Stratification Analysis of Certain Nakayama Algebras

Transcription

1 Advaces ue athematcs, 5, 5, ublshed Ole Decembe 5 ScRes Statfcato Aalyss of Ceta Nakayama Algebas José Fdel Heádez Advícula, Rafael Facsco Ochoa de la Cuz Depatmet of athematcs, Havaa Uvesty, Havaa, Cuba Receved 7 Octobe 5; accepted Decembe 5; publshed 4 Decembe 5 Copyght 5 by authos ad Scetfc Reseach ublshg Ic Ths wok s lcesed ude the Ceatve Commos Attbuto Iteatoal Lcese (CC BY) Abstact Ou pupose these otes s to peset a esult fo a specfc Nakayama Algeba I essece, t affms that fo ay ode of smple modules, the cyclc Nakayama Algebas wth elatos ad KQ = (e Λ ) ae ot stadadly statfed o costadadly statfed Keywods Stadadly Statfed, Cyclcal Nakayama Algebas, Ifte ojectve Dmeso Itoducto The am of ths pape s to peset a sees of esults obtaed elato to a patcula class of Nakayama algebas We wll beg by ecallg the fudametal otos ad esults of stadaly statfed ad almost heedtay algebas theoy, whch wll be ou ma tool The cocept of stadadly statfed algebas emeged as a atual geealzato of quas-heedtay algebas The class of quas-heedtay algebas was toduced by Cle, ashall ad Scott coecto wth the study of hghest weght categoes asg the epesetato theoy of semsmple complex Le algebas ad algebac goups We peset ou fst esult, whch allows to obta the ma theoem of the atcle as a mmedate cosequece Theoem Let Λ be a algeba, such that all o tval quotet of decomposable pojectve has fte pojectve dmeso, the Λ s ot a stadadly statfed algeba fo ay ode of smple modules, uless F ( ) s the subcategoy oj ( Λ ) Late, we toduce some otos of useal algebas ad useal modules I secto 4, we toduce Nakayama algebas, also kow as useal geealsed Algebas that ae studed by Tadas Nakayama [] How to cte ths pape: Advícula, JFH ad de la Cuz, RFO (5) Statfcato Aalyss of Ceta Nakayama Algebas Advaces ue athematcs, 5,

2 J F H Advícula, R F O de la Cuz I hs shot otes, as Nakayama called hs publcato, t was poposed to make some obsevatos to hs pevous publcato about Fobeusas Algebas, whose fst pat was publshed 99 at Aals of athematcs We coclude by pesetg a specal class of Nakayama algebas, fo whch s the ma esult of ths pape that we quote below: Theoem Thee s o smple ode of smple modules fo whch the cyclcal Nakayama Algebas wth ela- tos ad KQ = (e Λ ) ae stadadly statfed o costadadly statfed ojectve Dmeso, Ijectve Dmeso ad Global Dmeso The followg cocepts wll allow us to defe the otos of pojectve dmeso, jectve dmeso ad global dmeso; whch we wll be vey useful demostatg the fudametal esult of ths pape Defto Let be a Λ-module A pojectve esoluto of s a complex wht = fo < modλ : whee s a pojectve module fo It should also satsfy that the map s a epmophsm ad Ke = Im It s possble show that beg Λ a K-algeba, t follows that evey modλ has a pojectve esoluto modλ oe geeally, f a abela categoy A has eough pojectves, the evey object A has a pojectve esoluto Defto Let be a Λ-module A mmal pojectve esoluto of s a pojectve esoluto of such that, the homomophsm : Ke s a pojectve cove ad : s a pojectve cove of Dually we defe cocepts jectve esoluto ad mmal jectve esoluto Is possble also pove that f Λ s a fte dmesoal algeba the all module modλ has a mmal pojectve esoluto ad mmal jectve esoluto modλ The cocepts of pojectve dmeso, jectve dmeso ad global dmeso fo a Λ-module ae as follows Defto ojectve dmeso of Λ-module s the umbe pd = such that thee s a pojectve esoluto, : of legth ad does ot have pojectve esoluto of legth m If does ot admt a fte pojectve esoluto, the by coveto the pojectve dmeso s sad to be fte Dually t has the jectve dmeso of a Λ-module Defto 4 Let Λ be a fte dmesoal K-algeba The global dmeso ( gld Λ) s the supemum of the set of pojectve dmesos of all Λ modules, e { } { } gldλ= : sup pd ; wht a Λ - module = sup d ; wht a Λ-module Stadaly Statfed Algebas ad Quas-Heedtay Algebas Let R be a commutatve At g ad Λ a basc At algeba ove R As futhe we assume full subcategoes of modλ, uless othewse stated We cosde K-algebas of fte dmeso basc ad decomposable, KQ whee K s a algebacally closed feld ad by the Gabel theoem, λ =, whee Q s a fte quve ad I s a I admssble deal The pcpal esults of ths secto ca be fd []-[8] Defto 5 Let Λ be a At algeba ad θ = { θ(, ), θ( ) } modλ such that Ext ( θ( j), θ ( ) ) =, j We deote: F ( θ ) the class of moda fo wch thee s a cha of submodules wth θ( k ) θ F θ ( ) θ X the subcategoy o modλ of modules ae dect summads of modules ( ) 85

