REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 1. INTRODUCTION

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1 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS LOUIZA FOULI AND KUEI-NUAN LIN ABSTRAT. We study the defining equations of the Rees algebras of square-free onoial ideals in a polynoial ring over a field. We deterine the defining equations of the Rees algebra of an ideal I in the case I is a square-free onoial ideal such that each connected coponent of the line graph of the hypergraph corresponding to I has at ost 5 vertices. Moreover we show in this case that the non-linear equations are arising fro even closed walks of the line graph and we also give a description of the defining ideal of the toric ring when I is generated by square-free onoials of the sae degree. Furtherore we provide a new class of ideals of linear type. We show that when I is a square-free onoial ideal with any nuber of generators and the line graph of the hypergraph corresponding to I is the graph of a disjoint union of trees and graphs with a unique odd cycle then I is an ideal of linear type.. INTRODUTION The ain proble of interest in this article is the question of deterining the defining equations of the Rees algebra of a square-free onoial ideal in a polynoial 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 R (I) = R[It] = I i t i R[t] where t is an indeterinate. The Rees algebra of an ideal encodes i 0 any of the analytic properties of the variety defined by I and it is the algebraic realization of the blowup of a variety along a subvariety. The blowup of Spec(R) along V (I) is the projective spectru of the Rees algebra R (I) of I. This iportant construction is the ain tool in the resolution of singularities of an algebraic variety. Fro the algebraic point of view the Rees algebra of an ideal facilitates the study of the asyptotic behavior of the ideal and it is essential in coputing the integral closure of powers of ideals. The Rees algebra of an ideal I in a Noetherian ring can be realized as a quotient of a polynoial ring and hence once the defining ideal of R (I) is deterined it is straightforward to copute the integral closure R (I). It is well known that R (I) = I i t i and one obtains I i = [R (I)] i. i 0 Another reason for exploring the Rees algebra of an ideal is to deterine the defining ideal of toric rings. Recall that a toric ideal is the ideal of relations of a onoial subring of a polynoial ring see Section for a detailed definition. For a Noetherian local ring (R) and I an R-ideal the special fiber ring F (I) of I is defined to be F (I) = R (I)/IR (I) R/. When R is a polynoial ring over a field k and I is generated by square-free onoials of the sae degree then the defining ideal of the special fiber ring F (I) is the toric ideal associated to I. Notice that in this case F (I) = k[ f... f n ] where f... f n is a inial onoial generating set of I. There are any questions 00 Matheatics Subject lassification. 3A30 3A0 3A5 05E40 05E Key words and phrases. Rees algebras square-free onoial ideals relation type ideals of linear type graph.

2 L. FOULI AND K. N. LIN arising fro the study of toric varieties as these have applications in cobinatorics and in geoetry for exaple see [ ]. Once we understand R (I) then we can obtain a concrete description of the toric ring corresponding to a square-free onoial ideal generated in the sae degree directly fro R (I). The proble of finding an explicit description of the defining ideal of the Rees algebra has been addressed by any authors see for exaple [ ]. In general finding the generators for the defining ideal of the Rees algebra of an ideal is difficult. Recently Kustin Polini and Ulrich in [] studied the defining equations of the Rees algebra of height ideals generated by fors of the sae degree in k[xy] where k is any field. Furtherore in [6] ox Kustin Polini and Ulrich provide a detailed study of the singularities of rational curves in P by studying the defining equations of the Rees algebra. We consider the following construction for the Rees algebra. Let R be a Noetherian ring and let I be an R-ideal. Let f... f n be a inial generating set for I. onsider the polynoial ring S = R[T...T n ] where T...T n are indeterinates. Then there is a natural ap φ : S R (I) = R[It] that sends T i to f i t. Let J = kerφ be the defining ideal of R (I). Then R (I) S/J and J = J i is a graded ideal. A inial 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 syetric algebra Sy(I) of I and it is generated by linear fors in the T i. The generators of J arise fro the first syzygies of I. When J = J then I is called an ideal of linear type. In this case R (I) Sy(I) and the defining ideal of R (I) is well understood. Furtherore there is a hooorphis ψ : k[t...t n ] F (I) that sends T i to f i where k is the residue field of R. Then F (I) k[t...t n ]/H where H = kerψ. 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 for the special fiber ring i.e. J = SJ + SH. In the case of square-free onoial ideals generated in degree naely edge ideals Villarreal showed that they are all ideals of fiber type [34 Theore 3.]. Moreover Villarreal gave an explicit description of the defining equations of the Rees algebra of any edge ideal [34 Theore 3.]. It is also worth noting that Villarreal exhibited an exaple to show that his techniques do not extend for onoial ideals generated in degree higher than [34 Exaple 3.]. His exaple is a squarefree onoial ideal generated in degree 3 that is not of fiber type. We reark that in general a square-free onoial ideal generated in degree greater or equal to 3 is not necessarily of fiber type. Therefore as the degree of the generators exceeds the coplexity of the proble increases. In Section we develop a series of leas that allow us to deterine conditions under which an eleent in the defining ideal of the Rees algebra becoes redundant and hence not needed in the defining equations. One of our essential tools is the classic result of Taylor in which she deterines a non-inial generating set for the defining ideal of the Rees algebra of any onoial ideal [30]. We construct a graph associated to onoial ideals which is also known as the line graph of the corresponding hypergraph. The onoial generators are the vertices for this graph and the edges

