Combinatorics of syzygies for semigroup algebras

Collec . Ma h. 49, 2–3 (1998), 239–256 c 1998 Uni e si a de Ba celona Combina o ics o syzygies o semig oup algeb as Emilio B iales∗and Pila Pis´ on∗ Depa amen o de ´ Algeb a, Facul ad de Ma em´ a icas, Apa ado 1160, 41080 Se illa (Spain) E-mail: [email p o ec ed] [email p o ec ed] An onio Campillo†and Ca los Ma iju´ an† Depa amen o de ´ Algeb a, Geome ´ ıa y Topolog´ ıa, Facul ad de Ciencias, P ado de la Magdalena s/n, 47005-Valladolid (Spain) E-mail: [email p o ec ed]a.es [email p o ec ed]a.es Dedica ed o he memo y o P o esso Fe nando Se ano Abs ac We desc ibe how he g aded minimal esolu ion o ce ain semig oup algeb as is ela ed o he combina o ics o some simplicial complexes. We ob ain cha ac- e iza ions o he Cohen-Macaulay and Go ens ein condi ions. The Cohen- Macaulay ypeis compu ed om combina o ics. Asan applica ion, wecompu e explici ly he g aded minimal esolu ion o monomial bo h a ine and simplicial p ojec i e su aces. In oduc ion The mo i a ion o his pape is o s udy he ela ionships be ween he gene a o s o he ideals defining monomial a ie ies, i.e., affine a ie ies pa ame e ized by mono- mial equa ions. Mo e p ecisely, one wan s o s udy he minimal esolu ion o he ∗Pa iallysuppo edbyDGICYTPB94-1435andpa iallysuppo edbyJun adeAndaluc´ıaAyuda a G upos 1144. †Suppo ed by DGICYT PB94 1111-C02-01. 239 Collec anea Ma hema ica (elec onic e sion): h p://www.ma .ub.es/CM 240 B iales, Campillo, Ma iju´ an and Pis´ on algeb a o a fini ely gene a ed semig oup iewed as a module o e a polynomial ing. The module s uc u e co esponds o he choice o a gene a o sys em o he semig oup. I is well known ha such defining ideals (also called o ic ideals, see o in- s ance [16]) a e gene a ed by binomials and ha one can de i e some combina o ial me hods o cons uc minimal sys ems o binomial gene a o s ([10], [9], [2], [13], [11], [6], [7], [3], [14], [16]). One can see (see [5] and 2.1 below), how he g aded minimal esolu ion o he semig oup algeb a is ela ed o he combina o ics o some simplicial complexes associa ed o he semig oup elemen s o commu a i e cancella i e fini ely gene a ed semig oups Swi h S∩(−S)={0}. In pa icula , when he algeb a is Cohen-Macaulay (see [15], [17] and 2.2 below o cha ac e iza ion o ha condi ion) he Cohen-Macaulay ype can be compu ed om combina o ics (2.2) and he Go en- s ein case can be cha ac e ized by a symme y p ope y on S(2.3) simila o he well known case o nume ical semig oups due o Kunz ([12]). Me hods o compu e he homology o hese complexes a e de eloped in [4] using u he combina o ics. In sec ion 3, we gi e, in e ms o he homology o abo e simplicial complexes, an explici cons uc ion o he minimal g aded esolu ion o he algeb a o such a semig oup. Thus, one concludes ha , in p ac ice, combina o ics can be applied o cons uc minimal sys ems o gene a o s no only o he ideal bu also o modules o highe