- Email: [email protected]
On Certain Algebras of Reduction Number One
206, 113]128 Ž1998. JA977414
JOURNAL OF ALGEBRA ARTICLE NO.
On Certain Algebras of Reduction Number One Margherita BarileU Dipartimento di Matematica, Uni¨ ersita ` degli Studi di Bari, Via Orabona 4, 70125, Bari, Italy
and Marcel Morales Uni¨ ersite´ de Grenoble I, Institut Fourier, Laboratoire de Mathematiques associe´ au CNRS, ´ URA 188, B.P. 74, 38402 Saint-Martin D’Heres ` Cedex, and IUFM de Lyon, 5 rue Anselme, 69317 Lyon Cedex, France Communicated by Craig Huneke Received April 1, 1997
INTRODUCTION In this paper we consider a large class of coordinate rings of certain unions of projective scrolls. This class appears in several recent works studying properties of the special fibre F Ž I . of an ideal I in a local ring w5]7x. We prove that the reduction number of these algebras is always equal to one w15x. We also prove that these reduced algebras are Cohen]Macaulay of minimal degree, so that the above assertion also follows from a theorem by w4x. Our methods, however, are constructive, and we can explicitly describe a Noether subalgebra. This can be used to find explicit reductions for the ideal I Žsee Examples 3.7.. As an application we describe the special fibre F Ž I . when I is the defining ideal of a projective monomial variety of codimension 2, and prove a conjecture contained in w5, 6x: we show that the Rees algebra of I is defined by relations of degree two at most. * Supported by an Institutional Fellowship from the Commission of the European Communities, Contract ERBCHBG CT 94-0540. 113 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
114
BARILE AND MORALES
1. THE IDEAL ASSOCIATED TO A BARRED MATRIX Let K be an algebraically closed field. Let T s Ta <1 F a F n4 be a set of variables over K. We consider the barred matrix N s Ž B1 < B2 < . . . < Bs1 5 Bs1q1 < . . . < Bs 2 5 . . . 5 Bs ry 1q1 < . . . < Bs r . , where for all n s 1, . . . , sr , Bn s
ž
Tinq1
Tinq2
???
???
Tinq1
Tinq2
Tinq3
???
Tinq1
Tjn
/
.
We call Bn the n th small block of N . For all i s 1, . . . , r, the submatrix Bi s Ž Bs iy 1q1 < Bs iy 1q2 < . . . < Bs i . will be called the ith big block of N . We suppose that different indices correspond to different variables, and that the entires of each big block are pairwise distinct. Moreover we assume that the indices jn are pairwise distinct and for all n s 1, . . . , sr , the index jn verifies one of the following two conditions: Ži. Tj does not appear anywhere else in the matrix N , or n Žii. there exists a unique m , n - m F sr , such that Bn and Bm belong to different big blocks and Tjn s Tim q1. In other words, all the entries of N appear one time only, except for the last entry of the second row of each small block, which can appear a second time as the first entry of the first row of a small block belonging to one of the following big blocks. Let J be the ideal of K w T x generated by the 2 = 2-minors of every big block of N and by every product TaTb , where Ta is in the first row of Bi , and Tb is in the second row of Bj for some indices i and j, i - j. The latter will be called trans¨ ersal products. We shall say that J is the ideal associated to the barred matrix N . We introduce the following notation. For all i let Mi be the set of 2 = 2-minors of the big block Bi . Let Pi be the set of entries of the first rows of the big blocks B1 , . . . , Biy1 and Di the set of entries of the second rows of the big blocks Biq1 , . . . , Br . Set P1 s B and Dr s B. The following result generalizes Proposition 5.1 in w7x. The technique used in the proof is similar. PROPOSITION 1.1.
For all i, 1 F i F r, let Ji s Ž Mi , Pi , Di .. Then r
Js
F Ji is1
is a primary decomposition of the ideal J.
ALGEBRAS OF REDUCTION NUMBER ONE
115
Proof. The inclusion : is trivial. We prove the other inclusion. We use induction on i to show that every r
fg
F Ji is1
can be written in the form f s fi q di ,
where f i g J , d i g Ž Di . ,
for all i, 1 F i F r. Taking i s r then yields the required result. Let i s 1. Since f g J1 , we have that f s f 1 q d1 ,
for some f 1 g Ž M1 . : J , d1 g Ž D 1 . .
Now let i G 2, and suppose that f s f iy1 q d iy1
for some f iy1 g J , d iy1 g Ž Diy1 . .
Ž 1.
Since f g Ji , and f iy1 g Ji , one has that d iy1 g Ji , whence d iy1 s m i q
Ý
lT T q
TgPi
Ý
gT T ,
TgD i
for some m i g Ž Mi . , l T , g T g K w T x . lXT
lYT ,
lXT
Ž 2.
lYT
One can write l T s q where g Ž Diy1 ., where does not contain any of the variables of Diy1. Since Mi : Ž Diy1 ., Di : Diy1 , Eq. Ž2. implies that ÝT g P i lYT T g Ž Diy1 .. But by property Žb. one has that Pi l Di s B, so that ÝT g P i lYT T s 0. On the other hand by definition
Ý
lXT T g J.
