Rees algebras of square-free monomial ideals

0.22 MB·English

by  Louiza Fouli

REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS
LOUIZAFOULIANDKUEI-NUANLIN
ABSTRACT. WestudythedefiningequationsoftheReesalgebrasofsquare-freemonomialidealsin
3
apolynomialringoverafield. Weproposetheconstructionof agraph,namelythegeneratorgraph
1
of amonomial ideal, where themonomial generators serve asverticesfor thegraph. When I isa
0
square-freemonomialidealsuchthateachconnectedcomponent ofthegeneratorgraphof I hasat
2
most 5 verticesthen I has relationtype at most 3. In general, weestablish the defining equations
n
of the Reesalgebra inthiscase and givea combinatorial interpretationof them. Furthermore, we
a
providenewclassesofidealsoflineartype. WeshowthatwhenI isasquare-freemonomialideal
J
andthegeneratorgraphof I isthegraphofadisjointunionoftreesandgraphswithauniqueodd
8
cycle,thenIisanidealoflineartype.
1

]
C
A
1. INTRODUCTION
.
h
t Themainproblem ofinterest inthisarticle isthequestion ofdetermining thedefining equations
a
m of the Rees algebra of a square-free monomial ideal in a polynomial ring over a field. Let R be a
[ Noetherian ring and let I be an ideal. The Rees algebra, R(I), is defined to be the graded algebra

2 R(I)=R[It]= ⊕Iiti ⊂R[t], where t is an indeterminate. The Rees algebra of an ideal encodes
v i≥0
7 many of the analytic properties of the variety defined by I and it is the algebraic realization of the
2
blowupofavarietyalongasubvariety. TheblowupofSpec(R)alongV(I)istheprojectivespectrum
1
3 of the Rees algebra, R(I), of I. This important construction is the main tool in the resolution of
.
5 singularities ofanalgebraic variety.
0
2 Fromthealgebraic point ofview theReesalgebra ofanideal facilitates the study oftheasymp-
1
totic behavior of the ideal and it is essential in computing the integral closure of powers of ideals.
:
v
The Rees algebra of an ideal I in a Noetherian ring can be realized as a quotient of a polynomial
i
X ring and hence once the defining ideal of R(I) is determined, it is straightforward to compute the
r
a integralclosure, R(I). ItiswellknownthatR(I)= ⊕Iiti andoneobtainsIi=[R(I)].
i
i≥0
Weconsider thefollowing construction fortheRees algebra. LetRbe aNoetherian ring and let
I bean R-ideal. Let f ,...,f beaminimal generating set for I. Consider the polynomial ring S=
1 n
R[T ,...,T ],whereT ,...,T areindeterminates. Thenthereisanaturalmapf :S−→R(I)=R[It]
1 n 1 n
¥
that sends T to ft. Let J =kerf be the defining ideal of R(I). Then R(I)≃S/J and J = J
i i L i
i=1
is a graded ideal. A minimal generating set for J is often referred to as the defining equations of
the Rees algebra. Also, J is known as the ideal of linear relations as it is the defining ideal of the
1
symmetric algebra, Sym(I), of I and it is generated by linear forms in the T. Thegenerators of J
i 1
2010MathematicsSubjectClassification. 13A30,13A02,13A15.
Keywordsandphrases. Reesalgebras,square-freemonomialideals,relationtype,idealsoflineartype,graph.
1
2 L.FOULIANDK.N.LIN
arise from the first syzygies of I. When J =J then I is called an ideal of linear type. In this case
1
R(I)≃Sym(I)andthedefiningidealofR(I)iswellunderstood.
The problem of finding an explicit description of the defining equations of R(I) has been ad-
dressed by many authors, see for example [2, 3, 4, 5, 6, 7, 8, 9, 16, 19, 20, 21, 25]. In general,
finding thegenerators forthedefining ideal oftheReesalgebra ofanideal isadifficultproblem to
address.