3 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 3 are deterined by the greatest coon divisor aong the generators see onstruction.6. In the ain result in this section we show that when I is a square-free onoial ideal such that the line graph of the corresponding hypergraph is the disjoint union of graphs with at ost 5 vertices then one can describe the defining equations of the Rees algebra of I Theore.0. Furtherore we show that the non-linear part of the defining equations of R (I) arise fro even closed walks of the line graph of the corresponding hypergraph Theore.0. Moreover in Exaple. we give a concrete explanation of how one obtains the defining equations for the ideal in [34 Exaple 3.]. As entioned above one can deterine the toric ideal of a square-free onoial ideal generated in the sae degree directly fro the defining equations of the Rees algebra. Villarreal provided a concrete description of the toric ideal in the case of square-free onoials generated in degree [34 Proposition 3.]. We are then able to give a generalization of Villarreal s result in the case of square-free onoial ideals generated in any degree orollary.3. 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 coplete intersection ideals [3]. Many authors have worked on establishing other classes of ideals of linear type see for exaple [ ]. In the case of square free onoial ideals generated in degree Villarreal gave a coplete characterization of ideals of linear type. More precisely 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 [34 orollary 3.]. In this work we establish new classes of square-free onoial ideals that are of linear type. We prove that when I is an ideal generated by square-free onoials and the line graph of the hypergraph corresponding to I is a disjoint union of graphs of trees and graphs with a unique odd cycle then I is an ideal of linear type Theore 3.4. Our results have a siilar flavor as those of Villarreal in [34]. In soe sense these results extend the work of Villarreal even though we do not fully recover his results. Nonetheless our techniques allow us to consider square-free onoial ideals without any restrictions on the degrees of the generators nor any restrictions on the nuber of inial generators. We conclude with soe final rearks. In [3] onca and De Negri introduced the notion of onoial ideals generated by M-sequences. They showed that all ideals generated by an M-sequence are ideals of Gröbner linear type and in particular of linear type. In [9] Soleyan and Zheng introduced the class of onoial ideals of forest type and showed that in the case of square-free onoial ideals the class of onoial ideals of forest type coincides with the class of onoial ideals generated by M-sequences. It is straightforward to establish that an ideal whose line graph is a graph of a forest is an ideal of forest type and hence aking it an ideal generated by an M- sequence. Hence by [3 Theore.4] these ideals are of (Gröbner) linear type. However an ideal whose line graph is the graph of an odd cycle is not an ideal of forest type but nonetheless it is an ideal of linear type. Hence the class of ideals of linear type that we uncovered in Theore 3.4 are not necessarily generated by M-sequences and thus our result is incoparable to the results of onca and De Negri in [3].

4 4 L. FOULI AND K. N. LIN. THE DEFINING EQUATIONS OF THE REES ALGEBRA Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let f... f n be a inial onoial generating set of I and let R (I) denote the Rees algebra of I. Then R (I) = R[ f t... f n t] R[t] and there is an epiorphis φ : S = R[T...T n ] R (I) induced by φ(t i ) = f i t. Let J = kerφ. We note that J = J i is a graded ideal of S. As entioned in the Introduction a inial generating set for the ideal J is referred to as the defining equations of the Rees algebra of I. Definition.. Let I be a onoial ideal in a polynoial ring R over a field k. Let f... f n be a inial onoial generating set of I. Let I s denote the set of all non-decreasing sequences of integers α = (i...i s ) {...n} of length s. Then f α = f i f is is the corresponding product of onoials in I. We also let T α = T i T is be the corresponding product of indeterinates in S = f R[T...T n ]. For every αβ I β s we consider the binoial T αβ = gcd( f α f β ) T f α α gcd( f α f β ) T β. The following is in [30] and we cite it here for ease of reference. Theore.. ([30]) Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let f... f n be a inial onoial generating set of I and let R (I) be the Rees algebra of I. Then R (I) S/J where S = R[T...T n ] is a polynoial ring T...T n are indeterinates and J = SJ + S ( J i ) such that J s = {T αβ for αβ I s }. The following reark states soe properties for the greatest coon divisor aong square-free onoials. The proofs are oitted as they are eleentary. Reark.3. Let abcd...b s be square-free onoials in a polynoial ring R over a field. Let n l r be positive integers. Then (a) gcd(a n b ) = gcd(ab) in{n} (b) gcd(a n b s ) = gcd(a s b s ) for all n s (c) gcd(a n b c l ) = gcd(ab)in{n} gcd(ac) in{nl} for soe R. The ain goal in this section is to deterine conditions under which for sequences αβ of length s we have T αβ SJ + S ( s J i ). In other words we are interested in finding conditions on the sequences αβ such that the generator T αβ is redundant in the defining ideal of the Rees algebra. The following two leas follow directly fro Definition.. Le.4. Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let α = (...a s ) β = (...b s ) I s be two sequences of length s where I s is as in Definition.. Suppose that a i = b j for soe i and soe j. Then T αβ SJ + S ( s J i ).

5 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 5 Proof. Without loss of generality we ay assue that =. Let α = (...a s ) and β = (...b s ). Then gcd( f α f β ) = f a gcd( f α f β ) = f b gcd( f α f β ). Notice that f a = f b and T a = T b. Then T αβ = f β gcd( f α f β ) T α f α s = T a [T α β ] SJ + S ( J i ). f β gcd( f α f β ) T β = T a [ gcd( f α f β ) T α f α gcd( f α f β ) T β ] Le.5. Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let α β I s be two sequences of length s where I s is as in Definition.. Suppose that α = (α...α ) and β = (β...β ) where α β I and s. Then T αβ SJ + S ( J i ). Proof. First we observe that s is a ultiple of. Let l be an integer such that s = l. Notice that f α = fα l and f β = fβ l. Also by Reark.3 (a) we have that gcd( f α f β ) = gcd( f α f β ) l. Let f β T α a = gcd( f α f β ) b = f α T β gcd( f α f β ) and let A = (al + a l b ab l + b l ). Notice that ab S and thus A S. Then f β T αβ = gcd( f α f β ) T f α α gcd( f α f β ) T β = a l b l = (a b)(a l + a l b ab l + b l ) = T a A SJ + S ( J i ). We observe the following properties for greatest coon divisors aong onoials. Again we oit the proofs as they are eleentary. Reark.6. Let abcde f be onoials in a polynoial ring R over a field. (a) gcd(abc) = gcd(ab)gcd(ac) for soe R. (b) Suppose that gcd(a c) =. Then gcd(ab c) = gcd(b c). The following Lea plays an iportant role in the rest of this article as it allows us to show that certain expressions T αβ are redundant in the defining ideal of the Rees algebra. Le.7. Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let α β I s be two sequences of length s where I s is as in Definition.. Suppose α = (α...α ) and β = (β...β ) with α i β i I si and s s = s. Suppose that for all i there exist i R such that gcd( f α f β ) = gcd( i f α j f β )gcd( j= k=i where epty products are taken to be. Then [ T αβ = A i j=i+ T α j i k= f αk k=i+ f βk )gcd( f αi f βi ) i T βk ]T αi β i SJ + S ( r J i )

