Minimal Resolu ions o La ice Ideals and
In ege Linea P og amming
Emilio B iales-Mo ales
Dp o de ´
Algeb a
Uni e sidad de Se illa
E-mail: [email p o ec ed] ∗
An onio Campillo-L´opez
Dp o de ´
Algeb a Geome ´ıa y Topolog´ıa
Uni e sidad de Valladolid
E-mail: [email p o ec ed]
Pila Pis´on-Casa es
Dp o de ´
Algeb a
Uni e sidad de Se illa
E-mail: [email p o ec ed] †
Albe o Vigne on-Teno io
Dp o de Ma em´a icas
Uni e sidad de C´adiz
E-mail: [email p o ec ed] ‡
Dedica ed o P o esso J.L. Vicen e on his six ie h bi hday.
Abs ac
A combina o ial desc ip ion o he minimal ee esolu ion o a la ice ideal
allows us o he connec ion o In ege Linea P og amming and Algeb a. The
non null educed homology spaces o some simplicial complexes a e he key. The
ex emal ays o he associa ed cone educe he numbe o a iables.
Keywo ds : Resolu ions, simplicial complex, syzygy, la ice ideal, eg-
ula i y, In ege Linea P og amming, Hilbe bases, G ¨
obne bases
2000 Ma hema ics Subjec Classi ica ion:P ima y 13D02, 14M25; Seconda y 13P10,
68W30, 90C27
In oduc ion
The objec i e o his pape is o desc ibe how In ege Linea P og amming allows us
o ob ain he minimal ee esolu ion o a la ice ideal, I, om he gene a o s o he
semig oup, S, which pa ame izes he associa ed algeb aic a ie y.
Conc e ely, Hilbe bases o some diophan ine sys ems a e employed. These bases
a e he solu ion o he ypical In ege Linea P og amming P oblem, bu he mini-
∗Suppo ed by MCyT Spain, BFM2000-1523, and Jun a de Andaluc´ıa, FQM304.
†Suppo ed by MCyT Spain, BFM2000-1523, and Jun a de Andaluc´ıa, FQM304.
‡Suppo ed by MCyT Spain, BFM2000-1523, and Jun a de Andaluc´ıa, FQM304.
2Algeb aic Geome y and Singula i ies. Se illa, Sep embe 19-22, 2001
mali y wi h espec o a cos map is no imposed. Recall ha his ypical p oblem
is:
min{c·x|Ax =b, x ∈Nn},
whe e Ais an in ege ma ix, ban in ege ec o and ca eal ec o (c·xis he cos
map).
Anybody who has sol ed linea diophan ine equa ions in non nega i e in ege s, e en
wi h he mo e ecen me hods (see [18], [20], [23], [44] and [46]), knows ha only in he
case o a ew a iables he p oblem is ac able. I is well-known ha his p oblem is
NP-comple e (see o example [37]). The e o e, om he compu a ional iewpoin , ou
desc ip ion is no p ac ical in o de o ob ain he minimal ee esolu ion. Howe e ,
he me hod can be used o he con a y. Ou desc ip ion allows he unde s anding
o he ela ion be ween he syzygies o he ideal and In ege Linea P og amming.
One can compu e wi h G ¨obne bases using o example he Sch eye Theo em and
i s imp o emen s (see [33]), and look o applica ions o In ege P og amming. This
philosophy comes om [19] and [46], and p o ides a lo o applica ions in [50]. Fo
example, he ypical In ege Linea P og amming P oblem can be sol ed compu ing
he educed G ¨obne basis o an associa ed la ice ideal. O o ins ance, he G a e
basis ([28]) o an ideal can be ob ained om a educed G ¨obne basis o i s Law ence
li ing, which is i s unique minimal gene a ing se . Ne e heless, a he momen his
philosophy has only been employed in he case o he ideal I, bu no he syzygies
o he highe o de ( he ideal can be conside ed as he syzygies o o de ze o). Ou
desc ip ion yields he gene aliza ion.
As in [30] and [48], he combina o ial objec s we use a e simplicial complexes. Con-
c e ely, o any elemen o he semig oup S, we associa e wo simplicial complexes.
