LIAISON OF MONOMIAL IDEALS

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1 LIAISON OF MONOMIAL IDEALS CRAIG HUNEKE AND BERND ULRICH Abstract. We give a simple algorithm to ecie whether a monomial ieal of nite colength in a polynomial ring is licci, i.e., in the linkage class of a complete intersection. The algorithm proves that whether or not such an ieal is licci oes not epen on whether we restrict the linkage by only allowing monomial regular sequences, or homogeneous regular sequences, or arbitrary regular sequences. We apply our results on monomial ieals to compare when an ieal is licci versus when its initial ieal in some term orer is licci. We also apply an iea of Migliore an Nagel to prove that monomial ieals of nite colength are always glicci, i.e., in the Gorenstein linkage class of a complete intersection. However, our proof requires the use of non-homogeneous Gorenstein links.. Introuction Let R be a commutative Noetherian ring, an let I an J be two proper ieals in R. These ieals are sai to be irectly linke if there exists a regular sequence f ; : : : ; f g containe in I \ J such that (f ; : : : ; f g ) : I = J an (f ; : : : ; f g ) : J = I. We say I an J are in the same linkage class (or liaison class) if there exists a sequence of ieals I = I 0 ; : : : ; I n = J such that I j is irectly linke to I j+ for 0 j n, the case n = 2 being referre to as ouble linkage. Such a sequence of links connecting I an J is far from unique. We call the ieal I licci if I is in the linkage class of a complete intersection, i.e., of an ieal generate by a regular sequence. In a similar manner, at least when R is regular, we say that I an J are Gorenstein irectly linke if there exists an ieal K I \ J such that R=K is Gorenstein, K : I = J, an K : J = I; the last equality is actually a consequence of the previous one in case I is unmixe an has the same height as K. The Gorenstein linkage class of I is ene by making this relation an equivalence relation as above, an I is sai to be glicci if it is in the Gorenstein linkage class of a complete intersection. Finally, by a Gorenstein ouble link we mean a sequence of two irect Gorenstein links. This paper stuies when monomial ieals in polynomials rings are licci or glicci. Our main theorem, Theorem 2.6, gives a simple algorithm to ecie whether a monomial ieal of nite colength is licci. This theorem is one of the few instances where one has not only necessary, but also sucient conitions for an ieal to be licci. In Theorem 3.2 we compare when an Date: November 25, Both authors were partially supporte by the National Science Founation, grants DMS an DMS , respectively. The secon author also thanks the University of Kansas for generous support while this work was one.

2 2 CRAIG HUNEKE AND BERND ULRICH ieal of nite colength is licci to when its initial ieal with respect to some term orer is licci. In Theorem 4.2 we prove that any monomial ieal of nite colength is glicci. For basic information on linkage we refer the reaer to [6], [8], an [3]. 2. Licci Monomial Ieals We begin by establishing some notation. We will always write S = k[x ; : : : ; x ] for a polynomial ring over a el k an m for its homogeneous maximal ieal ( x ; : : : ; x ). By a monomial in S we mean an element of the form x a x a. We simplify this notation by using capital letters to enote -tuples of non-negative integers, A = (a ; : : : ; a ), an writing x A = x a x a. A monomial ieal is an ieal generate by monomials. Every m- primary monomial ieal I can be written uniquely in stanar form I = (x a ; : : : ; xa )+ I #, where I # is generate by monomials that together with fx a ; : : : ; xa g generate I minimally. Notice that I # = 0 if an only if I is a complete intersection. We will use the fact that if x B = x b x b =2 (x a ; : : : ; xa ), then (xa ; : : : ; xa ) : xb = (x a b ; : : : ; x a b ). In particular, any m-primary monomial almost