In Section 2, we develop a series of lemmas that allow us to determine conditions under which
an element in the defining ideal of the Rees algebra becomes redundant and hence not needed in
the defining equations. One of our essential tools is the result of Taylor, where she determines a
non-minimalgeneratingsetforthedefiningidealoftheReesalgebraofamonomialideal[22]. One
ofourmainresults inthis section giveanexplicit description ofthe defining equations oftheRees
algebra of an ideal generated by n≤5 square-free monomials, Theorems 2.12, 2.15. Recall that
s
the relation type of an ideal I is defined to be rt(I)=min{s|J = J}, where J is the defining
L i
i=1
ideal of R(I). In otherwords, the relation type of I isthe largest degree (in the T)ofany minimal
i
generator of J. Inparticular, when I isoflineartypeitisofrelation type1. Therelation typeof an
idealhasbeenexploredinvariousarticles, seeforinstance [1,10,17,23,27]. InTheorem2.12we
establishthatforasquare-freemonomialidealgeneratedbyn≤5monomials,therelationtypeisat
mostn−2. Wealsoprovideaclassofexamplesofsquare-free monomialidealsgeneratedbyn>4
monomials, Example 2.13, for which the relation type is at least 2n−7. Notice that when n=5
then 2n−7=n−2, and hence showing that the bound in Theorem 2.12 is attained. We remark
thatwedonotimposeanyrestrictionsonthedegreesofthemonomialgeneratorsofthesquare-free
monomialidealsunderconsideration.
Weconstructthegeneratorgraphassociatedtomonomialideals,whichisalsoknownastheline
graphofthehypergraph. Themonomialgenerators aretheverticesforthisgraph andtheedgesare
determinedbythegreatestcommondivisoramongthegenerators, seeConstruction 2.16. Thiscon-
struction enables us toextend Theorem 2.12 and allow more freedom on the number ofgenerators
of the ideal. We show that when I is a square-free monomial ideal such that its generator graph is
thedisjointunionofgraphswithatmost5vertices, thenrt(I)≤3,Theorem2.20. Furthermore, we
show that the non-linear part of the defining equations of the Rees algebra, R(I), of I arise from
evenclosedwalksofthegeneratorgraph,Theorem2.20.
Recall that for a Noetherian local ring (R,m) and I an R-ideal, the special fiber ring, F(I), of
¥
I is defined to be F(I) = gr (I)⊗R/m, where gr (I) = R[It]/IR[It] = Ii/Ii+1. When R is a
R R L
i=0
polynomial ring over a field k and I is an R-ideal then F(I) is defined to be k[f ,...,f ], where
1 n
f ,...,f isaminimalgenerating setforI. Therefore, thereisahomomorphism y :k[T ,...,T ]→
1 n 1 n
F(I) that sends T to f. Let H =kery and F(I)≃k[T ,...,T ]/H. Let S=R[T ,...,T ] and let
i i 1 n 1 n
J be the defining ideal of the Rees algebra, R(I)≃S/J, of I. An ideal I is called an ideal of fiber
type if the defining ideal J of R(I) is obtained by the linear relations and the defining equations
forthe special fiberring, i.e. J =SJ +SH. In the case of edge ideals, Villarreal showed that they
1
REESALGEBRASOFSQUARE-FREEMONOMIALIDEALS 3
are all ideals of fiber type, [26, Theorem 3.1]. Moreover, Villarreal gave an explicit description of
the defining equations of the Rees algebra of any square-free monomial ideal generated in degree
2, [26, Theorem 3.1]. It is also worth noting that Villarreal exhibited an example to show that his
techniquesdonotextendformonomialidealsgeneratedindegreehigherthan2,[26,Example3.1].
His example is a square-free monomial ideal generated in degree 3 that is not of fiber type. We
remark that in general, a square-free monomial ideal generated in degree greater or equal to3 is
not of fibertype. In Example 2.22 wegive a concrete explanation of how one obtains the defining
equations fortheidealin[26,Example3.1].
In Section 3, we concentrate on the special class of ideals of linear type. The first well known
class of ideals of linear type are complete intersection ideals, [18]. Huneke and Valla showed
that ideals generated by d-sequences are also ideals of linear type, [14], [24]. Later on, Herzog-
Simis-Vasconcelos and Herzog-Vasconcelos-Villarreal used strongly Cohen-Macaulay ideals and
certain conditions on local numbers of generators to describe otherclasses of ideals of linear type,
[11,12,13]. Hunekealsoshowedthattheidealofall(n−1)×(n−1)minorsofagenericmatrixin
R=Z[X ]isalsoanidealoflineartype,[15]. Inthecaseofsquare–freemonomialidealsgenerated
i,j
indegree 2, Villarreal gave acomplete characterization of ideals oflinear type. Moreprecisely, he
showed that such an ideal I is of linear type if and only if the graph of I is the disjoint union of
graphs of trees or graphs that have a unique odd cycle, [26, Corollary 3.2]. In this work we find
newclassesofsquare-free monomialidealsthatareoflineartype. Weprovethatwhen I isanideal
generated by square-free monomials and the generator graph of I is a disjoint union of graphs of
treesandgraphs withaunique oddcyclethen I isanideal oflineartype, Theorem 3.4. Ourresults
have a similar flavor as those of Villarreal in [26]. In some sense these results extend the work of
Villarreal, eventhough wedonotfully recoverhisresults. Nonetheless, ourtechniques allow usto
considersquare-free monomialidealswithoutanyrestrictions onthedegreesofthegenerators.