6 6 L. FOULI AND K. N. LIN where r = ax{s...s } and A i R for all i. Proof. Notice that f α = f αi and f β = f βi. Then we have T αβ = = f β gcd( f α f β ) T α i i f α j j= f α gcd( f α f β ) T β f βk k=i+ j=i+ gcd( i f α j f β )gcd( j= k=i i T α j T βk k= f αk k=i+ f βk ) ( ) f βi gcd( f αi f βi ) T f αi α i gcd( f αi f βi ) T β i. Finally we note that A i = i i j= f α j f βk k=i+ gcd( i f α j f β )gcd( j= k=i f αk k=i+ R. f βk ) Next we give a list of conditions that when satisfied by two sequences αβ then the generator T αβ is not a inial generator in the defining ideal of the Rees algebra of the corresponding ideal. Proposition.8. Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let α β I s be two sequences of length s where I s is as in Definition.. Let α = (...a s )β = (...b s ). Suppose that after soe reordering there exist integers kl with kl s such that gcd( f ai f b j ) = for every i l and every k+ j s. We further assue that gcd( f au f bv ) = for every l + u s and every v k. Then T αβ SJ + S ( r J i ) where r = ax{k k l s ks l}. Proof. Suppose that l = k and write α = (α α ) and β = (β β ) where α = (...a k ) α = (a k+...a s ) β = (...b k ) and β = (b k+...b s ). By our assuptions we have gcd( f αi f β j ) = for i j. Hence by Reark.6 we have gcd( f α f β ) = gcd( f α f β )gcd( f α f β ) = gcd( f α f β )gcd( f α f β ) gcd( f α f β ) = gcd( f α f β )gcd( f α f β ) = gcd( f α f β )gcd( f α f β ). The result follows fro Le.7 with =. Without loss of generality we ay now assue that l < k. We write α = (α α α 3 ) and β = (β β β 3 ) with α = (...a l ) α = (a l+...a k ) α 3 = (a k+...a s ) β = (...b l ) β = (b l+...b k ) and β 3 = (b k+...b s ). Then gcd( f α f β3 ) = and gcd( f αi f β j ) = for i = 3 and j =. Hence by Reark.6 we have

7 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 7 gcd( f α f β ) = gcd( f α f β )gcd( f α f β f β3 ) = gcd( f α f β )gcd( f α f β f β3 ) gcd( f α f β ) = gcd( f α f β )gcd( f α f α3 f β ) = gcd( f α f β )gcd( f α f α3 f β3 )gcd( f α f α3 f β f β ) = gcd( f α f β )gcd( f α f α3 f β3 )gcd( f α f α3 f β ) = gcd( f α f β )gcd( f α f α3 f β3 )gcd( f α f β ) and gcd( f α f β ) = gcd( f α f α f β )gcd( f α3 f β ) = gcd( f α f α f β )gcd( f α3 f β3 ). 3 3 Therefore we ay apply Le.7 with = 3. The next three Leas deal with possible repetitions in the sequences αβ I s and how that affects the ter T αβ. Le.9. Let R be a polynoial ring over a field and let I be a square-free onoial ideal in R. Let α β I s be two sequences of length s where I s is as in Definition.. Suppose that α = (... ) and β = (...b s ). Then T αβ SJ + S ( s J i ). Proof. Notice that f α = f s and f β = f b f bs. Let β = (...b s ). We also note that gcd( f s f b ) = gcd( f a f b ) by Reark.3 (a). Then gcd( f α f β ) = gcd( f a s f b )gcd( fa s f β ) = gcd( f f b )gcd( fa s f β ) for soe R by Reark.3 (b) and (c). Hence T αβ SJ + S ( s J i ) by Le.7. Le.0. Let R be a polynoial ring over a field and let I be a square-free onoial ideal in R. Let α β I s be two sequences of length s where I s is as in Definition.. Suppose that α = ( ) and β = ( ). Then T αβ SJ + SJ. Proof. We will proceed by induction on s. If s = then there is nothing to show. Suppose that s >. Suppose that there are l i distinct copies of a i in α and k i distinct copies of b i in β for i =. Then f α = f l and f β = f k f k. For siplicity we write α = (a l a l ) and β = (b k bk ). If k = k and l = l then the result follows fro Le.5. Thus we ay assue without loss of generality that k > k and l l. We will show gcd( f α f β ) = gcd( f l f k f k )gcd( f a f b ) G = gcd( f l f k f k )gcd( f l f b ) H