The elemen s in he semig oup ep esen he deg ees o he syzygies, in ac , he min-
imal ee esolu ion is S-g aded. The s udy o he non null educed homology spaces
o he simplicial complexes p o ides he concep o i- iangula ion. This concep is
he key in o de o unde s and he ela ion be ween In ege Linea P og amming and
Algeb a, conc e ely, be ween Hilbe bases and i h syzygies.
By means o a pa i ion o he gene a ing se o S, he numbe o a iables is educed
o he numbe o ex emal ays o he associa ed cone. This is ano he possible poin
o con inue esea ching. A gene a o o e each ex emal ay is chosen. Fixing he
a en ion on his subse o gene a o s, a new esolu ion is conside ed, he minimal
ee esolu ion o Io e a polynomial ing wi h only he a iables co esponding o
hese gene a o s.
We begin in sec ion 1 wi h he de ini ion o he algeb aic objec s we employ. In
sec ion 2 we gi e he combina o ial desc ip ion o he wo minimal ee esolu ions.
Sec ion 3 is dedica ed o he i- iangula ions in a simplicial complex and some appli-
ca ions. The exposi ion o how o compu e bo h esolu ions wi h G ¨obne bases is in
sec ion 4. All hese sec ions include he esul s we ha e al eady ob ained using he
echniques his pape desc ibes. Fo de ails he eade may also wan o consul he
e e ence joined o he conc e e esul .
3
Ano he possible applica ion o ou desc ip ion is in To ic Geome y. The no mal
o ic a ie ies [26], and mo e gene ally, he non-no mal o ic a ie ies [27] and [50],
appea as algeb aic a ie ies whose ideals a e la ice ones. Among ou esul s can be
ound desc ip ions o he egula i y o hese ideals as well as uppe bounds o he
deg ee o hei gene a o s. I is expec ed ha he e is some ela ion be ween hese
esul s and he conje u es o [24] and [50] (see also [49] and [39]). Fo a su ey o he
mode n de elopmen s in he heo y o o ic a ie ies see [21]. Some applica ions o
his heo y o he A i hme ic and In ege P og amming can be ound in [17].
The hull esolu ion is ano he ee esolu ion o a la ice ideal. This esolu ion is a
gene aliza ion o he esul s o gene ic la ice ideals in [2] and [38]. I was in oduced
in [5] using In ege P og amming. The s udy o he minimali y o his esolu ion is
a cu en esea ch objec i e (see [3] whe e he case o unimodula la ice ideals is
conside ed, and [36] o he monomial cu es in he a ine space o low dimension).
On he o he hand, i is known ha any binomial ideal is an in e sec ion o cellula
ideals [25]. The cellula ideals a e closely ela ed o he la ice ideals. Using he
cellula decomposi ion o a binomial ideal, i is possible o ob ain in o ma ion abou
he binomial ideal om he p ope ies o he la ice ideals ( o example, p ima y
decomposi ion o nilpo ence index, see [34] and [35]).
1 The wo minimal ee esolu ions associa ed wi h
a la ice ideal
Le kbe a commu a i e ield and k[X] = k[X1, . . . , Xn] he polynomial ing in n
inde e mina es, and he ideal m= (X1, . . . , Xn).
Le L ⊂ Znbe a la ice. The ideal o he la ice Lis
IL=hXu+−Xu−|u∈ Li,
whe e u=u+−u−,u+, u−∈Nn,ha e disjoin suppo .
Le Sbe a cancella i e commu a i e semig oup, wi h ze o elemen and gene a ed
by nelemen s Λ = {m1, . . . , mn}. Thus, Sis a subsemig oup o a ini ely gene a ed
abelian g oup. Deno e G(S) he smalles g oup con aining S. The semig oup k-
algeb a is k[S] = Lm∈Skχm,(χm·χm0=χm+m0). The ideal o S ela i e o Λ is
ke (ϕ0), whe e ϕ0is he k-algeb a mo phism
ϕ0:k[X]−→ k[S]
de ined by ϕ0(Xi) = χmi.No ice ha ϕ0is su jec i e, and hence k[S]≃k[X]/ke (ϕ0).
I ILis he ideal o he la ice L ⊂ Zn, hen ILis he ideal o he subsemig oup o
Zn/Lgene a ed by {e1+L, . . . , en+L}, whe e he ei’s a e he uni ec o s.
On he o he hand, he ideal o any semig oup S ela i e o a gene a ing se Λ is
he ideal o he la ice {u= (u1, . . . , un)∈Zn|Puimi= 0}(see [52] o de ails).
4Algeb aic Geome y and Singula i ies. Se illa, Sep embe 19-22, 2001
F om now on, we ix a la ice Lo equi alen ly a semig oup S. Assume ha
L ∩ Nn= (0), o equi alen ly S∩(−S) = (0). Le Ibe he ideal ela i e o a ixed
gene a ing se Λ = {m1, . . . , mn}o S. No ice ha Iis S-g aded because ϕ0is an
S-g aded mo phism o deg ee ze o, conside ing k[S] wi h he na u al S-g ading and
k[X] as an S-g aded ing, assigning he deg ee mi o Xi. The condi ion S∩(−S) = (0)
says ha k[X]m, he homogeneous elemen s o deg ee m∈Sin k[X], is a k- ec o
space o ini e dimension (see [8]).
Ano he applica ion o he condi ion S∩(−S) = (0), is Nakayama’s lemma o S-
g aded k[X]-modules (see [8]). Thus, he e exis s an S-g aded minimal ee esolu ion
o k[S], which is unique up o isomo phism. We deno e such a esolu ion by
0→k[X]bpϕp
→ · · · → k[X]b2ϕ2
→k[X]b1ϕ1
→k[X]ϕ0
→k[S]→0,
and le Ni= ke (ϕi) be he i h module o syzygies 0 ≤i≤p(N0=I).
No ice ha
bi+1 = dim(Ni/mNi),
whe e Ni/mNiis conside ed as a k- ec o space. Mo eo e , since his space is S-
g aded, i Vi(m) := (Ni/mNi)m, whe e m∈S, hen
bi+1 =X
m∈S
dimVi(m).
The Auslande -Buchbaum heo em gua an ees ha
p=n−dep hk[X]k[S],
whe e dep hk[X]k[S] is he dep h o k[S] as k[X]-module. I is known ha dep hk[X]k[S]
is bounded by dimk[S], which is he ank o he abelian g oup G(S). In he case he
bound is eached, k[S] is a Cohen-Macaulay ing. Thus, his case will be called Cohen-
Macaulay case. On he o he hand, i S6={0}, i is sa is ied ha dep hk[X]k[S]≥1.
Assume ha ank(G(S)) = d, le V=G(S)NZQ, and le C(S) be he cone
gene a ed by he image ¯
S, o Sin V. The cone C(S) is s ongly con ex because
S∩(−S) = (0).Thus, i is he numbe o ex emal ays o C(S), hen ≥d. This
implies ha he e exis s a se E⊂Λ wi h ]E = , such ha C(E) = C(S),whe e
C(E) is he cone in Vgene a ed by E. Fix such a se E.
The Ape y se Qo S ela i e o Eis de ined as
Q={q∈S|q−e6∈ S, ∀e∈E}.
Deno e k[E] he subalgeb a o k[S],
k[E] = M
m∈SE
kχm,
whe e SEis he subsemig oup o Sgene a ed by E. Le k[XE] be he polynomial
ing in he inde e mina es associa ed wi h E.k[XE] can be p ojec ed o e k[E], i
is enough o associa e o he inde e mina e Xi he symbol χmi, o any mi∈E.
5
k[S] is a k[E]-module, and he e o e also a k[XE]-module. The se
{χq|q∈Q},
is a minimal sys em o gene a o s o k[S] as k[E]-module, and he e o e, also as k[XE]-
module. Since k[E]⊂k[S] is an in eg al ex ension and k[S] is ini ely gene a ed as a
k[E]-algeb a, k[S] is ini ely gene a ed as a k[E]-module. So k[S] is a ini ely gene a ed
k[E]-module, and Qis a ini e se . Suppose ha β0=]Q,Q={q1, . . . , qβ0}, and
conside
Φ0:k[XE]β0−→ k[S]
de ined by Φ0(ei) = χqi,1≤i≤β0. We can conside he S-g aded minimal esolu ion
o k[S] as k[XE]-module
0→k[XE]βqΦq
→ · · · → k[XE]β2Φ2
→k[XE]β1Φ1
→k[XE]β0Φ0
→k[S]→0,
which is unique excep isomo phisms. Le Mi= ke (Φi) be he i h module o syzygies
o k[S] as k[XE]-module, 0 ≤i≤q. As be o e, by S-g aded Nakayama’s lemma, we
ob ain
βi+1 =X
m∈S
dimWi(m),
whe e Wi(m) := (Mi/mEMi)mis conside as a k- ec o space, and mEis he ideal o
k[XE] gene a ed by he inde e mina es o XE(Xisuch ha mi∈E).
The Auslande -Buchbaum heo em gua an ees ha
q= −dep hk[XE]k[S],
whe e dep hk[XE]k[S] is he dep h o k[S] as k[XE]-module. Using he combina o ial
desc ip ions o he abo e wo esolu ions in he ollowing sec ion, and he heo em
4.1 in [13], one ge s ha
dep hk[X]k[S] = dep hk[XE]k[S].
The e o e, p≥q.
Now, we will call he S-g aded minimal ee esolu ion o k[S] as k[X]-module he
long esolu ion, and he sho esolu ion he S-g aded minimal ee esolu ion o k[S]
as k[XE]-module.
2 Combina o ial desc ip ion o he esolu ions
Assume ha S6= (0), and conside he S-g aded minimal ee esolu ion,
0→k[X]bpϕp
→ · · · → k[X]b2ϕ2
→k[X]b1ϕ1
→k[X]ϕ0
→k[S]→0.

6Algeb aic Geome y and Singula i ies. Se illa, Sep embe 19-22, 2001
Fo any m∈S(o e en m∈G(S)) we de ine (inspi ed in some g aphs o [47]) he
simplicial complex :
∆m={F⊂Λ|m−nF∈S},
whe e nF=Pm0∈Fm0. Le e
Hi(∆m) be he k- ec o space o he i h educed homol-
ogy o ∆m, and ˜
hi(∆m) = dim( ˜
Hi(∆m)).
The e exis s an e ec i e isomo phism
(∗)˜
Hi(∆m)≃Vi(m),
o any m∈Sand o any i, 1 ≤i≤n−2, ( o de ails see [14],[16] and [7], o also [1]).
These isomo phisms a e a b idge be ween Combina o ics and Algeb a. Fo example,
no ice ha he numbe s biin he long esolu ion can be desc ibed by he ollowing
o mula
bi+1 =X
m∈S
˜
hi(∆m).
Ano he example, k[S] is Cohen Macaulay i and only i one has e
Hn−d(∆m) = 0 o
e e y m∈S, whe e d= ank G(S). I k[S] is Cohen Macaulay hen he Cohen
Macaulay ype τk[X]o k[S] is gi en by
τk[X]=X
m∈Se
hn−d−1(∆m).
Thus, in pa icula , k[S] is Go ens ein i and only i k[S] is Cohen Macaulay and i
e
Hn−d−1(∆m)6= 0 exac ly o one m o which, mo eo e , one has e
hn−d−1(∆m) = 1.
The o mula o τk[X] ollows om he ac ha τk[X]=bn−din he Cohen Macaulay
case. Mo eo e , i is possible o gene alize he well known cha ac e iza ion o Go en-
s einess o nume ical semig oups due o Kunz [32]. To s a e he esul , no ice ha o
m∈G(S)−S, ∆mis he emp y simplicial complex and he e o e one has ˜
Hi(∆m) = 0
o such an mand i≥ −1. Also no ice ha ∆0is he only complex among he ∆m’s
wi h he p ope y ˜
H−1(∆m)6= 0 (in ac i is a one dimensional space). Finally se
˜
Hi(∆m) = 0 o i∈Z,i < −1, and m∈G(S). F om he symme y o he g aded
esolu ion in he Go ens ein case, i Ris Go ens ein and le m∈Sbe he elemen
such ha ˜
Hn−d−1(∆m)6= 0, hen o any couple o elemen s m1, m2∈G(S) wi h
m1+m2=mand i∈Zone has
˜
Hi(∆m1)≃˜
Hn−d−i−2(∆m2)
(see [7] o de ails).
In he case o a nume ical semig oup, le cbe he leas elemen , such ha m∈S
o any m≥c.˜
Hi(∆m) = 0 o any m≥c+nΛ−1 and any i, because ∆mis he ull
simplex. The e o e, i Sis symme ic, he abo e isomo phism p o ides a symme ic
p ope y on he ma ix {˜
hi(∆m)}i,m.(This pa icula case was p o ed in [14])
Ano he impo an applica ion o hese isomo phisms is he cons uc ion o minimal
gene a ing se s o syzygies. No ice ha
S(i) := {m∈S|e
Hi(∆m)6= 0}, n −2≥i≥0,
7
is he se o S-deg ees o he minimal i-syzygies. The no he ian p ope y gua an ees
ha S(i) is a ini e se , he e o e he ollowing cons uc ion p o ides a me hod o
compu ing a minimal gene a ing se o Ni.
CONSTRUCTION:
STEP 1: Compu e S(i).
STEP 2: Fo any m∈S(i), ake he images o he elemen s in a basis o he i h
educed homology space ˜
Hi(∆m) by he isomo phism.
S ep 1 is comple ely sol ed in [12], bu he pa ial solu ion o i= 0 appea s in [8],
and o i= 1 in [43]. S ep 2 is sol ed wi h an algo i hmic me hod in [7] (Rema k 3.6).
The case i= 0 co esponds o he ideal I=N0. In his case, s ep 1 is equi alen
o de e mine he elemen m∈Ssuch ha ∆mis non-connec ed. These elemen s
a e cha ac e ized by he concep o o be m-isola ed ([16]) gi en by h ee a i hme ical
condi ions. Conc e ely:
Le m∈S, and le B={i1, ..., ip} ⊂ C⊂Λ, C6= Λ. We shall say Bis m-isola ed
om Λ −Ci :
1. I is possible o w i e
m=
p
X
j=1
γijnij=X
6∈C
ρ n ,
whe e γij,ρ ∈N, and 0 < γij o any j, 1 ≤j≤p.
2. I he e exis s m0∈Ssuch ha i is possible o w i e
m0=
p
X
j=1
γ0
ijnij=X
6∈B
ρ n ,
wi h γ0
ij,ρ ∈N,γ0
ij6= 0, and he e exis s 6∈ Csuch ha ρ 6= 0, hen
(γ0
i1, ..., γ0
ip)6<(γi1, ..., γip).
3. I B0={l1, ..., ls} ⊂ Band he e exis s m0∈Ssuch ha i is possible o w i e
m0=
s
X
j=1
γ0
ljnlj=X
6∈B0
ρ n ,
wi h γ0
lj,ρ ∈N, and he e exis s 6∈ Csuch ha ρ 6= 0, hen
(γ0
l1, ..., γ0
ls)6≤ (γl1, ..., γls).
We ob ain he ollowing esul :
Theo em 1 ([16]) Le m∈S, he ollowing condi ions a e equi alen s:
1: ∆mis non-connec ed ( ˜
H0(∆m)6= 0).
2: The e exis s C⊂Λ, such ha :
8Algeb aic Geome y and Singula i ies. Se illa, Sep embe 19-22, 2001
•C=∪g
j=1Tj.
•Tjis m-isola ed om Λ−C, o any j.
•Tj∩Tj+1 6=∅, o any j, 1≤j≤g−1.
This cha ac e iza ion allows us o ind he pa icula solu ions gi en o ew gene a-
o s in he nume ical case in [29] (n=3), [6] and [40] (n=4), and [15] (n=5). Mo eo e ,
by means o new combina o ial elemen s, he heo em yields an algo i hm. Conc e ely,
he e ices o some ladde s, o equi alen ly, he Hilbe bases o some diophan ine
sys ems a e used. (See [8] o de ails)
The case i= 1 is sol ed in [43] by cons uc ion o a ini e se con aining S(1).
This se is ob ained a e s udying he non-null spaces ˜
H1(∆m)6= (0).The concep o
F-ca i y in ∆mallows us o associa e wi h Ssome diophan ine sys ems. The Hilbe
bases o hese sys ems p o ide a check ini e se . This echnique is gene alized in [12]
o i≥2 . A new concep is necessa y, he i- iangula ion in ∆m.