3 J F H Advícula, R F O de la Cuz I the followg we cosde that Λ deote a K -algeba togethe wth a fxed odeg o a complete set e= ( e, e,, e ) of pmtve othogoal dempotets (gve by the atual odeg of dces) Note that cosde the system e s equvalet to cosde a ode establshed of set of all smple Λ-modules ot somophc Λ to S e (we kow to be Λ a At algeba has a fte umbe of Λ-modules) ad Λ Defto 6 Let be a Λ-module A omal sees s a sequece of submodules t = = + The umbe t s called the legth of the sees The quotets ae called factos of the sees A sees of composto s a omal sees whose factos ae smple modules, e, a omal see whch ca ot be efed to aothe logest If X s a Λ-module, we deote by [ X : S ] the umbe of factos somophc to S composto sees X, e, the multplcty of S as composto facto of X Fo, let S be the smple Λ-module, whch s the smple top of the decomposable pojectve = Λe Defto 7 Stadad module, fo, s the maxmal facto module of wthout composto factos S j, fo j Dually fo, module coestda s the maxmum submodule Q wthout composto factos S j, fo j Let be the full subcategoy cosstg of all I smla way, we toduce, ad so o Note that the above defto mples that : S j = to j ad module = Dually, t has that : S j = to j ad module =Q Defto 8 A algeba Λ s sad stadadly statfed f Λ F ( ) If addto to that, the edomophsm g of each stadad module s smple the we say that algeba s quas-heedtay (e stadadly statfed algebas geealze the cocept of quas-heedtay algebas whee we eque the addtoal codto that the stadad modules ae Schu modules) Dually, f DΛ F ( ) we say that Λ s costadaly statfed Note that f Λ s stadadly statfed the pojectve modules ae F ( ) I addto, f Λ s quasheedtay the jectve modules ae F ( ) ad = =S The followg example wll allow us to udestad the theoy dscussed above Example Let Λ be the algeba gve by the followg quve wht elatos I = βα, αγβ, αγ ; We have to: : : : ad : : I I I It s ot dffcult to check that ths algeba s stadadly statfed ad costadadly statfed oly at odes fo espectve smple modules gve below: To ode,, = { S,, } all ae flteed To ode,, = { S, I, I } all I ae flteed We deote by < ( Λ ) the full subcategoy of modλ defed by modules of fte pojectve dmeso The followg esult s [8] whch wll be of geat utlty oposto Let Λ be a stadadly statfed algeba, the F ( ) < The followg theoem s the fst esult that peset us ths pape It wll allow us to obta, as a mmedate cosequece, ou ma esult Theoem 4 Let Λ be a algeba, such that all o tval quotet of decomposable pojectve has fte pojectve dmeso, the Λ s ot stadadly statfed algeba fo ay ode of smple modules, uless F ( ) oj Λ s the subcategoy ( ) : 85