8 8 L. FOULI AND K. N. LIN for soe GH R. Then T αβ SJ SJ by Le.7 and the induction. Notice that we have the following equalities: gcd( f α f β ) = gcd( f l f k f k )gcd( f a f k f k ) = gcd( f l f k f k )gcd( f a f k f k )gcd( f a f b ) D gcd( f α f β ) = gcd( f l f k f k )gcd( f l f b ) E = gcd( f l f k f k )gcd( f a f k f k )gcd( f l f b ) EF for soe DEF R by Reark.6 (a). We will show that for any integer t and any variable x R if x t gcd( f a f k f k ) then x t D and x t EF. This is equivalent to showing that for any variable x R and any integer u if x gcd( f a f k f k ) and x u gcd( f α f β ) then x u gcd( f l f k f k )gcd( f a f b ) and x u gcd( f l f k f k )gcd( f l f b ). This follows iediately since s = l + l = k + k > l and l s > k. Le.. Let R be a polynoial ring over a field and let I be a square-free onoial ideal in R. Let α β I s be two sequences of length s 4 where I s is as in Definition.. Suppose that α = ( a 3...a 3 ) and β = ( ). Then T αβ SJ + S (J J 3 ). Proof. We will show T αβ SJ + S ( s J i ) and by induction on s 4 we conclude that T αβ SJ + S (J J 3 ). Suppose that there are l i distinct copies of a i in α and k i distinct copies of b i in β. Then l + l + l 3 = k + k = s 4. Without lost of generality we assue that l l l 3 and k k. Since s 4 we have l and k. We write α = (a l a l a l 3 3 ) and β = (b k bk ). We then have three possible scenarios. We clai the following: (i) If l + l > k then gcd( f α f β ) = gcd( f a f a f b f b )gcd( f α f β )/A = gcd( f α f b f b )gcd( f α f β )/B for soe AB R and α = (a l a l a l 3 3 ) and β = (b k b k ). (ii) If l > k then gcd( f α f β ) = gcd( f a f b )gcd( f α f β )/ = gcd( f α f b )gcd( f α f β )/D for soe D R and α = (a l a l a l 3 3 ) and β = (b k b k ). (iii) If l + l k and l k then gcd( f α f β ) = gcd( f s/3 f s/3 f s/3 a 3 f s/3 f s/3 ) = (gcd( f a f a f a3 f f b )) s/3.

9 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 9 Once we obtain the above clais then the result will follow by Le.7. To establish clai (i) we notice that we have the following equalities: gcd( f α f β ) = gcd( f a f a f β )gcd( f α f β )/E = gcd( f a f a f b f b )gcd( f a f a f β )gcd( f α f β )/EF = gcd( f α f b f b )gcd( f α f β )/G = gcd( f α f b f b )gcd( f α f β )gcd( f a f a f β )/GH for soe EFGH R. It is enough to show that for any variable x R and any positive integer t if x t gcd( f a f a f β ) then x t EF and x t GH. This is equivalent to showing that if x u gcd( f α f β ) and x gcd( f a f a f β ) then x u gcd( f a f a f b f b )gcd( f α f β ) and x u gcd( f α f b f b )gcd( f α f β ) for any positive integer u. But this follows iediately since l +l > k k and l +l < l +l +l 3 = s = k + k. For clai (ii) we notice that we have the following equalities: gcd( f α f β ) = gcd( f a f β )gcd( f α f β )/K = gcd( f a f b )gcd( f a f β )gcd( f α f β )/KL = gcd( f α f b )gcd( f α f β )/M = gcd( f α f b )gcd( f α f β )gcd( f a f β )/MN for soe KLMN R. The clai follows since l > k. Finally to establish the last clai we note that the assuption l k is equivalent to l + l 3 k and thus l + l 3 k l + l. Hence l 3 l l l 3 which iplies l i = s/3 and s/3 k. Furtherore s/3 k s/3 and hence k = s/3 and k = s/3. The next Theore is one of the ain results of this section. We will use all the inforation we obtained about how various conditions on two sequences αβ I s affect the ter T αβ to obtain a bound on the relation type of I as well as a description of the defining equations of the Rees algebra. Recall that the relation type of an ideal I is defined to be rt(i) = in{s J = s J i } where J is the defining ideal of R (I). In other words the relation type of I is the largest degree (in the T i ) of any inial generator of J. In particular when I is of linear type it is of relation type. The relation type of an ideal has been explored in various articles see for instance [ ]. Theore.. Let R be a polynoial ring over a field and let I be a square-free onoial ideal generated by n square-free onoials in R. When n 5 then R (I) = S/J where S = R[T...T n ] and J = SJ + S ( n J i ). In particular rt(i) n. Proof. By Theore. it suffices to show that given two sequences αβ I s of length s > n then T αβ SJ + S ( n J i ). Suppose that there are l i distinct copies of a i in α and k i distinct copies of b i in β and write α = (a l...a l ) and β = (b k...bk t t ).

10 0 L. FOULI AND K. N. LIN By Le.4 we ay assue a i b j for all i j. Also by Le.9 we ay assue < 3 and < t since n 5. We are now left with the following cases: (i) Suppose that α = (a l a l ) and β = (b k bk ) where l + l = k + k 3. The result follows fro Le.0. (ii) Suppose that α = (a l a l a l 3 3 ) and β = (b k bk ) where l + l + l 3 = k + k 4. The result follows fro Le.. The following exaple provides a class of square-free onoial ideals generated by n > 4 square-free onoials for which the relation type is at least n 7. Notice when n = 5 one has n 7 = n. In particular this establishes that the bound given in Theore. is sharp. We also note that the ideals in the following exaple are not of fiber type. Exaple.3. Let R = k[yx...x n zu...u n ] be a polynoial ring over a field k with n a positive integer such that n > 4. Let I = ( f... f n ) where f = ( n x i )z f i = x i y( n u j ) for all j= j i i =...n f n = ( n F = T n 4 n x i )y and f n = ( n u i )z. onsider T i Tn n 3 T n n 4 and G = zt n 5 n T i yt n 5 n T n. It is clear that F and G are in the defining ideal of the Rees algebra of I. Moreover F and G are irreducible and thus F SJ n 7 \ S ( n 8 J i ) and G SJ n 8 \ S ( n 9 J i ). Hence the relation type of I is at least n 7. Finally we note that G is not in the defining ideal of the special fiber of I and therefore I is not an ideal of fiber type. The next lea establishes conditions for when various generators of the defining ideal of the Rees algebra as in Theore. are irredundant. Le.4. Let R be a polynoial ring over a field and let I be a square-free onoial ideal in R. Let Let α β I s be two sequences of length s 4 where I s is as in Definition.. Suppose that α = (...a s ) β = (... ) where a i b j for all i j. Suppose that for all i there exists a variable x i R such that x i f a j for all j i and x i f b x i f ai and x i f b. Furtherore suppose that for all i there exists a variable z i R such that z i gcd( f ai f b ) z i f a j for all j i and z i f b. Then T αβ SJ s \ S ( s J i ). s j= j i Proof. We write f ai = z i h i ( x j ) for all i =...s f b = k ( s x i ) and f b = k ( s z i ) for soe square-free onoials h...h s k k R such that x i h j x i k l z i h j and z i k l for all i