complete intersection ieal is irectly linke to a complete intersection by a monomial regular sequence. We will nee the following theorem that is a special case of a main result in [3]: Theorem 2.. Let S = k[x ; : : : ; x ] be a polynomial ring over a el k an let I be an m- primary ieal generate by homogeneous polynomials of egrees at least. If m ( )( ) I then I m is not licci in S m. Proof. The maximal last shift in a minimal homogeneous free S-resolution of S=I is at most ( ). Now [3, 5.3(a)] implies that I m cannot be licci. (The essential point in applying [3, 5.3(a)] is that the maximal last shift in a minimal homogeneous free S-resolution of S=I is at most times the minimal egree of a generator of I.) Theorem 2.4 below gives a necessary conition for an m-primary monomial ieal to be licci. The conition is rather strong an was surprising to us. To prove this theorem we nee Proposition 2.3, which basically says that an ieal J is licci if an only if J + ys is licci for y a regular element on S an S=J. This result is not unexpecte, but oes not seem to be in the literature. Its proof requires the use of universal linkage as evelope in [3]. We briey review the enition. Let ( R; m) be a local Gorenstein ring an let I be an unmixe ieal of height g > 0 in R. Fix a generating sequence f ; : : : ; f n of I. Let x ij be variables for i g an j n, an write R(X) for P the ring R[fx ij g] mr[fxij g]. In n R(X) consier the regular sequence ; : : : ; g where i = j= x ijf j. We ene the rst universal link L (I) of I to be the ieal ( ; : : : ; g )R(X) : IR(X) in R(X). Inuctively we set L n (I) = L (L n (I)) for n > as long as L n (I) is not the unit ieal, an call this ieal the nth universal link. Write L 0 (I) = I. Although these enitions apparently epen upon generating sets, it turns out that universal links are essentially unique (see [3, 2.(b)] for a precise statement). One of the basic facts about universal linkage says that I is licci if an only if L n (I) is generate by a regular sequence for some n 0, at least when R has an innite resiue el (see [4, 2.9]).

3 LIAISON OF MONOMIAL IDEALS 3 Lemma 2.2. Let R be a local Gorenstein ring an let J be an ieal such that R=J is Cohen- Macaulay. Let y 2 R be regular on R an R=J. Then L ((J; y)) = (K; z), where K is an ieal in R(X) irectly linke to JR(X) an z 2 R(X) is regular on R(X) an R(X)=K. Proof. Fix a generating sequence f ; : : : ; f n of J. Set f n = y an ene i in R(X) as above. Let z = g. We may write ( ; : : : ; g )R(X) = ( ; : : : ; g ; z)r(x), where ; : : : ; g form a regular sequence containe in JR(X). Set K = ( ; : : : ; g )R(X) : JR(X). We have that L ((J; y)) = ( ; : : : ; g )R(X) : (J; y)r(x) = ( ; : : : ; g ; z)r(x) : (J; z)r(x) = (K; z), where the last equality hols by [2, 2.2]. Finally, the element z is regular on R(X) an R(X)=K because ; : : : ; g ; z form an R(X)-regular sequence. Proposition 2.3. Let R be a local Gorenstein ring with innite resiue el. Let J be an ieal such that R=J is Cohen-Macaulay, an let y 2 R be regular on R an R=J. Then (J; y) is licci if an only if J is licci. Proof. Assume that (J; y) is licci. Accoring to [3, 2.7(b)], L n ((J; y)) is generate by a regular sequence for some n 0. By repeate use of Lemma 2.2, L n ((J; y)) = (K; z) for some ieal K in the linkage class of JR(X) an some z 2 R(X) which is regular on R(X) an R(X)=K. Hence K is generate by a regular sequence, showing that JR(X) is licci in R(X). Then J is licci in R by [4, 2.2], which states that the property of being licci escens from at local extensions of local Gorenstein rings with innite resiue els. Conversely, if J is licci, then L n (J) is generate by a regular sequence for some n 0, an it is clear that y is regular moulo L i (J) for every 0 i n. By [2, 2.2] it then follows that (J; y)r(x) is in the same linkage class as ( L n (J); y)r(x). Using [4, 2.2] again we obtain that (J; y) is licci. Theorem 2.4. Let