4 oof It s clea that ( ) J F H Advícula, R F O de la Cuz F Futhemoe s quotet of pojectve As we assume that all decomposable pojectve quotet has fte pojectve dmeso the pd = theefoe F ( ) < so Λ s stadadly statfed ay ode of smple modules 4 Nakayama Algebas Thoughout, Λ s assumed to be a fte dmesoal K-algeba, defed ove a algebacally closed feld K The pcpal esults of ths secto ca be fd [] [9] [] Defto 9 Let be a Λ-module Radcal sees s defed as follows: ad ad We agee to deote by l ( ) the adcal sees legth of We ca defe ductvely soclo sees fo module as: soc = :, + : = π soc soc soc whee π : s the quotet applcato, e soc + soc soc soc soc We deote sl ( ) the soclo sees leght of Note that f, ad ad futhemoe t has to dmk < Ths clealy mples that the adcal ad = adλ, the ad ( ad ) l l Λ sees s fte How ( ) Λ= Λ ad ( ) ( ) We ca obseve that dmk <, the soc f ad the soclo sees soc soc s ft oposto 5 Let modλ, the sl ( ) = l ( ) Defto Let modλ, the Loewy leght of s defed by ll( ) : = l( ) = sl( ) Is ecessay toduce ew oto fo defe the Nakayama Algebas Defto Let be a Λ-module We say that s useal f possesses exactly oe composto sees Lemma 4 Let be a Λ-module Next codtos ae equvalets s useal; Radcal sees of s a composto see; Soclo sees of s a composto see; 4 l( ) = ll( ) Defto Let Λ be a K = - algeba Λ s ght seal f all ght decomposable pojectve s a useal Λ-module Dually defe us the left decomposable pojectve oto If Q Λ deote the udelyg quve of Λ the, Theoem 6 A basc K-algeba Λ s left seal zqueda f ad oly f fo each vetex α Q Λ thee s at most oe aow that stats α Coollay A basc K-algeba Λ s ght seal f ad oly f fo each vetex α Q Λ thee s at most oe aow that eds α Defto The algeba Λ s a Nakayama Algeba f evey pojectve decomposable ad evey jectve decomposable Λ-module s useal It s possble to chaacteze Nakayama Algebas though ts udelyg quve Theoem 7 A basc ad coected algeba Λ s a Nakayama Algeba f ad oly f Q Λ t has oe of the foms: 85

5 J F H Advícula, R F O de la Cuz o (cyclcal) Q : oof Immedately of Theoem 6 ad Coollay 5 a Result Let Λ be a Nakayama algeba We deote Λ the Nakayama algeba wth cyclcal udelyg quve Q Λ wth elatos ad KQ = I [], t shows that both Λ ad Λ ae ot stadaly statfed o costadaly statfed to ay ode of the smple modules, whch motvates us to pove the followg geealzato of these esults Theoem 8 Thee s o smple ode of smple modules fo whch the cyclcal Nakayama Algebas wth ela tos ad KQ = (e Λ ) ae stadadly statfed o costadadly statfed oof It s easy to see that evey pojectve module has the same legth ad we also kow that has a oly oe composto sees Let be a quotet of pojectve module ad cosde the followg shot exact sequece, Note that the legth of Λ-module = ke s stct less tha the followg shot exact sequece s cosdeed,, theefoe s ot pojectve Now whch, aga, we ote us that the legth of Λ-module = ke s stct less tha, theefoe s ot pojectve Iductvely, gve a module we choose a pojectve ad a sujecto : Let = ke, ad let be the composte Sce ( ) = = ke, ths cha complex s a pojectve esoluto of, pd = The, though Theoem, the esult s cocluded Commetay 5 Geeally Nakayama algebas that ae ot as we saw Example Λ may be stadaly statfed o ot to be, Ackowledgemets We thak the Edto ad the efeee fo the commets 854

6 J F H Advícula, R F O de la Cuz Refeeces [] Nakayama, T (94) Note o U-seal ad Geealzed U-seal Rgs athematcal Isttute, Osaka Impeal Uvesty, Japaese, [] Rgel, C (99) The Categoy of odules wth Good Fltatos ove a Quas-Heedtay Algebas Has Almost Splt Sequeces athematsche Zetschft, 8, 9- [] X, CC () Stadadly Statfed Algebas ad Cellula Algebas athematcal oceedgs of the Cambdge hlosophcal Socety,, 7-5 [4] Dlab, V (994) Quas-heedtay Algebas Appedx Fte Dmesoal Algebas by Dozd, Y ad Kcheko, V, Spge-Velag, -44 [5] Dlab, V ad Rgel, C (998) Quas-Heedtay Algebas Caleto athematcal Sees, 4 [6] Dlab, V ad Rgel, C (99) Auslade Algebas as Quas-Heedtay Agebas Depatmet of athematcs ad Statstcs, Caleto Uvesty ad Fakultät fü athematk, Uvestät Belefeld, - [7] Heádez, JF (4) Sobe as algebas estadamete estatfcadas hd Thess eseted o athematcs ad Statstcs Isttute of Sao aulo Uvesty, Basl (I otuguese) [8] latzeck, I ad Rete, I () odules of Fte ojectve Dmeso fo Stadadly Statfed Algebas Commucatos Algeba, 9, [9] Ssoda, G (7) Nakayama Algebas Uvesty of Sao aulo, Basl, -8 [] Ta, R () Auslade Algebas of Self-Ijectve Nakayama Algebas Depatmet of athematcs, Hube Uvesty, Cha, [] Ochoa, RF (4) Statfcato Aalyss of Ceta Nakayama Algebas aste s Thess eseted o Faculty of athematcs ad Compute Sceces of Havaa Uvesty, Cuba 855