11 j l. Then we have f α = s M = f α gcd( f α f β ) = REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS x s i s s z i s h i gcd( s h i k s k ) h i and f β = k s k ( s and N = x s i s f β gcd( f α f β ) = z i ). Let k s k gcd( s h i k s k ) One can observe iediately that x i M and z i N for all i. Let α and β be any proper subsequences f α of α and β. Then there exists either x i or z j such that x i gcd( f α f β ) or z f α j gcd( f α f β ) by f α our assuptions. Therefore gcd( f α f β ) M and siilarly f β N. This shows that gcd( f α f β ) T αβ SJ s \ S ( s J i ). The following Theore that establishes precisely the defining equations for the Rees algebra of a square-free onoial ideal with up to five generators.. Theore.5. Let R be a polynoial ring over a field and let I be a square-free onoial ideal generated by n square-free onoials in R. Suppose that n 5. Suppose that there does not exist a pair of sequences αβ I s of length s as in Definition. such that the conditions of Le.4 are satisfied. Then I is an ideal of linear type. Proof. onsider a pair of sequences αβ as in Definition. of length s such that α = (...a s ) β = (... ) where a i b j for all i j. Suppose that one of the following conditions is not satisfied: for all i there exists a variable x i R such that x i f a j for all j i and x i f b x i f ai and x i f b and for all i there exists a variable z i R such that z i gcd( f ai f b ) z i f a j for all j i and z i f b. We will show that T αβ SJ + S ( s J i ) and by induction T αβ SJ. By the Theore. and Leas.4.9 it is enough to consider sequences of length or 3 i.e. either α = ( ) and β = ( ) or α = ( a 3 ) and β = ( ). The proof for the first case can be treated as a special case of the second and thus we will only consider the case with α = ( a 3 ) and β = ( ). Without lost of generality we will show that if there does not exist a variable x R such that x gcd( f a f a3 ) x f b x f a and x f b then for soe AB R. gcd( f a f a f a3 f f b ) = gcd( f f b )gcd( f α f b f b ) A = gcd( f f β )gcd( f a f a3 f b f b ) B Siilarly if there does not exist a variable z R such that z gcd( f a f b ) z f a j for j = 3 and z f b then

12 L. FOULI AND K. N. LIN gcd( f a f a f a3 f f b ) = gcd( f f b )gcd( f a f a3 f β ) = gcd( f α f b )gcd( f a f a3 f b f b ) D for soe D R. Hence we ay apply Le.7 to obtain the result. For the first part we notice that gcd( f a f a f a3 f f b ) = gcd( f α f b )gcd( f α f b f b ) E = gcd( f f b )gcd( f a f a3 f b )gcd( f α f b f b ) EF gcd( f a f a f a3 f f b ) = gcd( f f β )gcd( f a f a3 f β ) G = gcd( f f β )gcd( f a f a3 f b )gcd( f a f a3 f b f b ) GH for soe EFGH R. It is enough to show for any variable x R if x gcd( f a f a3 f b ) then x EF and x GH. This is equivalent to showing that for any integer u if x u gcd( f α f β ) then x u gcd( f a f b )gcd( f α f b f b ) and x u gcd( f a f β )gcd( f a f a3 f b f b ). This is straightforward to verify and the only case that is not trivial is when x f a and x gcd( f a f a3 ). Then x f b by assuption. Hence x gcd( f α f b f b ) and x gcd( f a f a3 f b f b ). We can use a siilar arguent for the second part and the only case that is not trivial is when x gcd( f a f b f b ) and x f b. Then x f b and x f a or x f a3 by assuption. Hence x gcd( f α f β ) x gcd( f a f a3 f β ) and x gcd( f a f a3 f b f b ). We now introduce a construction that is also known as the line graph of the hypergraph corresponding to a onoial ideal. onstruction.6. Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let f... f n be a inial onoial generating set of I. We construct the following graph G(I): For each f i we associate a vertex y i to it. The edges of this graph are {y i y j } where gcd( f i f j ). We call the graph G(I) the line graph of I. The purpose of introducing the line graph of the hypergraph corresponding to a onoial ideal is to utilize the graph structure in order to deterine cobinatorial conditions that deterine the defining equations of the Rees algebra. We first observe the following. Reark.7. With the sae assuptions as in Le.4 the sequences αβ correspond to an even closed walk in G(I). Indeed the following {y b {y b y a }y a {y a y b }y b {y b y a }y a {y a y b }y b... y b {y b y as }y as {y as y b }y b {y b y as }y as {y as y b }y b } is an even closed walk and T αβ SJ s \ S ( s J i ).