o de syzygies. Such a cons uc ion was fi s conside ed and applied in [1]. In sec ion 4, we apply he abo e cons uc ion o compu e explici ly he g aded minimal esolu ion o monomial bo h affine and simplicial p ojec i e su aces. 1. The minimal esolu ion Le Sdeno e a commu a i e semig oup wi h a ze o elemen 0 ∈S. The associa ed abelian g oup is a pai (G(S),i) whe e G(S) is an abelian g oup and i:S→G(S) a semig oup homomo phism such ha , o any o he such pai (H,j) one has a unique g oup homomo phism ϕ:G(S)→Hsuch ha ϕ◦i=j. The associa ed abelian g oup G(S) exis s and i is unique dis ega ding isomo phism, and is a fini ely gene a ed g oup i Sis a fini ely gene a ed semig oup. The map iis injec i e i and only i Sis cancella i e, i.e., i m+n=m+n,m, n, n∈S, implies n= n. Equi alen ly, Sis cancella i e i and only i i is isomo phic o an addi i e subsemig oup o some abelian g oup. Fo he pu pose o his pape , we will say ha a semig oup Sis combina o ially fini e (c. .) i o any m∈S he e a e only a fini e numbe o exp essions o ype m=m1+···+mqwi h q∈Nand mi∈S−{0}. P oposi ion 1.1 in [3] cha ac e izes Combina o ics o syzygies o semig oup algeb as 241 he p ope y c. . o fini ely gene a ed cancella i e semig oups, and p o ides he Nakayama lemma o S-g aded modules (P oposi ion 1.4 in [3]). Assume Sis cancella i e and combina o ially fini e, and le A=⊕m∈SAmbe a commu a i e ing g aded o e S. Le P=⊕m∈S−{0}Ambe i s i ele an ideal. Thus, i N=⊕m∈SNmis a g aded A-module and Gis a subse o homogeneous elemen s in N, hen Gis a sys em o gene a o s o he module Ni and only i he classes module PN o he elemen s o Ga e a sys em o gene a o s o he A0=A/P- module N/NP. In pa icula , when A0is a field, he minimal se s o homogeneous gene a o s o Nall ha e he same ca dinali y and a e exac ly hose subse s Ggi ing ise o bases o he ec o space N/PN. F om now on, Sdeno e a semig oup ha is: fini ely gene a ed, combina o ially fini e, cancella i e, and commu a i e ( .g.c. .c.c. in sho ). Le us fix a sys em o gene a o s n1, ..., n o Swi h ni∈S−{0}. Le also fix a commu a i e field k. Associa ed wi h his si ua ion one has he algeb a o S, i.e. he ec o space R= m∈S Rm,R m:= k{m}, endowed wi h a mul iplica ion which is k-linea and such ha {m}·{n}:= {m+ n} o he symbols {m},{n}o m, n ∈S. The choice o gene a o s p o ides an S-g adua ion o e he polynomial algeb a in inde e mina es A:= k[X1, ..., X ], assigning he weigh ni o a iable Xi. Tha is, A= m∈S Am, whe e Amis he ec o subspace o Agene a ed by all he monomials Xl1 1···Xl wi h  i=1 lini=m. The condi ion c. . means p ecisely ha he ec o spaces Am a e fini ely dimensional. One has he su jec i e S-g aded k-algeb a homomo phism ϕ0:A→R which akes Xi o he symbol {ni}, gi ing on Ran S-g aded A-module. By he S-g aded Nakayama lemma and using ecu ence, one cons uc s S- g aded k-algeb a homomo phisms ϕj+1 :Abj+1 →Abj co esponding o a choice o a minimal se o homogeneous