S = k[x ; : : : ; x ] be a polynomial ring over a el k an let I be an m-primary monomial ieal. If I # has height at least two then I m is not licci in S m. In particular, I is not licci. Proof. We may assume that k is innite an we write I = (x a ; : : : ; xa ) + I #. Assume that I # has height at least two. If I contains a variable, say x, we may write I = I 0 S + (x ), where I 0 is a (x ; : : : ; x )-primary ieal in k[x ; : : : ; x ]. Clearly x is regular on S an on S=I 0 S. Accoring to Proposition 2.3, if I m is licci, then so is I 0 S m. By [4, 2.2] we obtain that I 0 k[x ; : : : ; x ] (x ;::: ;x ) is also licci. Since I # = I # 0 S, I # 0 has height at least two. Inucting on the number of variables proves that I 0 k[x ; : : : ; x ] (x ;::: ;x ) is not licci, an hence neither is I m. Thus we may assume that a i 2 for i. Let T be the polynomial ring S[y ; : : : ; y ] with homogeneous maximal ieal n. We ene a map on the set of monomials in S to the set of monomials in T by rst specifying its

4 4 CRAIG HUNEKE AND BERND ULRICH action on pure powers of the variables, (x n i ) = 8 < : if n = 0 y n i x i if n < a i y n 2 i x 2 i if n a i : We then exten to a map on the set of all monomials by setting (x C ) = (x c ) (x c ): Notice that is not multiplicative, but that it preserves ivisibility. Finally, for K any monomial ieal in S, we ene a monomial ieal e K in T by applying to the monomial generators of K an letting e K be the ieal in T generate by their images. Consier the epimorphism of S-algebras : T! S mapping y i to x i. Notice that ( K) e = K an that the kernel of this map is generate by the T -regular sequence y x ; : : : ; y x. We claim that this sequence is regular on the quotient ring T = K e as well. The claim is equivalent to the vanishing of the rst Koszul homology H (y x ; : : : ; y x ; T = e K). This homology is Tor T (S; T = e K). Thus it suces to prove that generating relations on the monomial minimal generators fx C i g of K lift via to relations on the corresponing monomial generators f (x C i )g of e K. Inee, a set of generating syzygyies for K can be obtaine as follows: let x C an x D be any two monomial minimal generators of K, an let x E be the greatest common ivisor of x C an x D. The syzygy moule is generate by the syzygies given by xd x C = xc x D. This relation lifts to (xd ) x E x E (x E ) (xc ) = (xc ) (x E ) (xd ). Notice that (xd ) an (xc ) are monomials in T because the map preserves ivisibility, an that (x E ) (x E ) we have inee obtaine a lift because the map is multiplicative. Thus in the language of [3, 2.2(a)], the pair ( T; K) e is a eformation of ( S; K). Hence accoring to [3, 2.6], if I m is licci then so is In e. In particular, the further localization ei mt woul be licci as well. Set J = I #. Since (T; J) e is a eformation of ( S; J) an J e is homogeneous, we also obtain ht J e = ht J 2. Now write k 0 = k(y ; : : : ; y ), S 0 = k 0 [x ; : : : ; x ], m 0 = (x ; : : : ; x )S 0, I 0 = IS e 0 an J 0 = JS e 0. Notice that T mt = S 0 an hence e m 0 ImT = I 0 0, reucing us to prove that I 0 cannot m m 0 be licci. By the enition of the map, I 0 = (x 2 ; : : : ; x2) + J 0 an J 0 is generate by squarefree monomials of egrees at least 2. Moreover, ht J 0 = ht JS e 0 ht J e 2 by the above. It follows that J 0 contains every squarefree monomial of egree. Inee if x x x i is not in J 0 then J 0 cannot contain a squarefree monomial not ivisible by x i. Thus x i ivies every squarefree monomial in J 0 an hence every monomial in J 0. This forces J 0 to have height at most one, a contraiction. Therefore I 0 contains every monomial of egree. As m 0 I 0, Theorem 2. now shows that I 0 m0 cannot be licci. Lemma 2.5. Let S = k[x ; : : : ; x ] be a polynomial ring over a el k an let I be an m- primary monomial ieal. If I # = x B K for some monomial x B = x b x b an a monomial