13 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 3 We now introduce the notion of a subgraph induced by two sequences. Definition.8. Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let α β I s be two sequences of length s where I s is as in Definition.. Let α = (a l a l...a l ) and β = (b r...br t t ) where l i and r i are the nuber of distinct copies of a i and b i respectively. Let α = (...a ) and β = (b i...b ir ) where {b i...b ir } = {...b t } \ {...a }. The last condition ensures that a i b ik for all ii k. onsider K = ( f a... f a f bi... f bir ) and the line graph G(K) of K. Notice that K I and that G(K) is a subgraph of G(I) since a i b ik for all ii k. We call G(K) the graph induced by α and β. Reark.9. Let R be a polynoial ring over a field and let I be a onoial ideal in R. Let αβ I s be two sequences of length s where I s is as in Definition.. Let G be the graph induced by α and β. If G is a disconnected graph then T αβ SJ + S ( s J i ) by Proposition.8. The following Theore is an extension of Theore.. We use the line graph to give a description for the defining equations of the Rees algebra. Theore.0. Let R be a polynoial ring over a field and let I be a square-free onoial ideal generated by n square-free onoials in R. Suppose that the line graph of I is the disjoint union of graphs with at ost 5 vertices. Let R (I) = S/J where S = R[T...T n ]. Then J = SJ +S (J J 3 ) and in particular rt(i) 3. Furtherore if T αβ S (J J 3 ) \ SJ then the subgraph induced by αβ is an even closed walk of G(I). Proof. By Reark.9 we ay assue that G(I) is connected. Then the first part follows iediately by Theore.. The last part follows fro Theore.5 Le.4 and Reark.7. The next exaple illustrates the result of Theore.0. Exaple.. Let R = k[x...x 7 ] be a polynoial ring over a field k. Let I = ( f... f 5 ) where f = x x x 3 f = x x x 4 x 7 f 3 = x x 3 x 6 f 4 = x 4 x 5 x 6 and f 5 = x x 3 x 5. Let R (I) = S/J where S = R[T...T 5 ]. Let α = (4) and β = (35). The subgraph induced by αβ is the even closed walk {y {y y }y {y y }y {y y 3 }y 3 {y 3 y 4 }y 4 {y 4 y 5 }y 5 {y y 5 }} of the line graph of I shown below: y = x x x 4 x 7 y 3 = x x 3 x 6 y = x x x 3 y 4 = x 4 x 5 x 6 y 5 = x x 3 x 5 Then by Theore.0 we have that T αβ = T T 3 T 5 x 7 T T 4 SJ 3 \ (SJ + SJ ) since it is irreducible.

14 4 L. FOULI AND K. N. LIN The following exaple coes fro [34 Exaple 3.]. Villarreal uses this exaple to show that his ethods do not extend in the case of square-free onoial ideals generated in degree higher than. In light of Theore.0 we now give an explicit description of the defining equations of the Rees algebra for this exaple. Exaple.. Let R = k[x...x 7 ] where k is a field and let I be an ideal of R generated by f = x x x 3 f = x x 4 x 5 f 3 = x 5 x 6 x 7 f 4 = x 3 x 6 x 7. Then the defining ideal of R (I) is inially generated by the binoials: x 3 T 3 x 5 T 4 x 6 x 7 T x x T 4 x 6 x 7 T x x 4 T 3 x 4 x 5 T x x 3 T x 4 T T 3 x T T 4. Notice that the only binoial that is not linear is x 4 T T 3 x T T 4 and it coes fro the unique even cycle of G(I) which in this case is the graph of a square. Before concluding this section we turn our attention to toric ideals. Recall that a toric ideal is the ideal of relations of a onoial subring of a polynoial ring. Let A = {...a n } N c \ {0} and let A be a atrix with coluns a i and suppose that A has rank c. Let k be a field and let S = k[t...t n ] and let φ : S k[t...t c ] be a ap defined by φ(x i ) = t a i. Then kerφ is a prie ideal and it s called the toric ideal associated to A. In particular when I is generated by square-free onoials of the sae degree then the defining ideal of the special fiber ring F (I) = R (I) k is the toric ideal associated to I. Notice that in this case F (I) = k[ f... f n ] where f... f n is a inial onoial generating set of I. In light of Theore.0 we can give a concrete description of such toric ideals. orollary.3. Let R be a polynoial ring over a field k let f... f n be square-free onoials of the sae degree and let I = ( f... f n ). Suppose that the line graph of I is the disjoint union of graphs with at ost 5 vertices. Let F (I) = k[ f... f n ] be the onoial subring of R generated by f... f n. Then F (I) k[t...t n ]/J and the toric ideal J corresponding to I is generated by {T α T β αβ I s with s = 3 and f α = f β where αβ induce an even closed walk in G(I)}. 3. SQUARE-FREE MONOMIAL IDEALS OF LINEAR TYPE In this section we turn our attention to ideals of linear type. Villarreal showed that when I is the edge ideal of a graph then 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 [34 orollary 3.]. Inspired by this result we use the line graph associated to any onoial ideal as in onstruction.6 in order to obtain siilar results as Villarreal for any square-free onoial ideal without restrictions on the degrees of the generators. A natural first class to consider is ideals with line graph the graph of a forest. Proposition 3.. Let R be a polynoial ring over a field and let I be a square-free onoial ideal in R. We further assue that the line graph G(I) is the graph of a forest. Then I is an ideal of linear type.

15 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 5 Proof. Let f... f n be a inial onoial generating set for I. We show that for all sequences αβ I s of length s where I s is as in Definition. we have T αβ SJ + S ( s J i ) and by induction it follows that T αβ SJ. Since G(I) is the graph of a forest then every subgraph of G(I) is also a forest. By Proposition.8 we ay assue that G(I) is connected i.e. it is the graph of a tree. Notice that for every graph of a tree there exists a vertex that is only connected to one other vertex. Suppose that s. Let α = (a l a l...a l ) and β = (b r...br t t ) where l i and r i are the nuber of distinct copies of a i and b i respectively. Notice that if a i = b j for soe i and soe j then the result follows by Le.4 and the induction hypothesis. Hence we ay assue that a i b j for all i j. Let α = (...a ) and β = (...b t ). Then the graph G induced by α β is a subgraph of G(I) by Definition.8. Hence G is the graph of a forest. If G is disconnected then the result follows by Proposition.8 and the induction hypothesis. Therefore we ay assue that G is connected and hence the graph of a tree. Thus without loss of generality we ay assue that y a is only connected to y b. If l r then gcd( f a f r f r t b t ) = and hence T αβ SJ + S ( s J i ) by Proposition.8. So we ay assue that l < r. Then we clai that (a) gcd( f α f β ) = gcd( f l f l )gcd( f l a f β ) (b) gcd( f α f β ) = gcd( f α f l )gcd( f l k a f r l f r f r t A for soe A R. Notice that by the clai and Le.7 it follows that T αβ SJ + S ( s Therefore it reains to prove the clai. For part (a) notice that gcd( f α f β ) = gcd( f l f β )gcd( f l a f β ) = ( f l f l )gcd( f l a f β ) where the equalities follow fro the fact that gcd( f a f ai ) = gcd( f a f b j ) = for any i > j > and Reark.6 (b). For part (b) notice that b t ) J i ). gcd( f α f β ) = gcd( f α f l )gcd( f α f r l f r f r t b t )/B = gcd( f α f l )gcd( f l f r l f r f r t b t )gcd( f l a f r l f r f r t b t )/B = gcd( f α f l )gcd( f l f r l )gcd( f l a f r f r t b t )/B for soe B R since gcd( f l f r f r t b t ) =. It is enough to show that gcd( f l f r l ) B i.e. for any variable x R and any integer v if x v gcd( f l f r l ) then x v B. This is equivalent to showing that for any variable x R if x gcd( f l f r l ) and x u gcd( f α f β ) for soe integer u then x u gcd( f α f l )gcd( f l a f r l f r f r t b t ). If x gcd( f l f r l ) and x u gcd( f α f β ) then u l by the fact that y a is only connected to y b and l < r. Therefore x u gcd( f α f l ).