2. THE DEFINING EQUATIONS OF THE REES ALGEBRA
Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let f ,...,f be a
1 n
minimal monomial generating set of I and let R(I) denote the Rees algebra of I. Then R(I)=
R[f t,...,f t]⊂R[t]andthereisanepimorphism
1 n
f :S=R[T ,...,T ]−→R(I)
1 n
¥
induced by f (T)= ft. Let J =kerf . Wenote that J = J is agraded ideal of S. Asmentioned
i i L i
i=1
in the Introduction, aminimal generating set forthe ideal J is referred toas the defining equations
oftheReesalgebraofI.
Definition 2.1. Let I be a monomial ideal in a polynomial ring R over a field k. Let f ,...,f be
1 n
a minimal monomial generating set of I. Let I denote the set of all non-decreasing sequences of
s
integers a =(i1,...,is)⊂{1,....,n} of length s. Then fa = fi1···fis is the corresponding product
4 L.FOULIANDK.N.LIN
ofmonomialsinI. WealsoletTa =Ti1···Tis bethecorresponding productofindeterminatesin S=
R[T1,...,Tn]. Forevery a ,b ∈Is weconsiderthebinomial Ta ,b = gcd(ffba ,fb )Ta −gcd(ffaa ,fb )Tb .
Thefollowingisin[22]andweciteithereforeaseofreference.
Theorem 2.2. ([22]) Let R be a polynomial ring over a field and let I be a monomial ideal in R.
Let f ,...,f be a minimal monomial generating set of I and let R(I) be the Rees algebra of I.
1 n
Then R(I)≃S/J, where S=R[T ,...,T ] is a polynomial ring, T ,...,T are indeterminates, and
1 n 1 n
¥
J=SJ1+S·(S Ji)suchthatJs ={Ta ,b | fora ,b ∈Is}.
i=2
Thefollowingremarkstatessomeproperties forthegreatestcommondivisoramongsquare-free
monomials. Theproofsareomittedastheyareelementary.
Remark2.3. Leta,b,c,d,b ,...,b besquare-free monomials inapolynomial ringRover afield.
1 s
Letn,m,l,r bepositive integers. Then
(a) gcd(an,bm)=gcd(a,b)min{m,n},
(b) gcd(an,b ···b )=gcd(as,b ···b )foralln≥s,
1 s 1 s
gcd(a,b)min{n,m}gcd(a,c)min{n,l}
(c) gcd(an,bmcl)= ,forsomeC∈R.
C
Themaingoalinthissectionistodetermineconditionsunderwhichforsequences a ,b oflength
s−1
s≥2wehaveTa ,b ∈SJ1+S·(S Ji). Inotherwords,weareinterested infindingconditions onthe
i=2
sequences a ,b such that the generator Ta ,b is redundant in the defining ideal of the Rees algebra.
ThefollowingtwolemmasfollowdirectlyfromDefinition2.1.
Lemma 2.4. Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let a =
(a ,...,a ),b =(b ,...,b )∈I betwosequences oflength s≥2,whereI isasinDefinition 2.1.
1 s 1 s s s
s−1
Suppose thatai=bj forsomeiandsome j. ThenTa ,b ∈SJ1+S·(S Ji).
i=2
Proof. Without loss of generality we may assume that a = b . Let a =(a ,...,a ) and b =
1 1 1 2 s 1
(b2,...,bs). Thengcd(fa ,fb )= fa1gcd(fa 1,fb 1)= fb1gcd(fa 1,fb 1). Noticethat fa1 = fb1 andTa1 =
T . Then
b1
fb fa fb fa
Ta ,b = gcd(fa ,fb )Ta −gcd(fa ,fb )Tb =Ta1[gcd(fa 1,fb )Ta 1−gcd(fa 1,fb )Tb 1]
1 1 1 1
s−1
= Ta1[Ta 1,b 1]∈SJ1+S·([Ji). (cid:3)
i=2
Lemma2.5. LetRbeapolynomialringoverafieldandletIbeamonomialidealinR. Leta ,b ∈I
s
be two sequences of length s≥2, where I is as in Definition 2.1. Suppose that a =(a ,...,a )
s 1 1
m
andb =(b 1,...,b 1),wherea 1,b 1∈Im andm≤s. ThenTa ,b ∈SJ1+S·(S Ji).
