Minimal resolutions and other minimal models

Publicacions Ma emá iques, Vol 37 (1993), 285-303 . Abs ac MINIMAL RESOLUTIONS AND OTHER MINIMAL MODELS AGUSTÍ RoIG In many si ua ions, minimal models a e used as ep esen a i es o homo opy ypes . In his pape we s a e his ac as an equi - alence o ca ego ies . This equi alence ollows om an axioma ic de ini ion o minimal objec s . We see ha his de ini ion includes examples such as minimal esolu ions o Eilenbe g-Nakayama- Ta e, minimal ibe spaces o Kan and A-minimal A-ex ensions o Halpe in . Fo he i s one, his is done by gene alizing he cons uc ion o minimal esolu ions o modules o complexes . The o he s ollow by a ca ac e iza ion o minimal objec s in bi ib ed ca ego ies . In oduc ion Minimal models appea in di e en si ua ions as ep esen a i e o ho- mo opy ypes . We claim ha his is an in insic p ope y o hose objec s called "minimal", and does no depend on he pa icula cons uc ion employed in each ca ego y . To his end, we p opose an axioma ic de ini- ion o minimali y, inspi ed by he de ini ion o minimal (R, )-algeb as o [H-TI . Ou esul is ha he homo opy ca ego y o a model ca ego y and he ca ego y o i s minimal objec s and homo opy classes o maps a e equi alen (Theo em 1 .17) . We also p o e ha ou axioma ic de ini ion p oduces he objec s ha a e commonly known as "minimals" . In pa icula , we see ha he min- imal esolu ions o Eilenbe g-Nakayama-Ta e a e minimal objec s o a ca ego y o cochain complexes . This is done by gene alizing he cons uc- ion o hese minimal esolu ions o modules o complexes ; Le ., g aded modules wi h di e en ial (Theoem 2 .4 and 2 .5) . We also e i y ha minimal ib e spaces o [Kan] and A-minimal A-ex ensions o [Hal] a e minimal objec s o bi ib ed ca ego ies (see [Roigl]) in e ms o ou de - ini ion (Theo em 3 .2) . O he examples may be ound in [Roig2] and [Roig3] . 286  A . Role 1 am g a e ul e F ancisco Guillén, Vicen e Na a o Azna , Pe e Pascual-Gainza and Daniel Tan é o help ul discussions and sugges ions . 1 am also g a e ul e he e e ee o his ema ks and c i icisms . 1 . Minimal ob jec s and he homo opy ca ego y Le Cbe a ca ego y and S C mo C a class o mo phisms o C . We will o en deno e he ac ha a mo phism s : a --> b o C is in S by s : a -> b and we will loosely say ha s is a S-quasi-isomo phism (quism) and a is S-quasi-isomo phic o b . We will also say ha a is a S-le model o b o ha b is a S- igh model o a . De ini ion 1 .1 . An objec m o C is S-le minimal i , o all s x -3 m E S, he e exis s a sec ion s' : m - x ; Le ., ss' = 1 . A S-le minimal model o an objec a E obj C is a S-le model m --> a wi h m a le minimal objec . By in e ing a ows in he p e ious de ini ion we ha e he no ions o S- igh minimal objec and S- igh minimal model (see [Roig3]) . F om now en we suppose ha he class S is ixed o e e y ca ego y and con- side only he "le pa " o he heo y, unless o he wise s a ed . So we will simply say minimal objec , model and minimal model . The ela i o e sion o de ini ion 1 .1 is he ollowing (c . [B-G], [Hall and [H-T]) : le C be a ca ego y, a an objec o C and S C mo C . Le a S be he class o mo phisms o he ca ego y o objec s unde a, a C, made up by he commu a i e iangles o C in which s E