16 6 L. FOULI AND K. N. LIN The following lea allows us to handle the induction step in the case of odd cycles in order to prove that when the line graph of a square-free onoial ideal is the graph of an odd cycle then the ideal is of linear type. Lea 3.. Let R be a polynoial ring over a field and let I be a square-free onoial ideal in R. Let α β I s be two sequences of length s 4 where I s is as in Definition.. Let α = (a l a l...a l ) and β = (b r...br t t ) where l i and r i are the nuber of distinct copies of a i and b i respectively. Suppose that l < r l < r and that the graph G(I) is the graph of an odd cycle of length at least 5. We further assue that y a is connected to y b and y a only and y a is connected to y a and y b only. Then T αβ SJ + S ( s J i ). Proof. Let α = (a l a l ) and β = (b r l b r l b r br t t ). Using Reark.6 and Reark.3 we have that since l < r and l < r. gcd( f α f β ) = gcd( f l f β )gcd( f α f β ) = gcd( f l f l )gcd( f α f β ) We clai that gcd( f α f β ) = gcd( f α f l )( f α f β ) for soe R. Then the result will follow by Le.7. Notice that gcd( f α f β ) = gcd( f α f l )gcd( f α f β ) D = gcd( f α f l )gcd( f α f β )gcd( f l f β ) DE = gcd( f α f l )gcd( f α f β )gcd( f l f r l f r l DE where DE R and the third equality follows fro the fact that y a and y a are only connected to ) y b and y b respectively. It is enough to show that for any variable x R and any integer v if x v gcd( f l f r l f r l ) then x v DE. This is equivalent to showing that if x gcd( f l f r l f r l and x u gcd( f α f β ) then x u gcd( f α f l )gcd( f α f β ). If x gcd( f l f r l f r l ) then x can not divide both f a and f a since otherwise {y a y a y b } or {y a y a y b } will for a 3-cycle in G(I). Siilarly x can not divide both f b and f b. Therefore without loss of generality we ay assue x f a and x f b. Since l < r and y a is not connected to y ai for all i > we obtain that if x u gcd( f α f β ) then u l. Therefore x u gcd( f α f l ). In the next Proposition we are able to handle the case where each of the connected coponents of the line graph has a unique odd cycle. Proposition 3.3. Let R be a polynoial ring over a field and let I be a square-free onoial ideal in R. We further assue that line graph G(I) of I is the disjoint union of graphs with a unique odd cycle. Then I is an ideal of linear type. )

17 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 7 Proof. By Reark.9 it suffices to consider the connected coponents of G(I). Hence we ay assue that G(I) is connected and it has a unique odd cycle. We show that for all sequences αβ I s of length s where I s is as in Definition. we have T αβ SJ +S ( s J i ). By induction it will follow that T αβ SJ. Let α = (a l a l...a l ) and β = (b r...br t t ) where l i and r i are the nuber of distinct copies of a i and b i respectively. Using Le.4 we ay assue without loss of generality that a i b j for all i and j. Let α = (...a ) and β = (...b t ). Then the graph G induced by α and β is a subgraph of G(I) by Definition.8. By Proposition.8 we ay assue that G is a connected graph. Notice that if there exists a vertex y i that is only connected to one of the y j then T αβ SJ + S ( s J i ) by the proof of Proposition 3.. Hence we ay assue that G is the graph of an odd cycle. Therefore every vertex is only connected to two other vertices. Let l = +t. Since G is the graph of an odd cycle then l is odd and hence t. We proceed by induction on l. If l = 3 then G(I) is the graph of a triangle and the result follows fro Le.4 and Le.9. Suppose that l 5. Using Le.9 we ay assue that and t. Notice that y a is connected to two other vertices. Without loss of generality we ay assue that we have the following possible cases: (i) y a is connected only to y a and y a3 (ii) y a is connected only to y a and y b (iii) y a is connected only to y b and y b. We handle each case separately. For case (i) we note that since y a is connected only to y a and y a3 then using k = l in Proposition.8 and induction we obtain the result. For case (ii) suppose that y a is connected only to y a and y b. If l r then using k = l in Proposition.8 and induction we obtain the result. If l < r then we consider y a. Notice that y a can not be connected to y b since then {y a y a y b } will for a triangle which is a contradiction since l 5. Hence either y a is connected to y a3 or y a is connected to y bi for soe i. If y a is connected to y a3 then case (i) yields the result. Suppose that y a is connected to y bi for soe i. Without loss of generality suppose that i =. I r then we ay use k = l in Proposition.8 and induction. I < r and since l < r we can use Lea 3. and induction. Finally for case (iii) suppose that y a is connected only to y b and y b. If every y ai is connected to y bi and y bi for i i then either = t which is ipossible since l is odd or there exist i j such that y bi is connected to y b j. Then by switching the role of α and β we are in either case (i) or case (ii). We are now ready to state the ain theore of this section. Theore 3.4. Let R be a polynoial ring over a field and let I be a square-free onoial ideal in R. We further assue that the line graph of I G(I) is the disjoint union of graphs of trees and graphs with a unique odd cycle. Then I is an ideal of linear type.

