.) hen ( ). Snce ha ( ) ( ) : S Um c hee e S b b uch ha b and hence ( ). b Le ( ) : b and ); ( : S (7.) hen ( ). Moeoe le ( ); : S (7.)
MRIX COMPUIONS ON PROJCIV 9 hen ( ). Fnall le : ( S); (7.5) hen e and U : ( S). Fo an effece calculaon of a ba of M we a eablhng an algohm fo o calculae he elemena ma U n he Popoon 7.: lgohm fo compung U n Popoon 7. INPU: n unmodula able column eco [ ] oe S. OUPU: n elemena ma U M ( S) uch ha U. DO:. Compue a a S uch ha (7.) hold.. Compue he ma gen n (7.).. Calculae he elemen b b S wh he pope ha b wh a fo.. Defne : ( ) b fo and compue he mace and gen n (7.)-(7.5). RURN: U :. We wll lluae below h algohm. ample 7.. Fo h eample we conde he quanum Wel algeba ( ). Recall h k -algeba geneaed b he aable J a b wh he elaon (dependng upon paamee a b k ): e
CLUDI GLLGO a b a a ab b b. When a b we hae ha ( J ) ( k) fo an feld k (ee [] fo moe popee). I no dffcul o how ha ( ) σ(k J a b [ ]). ake k Q a and b. hu he elaon n h ng ae gen b:. ( ( J )) wll denoe he goup geneaed b all elemena mace of ze oe ( J ). hen [ ] Le [ ] u uch ha u wheeb Umc ( ( J )). Moeoe he column eco [ ] ha a lef nee u [ ] o a able unmodula column. In h cae a a a and he ma gen b
MRIX COMPUIONS ON PROJCIV. Wh h elemena ma we ge [ ]. If we defne ( )( ) ( ) : : and we oban [ ]. Fnall f we defne ( ( )) J ( ( )) J and ( ( )) : J U hen we hae. e U
CLUDI GLLGO he poof of heoem 7. allow u o eablh an algohm o compue a ba fo M when M a abl fee module gen b a mnmal peenaon S f S f M (7.6) wh g : S S uch ha g f and ank( M ) ( S). S lgohm fo compung bae INPU: F m( ) uch ha F M f afe ( S). ha a gh nee G M and OUPU: ma U M ( S) uch ha UG [ I ] ; b Lemma. he e {( ) ( ) ( ) ( U U ) } a ba fo M whee ( U ) ( j) denoe he j-h column of U fo j. INIILIZION: V I. WHIL < DO:. Denoe b S he column eco gen b akng he la ene of he -h column of VG.. ppl he peou algohm o compue L ( S) uch ha L e. I. Defne he ma U : ( S) L fo > and U : L... RURN U : PUV whee P an adequae elemena ma. ample 7.. Le be he quanum Wel algeba ( ) J a b condeed n ample 7. wh k Q a and b. In ode o 6 lluae he peou algohm ake M Im( F ) whee
MRIX COMPUIONS ON PROJCIV. F Ung he algohm decbed n Coolla.5 he degle ode oe Mon() wh > and he OPRV ode on ( ) Mon 6 wh e e > poble o how ha F ha a gh nee gen b:. G Hence we hae he followng mnmal peenaon fo M: 6 π M F (7.7) whee π he canoncal pojecon. hu M a abl fee -module wh ank(m). Snce lkdm() (ee [] heoem.) hen ( ) and b he heoem 7. M fee wh dmenon equal o ank(m). We wll ue he peou algohm fo compung a ba of M.
CLUDI GLLGO Sep. Le 6 I V and he f column of VG.e. [ ] hen ( ) Um c 6 and [ ] u uch ha. u Noe ha [ ] all unmodula. pplng o he f algohm of he cuen econ we hae ha I 6 ( ) ( ) ( ) and
MRIX COMPUIONS ON PROJCIV 5. We can check ha : U ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6 and. G U
CLUDI GLLGO 6 Sep. Make : U V and le be he column eco gen b akng he la fe ene of he nd column of ; VG.e. [ ]. Noe ha [ ] u afe u hu ( ). 5 Um c Moeoe [ ] unmodula wh [ ] u uch ha u and hence able. Ung he algohm a he begnnng of h econ we hae ha I 5 ( ) ( ) ( ) and.
MRIX COMPUIONS ON PROJCIV 7 Makng he epece calculaon we hae ha ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) : L and. 5 L e Defne ; : L U hen. G U U Fnall f hen : UG P
CLUDI GLLGO 8 whee. : U P U U hu a ba fo M gen b { ( ( )) ( ( )) U U π π ( ( )) ( ( ))} 6 5 U U π π wh () U denong he anpoe of -h ow of he ma U fo 5 6.e. ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) U ( ) ( )( ) 5 6 5 5 U ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ). 6 5 U U
MRIX COMPUIONS ON PROJCIV 9 Refeence [] F. Chzak. Quada and D. Robez ffece algohm fo paamezng lnea conol em oe oe algeba ppl. lgeba ngg. Comm. Compu. 6(5) (5) 9-76. []. Cluzeau and. Quada Facong and decompong a cla of lnea funconal em Ln. lg. and ppl. 8() (8) -8. [] P. Cohn Fee Ideal Rng and Localzaon n Geneal Rng Cambdge Une Pe 6. [] H. Fuja Global and Kull dmenon of quanum Wel algeba Jounal of lgeba 6() (999) 5-6. [5] C. Gallego Ma Mehod fo Pojece Module Oe σ -PBW enon Ph.D. he Unedad Naconal de Colomba Bogoá 5. [6] C. Gallego and O. Lezama Göbne bae fo deal of kew PBW eenon Comm. n lgeba 9() () 5-75. [7] C. Gallego and O. Lezama Ma appoach o noncommuae abl fee module and Heme ng lgeba and Dcee Mahemac 8() () -9. [8] C. Gallego and O. Lezama d-heme ng and kew PBW eenon São Paulo Jounal of Mahemacal Scence (F onlne: 5 ugu 5) -. [9] C. Gallego and O. Lezama Pojece module and Göbne bae fo kew PBW eenon (o appea n lgebac and Smbolc Compuaon Mehod n Dnamcal Sem n he Spnge ee dance n Dela and Dnamc). [] J. Gago-Vaga Bae fo pojece module n n ( k ) J. Smb. Comp. 6 () 85-85. []. Y. Lam See Poblem on Pojece Module Spnge Monogaph n Mahemac Spnge 6. []. Y. Lam Lecue on Module and Rng Spnge Gaduae e n Mahemac 89 999. [] O. Lezama Ma and Göbne Mehod n Homologcal lgeba oe Commuae Polnomal Rng Lambe cademc Publhng. [] O. Lezama and M. Ree Some homologcal popee of kew PBW eenon Comm. n lgeba () -. [5]. Quada and D. Robez Compuaon of bae of fee module oe he Wel algeba J. Smb. Comp. (7) -. [6] J.. Saffod Module ucue of Wel algeba J. London Mah. Soc. 8 (978) 9-. g