i=2
REESALGEBRASOFSQUARE-FREEMONOMIALIDEALS 5
Proof. First we observe that s is a multiple of m. Let l be an integer such that s = lm. Notice
that fa = fal1 and fb = fbl1. Also, by Remark2.3(a)wehave thatgcd(fa ,fb )=gcd(fa 1,fb 1)l. Let
fb Ta fa Tb
a= 1 1 , b= 1 1 , and let A=(al−1+al−2b+...+abl−2+bl−1). Notice that
gcd(fa ,fb ) gcd(fa ,fb )
1 1 1 1
a,b∈SandthusA∈S. Then
Ta ,b = gcd(ffba ,fb )Ta −gcd(ffaa ,fb )Tb =al−bl
m
= (a−b)(al−1+al−2b+...+abl−2+bl−1)=Ta1,b1A∈SJ1+S·([Ji). (cid:3)
i=2
Weobservethefollowingpropertiesforgreatestcommondivisorsamongmonomials. Again,we
omittheproofsastheyareelementary.
Remark2.6. Leta,b,c,d,e,f bemonomialsinapolynomial ringRoverafield.
gcd(a,b)gcd(a,c)
(a) gcd(a,bc)= ,forsomeC∈R.
C
(b) Supposethatgcd(a,c)=1. Thengcd(ab,c)=gcd(b,c).
The following Lemma plays an important role in the rest of this article as it allows us to show
thatcertainexpressions Ta ,b areredundant inthedefiningidealoftheReesalgebra.
Lemma 2.7. Let R be a polynomial ring over a field and let I be a monomial ideal in R. Let a ,
b ∈I betwosequences oflengths≥2,whereI isasinDefinition 2.1. Suppose a =(a ,...,a )
s s 1 m
and b =(b ,...,b ) with a , b ∈I and s +...+s =s. Suppose that for all 1≤i≤m, there
1 m i i si 1 m
existC ∈Rsuchthat
i
i−1 m m
gcd((cid:213) fa ,fb )gcd((cid:213) fa , (cid:213) fb )gcd(fa ,fb )
j k k i i
j=1 k=i k=i+1
gcd(fa ,fb )= ,
C
i
whereemptyproducts aretakentobe1. Then
m m i−1 r
(cid:229) (cid:213) (cid:213)
Ta ,b = Ai Ta j Tb k Ta i,b i ∈SJ1+S·([Ji),
i=1" j=i+1 k=1 # i=2
wherer=max{s ,...,s }andA ∈Rforalli.
1 m i
m m
Proof. Noticethat fa = (cid:213) fa and fb = (cid:213) fb . Thenwehave
i i
i=1 i=1
fb fa
Ta ,b = gcd(fa ,fb )Ta −gcd(fa ,fb )Tb
i−1 m m i−1
= (cid:229)m  Cij(cid:213)=1fa jk=(cid:213) i+1fb k j=(cid:213) i+1Ta jk(cid:213)=1Tb k fb i Ta − fa i Tb .
i−1 m m gcd(fa ,fb ) i gcd(fa ,fb ) i
i=1gcd((cid:213) fa ,fb )gcd((cid:213) fa , (cid:213) fb )(cid:18) i i i i (cid:19)
 j=1 j k=i k k=i+1 k 
 
 
6 L.FOULIANDK.N.LIN
i−1 m
Ci (cid:213) fa j (cid:213) fb k
Finally,wenotethatA = j=1 k=i+1 ∈R. (cid:3)
i
i−1 m m
gcd((cid:213) fa ,fb )gcd((cid:213) fa , (cid:213) fb )
j k k
j=1 k=i k=i+1
Next we give a list of conditions that when satisfied by two sequences a ,b then the generator
Ta ,b isnotaminimalgeneratorinthedefiningidealoftheReesalgebraofthecorresponding ideal.