5 LIAISON OF MONOMIAL IDEALS 5 ieal K with 0 6= K 6= S, then the ieal I 0 = (x a b ; : : : ; x a b ) + K is obtaine from I by a ouble link ene by the monomial regular sequences x a ; : : : ; xa an x a b ; : : : ; x a b. Proof. It suces to prove that ( x a ; : : : ; xa ) : I = (xa b ; : : : ; x a b ) : I 0. This follows from the chain of equalities ( x a ; : : : ; xa ) : I = (xa ; : : : ; xa ) : xb K = ((x a ; : : : ; xa ) : x B ) : K = (x a b ; : : : ; x a b ) : K = (x a b ; : : : ; x a b ) : I 0 : Let S = k[x ; : : : ; x ] be a polynomial ring over a el k, let I be an m-primary monomial ieal, an consier the stanar form I = (x a ; : : : ; xa ) + I #. We set I f0g = I. If I is not a complete intersection then I # can be written uniquely as I # = x B K, where x B = x b x b is a monomial an K a monomial ieal of height at least two. We ene I fg = (x a b ; : : : ; x a b ) + K: If on the other han I is a complete intersection we set I fg = S. For n > we ene inuctively I fng = (I fn g ) fg provie I fn g 6= S. Observe that the representation of I fg 6= S above may not be in stanar form since K coul contain a pure power of a variable; in fact this happens exactly when (I fg ) < (I). Also notice that accoring to Lemma 2.5, the ieals I an I fng 6= S are linke by a sequence of 2 n links ene by monomial regular sequences. If A R 0 is a nite set of points we can ene a set A fg R 0 obtaine from A by removing the points on the coorinate axes an then translating the remaining set until each coorinate hyperplane contains a point of the set. Iterating one enes A fmg. The set A = fcg of exponents of the minimal monomial generators x C of I can be reuce to the empty set by this proceure, i.e., A fmg = ; for some m if an only if I fng = S for some n. It is this conition that characterizes the licci property: Theorem 2.6. Let S = k[x ; : : : ; x ] be a polynomial ring over a el k an let I be an m-primary monomial ieal. The following conitions are equivalent : () I can be linke to a complete intersection by a sequence of links ene by monomial regular sequences. (2) I can be linke to a complete intersection by a sequence of links ene by homogeneous regular sequences. (3) I m is licci in S m. (4) (I fng ) # has height at most one whenever I fng 6= S. (5) I fng = S for some n. Proof. () ) (2) ) (3): The implications are obvious. (3) ) (4): Accoring to Lemma 2.5 the ieals ( I fng ) m are licci as well. Now apply Theorem 2.4. (4) ) (5): Write for the sum of the egrees of homogeneous minimal generators of a homogeneous ieal. We use inuction on (I). We may assume that I fg 6= S. Since I # has height at most one it follows that x B 6= in the enition of I fg. Therefore (I fg ) < (I) an we may apply the inuction hypothesis to I fg.

6 6 CRAIG HUNEKE AND BERND ULRICH (5) ) (): By Lemma 2.5 we may replace I by I fn g to assume that I fg = S. But then I is an almost complete intersection an hence linke to a complete intersection by a monomial regular sequence. Notice that Theorem 2.6 immeiately implies the well known fact that I is licci if 2, as can be seen from conition (4). Corollary 2.7. Let S = k[x ; : : : ; x ] be a polynomial ring over a el k an let I an L be m-primary monomial ieals with I # = L #. Then I is licci if an only if L is licci. Proof. The ieals I fg an L fg have the same monomial minimal generators except possibly for the pure powers. Hence ( I fg ) # = (L fg ) #, an inuctively (I fng ) # = (L fng ) # whenever either ieal is ene. Now use conition (4) of Theorem 2.6. Remark 2.8. Let S = k[x ; : : : ; x ] be a polynomial ring over a el k an let I be an m-primary ieal generate by at most 5 monomials. Then I is licci. Inee since I is licci if it is an almost complete intersection or if 2, we may assume that = 3. Hence I # is generate by at most two monomials, an the same is true for ( I fng ) # as long as I fng 6= S. The two monomials generating ( I fng ) # must have a common factor 6=. Therefore conition (4) of Theorem 