TgPi
Thus f i s m i q ÝT g P i lXT T and d i s ÝT g D i g T T provide the required decomposition of f. For all i let c i be the number of columns of Bi . One has that codim Ž Mi . s c i y 1. For the generalities on rational normal scrolls see w4x or w8x. Remark 1.2. Note that the ideal Ž Mi . defines a scroll not in P n, but in its space of immersion. The latter is the linear subspace of P n defined by the vanishing of the variables in Pi j Di . We are going to show that J has minimal degree. We now give some immediate properties of the sets Pi and Di , which will be useful in the proofs of the next results.
116
BARILE AND MORALES
Ža. One has that B s P1 : P2 : ??? : Pr , and D 1 = D 2 = ??? = Dr s B. Žb. Pi l Di s B for all i s 1, . . . , r. Žc. < Piq1 < s Ýijs1 c j , and < Di < s Ý rjsiq1 c i , for all i s 1, . . . , r y 1. Žd. Mi : Ž Diy1 ., and Miy1 : Ž Pi . for all i s 2, . . . , r. COROLLARY 1.3.
Let c be the number of columns of N. Then codim Ji s c y 1
for all i s 1, . . . , c. In particular J is of pure codimension c y 1. Proof. By properties Žb. and Žc. one has the Pi j Di is a set of c y c i pairwise distinct variables. By construction of N none of these appears in the minors lying in Mi . Thus codim Ji s c i y 1 q c y c i s c y 1 for all i s 1, . . . , r. PROPOSITION 1.4.
Let J be the ideal defined abo¨ e. Then deg J s codim J q 1.
Proof. The degree of a rational normal scroll X is equal to the number of columns of the associated matrix. The degree is the same for all cylinders over X. Hence deg Ji s c i for all i s 1, . . . , r, so that r
deg J s
r
Ý deg Ji s
Ý c i s c.
is1
is1
The claim then follows from Corollary 1.3. Next we prove that J is connected in codimension 1. According to the definition given by Hartshorne w9x, for an equidimensional ideal this property follows from the condition ky1
codim Jk q
ž
F Ji is1
/
s codim Jk q 1
for k s 1, . . . , r .
Ž ).
PROPOSITION 1.5. The prime decomposition of J ¨ erifies condition Ž).. Proof. Let k g 1, . . . , r 4 . We show that ky1
Jk q
F Ji s Ž Pk , Dky1 . . is1
ALGEBRAS OF REDUCTION NUMBER ONE
117
Let
˜ky 1 s Ž B1 5 . . . 5 Bky1 . . B for all i s 1, . . . , k y 1 let J˜i s Ž Mi , Pi , D iq1 , . . . , D ky1 . , i.e., the intersection of Ji with the subring generated by the entries of ˜ky 1. Note that B Ji s J˜i q Ž D ky1 .
for i s 1, . . . , k y 1.
˜ky 1 one deduces that the intersection Applying Proposition 1.1 to B ky 1 ˜ J˜s F is1 J1 is generated by M1 , . . . , Mky1 and the transversal products of ˜ky 1. In particular J˜: Ž Pk .. It follows that B ky1
Jk q
F Ji s Ž Mk , Pk , Dk . q J˜q Ž Dky1 . : Ž Pk , Dky1 . , is1
where we used properties Ža. and Žd.. Since the opposite inclusion is obvious, this suffices to conclude. The following definition is due to Vasconcelos: DEFINITION 1.6 w16, Definition 1x. Let R be a finitely generated standard algebra over K. Let A s K w z1 , . . . , z l x ¨ R be a Noether normalization of R, and assume that all the z i are of degree 1. Let b1 , b 2 , . . . , bs 4 be a minimal set of homogeneous generators of R as an A-module. The number rA Ž R . s max deg bi < i s 1, . . . , s 4 is called the reduction number of R with respect to A. The minimum of rAŽ R . taken over all possible Noether normalizations A of R is called the Ž absolute. reduction number of R. From Propositions 1.4, 1.5, and w4, Theorem 4.2x it follows PROPOSITION 1.7. Let N be a barred matrix whose set of entries is a system T of ¨ ariables o¨ er the field K. Let J be the ideal of K w T x associated to N . Then K w T x is Cohen]Macaulay and r Ž K w T x rJ . s 1.
118
BARILE AND MORALES
2. THE NOETHER SUBALGEBRA In the sequel we present an explicit construction of a Noether normalization of K w T xrJ. First of all we introduce the barred matrix N 9 obtained by replacing the n th small block of N with the column Tinq1
ž / Tjn
.
We consider all sequences of the form Tin
1
Ž 1 F n 1 - n 2 - ??? - ns F r .
q1 , Tjn 1 , Tjn 2 , . . . , Tjn s
which verify the following conditions: Ži. Tj s Ti , for all k s 1, . . . , s, nk n kq 1 y 1 q1 Žii. the sequence is maximal with respect to Ži.. Note that Tj1 is the only entry of N that possibly does not appear in any of the above sequences. All the others occur one time exactly. We form the sum of the elements of each sequence: ns
t s Tin
q1 1
q
Ý
is n 1
Tj i .