gene a o s o he module Nj:= ke (ϕj)(b0=1,N0is he ideal ke (ϕ0) which will be deno ed by I). He e, 242 B iales, Campillo, Ma iju´ an and Pis´ on i miis he deg ee o he i- h gene a o o Nj, hen, he g ading on Abj+1 has as homogeneous elemen s he bj+1- uples whose en ies a e homogeneous elemen s in A he i- h one being o deg ee m−mi o each i. Thus, one ge s a minimal ee S-g aded esolu ion o he A-module Ro ype ···→Abj+1 ϕj+1 →Abj→···→Ab2ϕ2 →Ab1ϕ1 →Aϕ0 →R→0 whe e bj+1 := m∈SdimkVj(m) wi h Vj(m):=(Nj)m/(PNj)m. The abo e summa ion is fini e by he noe he ian p ope y on A. The dimension o he ec o space Vj(m) can be in e p e ed as he numbe o gene a o s o deg ee min a minimal sys em o gene a o s o j- h syzygy module Nj. The Auslande -Buchbaum heo em gua an ees ha bj= 0 o j>p= − dep hARand bp= 0. The dep h o Ris bounded by i s dimension as k-algeb a, his dimension being no hing bu he ank o he abelian g oup G(S). The bound is eached exac ly when Ris Cohen-Macaulay. On he o he hand, i S={0} he condi ion c. . yields a dep hARo a leas 1. We will assume S={0} h oughou he es o he pape . 2. Simplicial complexes and Koszul homology Le Sbe a .g.c. .c.c. semig oup and n1, ..., n a sys em o gene a o s o Swi h ni∈S−{0}. Se Λ := {1, ..., }and o each subse F⊂Λ, nF:= i∈Fni, (n∅= 0). Fo each m∈Sone has an abs ac simplicial complex (subcomplex o he simplex o pa s o Λ) gi en by ∆m:= F⊂Λ|m−nF∈S. We will conside he educed homology  H·(∆) o he complexes ∆ o P(Λ) wi h alues in a field k(fixed om now on o he es o he pape ). To fix no a ions, w i e dim F=ca d F−1 o a ace Fand choose he o ien a ion on each ace Fo ∆m aken he elemen s o Fin inc easing o de . Then,  Cj(∆m) is he k- ec o space gene a ed ( eely) by he j-dimensional aces o ∆mand,  Hj(∆m)∼ =(ke δj)/(Im δj+1) whe e δj(F)=  F∈∆m dim F=j−1 εFF·F o a j-dimensional ace Fand εFF=0,1,−1 a e he coefficien s gi en by he abo e choice o o ien a ions. No ice ha εFF= 0 i and only i F⊂ F. Combina o ics o syzygies o semig oup algeb as 243 The choice o he gene a o s in Sgi es us he sequences X1, ..., X o homo- geneous elemen s o he S-g aded algeb a A, as well as he sequence o symbols {n1}, ..., {n }which a e also homogeneous elemen s o R. Since X:= (X1, ..., X ) is a egula sequence in A, hen he Koszul complex o i is he exac sequence 0→ A →···→ j+1 A λj → j A λj−1 →···→A λ0 →A→k→0 which is S-g aded o deg ee 0 i one gi es o he elemen ei1∧···∧eij({ei} he s anda d basis o A ) he deg ee ni1+···+nij. No ice ha , acco ding o he choice o o ien a ions made abou he subsimplices o pa s o Λ, i o F={i0<···<i j}⊂Λ one w i es eF:= ei0∧···∧eij, hen λj is gi en by λj(eF)=  F∈∆m dim F=j−1 εFF XF XF eF whe e XFs ands o i∈FXi(X∅= 1). The Koszul homology o symbols {n}:= ({n1}, ..., {n })inRis ela ed o he homology o ∆m. No ice ha , since {ni}is homogeneous, again he Koszul complex is S-g aded and he e o e one has a g aded decomposi ion o he homology Kj{n},R = m∈S Kj{n},R m Fo each deg ee m, he defini ions o ∆mand he Koszul homology gi es iso- mo phisms Kj{n},R m∼ = Hj∆m Theo em 2.1 Le Sbe a .g.c. .c.c. semig oup. Fix a sys em o gene a o s n1, ..., n o Sin S−{0}and a commu a i e field kand conside he minimal S-g aded esolu ion o Rand he complexes ∆massocia ed o he choice o gene a o s. Then one has k- ec o space isomo phisms  Hj(∆m)∼ =Vj(m) o e e y m∈Sand j≥0. 