2.6 implies that I is licci. Discussion 2.9. Let S = k[x ; : : : ; x ] be a polynomial ring over a el k an let I be an m- primary monomial ieal. If I is licci then Theorem 2.6 an Lemma 2.5 show that the sequence of ouble links I; I fg ; : : : ; I fng ; : : : leas to a monomial almost complete intersection I fng, which is either a complete intersection or irectly linke to a complete intersection. Part (4) of Theorem 2.6 provies an algorithm for eciing when an m-primary monomial is licci. Furthermore, the following statements hol: () I I fg : : : I fng : : :. (2) The chain of ieals in () stabilizes at S if an only if I is licci. (3) The sequence of ouble links I; I fg ; : : : ; I fng ; : : : is the unique sequence of ouble links ene by using monomial regular sequences of smallest possible egrees at every step. Part () is obvious from the enition an (2) follows from Theorem 2.6. To see (3) we may assume that I is not a complete intersection. Consier the stanar form I = (x a ; : : : ; xa )+I #, an write I # = x B K with x B = x b x b a monomial an K a monomial ieal of height at least two. Accoring to Lemma 2.5 it suces to show that x a b ; : : : ; x a b is the monomial regular sequence of minimal egrees in ( x a ; : : : ; xa ) : I. Suppose without loss of generality that x n 2 (x a ; : : : ; xa ) : I for some n a. It follows that x B K (x a n ; x a 2 2 ; : : : ; xa ). However, no minimal monomial generator of xb K lies in (x a 2 2 ; : : : ; xa ) by enition of I # = x B K. We conclue that x B K (x a n ), hence x B 2 (x a n ) as K has height at least two. Therefore b a n, which gives n a b. This proves part (3). Since Theorem 2.6 gives a complete characterization of when an m-primary monomial ieal is licci, a natural question becomes the following: when is an arbitrary m-primary ieal in the

7 LIAISON OF MONOMIAL IDEALS 7 same linkage class as a monomial ieal? Obviously this is a broaer class than licci ieals, but we o not know any criterion for an ieal to be in the linkage class of a monomial ieal. However, there is an obstruction: if S = k[x ; : : : ; x ] is a polynomial ring over a el k an I a Cohen-Macaulay ieal in the linkage class of a monomial ieal, then ( S=I) m = T =(x), where T is a reuce local ring an x is regular on T. 3. Strong Liaison In this section we consier the consequences of Theorem 2.6 for non-monomial ieals. In particular we stuy the relationship between an ieal being licci an its initial ieal in some term orer being licci. Fix a polynomial ring S = k[x ; : : : ; x ] an a term orer >. When I is an ieal in S, we let in(i) enote the initial ieal of I with respect to >. We call a sequence of elements f ; : : : ; f n super regular if in(f ); : : : ; in(f n ) form a regular sequence. When f ; : : : ; f n generate a zero-imensional ieal, this conition means that after possibly reorering f ; :::; f n, one has in(f i ) = x a i i for some positive integers a ; : : : ; a n. We say I is strongly licci if I can be linke to an ieal generate by a super regular sequence in such a way that all the regular sequences use in the chain of links are super regular. Notice that in this language Theorem 2.6 implies that an m-primary licci monomial ieal is strongly licci. Finally, observe that a super regular sequence is a Grobner basis for the ieal it generates [, 5.5]. Lemma 3.. Let S = k[x ; : : : ; x ] with a xe term orer >, an let I an J be two zeroimensional ieals linke via the super regular sequence f ; : : : ; f. Then in(i) an in(j) are linke via the regular sequence in(f ); : : : ; in(f ). Proof. We rst prove that in( I) in(j) (in(f ); : : : ; in(f )). Let in(f) 2 in(i) an in(g) 2 in(j) for elements f 2 I an g 2 J. Then in(f) in(g) = in(fg) 2 in((f ; : : : ; f )) = (in(f ); : : : ; in(f )), where the last equality follows from the fact that f ; : : : ; f are a Grobner basis for (f ; : : : ; f ). Hence in(j) (in(f ); : : : ; in(f )) : in(i). To