Global dimension in Noetherian rings and rings with Gabriel and Krull dimension

Publicacions Ma emá iques, Vol 36 (1992), 189-195 . GLOBAL DIMENSION IN NOETHERIAN RINGS AND RINGS WITH GABRIEL AND KRULL DIMENSION A bs ac JUAN JACOBO SIMÓN PINCRO In his pape we compu e he global dimension o Noe he ian ings and ings wi h Gab iel and K ull dimension by aking a subclass o cyclic modules de e mined by he Gab iel il a ion in he la ice o he edi a y o sion heo ies . In oduc ion In his pape we exhibi a nice subclass o cyclic modules o compu e he global dimension o a ing (see [9], [12], [13], [15]) whose o igins a e in [3] and [11] . In he i s pa , he le global dimension o a noe he ian ing, R, is compu ed in e ms o he injec i e dimensions o he ollowing subclass o R-mod . I T-, < - o < ... < T, is he Gab iel il a ion in he la ice o he edi a y o sion heo ies o R, Le . R- o s [2], hen he subclass consis o all he cyclic ,,-coc i ical le R-modules whose injec i e dimension equals he injec i e dimension o e e y one o i s submodules, wi h p anging o e all he o dinals less han ,Q . Also, we ob ain some o he classical esul s o noe he ian ings as consequences o ou esul s . In he second pa we no e ha all ou esul s can be dualized . Th oughou his pape , R will deno e an associa i e ing wi h 1 and R-mod he ca ego y o all uni a y le R-modules . To sion classes and o sion heo ies will always be he edi a y ; all e minology conce ning o sion heo ies is quo ed om [2] . Gi en a nonze o M c R-mod, Id(M) and Pd(M) deno e espec i ely, he injec i e and p ojec i e dimensions o M, se ing Id(0) = Pd(0) = -oo . The le global dimension o R will be deno ed by lgl dim(R), and he Gab iel dimension G dim(R) . Fo u he de ails on each o hese dimensions we e e espec i ely o [13] and [2] . 19 0  J .J . SIMÓN PINERO 1 . Injec i e dimension The S ong Injec i e Dimension o a le R-module, M is de ined as Sid(M) = sup{Id(M%0 -+ M' - M is exac } . Following [11], gi en n E PNl, we will deno e by G,, he class o le R-modules M, wi h Sid(M) <_ n . We de ine Sid(M) = oo when o all n E IN, he e ex- is s a submodule M' C_ M such ha Id(M) >_ n (wi h he con en ion n< oo) . Obse e ha i he e exis M' C_ M wi h Id(M)= oo, hen Sid(M) = oo . We ema k [11] ha i R is a le noe he ian ing hen he classes L n (n = 0, 1 . . . . ) a e o sion classes and [11] i RR is he ing R aken as le R-module he Sid(RR) = lgl dim(R) . No e ha he e is a chain o o sion classes Co C_ L 1 C_ . . . C_ L n C . . . We add G, = {O} and ,C,,, = R-mod . Le un be he o sion heo y co esponding o En . No e ha M E R-mod is a,,- o sion ee i and only i o all submodules 0 =~ M' C_ M he e exis a submodule N C M' such ha Id(N) >n . Wi hin he abo e chain he e exis s a s ic ly inc easing subchain ; ha is, i no = -oo, hen Lno C ... C Cni C ... whe e he leng h o he subchain is a mos w . 1 .1 Examples . (i) In 7L we ha e ,C_,,, = L o C :,Cl = 7L-mod . u  Le K be a ield . Fo R =  K  K[X, Y]  we ha e G_ . C L o C ( . .)  0  K[X, Y] ,C l C £2 = R-mod . (iii) Le R be a commu a i e noe he ian egula local ing, wi h J maximal ideal . Suppose ha Id(R/J) = n, hen we ha e in his case ,C_ .= Co = . . . =,c,1 C Cn = R-mod . (i ) Le R be a le a inian le local ing [2] . Then R-mod has a chain as in (iii) . ( ) In [3], he e a e examples whe e all he inclusions in he chain a e p ope . ( i) In any le a inian ing which has a leas wo simple le R- modules wi h di e en injec i e dimensions (as Z12) he Gab iel il a ion has less e ms han he subchain . 1 .2 Lemma . Le R be a le noe he ian ing and L n , (j = -oo, 0, 1, . . .) a e m in he subehain such ha Lni =,A R-mod . Then he e exis s a Qnj -coc i ical le R-module M, such ha n, < Id(M) = Id(M1) o all submodules 0 =~ M' C M . GLOBAL DIMCNSION  19 1 P oo .. Since R is noe he ian and G,,, zA R-mod hen he e exis a Q ni -coc i ical le R-module M . (a) I Id(M) > nj (including oo) hen in each exac sequence 0 -> M' -+ M - M" - 0 we ha e Id(M") < nj and Id(M) > n ;, and so Id(M') = Id(M) ; hence M is he equi ed objec . (b) I Id(M) < n j hen since Sid(M) > n j hen he e exis s a submodule 0 =,~ M' C M such ha Id(M') >n j (including oo) and since M' is also a n ;-coc i ical we a e again in case (a), and M' is now he equi ed objec . 1 .3 P oposi ion . Le R be a le noe he ian ing, G nu (j = 0, 1, ... ) -n, :,A R-mod and m = min{Id(C) 1C is cyclic Qnj -coc i ical wi h Id(C) > nj} . Then m = nj+l . P oo .. By hypo hesis and Lemma 1 .2 i is clea ha G  ,, always exis s and ha G nu CG  , . Suppose ha he e exis s k E N1 such ha Cni C Gk C_ G  ,, . Since R is noe he ian [2] he e exis s a Q nj -coc i ical Qk - o sion le R-module M, and hence nj < Sid(M) <_ k . So, by he pa (a) in he p oo o Lemma 1 .2, he e exis s a cyclic submodule C C_ M such ha Id(C) = Sid(M) . Since C is also Qn j -coc i ical and Id(C) >n ; . Then, by he de ini ion o m we mus ha e Id(C) > m . Hence k >_ m and hus k = m . No e ha , in pa icula , i m = min{Id(S)JS E R-mod is simple} hen m=nl . 1 .4 Obse a ion . Since e e y subchain has a leas wo e ms, i is na u al o analize he s ep Qnj < Qnj+, . In each o he e s eps he e exis s a Qn j - coc i ical cyclic le R-module C, such ha Id(C) = Id(C') _ Sid(C) > nj+i o all submodules 0 :,A C C C . F om he e, Theo em C o B . Oso sky in [5] ollows immedia ely . In he nex heo em, we will see ha we can o ex ac a nice subclass o he class o cyclic le R-modules, o compu e he le global di iension . 1 .5 Theo em . Le R be a le noe he ian ing, such ha G dim(R) = Then lgl dim(R) = sup{Id(C) ¡C is cyclic, Id(C) = Id(C'), o all 0 jA C' C C and - ,,-coc i ical, wi h < /3} . P oo . Le _1 < ,, < . . . < - p be he Gab iel il a ion in R- o s and le lgl dim(R) = nk (o oo) . Fo any gi en j < k we ha e a s ep Qn ; < Qn ; +1 and by [2], he e exis s an o dinal a <_ ~3 which is leas wi h he p ope y ha ,, :9 Qn i . No e ha a is a successo . Then by Obse a ion 1 .4 and he ac ha ,,, :9 a no he e exis s a T,,_ I -coc i ical Q ni -coc i ical le R-module C, such ha Id(C) = Id(C') > nj+i o all 192  J .J . SIMÓN PINERO submodules 0 :~ C' C_ C . Se ing = a - 1 we ha e he esul in iew ha he choice o j was a bi a y . The nex co olla y is o pa icula impo an e inasmuch as he e exis a a abundan e o examples whe e he subchain is ini e . 1 .6 Co olla y . Le , R be a le .noe he ian ing such ha Gdim(R) = ,Q . Suppose ha ue ha e ini ely many e ms in he subchain . Then lgl dim(R) = sup{Id(C) ¡C is cyclic, Id(C) = Id(C') o all 0 =,4 C' C_ C and ,,-coc i ical u)he e p < Q is ixed} . P oo . . Conside he las s ep in he subchain, Qnj_, < Qnj = X . Then he e exis s a< ,3 such ha T C , :1 unj_l . He e, p = a - 1 . 1 .7 Obse a ion . Le R be a commu a i e noe he ian ing . We ake in his case he o iginal de ini ion o K ull dimension o e he p ime ideals o R . Suppose ha now, o all S E R-niod simple we ha e ha S E G  , o some n E IN ( ixed) . Le Jbeany maximal ideal o R, hen R/J is a simple le R-module and Supp(R/J) = J ; u he mo e, R/J E G n . Then, by [11] he local ing Ri is egula wi li K dim(Rj) <_ n . By [13] we lla e Sida,, (RJ) = lgl dim(Ri) < n, o all J . B,y [13] we ha e lgl dim(R) < n, lience R E  Tha is, we ha e jus p o ed ha o any commu a i e noe he ian ing, i u, j :~ X hen To :1 u n , and by Theo em 1 .5 we ha e he well-known esul : lgl dim(R) = sup{Id(S) ¡S is simple R-module } . 2 . P ojec i e dimension When we compu e he le global dimension as he sup emum o he p ojec i e dimensions o coc i ical and c i ical le R-modules wc lla e analogous esul s o he abo e ; u he mo e, we can elax he noe he ian condi ion o a he ing R . The S ong P ojec i e Dimension [11] is de ined as Spd(M) = sup{Pd(N) ¡M -> N -> 0 is exac } . We lla e o all n E N, he classes U,, = {M E R-mod ISpd(M) _< n} . In his case [11] U n a e o sion classes o a a a bi a y ing R, and lgl dim(R) = Spd(RR) . We deno e by pn he o sion heo y co esponding o U  , . Again, we ha e a chai a Uo C_ U C_ . . . C_ U n C_ . . . and adding U_ . a ad U,,, we can ake a s ic ly ascending subchain Uno C . . . C Uni C . . . wi h no = -oo . F om he e, we can do he duali a ion in a simila way o he i s pa and we can emo e he noe he ian condi ion . So, we will w i e o lly he p incipal esul . Fo p,,-coc i ical le R-modules C, such ha Pd(C) >_ nj+ , he consequence Pd(C) = Pd(C') o all submodules 0 7~ C C C will be emo ed in iew ha all p,,-coc i ical sa is y i . GLOBAL DIMENSION  19 3 2 .1 Theo em . Le , R be a, ing wi h Gab iel, dimension, suppose ha G dim(R) = 0 . Then lgl dim(R) = sup{Pd(C)1C is cyclic and T,,-coc i ical, wi h h< ,(3} . 2 .2 Examples . (i) In a non-noe he ian ing R, wi h Gab iel di nen- sion, he classes G n , a e no in gene al o sion classes, bu hey a e : Se e subca ego ies (see [5]) . E en i subchains can be ound, he esul s ha we ha e seen do no hold . Fo example, le S be (Z2) Nand R C S he sub ing gene a ed by (7Z2) '> oge he wi h 1 E S . Then R is a commu a i e boolean se nia inian he edi a y V- ing, ha ing as chai a G_,, C LO C G1 = R-mod . No e ha he o sion class gene a ed by C o is he same ha G . (ii) Le R be a ing wi h Gab iel dimension and nonsingula as le R-module (Z(j?R) = 0) . Then i R is no se ni simple we ha e lgl dim(R) = sup{Pd(C) ¡C is cyclic singula } . P oo . We shall p o e ha e e y C o Theo em 2 .1 adini s ano he singula module D such ha Pd(D) >_ Pd(C) . By [4], in e e y non- singula ing, cyclic uni 'o m nodules a e ei hc- " singula o nonsingula . Since i is clea when C is singula , we assu ne ha C is nonsingula . So ake a le ideal I o R such ha R/I = C . Because I is no la ge in R, he e is a le ideal 0 ~ J o R wi h I (D J la go in R . By aking D = R/I ® .I we ha e Pd(D) = 1 + Pd(I ® J) >_ 1 + Pd(I) _> Pd(C) and R/I is singula . (i ) I R is le se nia inian, lgl dim(R) = sup{Pd(S)JS is simple} (see-, " [8]) . Semi pe 'ec ings a e : se nia inian, o ins ance . ( ) Finally we e e o i kjec i e and p ojec i e dimension in le Fully Bounded Noe he ian (FBN) ings wi hou any o he assu ip ion (like he commonly used igh cohe ence o [12], [15]) . In his ings, he ( wo sided) p ime ideals o en ha e no hand desc ip io is a,nd we can see- : lnow ou classes wo k . Le R be a le FBN ing and ake G  ; z,~ R-mod (U, L , =, A R-mod) . By he esul s abo e, he e exis s a cyclic o, Lj -coc i ical (P,, ;-coc i ical) le R-module C . Take [14] he associa ed p ime ideal ass(C) and no e ha by [14] i x E C is such ha R - x =C hen ass(C) C_ l(x) ( he le annihila o o x) and hence, R/ass(C) 1 G,,, (R/ass(C) 0 U,,,,) and by [14] we ha e R/ass(C) is o, EJ - o sion ' ee (P,,,- o sio i ee) . Since R is le FBN he i, hc ; injec i e hulls E(R/ass(C)) = E(C) a nd hence he e exis s a copy o C, say C again, C C_ E(R/ass(C)) . Take K = C n R/ass(C) a ad no e ha lince K C C hen K has he p ope ies 19 4  J .J . SIMÓN PINCRO o modules in Theo ems 1 .5 and 2 .2 . Since R/ass(C) is a le o de in a simple ing [7], [14] hen o any x E K we ha e (R/ass(C)) - x R/ass(C) as le R/ass(C)-modules and hence as le R-modules . This implies ha Id(R/ass(C)) = Id(C) (Pd(R/ass(C) = Pd(C)) . So we ha e ha i R is a le FBN ing hen lgldim(R)=sup{Id(R/I)II E Spec(R)}=sup{Pd(R/I)II E Spec(R)} . a No e . The au ho hanks he e e ee o epo ing him abou he exis ence o [6] . Re e ences 1 .  M .F . ATIYAII AND I .G . 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Soc . 12 7 (1967), 136-149 . 10 .  I . PALMCR AND J .-E . ROOS, Fo mules explici es pou la dimension homologique des anneaux de ma ices gene alisées, C .R . Acad . Sci . Pa is, Sé . A-B 273 (1971), A1026-A1029 . 11 .  Z . PAPP, On he s ong injec i e (p ojec i e) dimension o modules, A ch . Ma h . 25 (1974), 354-360 . 12 .  J . RAINWATER, Global dimension o ully bounded noe he ian ings, Comm . Algeb a 15(10) (1987), 2146-2156 . 13 . J . ROTMAN, "An in oduc ion o Homological Algeb a," O lando, Academic P ess, 1979 . GLOBAL DIMENSION  19 5 14 .  BO STENSTRÓM, "Rings o quo ien s," Sp inge -Ve lag, Be lin, Hei- delbe g, New Yo k, 1975 . 15 .  M . TEPLY, Global dimensions o igh cohe en ings wi h le K ull dimension, Ball . Aus al . Ma h . Soc . 39, no 2 (1989), 215-223 . Depa amen o de Ma emá icas Uni e sidad de Mu cia 30071 Mu cia SPAIN P ime a ue sió ebulla el 8 de Juliol de 1991, da e a ue sió ebl da el 15 d'Oc ub e de 1991