S . Hypo hesis 1 .3 . c a b De ini ion 1 .2 . A mo phism a --> b o C is a minimal mo phism i i is a a S-minimal objec o a C . Fo he i s esul s, i is necessa y ha he class S e i ies he (i) isomo phisms o C a e in S, and (ii) i in he diag am o C, x ~ y ~ z, wo o he mo phisms { , g, g } a e in S, hen so is he hi d . hen MINIMAL RESOLUTIONS AND OTHER MINIMAL MODELS  287 These hypo hesis a e ul illed i , o ins an e, C is a model ca ego y and S = we is he class o i s weak equi alen es o i S is he class o mo phisms made in e ible by soma unc o H : C --> D . Thenwe can easily p o e (see [Roig3]) . P oposi ion 1 .4 . I m and n a e minimal objec s and s : m --> n is a quism, hen s is an isomo phism . De ini ion 1 .5 . Le C be a ca ego y wi h an ini ial objec e . An objec x o C is acyclic i i is quasi-isomo phic o e . Co olla y . An objec x o C is minimal and acyclic i and only i i is ini ial . P oo : An ini ial objec e is always minimal and, since 1, is in S by 1 .3(i), i is also acyclic . On he o he hand i x is minimal and acyclic he e is a mo phism in S x - ewhich is, by 1 .4, an isomo phism . The ollowing is a use ul c i e ium o minimali y . P oposi ion 1 .6 . I C admi s a class o objec s M such ha : (a) e e y objec has a model in M, and (b) e e y mo phism o S be ween objec s o M is an isomo phism, (1) he objec s o M a e minimal, and (2) e e y minimal objec o C is isomo phic o an objec o M . P oo . .- See [Roig3] . De ini ion 1 .7 . I C has a class o objec s M ha ul iles condi ions (a) and (b) o P oposi ion 1 .6, we will say ha C has enough minimals . We will deno e by HoC he homo opy ca ego y ; Le ., he ca ego y ob- ained om C by adjoining he in e sas o he mo phisms in S (see [Qui]) . I is also callad he localizad ca ego y, CS ([G-Z]), o he de i ed ca ego y, in he case whe e C is he ca ego y o complexes o an abelian ca ego y ([Ha ]) . I wo objec s o C a e isomo phic in HoC hey a e said " o ha e he same homo opy ype" . Fo he momen we will assume also Hypo hesis 1 .8 . S admi s a calculus o igh ac ions . This is he case when D is a model ca ego y wi h only ib an objec s, C = 7 D being he ca ego y ob ained iden i ying homo opymaps o 288  A . RoiG D and S C mo C being he image o he weak equi alences o D by he p ojec ion unc o D -- i D . We will de elop ou esul s unde hypo hesis 1 .8 and hen es a e hem "up o homo opy" o a model ca ego y . Finally, in ou pa icula examples, all objec s will be ib an . When S admi s a calculus o igh ac ions, he mo phisms o HoC om a o b can be ep esen ed by sequences o mo phisms o C as a+  1 - ----> b and he isomo phisms a e exac ly hose sequences whe e bo h mo phisms o (1) a e in S . Then, wi h his hypo hesis, wo minimal objec s o C which a e isomo phic in HoC a e necessa ily isomo phic in C : I in (1) a and b a e minimal, we ha e a sec ion o a <- - which is in S by (i) and (ii) o 1 .3 . Hence, i s composi ion wi h - -> b is also in S . Thus we ha e a quism be ween minimal objec s which, by 1 .4, is an isomo phism . The e o e, homo opy ypes become isomo phism classes when we use minimal objec s . We a e going o s a e his ac as an equi alence o ca ego ies . To begin wi h, we ha e a li ing p ope y . P oposi ion 1 .9 . Le p : a => b E S _and : m -> b a mo p_hism wi h m minimal . Then he e exis s a unique : m -> a such ha p = . P oo .