18 8 L. FOULI AND K. N. LIN Proof. The result follows iediately by Proposition 3. Proposition 3.3 and Proposition.8. We conclude this article with the following reark. Reark 3.5. Let R be a polynoial ring over a field and let I be a square-free onoial ideal generated by 3 square-free onoials in R. Then the fact that I is of linear type is already known but it also follows iediately fro Proposition 3. and Proposition 3.3. Acknowledgeents. We wish to thank Ali Alilooee and Yi-Huang Shen for their coents on an earlier version of the anuscript. REFERENES [] I. Aberbach L. Ghezzi H. T. Há Hoology ultipliers and the relation type of paraeter ideals Pacific J. Math. 6 (006) 39. [] L. Busé Eliination theory in codiension one and applications INRIA 598 June [3] A. onca E. De Negri M-sequences graph ideals and ladder ideals of linear type J. Algebr (999) [4] A. onca J. Herzog N.V. Trung G. Valla Diagonal subalgebras of bigraded algebras and ebeddings of blow-ups of projective spaces Aer. J. Math. 9 (997) [5] A. orso U. Nagel Monoial and toric ideals associated to Ferrers graphs Trans. Aer. Math. Soc. 36 (009) [6] D. ox A. Kustin. Polini B. Ulrich A study of singularities on rational curves via syzygies preprint Me. Aer. Math. Soc. (03) no. 045 x+6 pp. [7] D. ox J. Little H. Schenck Toric varieties Graduate Studies in Matheatics 4. Aerican Matheatical Society Providence RI 0 xxiv+84 pp. [8] V. Gasharov N. Horwitz I. Peeva Hilbert functions over toric rings Michigan Math. J. 57 (008) [9] A. Geraita A. Giigliano Generators for the defining ideal of certain rational surfaces Duke Math. J. 6 (99) [0] A. Geraita A. Giigliano B. Harbourne Projectively noral but superabundant ebeddings of rational surfaces in projective space J. Algebr69 (994) [] A. Giigliano On Veronesean surface Proc. Konin. Ned. Akad. van Wetenschappen Ser. A 9 (989) [] A. Giigliano A. Lorenzini On the ideal of veronesean surfaces anad. J. Math. 43 (993) [3] H.T. Hà Box-shaped atrices and the defining ideal of certain blowup surfaces J. Pure Appl. Algebr67 (00) [4] H.T. Hà On the Rees algebra of certain codiension two perfect ideals Manuscripta Math. 07 (00) [5] S. Huckaba On coplete d-sequences and the defining ideals of Rees algebras Math. Proc. abridge Philos. Soc. 06 (989) [6] J. Herzog A. Siis W. V. Vasconcelos Approxiation coplexes of blowing-up rings J. Algebra 74 (98) [7] J. Herzog A. Siis W. V. Vasconcelos Approxiation coplexes of blowing-up rings. II. J. Algebra 8 (983) [8] J. Herzog W. V. Vasconcelos R. H. Villarreal Ideals with sliding depth Nagoya Math. J. 99 (985) [9]. Huneke Deterinantal ideals of linear type Arch. Math. (Basel) 47 (986) [0]. Huneke On the syetric and Rees algebra of an ideal generated by a d-sequence J. Algebra 6 (980) [] A. Kustin. Polini B. Ulrich Rational noral scrolls and the defining equations of Rees algebras J. Reine Angew. Math. 650 (0) [] Y-H. Lai On the relation type of systes of paraeters J. Algebr75 (995) [3] A. Micali Sur les algèbres universelles Ann. Inst. Fourier 4 (964) [4] S. Morey Equations of blowups of ideals of codiension two and three J. Pure Appl. Algebr09 (996) 97. [5] S. Morey B. Ulrich Rees algebras of ideals with low codiension Proc. Aer. Math. Soc. 4 (996) [6] H. Ohsugi T. Hibi Toric ideals generated by quadratic binoials J. Algebr8 (999) no. 509?57. [7] E. Reyes. Tatakis A. Thoa Minial generators of toric ideals of graphs Adv. in Appl. Math. 48 (0) [8] A. Siis N. Trung G. Valla The diagonal subalgebra of a blow-up algebra J. Pure Appl. Algebr5 (998)

19 REES ALGEBRAS OF SQUARE-FREE MONOMIAL IDEALS 9 [9] J. A. Soleyan X. Zheng Monoial ideals of forest type o. Algebra 40 (0) [30] D. Taylor Ideals generated by onoials in an R sequence Ph. D. thesis University of hicago 966. [3] N. V. Trung Reduction exponent and degree bound for the defining equations of graded rings Proc. Aer. Math. Soc. 0 (987) [3] G. Valla On the syetric and Rees algebra of an ideal Manuscripta Math. 30 (980) [33] W. Vasconcelos Arithetic of Blowup Algebras in: London Matheatical Society Lecture Note Series vol. 95 abridge University Press abridge 994. [34] R. H. Villarreal Rees algebras of edge ideals o. Algebr3 (995) [35] H- J. Wang The relation-type conjecture holds for rings with finite local cohoology o. Algebr5 (997) DEPARTMENT OF MATHEMATIAL SIENES NEW MEXIO STATE UNIVERSITY LAS RUES NEW MEXIO USA E-ail address: lfouli@ath.nsu.edu DEPARTMENT OF MATHEMATIS AND STATISTIS SMITH OLLEGE NORTHAMPTON MASSAHUSETTS 0063 E-ail address: klin@sith.edu