244 B iales, Campillo, Ma iju´ an and Pis´ on P oo . Bo h R=A/I and k=A/P a e S-g aded A-modules, so one has S-g aded A-module isomo phisms To j A(R, k)∼ =To j A(k,R) o j≥0. In pa icula , deg ee by deg ee one has k- ec o space isomo phisms To j A(R, k)m∼ =To j A(k,R)m o e e y m∈Sand j≥0. Now, o compu e To j A(R, k)m, one can enso he minimal esolu ion o Rby k. This yields To j A(R, k)m∼ =Vj(m). In he same way, o compu e To j A(k,R)m, one can enso he Koszul complex o Xby R. Since he image o Xiin Ris no hing bu {ni}, one ge s he Koszul complex o {n}in Rand hence, one has To j A(k,R)m∼ = Hj(∆m). Thus he isomo phism  Hj(∆m)∼ =Vj(m) as equi ed.  Co olla y 2.2 Gi en he same assump ions as Theo em 2.1, he dep h o he semig oup algeb a Ris equal o −pwhe e pis he leas in ege such ha  Hp(∆m)=0 o e e y m∈S. In pa icula , Ris Cohen-Macaulay i and only i  H −s(∆m)=0 o e e y m∈S, whe e s= ank G(S).I Ris Cohen-Macaulay hen he Cohen-Macaulay ype τR o Ris gi en by τR= m∈S dimk H −s−1(∆m). Fu he mo e, Ris Go ens ein i and only i Ris Cohen-Macaulay and i  H −s−1(∆m)=0exac ly o one m o which dimk H −s−1(∆m)=1. P oo . The abo e ollows om he Auslande -Buchbaum heo em. The o mula o τR ollows om he ac ha τR=b −sin he Cohen-Macaulay case.  The ollowing esul eflec s in e ms o combina o ial symme y he Go ens ein condi ion on he ing R. This can be seen as a gene aliza ion o he well known cha ac e iza ion o Go ens einess o nume ical semig oups due o Kunz [12]. To s a e he esul , no ice ha ∆mmakes sense o min G(S). I is clea ha o m∈G(S)−S,∆ mis he emp y simplicial complex and ha he e o e  Hj(∆m) = 0 o such an mand j=−1,0,1,2. Also, no ice ha ∆0is he only complex among he ∆m’s wi h he p ope y ha  H−1(∆m)= 0 (in ac i is a one dimensional space). Finally, le us se Hj(∆m) = 0 o j∈Z,j<−1 and m∈G(S) Co olla y 2.3 Gi en he same assump ions a 2.1, assume ha he semig oup algeb a Ris Go ens ein and le n∈Sbe he elemen such ha  H −s−1(∆n)=0. Then o any pai o elemen s m, m∈G(S)wi h m+m=nand j∈Zone has  Hj(∆m)∼ = H −s−j(∆m). Combina o ics o syzygies o semig oup algeb as 245 P oo . This ollows om 2.1 and he symme y o he g aded esolu ion in he Go ens ein case.  Rema ks 2.4 (i) I o some n∈Sone has he isomo phisms in Co olla y 2.3, hen Ris Go en- s ein. In ac , by he symme y in 2.3 one has  Hj(∆m) = 0 o m∈Sand j> −s,soRis Cohen-Macaulay. Now, since  H−1(∆0)∼ =kand  H−1(∆m)=0 o m= 0, i ollows om he symme y ha  H −s−1(∆m) = 0 o m=nand  H −s−1(∆m)∼ =k, hence Ris Go ens ein. (ii) I Sis a nume ical semig oup, i.e. a subsemig oup o Nwi h N−Sfini e, hen he isomo phisms in 2.3 a e an equi alen condi ion o he ac ha Sis a symme ic semig oup, i.e. sa is ying he p ope y ha i m, m∈Zand such ha m+m=c−1, whe e cis he conduc o o S, hen ei he m∈So m∈S (see [5] o de ails). Thus, Co olla y 2.3 o nume ical semig oups is equi alen o he c i e ia by Kunz ha Ris Go ens ein i and only i Sis symme ic. (iii) Fu he cha ac e iza ions o Cohen-Macaulayness in combina o ial ways can be ound in [15], [17], [4]. 3. Compu ing syzygies om combina o ics In his sec ion we will pu o m he isomo phisms in Theo em 2.1 in an explici way. As a consequence, one can cons uc minimal sys ems o homogeneous gene a o s o he successi e syzygy modules only by aking he images o he base elemen s o he homology spaces  Hj(∆m). Fo j= 0 such isomo phisms a e no difficul o cons uc and hey we e al eady used in [3] o compu ing minimal sys ems o gene a o s o he ideal o he semig oup. As in he abo e sec ions, le Sbe a .g.c. .c.c. semig oup. Fix a commu a i e field kand a sys em o gene a o s n1, ..., n o Swi h ni∈S−{0}. Keep all he no a ions in he abo e wo sec ions and deno e by Lj=Abj,L = A ,L ,j(m) he deg ee mcomponen o L ⊗Lj,ϕ ,j :L ,j(m)→L ,j−1(m), 0 ≤j≤ −1, −1≤ ≤ −1 ( esp. λ ,j :L ,j →L −1,j,−1≤j≤ −1,0≤ ≤ −1), he linea map induced by ϕj( esp. λ ). No ice ha , as abo e ϕ ,j is no hing bu he deg ee mpa o he map IdL ⊗ϕj( esp. λ ⊗IdLj). Hence, one has λ ,j−1◦ϕ ,j =ϕ −1,j ◦λ ,j, o any j, wi h 0 ≤j, ≤ −1. Mo eo e , one has he exac sequences 0→L −1,j(m)λ −1,j →L −2,j(m)→···→L0,j(m)λ0,j →L−1,j(m) 246 B iales, Campillo, Ma iju´ an and Pis´ on o 0 ≤j≤ −1 and 0→L , −1(m)ϕ , −1 →L , −2(m)→···→L ,0(m)ϕ ,0 →L ,−1(m). Now, conside he ec o subspace T ,j(m)o L ,j(m) gi en by T ,j(m) = ke (λ ,j)∩ke (ϕ ,j )i j≥0 and ≥0, T ,−1(m) = ke (λ ,−1)∩Im(ϕ ,0) o ≥0, T−1,j(m)= Im(λ0,j)∩ke (ϕ−1,j) o j≥0. No ice ha one has ke (λ ,j ) = Im(λ +1,j) and ke (ϕ ,j) = Im(ϕ ,j+1) o j≥0 and ≥0. Lemma 3.1 (i) Fo ≥0,T ,−1(m)is canonically isomo phic o he zycle space  Z (∆m). (ii) Fo j≥0, one has T−1,j(m)=(Nj)m⊂Abjm=L−1,j(m). P oo . Since ϕ ,0is su jec i e, one has T ,−1(m)=ke (λ ,−1). Now, T ,−1(m)is canonically isomo phic o he o de chain ec o space and λ ,−1co esponds o he bounda y o he educed homology o ∆m. Thus, one has T ,−1(m)=  Z (∆m) which shows (i). Since ke (ϕ−1,j )=(Nj)m, o p o e (ii) i is enough o shows ha (Nj)mis included in Im(λ0,j)=(PAbj)m. Le a=(a1, ..., abj)∈(Nj)m. Then ais a syzygy o he S-g aded module Nj−1 ela i e o a minimal sys em o homogeneous gene a o s, so one should ha e al∈P o each l. This shows a∈(PAbj)mas equi ed.  In he sequel we will use he ollowing wo basic co espondences σ ,j =(λ ,j+1)◦(ϕ ,j+1)−1 o j≥−1, ≥0, γ ,j =(ϕ +1,j )◦(λ +1,j )−1 o j≥0, ≥−1. Since ϕ ,j+1 ( esp. λ ,j+1) is no necessa ily an injec i e map, he co espondence σ ,j ( esp. γ ,j) is seen as a mul i alued unc ion om Im(ϕ ,j+1) ( esp. Im(λ +1,j )) o L −1,j+1(m) ( esp. L +1,j−1(m)). Lemma 3.2 (i) The co espondence σ ,j akes T ,j(m) o T −1,j+1(m). (ii) The co espondence γ ,j akes T ,j(m) o T +1,j−1(m). Combina o ics o syzygies o semig oup algeb as 247 P oo . I ≥0, j≥−1 one has T ,j(m) = ke (λ ,j)∩Im(ϕ ,j+1) and T −1,j+1(m)= Im(λ ,j+1)∩ke (ϕ −1,j+1). Thus, in pa icula , σ ,j is defined on T ,j (m). On he o he hand, since each elemen o T ,j(m)isinke (λ ,j ), he image by λ ,j+1 o any in e se image in L ,j+1(m) o such an elemen belongs o Im(λ ,j+1)∩ke (ϕ −1,j+1)= T −1,j+1(m). This shows (i). In he same way, i j≥0, ≥−1 one has T ,j (m)=Im(λ +1,j )∩ke (ϕ ,j ) and T +1,j−1(m) = ke (λ +1,j−1)∩Im(ϕ +1,j). Again, in pa icula γ ,j is defined on T ,j(m). On he o he hand, since each elemen in T ,j(m)isinke (ϕ ,j ), he image by ϕ +1,j o any in e se image o such an elemen in L +1,j (m) belongs o ke (λ +1,j−1)∩Im(ϕ +1,j ). This shows (ii).  We now come o he main cons uc ion o he sec ion. I one fixes j≥0 and m∈ S, hen Lemmas 3.1 and 3.2 show ha one has he ollowing wo co espondences σj:= (σ0,j−1)◦(σ1,j−2)◦···◦(σj,−1):  Zj(∆m)→(Nj)m, γj:= (γj−1,0)◦(γj−2,1)◦···◦(γ−1,j):(Nj)m→ Zj(∆m). By composing σj( esp. γj) wi h he quo ien maps (Nj)m→(Nj)m/(PNj)m= Vj(m) ( esp.  Zj(∆m)→ Hj(∆m)) one ge s wo new co espondences σj: Zj(∆m)→Vj(m), γj:(Nj)m→ Hj(∆m). A p io i, he co espondences σjand γja e mul i alued unc ions. The nex heo em shows how hey a e, in ac , linea uni alued unc ions inducing he isomo phisms in Theo em 2.1. Theo em 3.3 Gi en he same assump ions and no a ions as abo e, o any m∈Sand j≥0 one has (1) The co espondences σjand γja e well defined k-linea maps. The map σj akes bounda ies o he educed homology o ze o. The map γj akes elemen s in (PNj)m o ze o. σj akes bounda ies o he educed homology o 0. γj akes elemen s in (PNj)m o 0. (2) The k-linea maps  Hj(∆m)→Vj(m)and Vj(m)→ Hj(∆m)induced, espec- i ely, by σjand γja e in e se o one ano he . 254 B iales, Campillo, Ma iju´ an and Pis´ on Fo his, no ice ha one has m∈Qi and only i Tm={∅}, and m∈D(1) i and only i Tm=Twhe e T=∅,{e1},{e2},{e3},{e1,e 2},{e1,e 3},{e2,e 3}. I m∈D(0), hen Tmshould be one o he ollowing se en complexes ( he only possible non connec ed complexes wi h h ee e ices). U1={∅,{e1},{e2},{e3}} U2={∅,{e1},{e2}} U3={∅,{e1},{e3}} U4={∅,{e2},{e3}} U5={∅,{e1},{e2},{e3},{e1,e 2}} U6={∅,{e1},{e2},{e3},{e1,e 3} U7={∅,{e1},{e2},{e3},{e2,e 3}} . Le us deno e by D(0)j he subse o D(0) consis ing o hose m∈Ssuch ha Tm=Uj. Le Q,D(1), D(0)jbe he espec i e p ojec ions o Q,D(1), D(0)jon he (1,2)-plane. Lemma 4.7 The p ojec ions Q→Q,D(1) →D(1),D(0)j→D(0)j o j=1,...,7a e bijec i e maps. P oo . The maps a e ob iously su jec i e. To p o e ha hey a e injec i e we will use he ollowing ac : each one o he simplicial complexes ∆ = {∅},T,U 1,...,U 7 has associa ed a ace J∈P(E) such ha e3∈J,J∈ ∆ and J−{e3}∈∆. Now, le ∆ be any one o he abo e nine simplicial complexes and le Hbe he se Q,D(1) o D(0)jwhich co espond o he complex ∆. Take (a, b)∈H. Deno e by he leas in ege such ha , i m=(a, b, ), hen Tm= ∆ (i.e. m∈H). Elemen s o Sin he fibe o (a, b) by he p ojec ion on he (1,2)-plane a e o ype m=(a, b, +λd) wi h λ∈Z.I λ<0, hen Tm= ∆ by he minimali y o .I λ>0 hen Tm=∆ as one has J∈ Tmand J∈Tm,Jbeing a ace associa ed o ∆ wi h he p ope y indica ed in he ac a he beginning o he p oo . This shows ha he p ojec ion H→His injec i e as equi ed.  Lemma 4.7 shows ha in o de o compu e he se s Q,D(1), D(0)j,i is sufficien o compu e he fini e se s Q,D(1), D(0)jand, o each elemen (a, b)on each one o hose se s, he alue o such ha he semig oup elemen (a, b, ) ealizes he co esponding simplicial complex {∅},T,Uj. Combina o ics o syzygies o semig oup algeb as 255 Lemma 4.8 Wi h assump ions and no a ions as abo e, o (a, b)∈S12 one has: (i) (a, b)∈Qi and only i l(a, b)≤l(a−d, b)and l(a, b)≤l(a, b −d). (ii) (a, b)∈D(1) i and only i l(a−d, b−d)≥l(a−d, b),l(a−d, b−d)≥l(a, b−d), and l(a−d, b −d)<∞. (iii) (a, b)∈D(0)1i and only i l(a, b)≤l(a−d, b)=l(a, b −d)≤l(a−d, b −d) and l(a−d, b)<∞. (i ) (a, b)∈D(0)2i and only i l(a, b)>l(a, b −d)=l(a−d, b)≤l(a−d, b −d). ( ) (a, b)∈D(0)3i and only i l(a, b)≤l(a−d, b)<l(a, b −d). ( i) (a, b)∈D(0)4i and only i l(a, b)≤l(a, b −d)<l(a−d, b). ( ii) (a, b)∈D(0)5i and only i l(a, b)≤l(a−d, b)=l(a, b−d)and l(a−d, b−d)< l(a−d, b). ( iii) (a, b)∈D(0)6i and only i l(a, b)≤l(a, b−d)≤l(a−d, b−d)and l(a−d, b)< l(a, b −d). (ix) (a, b)∈D(0)7i and only i l(a, b)≤l(a−d, b)≤l(a−d, b−d)and l(a, b−d)< l(a−d, b). The alue o such ha (a, b, )∈Q, D(1),D(0)j espec i ely in (i)-(ix) is gi en by ld−a−bwhe e lis gi en by (i) l(a, b), (ii) l(a−d, b−d)+2, (iii) l(a−d, b)+1, (i ) l(a, b), ( ) l(a−d, b)+1, ( i) l(a, b −d)+1, ( ii) l(a−d, b)+1, ( iii) l(a, b −d)+1, (ix) l(a−d, b)+1. P oo . Le (a, b, ) sa is y a+b+ =ld, hen (a, b, )∈Si and only i l≥l(a, b). Using he abo e ac , (i)-(ix) ollow by inspec ion case by case using he defini ion o l(a, b) and he same kind o a gumen s han in he p oo o Lemma 4.5.  Lemma 4.8 allows o compu e he se s Q,D(1), D(0) in a i hme ic e ms om he gene a o sys em o he semig oup. Again, by using (∗) and Rema k 4.3, one ge s he ollowing esul . Theo em 4.9 The syzygies o a simplicial p ojec i e monomial su ace can be de e mined om he knowledge o he semig oup gene a o s and he cha ac e is ic o he field. 256 B iales, Campillo, Ma iju´ an and Pis´ on Re e ences 1. A. A amo a and J. He zog, F ee esolu ions and Koszul homology, J. Pu e Appl. Algeb a 105(1) (1995), 1–16. 2. H. B esinsky, Binomial gene a ing se s o monomial cu es, wi h applica ions in A4,Rend. Sem. Ma . Uni . Poli ec. To ino 46(3) (1988), 353–370. 3. E. B iales, A. Campillo, C. Ma iju´an and P. Pis´on, Minimal Sys ems o Gene a o s o Ideals o Semig oups, J. Pu e Appl. Algeb a 127 (1998), 7–30. 4. A. Campillo and Ph. 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