Now, conside he S-g aded minimal ee esolu ion o k[S] as k[XE]-module
0→k[XE]β −1Φ −1
→ · · · → k[XE]β2Φ2
→k[XE]β1Φ1
→k[XE]β0Φ0
→k[S]→0.
This esolu ion can be desc ibed by means o o he simplicial complexes ([13]).
Conc e ely, i m∈S, le Tmbe he simplicial complex
Tm={F⊂E|m−nF∈S}.
Deno e e
Hi(Tm) he i h educed homology space o he simplicial complex Tm, and le
˜
hi(Tm) = dim( ˜
Hi(Tm)). The e exis s an isomo phism
(∗∗)˜
Hi(Tm)≃Wi(m),
o any m∈Sand o any i, 1 ≤i≤ −2 (see [41]).
As an applica ion o hese isomo phisms, i deno e
D(i) := {m∈S|e
Hi(Tm)6= 0},
we ob ain ha
βi+1 =X
m∈D(i)e
hi(Tm),0≤i≤ −2.
No ice ha , by he noe he ian p ope y, D(i) is ini e.
In [10] is shown how he se s D(i) can be ob ained gene alizing he echniques used
o compu ing S(i) in [12]. This p ocess will be ecalled in he ollowing sec ion.
Le A= Λ E, ]A =n− = . A i s applica ion o he abo e o mula is ha
S(i)⊂Ci, whe e
Ci={m∈S|m=m+nF,wi h m∈D( ) and F⊂A, ]F =i− , o some ≥ −1}
(see [13]). The e o e, in o de o de e mine he se S(i) i is enough o compu e D( )
o any ,−1≤ ≤min(i, −2).No ice ha his esul allows us o cons uc he
long esolu ion om he sho one.
9
3i-T iangula ions
The objec i e o his sec ion is o desc ibe how he se s S(i), 0 ≤i≤n−2, and D(i),
−1≤i≤ −2, can be ob ained sol ing diophan ine sys ems in non nega i e in ege s.
No ice ha D(−1) = Q, and since C(E) = C(S) o any elemen a∈A he e exis s
qa∈Nsuch ha
qa·a=X
e∈E
λe·e
wi h λe∈N.
Rema k 1 The e o e, in o de o ob ain he se Q, one can do:
1. Compu e he bounds qa,a∈A.
2. De e mine he elemen s m=Pa∈Aλa·a, wi h λa∈Nand λa< qa.
3. Check whe he he elemen s ma e in Qusing In ege Linea P og amming.
No ice ha one can compu e he se Qsol ing some diophan ine equa ions in non
nega i e in ege s, al hough his way is no p ac ical.
Fo sol ing he o he cases, we need he concep o i- iangula ion in a simplicial
complex. Le ∆ be an abs ac simplicial complex wi h e ices o e a ini e se V.
The educed i-homology o he simplicial complex ∆ is he k- ec o space
˜
Hi(∆) = ˜
Zi(∆)/˜
Bi(∆),
whe e ˜
Zi(∆) and ˜
Bi(∆) a e he spaces o cycles and bounda ies espec i ely.
Le i≥0 and F⊂ V. We will say ha τ={F1, . . . , F }is an i- iangula ion o F
i he ollowing p ope ies a e sa is ied:
1. ]Fj=i+ 1, ∀j= 1, . . . , .
2. F=S
j=1 Fj.
We will say ha τis an i- iangula ion o Fin ∆, i Fj∈∆, ∀j= 1, . . . , , and F /∈∆.
I ˜
Hi(∆) 6= 0, hen he e is c∈˜
Zi(∆) −˜
Bi(∆), c=P
j=1 λjFj, such ha τ=
{F1, . . . , F }is an i- iangula ion o Fin ∆, o F=S
j=1 Fj.
In he cases ∆ = ∆mo Tm,V= Λ o E espec i ely, i F⊂ V, and τ=
{F1, . . . , F }is an i- iangula ion o F, in ∆mo espec i ely in Tm, we can associa e
wi h τa diophan ine sys em solu ion. Conc e ely, le Gbe he ma ix whose columns
a e he chosen gene a o s o S,G:= (m1|. . . |mn)∈ M(d+s)×n(Z),conside ing he
16 Algeb aic Geome y and Singula i ies. Se illa, Sep embe 19-22, 2001
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