Proposition 2.8. LetRbeapolynomial ringoverafieldandletI beamonomialidealinR. Leta ,
b ∈I be two sequences of length s≥2, where I is as in Definition 2.1. Let a =(a ,...,a ),b =
s s 1 s
(b ,...,b ). Supposethataftersomereorderingthereexistintegersk,lwith1≤k,l≤s−1suchthat
1 s
gcd(f ,f )=1forevery1≤i≤landeveryk+1≤ j≤s. Wefurtherassumethatgcd(f ,f )=1,
ai bj au bv
r
for every l+1≤u≤s and every 1≤v≤k. Then Ta ,b ∈SJ1+S·(S Ji), where r =max{k,|k−
i=2
l|,s−k,s−l}.
Proof. Suppose that l =k and write a =(a ,a ) and b =(b ,b ), where a =(a ,...,a ), a =
1 2 1 2 1 1 k 2
(ak+1,...,as),b 1=(b1,...,bk),andb 2=(bk+1,...,bs). Byourassumptions,wehavegcd(fa i,fb j)=
1fori6= j. HencebyRemark2.6,wehave
gcd(fa ,fb )gcd(fa ,fb ) gcd(fa ,fb )gcd(fa ,fb )
gcd(fa ,fb ) = 1 2 = 1 1 2 ,
C C
1 1
gcd(fa ,fb )gcd(fa ,fb ) gcd(fa ,fb )gcd(fa ,fb )
gcd(fa ,fb ) = 1 2 = 1 2 2 .
C C
2 2
TheresultfollowsfromLemma2.7with m=2.
Without loss of generality we may now assume that l <k. We write a =(a ,a ,a ) and b =
1 2 3
(b ,b ,b ),witha =(a ,...,a ),a =(a ,...,a ),a =(a ,...,a ), b =(b ,...,b ), b =
1 2 3 1 1 l 2 l+1 k 3 k+1 s 1 1 l 2
(bl+1,...,bk),andb 3=(bk+1,...,bs). Thengcd(fa 1,fb 3)=1andgcd(fa i,fb j)=1fori=2,3and
j=1,2. HencebyRemark2.6wehave
gcd(fa ,fb )gcd(fa ,fb fb ) gcd(fa ,fb )gcd(fa ,fb fb )
gcd(fa ,fb ) = 1 2 3 = 1 1 2 3 ,
C C
1 1
gcd(fa ,fb )gcd(fa fa ,fb )
gcd(fa ,fb ) = 1 2 3
C
2
gcd(fa ,fb )gcd(fa fa ,fb )gcd(fa fa ,fb fb )
= 1 2 3 3 2 3 1 2
C′
2
gcd(fa ,fb )gcd(fa fa ,fb )gcd(fa fa ,fb )
= 1 2 3 3 2 3 2
C′
2
gcd(fa ,fb )gcd(fa fa ,fb )gcd(fa ,fb )
= 1 2 3 3 2 2 , and
C′
2
gcd(fa fa ,fb )gcd(fa ,fb ) gcd(fa fa ,fb )gcd(fa ,fb )
gcd(fa ,fb ) = 1 2 3 = 1 2 3 3 .
C C
3 3
ThereforewemayapplyLemma2.7withm=3. (cid:3)
REESALGEBRASOFSQUARE-FREEMONOMIALIDEALS 7
The next three Lemmas deal with possible repetitions in the sequences a ,b ∈I and how that
s
affectsthetermTa ,b .
Lemma 2.9. Let R be a polynomial ring over a field and let I be a square-free monomial ideal in
R. Let a , b ∈I be two sequences of length s≥2, where I is as in Definition 2.1. Suppose that
s s
s−1
a =(a1,...,a1)andb =(b1,...,bs). ThenTa ,b ∈SJ1+S·(S Ji).
i=2
Proof. Noticethat fa = fas1 and fb = fb1···fbs. Letb 1=(b2,...,bs). Wealsonotethatgcd(fas1,fb1)=
gcd(f ,f ),byRemark2.3(a). Then
a1 b1
gcd(fa ,fb ) = gcd(fas1,fb1)gcd(fas1,fb 1) = gcd(fa1,fb1)gcd(fas1−1,fb 1),
C C
s−1
forsomeC∈R,byRemark2.3(b)and(c). HenceTa ,b ∈SJ1+S·(S Ji),byLemma2.7. (cid:3)
i=2
Lemma2.10. LetRbeapolynomial ring overafieldandletI beasquare-free monomial idealin
R. Let a , b ∈I be two sequences of length s≥2, where I is as in Definition 2.1. Suppose that
s s
a =(a1,...,a1,a2,...,a2)andb =(b1,...,b1,b2,...,b2). ThenTa ,b ∈SJ1+SJ2.