prove equality, it suces to show that im k (S= in(j)) = im k S=((in(f ); : : : ; in(f )) : in(i)). But im k (S= in(j)) = im k (S=J ) = im k (S=(f ; : : : ; f )) im k (S=I) = im k (S= in((f ; : : : ; f ))) im k (S= in(i)) = im k (S=((in(f ); : : : ; in(f )) : in(i))). Theorem 3.2. Let S = k[x ; : : : ; x ] with a xe term orer >, an let I be a zeroimensional ieal. Then in(i) is licci if an only if I is strongly licci. Proof. Lemma 3. immeiately implies that if I is strongly licci then in( I) is licci. For the proof of the converse suppose that in( I) is licci. Theorem 2.6 shows that in this case in( I) can be linke to a complete intersection by a sequence of links only using monomial regular sequences. Let x a ; : : : ; xa be the rst such regular sequence containe in in( I). Choose f i 2 I such that in(f i ) = x a i. By construction i f ; : : : ; f is a super regular sequence in I. Setting J = (f ; : : : ; f ) : I, we use Lemma 3. to conclue that in( J) is the link of in(i)

8 8 CRAIG HUNEKE AND BERND ULRICH with respect to (x a ; : : : ; xa ). Inucting on the least number of links neee to link in( I) to a complete intersection via monomial linkage completes the proof. It is worth remarking that the conclusion that I is strongly licci if in( I) is licci relies on Theorem 2.6 an oes not follow irectly from stanar techniques of \Grobner eformation". Corollary 3.3. Let S = k[x; y] an let I be a zero-imensional ieal. Then licci with respect to every term orer. I is strongly Proof. It suces to prove in( I) is licci, which is well-known (see also the remark after Theorem 2.6). Remark 3.4. Notice that Theorems 2.6 an 3.2 give a complete characterization for when a zero-imensional ieal is strongly licci with respect to a given term orer. However, one might want to change either the variables or the term orer as the next example shows. Example 3.5. Let S = k[x; y; z] an I = (x 2 + y 2 ; y 2 + z 2 ; xy; xz; yz). Use the term orer revlex with the variables orere x > y > z. The initial ieal is in(i) = (x 2 ; y 2 ; xy; xz; yz; z 3 ) which is not licci since (in( I)) # has height two. Hence I is not strongly licci with respect to this orer. However, I is licci as it is a height three Gorenstein ieal. On the other han, one has I = ((x y) 2 ; (y z) 2 ; xy; (x y)z; yz), an changing variables to x 0 = x y, y 0 = y z an z 0 = z allows us to rewrite the ieal I = ((x 0 ) 2 ; (y 0 ) 2 ; (z 0 ) 2 x 0 y 0 ; x 0 z 0 ; x 0 y 0 + y 0 z 0 ). In revlex orer with z 0 > y 0 > x 0 these generators form a Grobner basis an the initial ieal is (( x 0 ) 2 ; (y 0 ) 2 ; (z 0 ) 2 ; x 0 z 0 ; y 0 z 0 ). This is a licci monomial ieal. The above theorem an example raise the question of whether or not zero-imensional licci ieals have licci initial ieals with respect to some term orer if in aition we allow a change of variables. Equivalently, are zero-imensional licci ieals strongly licci after a suitable change of variables an choice of term orer? This seems unlikely. A weaker question is whether there exists a monomial licci ieal with the same Hilbert function as any given m-primary homogeneous ieal I linke to a complete intersection by a sequence of links ene by homogeneous regular sequences. There is an \obvious" way to try to construct such a monomial ieal: starting with a sequence of links from I to (x ; : : : ; x ), simply go backwars by always using monomial regular sequences of the same egrees as the homogeneous regular sequences in the original linking sequence. The problem with this iea is that there may not be the appropriate pure powers in the monomial ieals obtaine via this algorithm. It is an interesting question whether or not the appropriate pure powers woul actually exist. 4. Glicci Mononial Ieals The next proposition can be foun in [5, 5.0] in the grae case; the local case given below follows easily from the same proof. We give an easy proof for the benet of the reaer.