- Because o he p ope ies o he calculus o ac ions, o e e y diag am in C, a he e exis s mo phisms p' : x => m and ' : x --> a such h ha p ' = p' . T hn hen, as m is minimal, p' has a sec ion s : m -~ x . Take = 's and so P = P , s = p , s = . _ Secondly, le g : m ---> a be ano he mo phism such ha pg = . Then p = pg . Bu pE S, and so_, by he calculus o ac ions, he e exis s : z --> m E S such ha = g and because m is m _ inima _ l, his mo phism has a sec ion s : m --> z, s = 1  , . So g = g s = s = . Co olla y . Two minimal models o an objec a a e isomo phic by a unique isomo phism o C/a . The nex p oposi ion will allow us o de ine he "minimal model unc- o ", when we ha e enough minimals . MINIMAL RESOLUTIONS AND OTHER MINIMAL MODELS  289 P oposi ion 1 .10 . Le : a -> b be a mo phism o C and Pa : ma a and Pb : mb - b wo minimal models . Then he e exis s a unique mo phism m : m a -> mb, ha ende s commu a i e he diag am m aPa ->a l b P oo . Applying he calculus o ac ions wo imes, we ha e a com- mu a i e diag am y ------- > x--,mb Pa  ' P b  -1 P,,  -IPb Pa  So, by he minimali y o ma, we ha e s : m a -> y such ha pes = 1  ,, a . We le m = 'p' s and ha e pbm = Pb 'P' S = P6Pá s = PaPb s = Pa- Le cp : m a -` mb be ano he mo phism ha makes commu a i e diag am (2) : PbW = Pa = pbm . Then, because Pb E S, by he calculus o ac ions, he e exis s : z -> m a E S such ha cp = m and, because ' m a is minimal, we ha e a sec ion s : m a-+ z o .  So, cp = cp s = m s = M . Rema k 1 .11 . m is no necessa ily he minimal model o in he sense o de ini ion 1 .2 . Fo ins an e, unless a is a minimal objec , m needs no o be an objec o a C . We ha e no assumed ye he exis en e o a minimal model o e e y objec . Le us suppose now ha C has enough minimals : o e e y aE obj C, we choose a minimal model which will be deno ed by Pa M(a) -> a . Le us also no e by C, (o Cmin when we will alk abou bi ib ed ca ego ies) he ull subca ego y o minimal objec s . Thenwe ha e Co olla y . The choice o a minimal model de ines a unc o M :C->C ' , by M(a) = m a and M( ) = m ; and a mo phism o unc o s p :M - ~ 1C 29 0  A . ROIG byp . :m a ~a . P oo . Al hough he e a e di e en choices o M(a) and M( ), he unici y o P oposi ions 1 .9 and 1 .10 makes he co espondence unc o- ial . FYom wha we ha e seen, i is clea ha he unc o M ca ies mo - phisms o S in o isomo phisms o C m . In ac , i in (2) E S, hen pa : m a ~ b is a minimal model o b . Hence, he e exis s an isomo - phism o C/b, cp : m a ` mb, which, because o he uniqueness is equal o m . The e o e, M ac o izes in a unique ashion h ough he localiza ion unc o ,y : C --> Cs . Tha is o say, we ha e he commu a i e diag am o unc o s M e s We will no e also by M his ac o iza ion and call i he minimal model unc o . Theo em 1 .12 .  