Proof. Wewillproceedbyinductionons. Ifs=2thenthereisnothingtoshow. Supposethats>2.
Suppose thatthereare l distinct copies ofa ina andk distinct copies ofb inb fori=1,2. Then
i i i i
fa = fal11fal22 and fb = fbk11fbk22. Forsimplicitywewritea =(al11,al22)andb =(bk11,bk22). Ifk1=k2 and
l =l thentheresultfollowsfromLemma2.5. Thuswemayassumewithoutlossofgeneralitythat
1 2
k >k andl ≥l .
1 2 1 2
Wewillshow
gcd(fa ,fb ) = gcd(fal11−1fal22,fbk11fbk22)gcd(fa1,fb1)
G
= gcd(fal11−1fal22,fbk11−1fbk22)gcd(fal11fal22,fb1)
H
forsomeG,H ∈R. ThenTa ,b ∈SJ1∪SJ2,byLemma2.7andtheinduction.
Noticethatwehavethefollowingequalities:
gcd(fa ,fb ) = gcd(fal11−1fal22,fbk11fbk22)gcd(fa1,fbk11fbk22)
C
= gcd(fal11−1fal22,fbk11fbk22)gcd(fa1,fbk11−1fbk22)gcd(fa1,fb1),
CD
gcd(fa ,fb ) = gcd(fal11fal22,fbk11−1fbk22)gcd(fal11fal22,fb1)
E
= gcd(fal11−1fal22,fbk11−1fbk22)gcd(fa1,fbk11−1fbk22)gcd(fal11fal22,fb1),
EF
forsomeC,D,E,F∈RbyRemark2.6(a). Wewillshowthatforanyintegertandanyvariablex∈R
ifxt|gcd(f ,fk1fk2),thenxt|CDandxt|EF. Thisisequivalenttoshowingthatforanyvariablex∈
a1 b1 b2
Randanyintegeru,ifx|gcd(fa1,fbk11fbk22)andxu|gcd(fa ,fb )thenxu|gcd(fal11−1fal22,fbk11fbk22)gcd(fa1,fb1)
8 L.FOULIANDK.N.LIN
and xu |gcd(fal11−1fal22,fbk11−1fbk22)gcd(fal11fal22,fb1). Thisfollows immediately since s=l1+l2=k1+
k >l andl ≥ s >k . (cid:3)
2 1 1 2 2
Lemma2.11. LetRbeapolynomial ring overafieldandletI beasquare-free monomial idealin
R. Let a , b ∈I be two sequences of length s≥4, where I is as in Definition 2.1. Suppose that
s s
a =(a1,...,a1,a2,...,a2,a3,...,a3)andb =(b1,...,b1,b2,...,b2). ThenTa ,b ∈SJ1+S·(J2∪J3).
s−1
Proof. We will show Ta ,b ∈SJ1+S·(S Ji) and by induction on s≥4, we conclude that Ta ,b ∈
i=2
SJ +S·(J ∪J ). Supposethattherearel distinctcopiesofa ina andk distinctcopiesofb inb .
1 2 3 i i i i
Thenl +l +l =k +k =s≥4. Without lostofgenerality, weassume that l ≥l ≥l ≥1and
1 2 3 1 2 1 2 3
k ≥k ≥1. Sinces≥4,wehavel ≥2andk ≥2. Wewrite a =(al1,al2,al3)andb =(bk1,bk2).
1 2 1 1 1 2 3 1 2
Wethenhavethreepossiblescenarios. Weclaimthefollowing:
(i) Ifl +l >k ,then
1 2 1
gcd(fa ,fb ) = gcd(fa1fa2,fb1fb2)gcd(fa 1,fb )/A
= gcd(fa ,fb1fb2)gcd(fa 1,fb 1)/B,
forsomeA,B∈R,anda =(al1−1,al2−1,al3)andb =(bk1−1,bk2−1).
1 1 2 3 1 1 2
(ii) Ifl >k ,then
1 2
gcd(fa ,fb ) = gcd(fa1,fb1)gcd(fa 1,fb )/C
= gcd(fa ,fb1)gcd(fa 1,fb 1)/D,
forsomeC,D∈R,anda =(al1−1,al2,al3)andb =(bk1−1,bk2).
1 1 2 3 1 1 2
(iii) Ifl +l ≤k andl ≤k ,then
1 2 1 1 2
s/3 s/3 s/3 2s/3 s/3
gcd(fa ,fb ) = gcd(fa1 fa2 fa3 ,fb1 fb2 )
= (gcd(f f f ,f2 f ))s/3.
a1 a2 a3 b1 b2
OnceweobtaintheaboveclaimsthentheresultwillfollowbyLemma2.7.