9 LIAISON OF MONOMIAL IDEALS 9 Proposition 4.. Let (R; m) be a local Gorenstein ring, J an K proper ieals of R, an x 2 m. Assume that the ring R=J is Cohen-Macaulay an generically Gorenstein an that the ieal J + xk is unmixe of height greater than ht J. Then J + xk an J + K are Gorenstein oubly linke. Proof. We write for images in R = R=J. Notice that x is a regular element on R an that K is an unmixe ieal of height one. Since the ring R is generically Gorenstein it has a canonical ieal, meaning an ieal! of positive height that is a canonical moule of R. Multiplying with an R-regular element we may assume that! K. Let H be an ieal in S with J H J + K an H =!. Since! an x! are canonical ieals, it follows that both R=H = R=! an R=J + xh = R=x! are Gorenstein rings. The element x being R-regular one also has x! : xk =! : K in R. Therefore back in R we obtain (J + xh) : (J + xk) = H : (J + K). As J + xh J + xk an H J + K an all four ieals are unmixe of the same height, we conclue that J + xk an J + K are Gorenstein oubly linke. The proof of the next theorem was inspire by the work of Migliore an Nagel in [7, 3.5], where they prove that strongly stable Cohen-Macaulay monomial ieals are glicci in the grae sense, i.e., using only homogeneous Gorenstein ieals in the links. The problem for such monomial ieals reuces at once to the m-primary case. Their proof can be generalize as follows: Theorem 4.2. Let S = k[x ; : : : ; x ] be a polynomial ring over an innite el k an let I be an m-primary monomial ieal. Then I m is glicci in S m. Proof. Write x = x. We use inuction on a, the smallest integer so that x a 2 I. If a = we are one by inuction on. Otherwise we may write I = I 0 S + xk with I 0 = I \ k[x ; : : : ; x ] an K a proper monomial ieal in S. By inuction on a it will suce to prove that I m an (I 0 S + K) m are Gorenstein oubly linke. To this en we wish to apply Proposition 4.. Thus write y = x a an let f j jj 2 Ng be a sequence of pairwise istinct elements in k. Similar to the proof of Theorem 2.4 we ene a map from the set of monomials in k[x ; : : : ; x ] to S. For i we set (x n i ) = an we exten this enition multiplicatively, ny j= (x i + j y); (x c x c ) = (xc ) (x c ): Finally, we ene J to be the ieal in S generate by the images ( x C ) of the monomials x C in I 0. Obviously I 0 S + ys = J + ys. Since y 2 xk it then follows that I = I 0 S + xk = J + xk an I 0 S +K = J +K. Hence it suces to prove that ( J +xk) m an (J +K) m are Gorenstein oubly linke. As in the proof of Theorem 2.4 one sees that the element y is regular on S=J. In particular S m =J m is a one-imensional Cohen-Macaulay ring. In view of Proposition 4.

10 0 CRAIG HUNEKE AND BERND ULRICH is remains to show that this ring is reuce, hence generically Gorenstein. Thus let p be a minimal prime of J. Let n be an integer so that x n i 2 I 0 for i, an consier the proucts containe in J, J 3 (x n i ) = ny j= (x i + j y): For every i, the ieal p contains at most one factor x i + j(i) y, because ; : : : ; n are pairwise istinct elements of k an p oes not contain y. Therefore in the ring S p we obtain J p ( (x n ); : : : ; (x n )) p = (x + j() y; : : : ; x + j( ) y) p : This shows that inee J p = p p. The special type of Gorenstein ouble linkage that arises in Proposition 4. has been ubbe Gorenstein biliaison. In this language the proof of Theorem 4.2 gives the stronger statement that I m is linke to a complete intersection in S m by a sequence of Gorenstein biliaisons. References. D. Eisenbu. Commutative Algebra with a View Towar Algebraic Geometry, Grauate Texts in Math., vol. 50, Springer, New York, C. Huneke an B. Ulrich. Divisor class groups an eformations, Amer. J. Math. 07 (985), C. Huneke an B. Ulrich. The structure of linkage, Annals Math. 26 (987), C. Huneke an B. Ulrich. Algebraic linkage, Duke Math. J. 56 (988), J. Kleppe, J. Migliore, R. Miro-Roig, U. Nagel, an C. Peterson. Gorenstein liaison, complete intersection liaison invariants an unobstructeness, Mem. Amer. Math. Soc. 732 (200). 6. J. Migliore. Introuction to Liaison Theory an Deciency Moules, Progress in Math., vol. 65, Birkhauser, Boston, J. Migliore an U. Nagel. Monomial ieals an the Gorenstein liaison class of a complete intersection, Compositio Math. 33 (2002), C. Peskine an L. Szpiro. Liaison es varietes algebriques, I, Invent. Math. 26 (974), 27{302. Department of Mathematics, University of Kansas, Lawrence, KS aress : huneke@math.ku.eu URL: Department of Mathematics, Purue University, West Lafayette, IN aress : ulrich@math.purue.eu URL: