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´
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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
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