Toestablish claim(i)wenoticethatwehavethefollowingequalities:
gcd(fa ,fb ) = gcd(fa1fa2,fb )gcd(fa 1,fb )/E
= gcd(fa1fa2,fb1fb2)gcd(fa1fa2,fb 1)gcd(fa 1,fb )/EF,
= gcd(fa ,fb1fb2)gcd(fa ,fb 1)/G
= gcd(fa ,fb1fb2)gcd(fa 1,fb 1)gcd(fa1fa2,fb 1)/GH,
forsomeE,F,G,H∈R. Itisenoughtoshowthatforanyvariable x∈Randanypositiveintegert if
xt |gcd(fa1fa2,fb 1),thenxt|EFandxt |GH.Thisisequivalenttoshowingthatifxu|gcd(fa ,fb )and
x|gcd(fa1fa2,fb 1),thenxu|gcd(fa1fa2,fb1fb2)gcd(fa 1,fb )andxu|gcd(fa ,fb1fb2)gcd(fa 1,fb 1)for
anypositiveintegeru. Butthisfollowsimmediatelysincel +l >k ≥k andl +l <l +l +l =
1 2 1 2 1 2 1 2 3
s=k +k .
1 2
REESALGEBRASOFSQUARE-FREEMONOMIALIDEALS 9
Forclaim(ii)wenoticethatwehavethefollowingequalities:
gcd(fa ,fb ) = gcd(fa1,fb )gcd(fa 1,fb )/K
= gcd(fa1,fb1)gcd(fa1,fb 1)gcd(fa 1,fb )/KL,
= gcd(fa ,fb1)gcd(fa ,fb 1)/M
= gcd(fa ,fb1)gcd(fa 1,fb 1)gcd(fa1,fb 1)/MN,
forsomeK,L,M,N ∈R. Theclaimfollowssincel >k .
1 2
Finallytoestablishthelastclaimwenotethattheassumptionl ≤k isequivalenttol +l ≥k
1 2 2 3 1
and thus l +l ≥ k ≥ l +l . Hence l ≥ l ≥ l ≥ l which implies l = s/3 and 2s/3 ≤ k .
2 3 1 1 2 3 1 2 3 i 1
Furthermore, s/3≥k ≥s/3andhencek =s/3andk =2s/3. (cid:3)
2 2 1
The next Theorem is one of the main results of this section. We will use all the information we
obtained about how various conditions on two sequences a ,b ∈Is affect the term Ta ,b to obtain a
boundontherelationtypeofI aswellasadescriptionofthedefiningequationsoftheReesalgebra.
Theorem 2.12. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
generated bynsquare-free monomialsinR. Whenn≤5,thenR(I)=S/J,whereS=R[T ,...,T ]
1 n
n−2
andJ=SJ +S·( J). Inparticular, rt(I)≤n−2.
1 S i
i=2
Proof. By Theorem 2.2, it suffices to show that given two sequences a ,b ∈I of length s>n−2
s
n−2
thenTa ,b ∈SJ1+S·(S Ji). Supposethatthereareli distinctcopiesofai ina andki distinctcopies
i=2
ofbi inb andwritea =(al11,...,almm),andb =(bk11,...,btkt).
ByLemma2.4wemayassumea 6=b foralli,j. AlsobyLemma2.9,wemayassume1<m≤3
i j
and1<t ≤2,sincen≤5. Wearenowleftwiththefollowingcases:
(i) Suppose that a =(al1,al2)and b =(bk1,bk2), where l +l =k +k ≥3. Theresult follows
1 2 1 2 1 2 1 2
fromLemma2.10.
(ii) Suppose that a =(al1,al2,al3)and b =(bk1,bk2), where l +l +l =k +k ≥4. Theresult
1 2 3 1 2 1 2 3 1 2
followsfromLemma2.11. (cid:3)
The following example provides a class of square-free monomial ideals generated by n > 4
square-free monomials for which the relation type is at least 2n−7. Notice when n=5 one has
2n−7=n−2. In particular, this establishes that the bound given in Theorem 2.12 is sharp. We
alsonotethattheidealsinthefollowingexamplearenotoffibertype.