The unc o M : Cs --> C m is an equi alen e o ca - ego ies . P oo . Le c : Cm -> CS be he composi ion o he inclusion C m - C and he localiza ion C --> Cs . We will show ha c is an equi alen e, quasi-in e se o M, by de ining isomo phisms : lc m ---> M and s ¿M ---> lc, Le m be minimal . Then, by co olla y o P oposi ion 1 .9, he e exis s an isomo phism ?7 m : m -> M (m) . On he o he hand, de ine Ea : ¿M(a) --> a as he isomo phism in C s induced by he choice in C o he minimal model o a, M(a) --> a . I is easy o check ha hese de ini ions a e na u al in m and a, espec i ely . Rema k 1 .13 . We can easily dualize he p e ious esul s aking in o accoun ha Hypo hesis 1 .3 a e sel -duals and eplacing he exis en e o calculus o igh ac ions by ha o le ac ions . Aswe ha e said, he exis en e o a calculus o igh ac ions is ull iled i we a e wo king in a model ca ego y . So, om now on, we will assume, ins ead o Hypo hesis 1 .3 and 1.8, Hypo hesis 1 .14 . C is a model ca ego y, S = we is he class o i s weak equi alen es and all he objec s o C a e ib an . Le us ecall ha , as a consequence, we ha e p ope ies (i) and (ii) o Hypo hesis 1 .3 and a calculus o ac ions in 7 C . To begin wi h he MINIMAL RESOLUTIONS AND OTHER MINIMAL MODELS  291 ansla ion o p e ious esul s in his se ing, we ha e a ela i e e sion o 1 .9 . P oposi ion 1 .15 . Le be a commu a i e diag am o C in which a is a minimal objec and i is a minimal mo phism . Then, i p is aweak equi alen e, he e exis s ~3 : b ~ x, unique up o homo opies, such ha p,C - ,0 and bi - a . P oo . Le ús ac o ize p in a co ib a ion j : x -> z and a ib a ion q : z --> y, bo h i ial ones . Take he pull-back o ~3 and q : q~ b T i ial ib a ions a e s able unde pull-backs, so q' is a i ial ib a ion, which we can conside a i ial ib a ion o a C om he mo phism in- duced by ja and i, y : a ---> c, o i . Bu i is a minimal mo phism, so he e exis s s : b , c such ha q's = 16 and si = y . On he o he hand, j is a i ial co ib a ion : so i has a homo opic in e se : z --> x . Then, ,Q's is he li ing we a e looking o : ,Qi = lo'si = ~3'-y = ja - a and pR = qj ,C's - qO's = ,Cq's = 0 . The uniqueness up o homo opy o /b is also an easy e i ica ion . Co olla y 1 . Le p : a Z b E we and _ : m --> b be a mo phism wi h m aminimal objec_ . Then he e exis s : m -> a, unique up o homo opies, such ha p - . P oo .. Take a = e in he p e ious p oposi ion . Co olla y 2 . Two minimal models o a E C a e isomo phic .  The isomo phism is unique up o homo opies o C/a . The es o he esul s admi analogous modi ica ions . 29 2  A . RoIG P oposi ion 1 .16 . Le : a -> b be a mo phism o C and pa : m a -> a and pb : mb -> b wo minimal models . Then he e exis s a mo phism m : m a ` mb, unique up o homo opies, ha ende s commu a i e up o homo opy he diag am byp . :m a -+a . ma Pa . a Le us assume ha C has enough minimals . Ci en aE obj C, le us choose a minimal model : pa : M(a) -> a . Since p e ious esul s a e s a ed "up o homo opy", he co espondence a H M(a) is no neces- sa ily unc o ial and he weak equi alen es Pa do no necessa ely de ine a mo phism o unc o s M -> lc . Ne e heless, we ha e a well-de ined unc o and a mo phism o unc o s i we ake as a ge he ca ego y 7 C,,, which has as objec s he minimal ones o C and as mo phisms he homo opy classes o mo phisms o C . Co olla y . The choice o a minimal model de ines a unc o M :C-+7 C,,, by M(a) = m a and M( ) = m , and a mo phism o unc o s P :M---> l,c We also ha e a unique ac o iza ion o M h ough he localiza ion unc o and Theo em 1 .12 now eads : Theo em 1 .17 . The unc o M : HoC --> 7 C  ,, is an equi alen e o ca ego ies . Rema ks 1 .18 . (1) All he abo e ac s a e independen o he classes o ib a ions and co ib a ions we choose . So, o a gi en class o weak equi alen es, all he possible model s uc u es sha e one class o co ib an objec s in common, i hey exis : he minimal ones . (2) The las heo em could also be deduced om he ac ha minimal mo phisms a e necessa ily co ib a ions (in a closed model ca ego y, a leas ) and hen applying [Qui, Theo em 1', Sec ion 1, chap e I], aking in o accoun ha we do no need all he co ib an objec s, bu only one ep esen a i e o each homo opy ype ( o ins an e, a minimal one) . (3) In o de o dualize p e ious esul s, we need only o subs i u e co ib a ion o ib a ion and ice- e sa . MINIMAL RESOLUTIONS AND OTHER MINIMAL MODELS  293 2 . Minimal esolu ions o complexes Le R be a uni a y commu a i e ing . A R-dg module is a g aded module M wi h a di e en ial o deg ee +1 ; Le ., a cochain complex . Le us ake as S he class o quasi-isomo phisms ; ha is o say, he mo phisms which induce an isomo phism in cohomology . Be o e we es ic ou sel es o local ings, le us examine some examples p oduced by ou de ini ion . P oposi ion 2 .1 . Le M be a R-dg module wi h ze o di e en ial . Then M is minimal i and only i M i is a p ojec i e module o each i . P oo .. Le M be a cochain complex, wi h a p ojec i e module in each deg ee, ze o di e en ial and X => M aquism . Then, o e e y i, we can choose a sec ion o Z'X , H'X = M i . These gi e us a mo phism o R-dg modules M --> X which is a sec ion o he quism abo e . Recip ocaly, assume ha M is minimal and le : X --> M i be an epimo phism o R-modules . Conside he mo phism o R-dg modules ...  > M i-2  o  -> Mi-1 ® ke (o ~ j ) X  o -~ Mi+l es . . . 11  1(1 0)  J-  11 > M i-2 0  Mi-1  ---- 0 -----> Mi  °  --> Mi+l (whe e he ho izon al a ows a e he di e en ials and j : ke y X is he inclusion) . Ob iously, i is a quism and, since M is minimal, i has a sec ion . In pa icula , has a sec ion and so Mi is a p ojec i e R-module . Co olla y . Le M be a R-module, conside ed as a homogeneous dg module wi h ze o di e en ial . Then M is minimal i and only i i is p ojec i e . P oposi ion 2 .2 . Le M be a R-dg module such ha H i M is a p o- jec i e R-module o each i . Then HM is a minimal model o M . P oo . Le Z i M -H i M be he na u al p ojec ion and s i : H i M -> Z i M a sec ion . Then s = (si) : HM -> M is a quism o R-dg modules and so HM is a model o M . Because o P oposi ion 2 .1 i is a minimal model . Co olla y 1 . I R has ze o global dimension, hen : (1) E e y R-dg module has a minimal model . (2) M is minimal i and only i M has ze o di e en ial . 300  A . Roic Example 3 .3 . Le R be a uni a y commu a i e ing, Adgc(R) he ca ego y o R-dgc algeb as and Adgc(R) 2 he ca ego y o mo phisms o R-dgc algeb as . The objec s o his ca ego y a e mo phisms a : A , B o R-dgc algeb as and he mo phisms commu a i e squa es B A C o Adgc(R) . We ake as P he domain unc o Adgc(R) 2 -+ Adgc(R) which sends a o A . Then he ibe ca ego ies a e he ca ego ies o A- dgc algeb as : Adgc(A) = A Adgc(R) . Fo each mo phism o R-dgc algeb as : A - C we ha e a ecip ocal image unc o and a di ec image unc o * : Adgc(C) ----> Adgc(A)  and  * : Adgc(A) ---> Adgc(C) de ined by * (l0) = /3 and * (a) = _ ®1 : C ` C ®A B . The ca esian mo phism ,(3 and he coca esian mo phism a a e he commu a i e squa es D 1 ) D  B  1® - ) C ®A B 1a = -(a) 1,3 A i C and C [Roigl] shows how o endow A wi h a model s uc u e om gi en s uc u es en £ and A ., . Ne e heless, in o de e alk o minimal objec s in A we only need a class o dis inguished mo phisms, which we de ine as ollows : suppose we a e gi en classes S£ C mo £ and S,, C mo A,, o e e y x E obj £ in such a way ha hey a e compa ible wi h ecip ocal images ; Le ., o e e y : x --> y E mo £, one has * (S 9 ) C S . . Then, pu S= {W : a -> b E mo AI Po, E SE and o,Pa E SP Q } Fo ins ance, a ca esian mo phism belongs o S i and only i i s p o- jec ion is in SE, and a ib e-mo phism co E mo A x belongs o S i and only i i is in S,, . I is clea ha his choice o S ag ees wi h ha o weak equi alences made in [Roigl, Theo em 5 .1], conside ing SE and S,, he weak equi alences o he model ca ego ies £ and A .,, espec i ely . In pa icula , i o he ca ego y Adgc(R) 2 we ake SAdgc(R) and SA he classes o quism o Adgc(R) and Adgc(A), espec i ely, he mo phism (1) is in S i and only i and g a e quism . MINIMAL RESOLUTIONS AND OTHER MINIMAL MODELS  301 Theo em 3 .4 . Le m E obj A . Then, m is a minimal objec o A i and only i m E obj Apm and Pm E £ a e minimal objec s . P oo . Le us assume ha m is a minimal objec o A . Le us show ha i is so in Apm . Le a : a -> m E Spm . Then u E S and, as m is minimal in A, he e exis s u' : m ---> a E mo A such ha uu' = l m . The only hing we ha e o p o e is ha a' E mo Apm and his ollows by e alua ing P on bo h sides o he las equali y, which gi es us Po,' = lpm . Le us see ha Pm E obj £ is also a minimal objec .  Le s : x --> Pm E S £ . Conside he ca esian mo phism ms : s*m -> m . I belongs o S because P(ms) = s . Then, since m is minimal in A, he e exis s p : m --> s*m such ha m s p = l m . Hence s(Pp) = lpm . Con e sely, le m be a minimal objec o Apm , Pm a minimal objec o £ and u : a -> m a mo phism o S . Then s = Pa : Pa -> Pm ESE and, as Pm is minimal, he e exis s : Pm --> Pa such ha s = lpm . Le us conside he sou ce ac o iza ion o a : o, = m'u' . Then s E SPa and so * (m') : *a ---> *s*m = m is in Sp m . Because o he minimali y o m in Apm , his mo phism has a sec ion cp : m --> * a and a (p : m -> a is a sec ion o a . Co olla y 1 . I o all x E obj £, A x and £ ha e enough minimals, hen A has enough minimals . P oo - Le us build a minimal model o a E obj A : ake a minimal model o Pa, which exis s by hypo hesis : s : m' -> Pa . Take a minimal model o s*a in . .4  ,, : p : m ---> s*a . Then a s p : m --> a is a minimal model o a in A . Co olla y 2 . Le A be endowed wi h a model ca ego y s uc u e such as in [Roig1, Theo em 5 .1] . Assume also ha , o e e y x E obj £, A, and £ ha e enough minimals . Then he inclusion Ami,, ---> A induces an equi alen e o ca ego ies 7 Amin = Ho A The ollowing esul could also be p o ed o any sui able ca ego y o mo phisms di ec ly om he de ini ions : Co olla y 3 . A mo phism o R-dgc algeb as : A -> B is a minimal objec o Adgc(R) 2 i and only i A is a minimal R-dgc algeb a and is a minimal mo phism . Adgc(R) 2 has enough minimals, o ins an e, i we ake R a ze o global dimension ing and es ic ou sel es o non-nega i e homologi- cally connec ed R-dgc algeb as (see [Hal, Chap e 9]) . A minimal model 302  A . RoIG o : A -> B in he ca ego y o mo phisms is cons uc ed as in Co ol- la y 1 : we ake a minimal model o A in Adgc(R), p : MA - A, and aminimal model o p : MA -> B in Adgc(MA) ; ha is o say, a commu a i e iangle o Adgc(R), Re e en es B whe e o, is a quism o R-dgc algeb as . In [Hal], MA -~ M pis called a A-minimal A-ex ension . Dualizing he p e ious esul s o igh minimal objec s and igh mod- els, we see ha in he ca ego y 0°Se o simplicial se s, aking S o be he mo phisms which induces isomo phisms in all he homo opy g oups, minimal Kan complexes a e he minimal objec s in e ms o ( he dual o ) ou De ini ion 1 .1 . This ollows om ( he dual o ) ou P oposi ion 1 .6 and [May, 9 .5 and 9 .7] . The same p oposi ion and [May, 10 .6, 10 .7 and 10 .13], aking in o accoun ha we do no peed he de o ma ion e ac s o be s ong, show ha he minimal Kan ib a ions wi h base B a e he minimal objec s o AlSe /B . These esul s and he dual o ou Theo em 3 .2, gi e us Co olla y 4 . Minimal ib e spaces cons i u e he minimal objec s o (AOSe ) 2 . He e, (AOSe ) Z is he ca ego y o mo phisms o simplicial se s, bi ib ed wi h he codomain unc o and minimal ib e space means aminimal Kan ib a ion p : E --> B wi h B a minimal Kan complex . This las co olla y could also be deduced om he dual o 1 .6 and [May, 10 .11 and 10 .16] . [B-G] BOUSFIELD, A . K . AND GUGENHEIM, V . K . A . M ., "On PL De Rham heo y and a ional homo opy ype," Mem . AMS 179, 1976 . [Ei] EILENBERG, S ., Homological dimensions and syzygies, Ann . o Ma hs . 64 (1956), 328-336 . [G-Z] GABRIEL, P . AND ZISMAN, M ., "Calculas o ac ions and ho- mo opy heo y," Sp inge , 1967 . [H-T] HALPERIN, S . ANDTANRÉ D ., Homo opie il ée e ib és C°°, Ill . J . o Ma hs . 34 (1990), 284-324 . MINIMAL RESOLUTIONS AND OTHER MINIMAL MODELS  303 [Ha ] HARTSHORNE, R ., "Residues and duali y," Sp inge LNM 20, 1966 . [Hal] HALPERIN, S ., "Lec u es on minimal models," Mem . SMF, nou- elle sé ie, 9/10, 1983 . [Kan] KAN, D ., Minimal ee CSS g oups, Ill . J . o Ma hs . 2 (1958), 449-476 . [Ma ] MATSUMURA, H ., "Commu a i e ing heo y," Camb idge Uni- e si y P ess, 1986 . [May] MAY, J . P ., "Simplicial objec s in algeb aic opology," Van Nos- and, 1967 . [Qui] QUILLEN, D . G ., "Homo opical Ageb a," Sp inge LNM 47, 1967 . [Roigl] RoIG A ., Model ca ego y s uc u es in bi ib ed ca ego ies, o appea in J . o Pu e and Applied Algeb a . [Roig2] RoIG, A ., Fo malizabili y o DG modules and mo phisms o DGC algeb as, o appea in Ill . J . o Ma hs . . [Roig3] RoIG, A ., Modéles minimaux e onc eu s dé i és, o appea in J . o Pu e and Applied Algeb a . [SGA1] GROTHENDIECK, A ., "Séminai e de Géome ie Algéb ique 1," Sp inge LNM 224, 1971 . [Ta e] TATE, J ., Homology o noe he ian ings and local ings, Ill . J . o Ma hs . 1 (1957), 14--27 . Depa amen de Ma emá ica Aplicada I, ETSEIB Uni e si a Poli écnica de Ca alunya A . Diagonal 647 08028 Ba celona SPAIN P ime a e sió ebuda el 29 de Se emb e de 1992, da e a e si6 ebuda el 8 de Gene de 1993