Example2.13. LetR=k[y,x ,...,x ,z,u ,...,u ]beapolynomial ringoverafieldkwithna
2 n−2 2 n−2
n−2 n−2
positiveintegersuchthatn>4. LetI=(f ,...,f ),where f =((cid:213) x)z, f =xy( (cid:213) u )forall
1 n 1 i i i j
i=2 j=2,j6=i
n−2 n−2
i=2,...,n−2, f =((cid:213) x)y,and f =((cid:213) u)z. Consider
n−1 i n i
i=2 i=2
n−2 n−2
F =Tn−4(cid:213) T −Tn−3Tn−4 andG=zTn−5(cid:213) T −yTn−5T .
1 i n−1 n 1 i n−1 n
i=2 i=2
10 L.FOULIANDK.N.LIN
It is clear that F and G are in the defining ideal of the Rees algebra of I. Moreover, F and G are
2n−8 2n−9
irreducible and thus F ∈SJ \S·( J) and G∈SJ \S·( J). Hence the relation type
2n−7 S i 2n−8 S i
i=1 i=1
ofI isatleast2n−7. Finally,wenotethatGisnotinthedefiningidealofthespecialfiberofI and
therefore I isnotanidealoffibertype.
The next lemma establishes conditions for when various generators of the defining ideal of the
Reesalgebra asinTheorem2.12areirredundant.
Lemma2.14. LetRbeapolynomial ring overafieldandletI beasquare-free monomial idealin
R. LetLeta ,b ∈I betwosequences oflengths≥4,whereI isasinDefinition 2.1. Suppose that
s s
a =(a ,...,a ), b =(b ,...,b ,b ), where a 6=b for all i,j. Suppose that for all i there exists a
1 s 1 1 2 i j
variable x ∈Rsuchthatx | f forall j6=iandx | f ,x ∤ f ,andx ∤ f . Furthermore, suppose
i i aj i b1 i ai i b2
thatforallithereexistsavariable z ∈Rsuchthatz |gcd(f ,f ),z ∤ f forall j6=iandz ∤ f .
i i ai b2 i aj i b1
s−1
ThenTa ,b ∈SJs\S·(S Ji).
i=1
s s s
Proof. We write f =zh( (cid:213) x ) for all i=1,...,s, f =k ((cid:213) x), and f =k ((cid:213) z), for
ai i i j b1 1 i b2 2 i
j=1,j6=i i=1 i=1
some square-free monomials h ,...,h ,k ,k ∈Rsuch that x ∤h , x ∤k , z ∤h , and z ∤k forall i,
1 s 1 2 i j i l i j i l
s s s s s
j,l. Thenwehave fa = (cid:213) xis−1 (cid:213) zi (cid:213) hi and fb =k1s−1k2((cid:213) xis−1 (cid:213) zi). Let
i=1 i=1 i=1 i=1 i=1
s
(cid:213) h
M= fa = i=1 i andN = fb = k1s−1k2 .
gcd(fa ,fb ) gcd((cid:213)s h,ks−1k ) gcd(fa ,fb ) gcd((cid:213)s h,ks−1k )
i 1 2 i 1 2
i=1 i=1
Onecanobserveimmediatelythatx ∤Mandz ∤Nforalli. Leta ′andb ′beanypropersubsequences
i i
of a and b . Then there exists either x or z such that x | fa ′ or z | fa ′ by
i j i j
gcd(fa ′,fb ′) gcd(fa ′,fb ′)
fa ′ fb ′
our assumptions. Therefore ∤ M, and similarly, ∤ N. This shows that
gcd(fa ′,fb ′) gcd(fa ′,fb ′)
s−1
Ta ,b ∈SJs\S·(S Ji). (cid:3)
i=1
Thefollowing Theorem that establishes precisely the defining equations for the Rees algebra of
asquare-free monomialidealwithuptofivegenerators.
Theorem 2.15. Let R be a polynomial ring over a field and let I be a square-free monomial ideal
generated by n square-free monomials in R. Suppose that n ≤ 5. Suppose that there does not
exist a pair of sequences a ,b ∈I of length s≥2 as in Definition 2.1, such that the conditions of
s
Lemma2.14aresatisfied. ThenI isanidealoflineartype.
Proof. Consider a pair of sequences a ,b as in Definition 2.1 of length s ≥ 2 such that a =
(a ,...,a ),b =(b ,...,b ,b ),where a 6=b foralli,j. Supposethatoneofthefollowing condi-
1 s 1 1 2 i j
tionsisnotsatisfied: foralli,thereexistsavariable x ∈Rsuchthatx | f forall j6=iandx | f ,
i i aj i b1