Let L be the set whose elements are all these sums, Tj1 if it does not appear in any of these sums, and all variables not appearing in N . EXAMPLE 2.1. In the ring K w T1 , . . . , T11 x consider the barred matrix Ns
ž
T1 T2
T2 T3
T3 T4
T5 T6
T6 T7
T7 T8
T7 T8
T9 . T10
T9 . T10
/
Then N9 s
ž
T1 T4
T5 T7
/
In this case the elements of L are T1 q T7 q T10 , T5 q T8 , T4 , T9 , T11 . Remarks and Notations. Ž1. For every variable Ta appearing in N 9 we shall denote: }by t ŽTa . the Žunique. element of L containing Ta ; }by t ŽTa .y the sum of all terms of t ŽTa . that precede Ta and appear in the same big block of N ;
ALGEBRAS OF REDUCTION NUMBER ONE
119
}by t ŽTa .yy the sum of the remaining terms of t ŽTa . preceding Ta . In a similar way we define t ŽTa .q and t ŽTa .qq. Ž2. If a variable Ta only appears above Žbelow. in N , then it also appears in N 9, and it is the first Žlast. term of t ŽTa .. This follows from condition Žii.. Each of the remaining terms of t ŽTa . appears one time below and one time above. The variables not appearing in N 9 are those appearing one time below and one time above in the same small block of N . Ž3. Each of t ŽTa .y and t ŽTa .q contains one summand at most. More precisely we have that } t ŽTa .y/ 0 if and only if Ta s Tjn for some n , and the n th small block is not the first left in a big block; } t ŽTa .q/ 0 if and only if Ta s Tinq1 for some n , and the n th small block is not the last right in a big block. We are now ready to prove our main result: THEOREM 2.2. Let A be the sub-K-algebra of K w T xrJ generated by the images mod J of the elements of L . Then the image mod J of the set of ¨ ariables T generates K w T xrJ as an A-module. Proof. For the sake of simplicity we shall keep the notation Ta for the image of Ta mod J. This will not cause any confusion, since in this proof all equalities will be written in K w T xrJ. It suffices to show that, for all entries Ta and Tb of N , the product TaTb is an element of Ý i ATi . We shall throughout suppose that there exist two indices m , n such that Ta appears in the m th column, Tb appears in the n th column of N , and m F n . We distinguish between several cases. For the sake of clearness, we structure our proof according to the top-down numbering. Ž1. Ta and Tb belong to different big blocks. In this case we shall say that Tb is right to Ta or that Ta is left to Tb . Ž1.1. Ta only appears below in N . Then Ta also appears in N 9 and t ŽTa .qs t ŽTa .qqs 0. Ž1.1.1. Tb appears below in N . Since every term of t ŽTa .yy and y t ŽTa . appears above in N 9, and is left to Tb , we have that t ŽTa .yy Tb s t ŽTa .y Tb s 0. Hence TaTb s t Ž Ta . Tb . Ž1.1.2. Tb only appears above in N . Then Tb certainly appears in N 9, and t ŽTb .yys t ŽTb .ys 0. Thus TaTb s t Ž Tb . Ta y t Ž Tb .
qq
q
Ta y t Ž Tb . Ta .
120
BARILE AND MORALES
Since every term in t ŽTb .qqq and t ŽTb .q appears below, and is right to Ta , we can apply the result in Ž1.1.1. to the last two summands. Ž1.2. Ta appears above in N . Ž1.2.1. Tb appears below in N . Then TaTb s 0. Ž1.2.2. Tb only appears above in N . Then Tb certainly appears in N 9, and t ŽTb .yys t ŽTb .ys 0. Since all terms of t ŽTb .q and t ŽTb .qq appear below, and are right to Ta , one has that Tat ŽTb .qs Tat ŽTb .qqs 0, hence TaTb s Tat Ž Tb . . Ž2. Ta and Tb belong to the same big block B, but to different small blocks. The subcases Ž2.1., Ž2.2., and Ž2.3. correspond to three different steps of an algorithm. At each step we replace Ta and Tb by new entries Ta and Tb , respectively, which also verify the assumption Ž2.. Let s be the number of small blocks in B. Suppose Ta and Tb appear in the a th and in the b th small block of B, respectively Ž a - b .. Let a and b be the indices of the small blocks of B occupied by Ta and Tb , respectively. Ž2.1. Ta appears above and Tb appears below in B. Consider the following submatrix of B:
ž
. . . Ta TaX TaY . . . TaX TaY . . .
... ...
... Tc
... ...
Td ...
. . . T b Y T bX T b Y T bX T b
... ...
... ...
/
.
One has that 0s
Ta TaX
T bX s TaTb y TaX TbX , Tb
and 0s
TaX TaY
T bY s TaX TbX y TaY TbY . T bX
Finally TaTb s TaY TbY . Hence the problem is reduced to finding the required presentation for TaY TbY . Take Ta s TaY , and Tb s TbY . If TaY s Tc apply Ž2.2.; if TbY s Td , apply Ž2.3.. Otherwise apply Ž2.1. once again. Ž2.2. Ta only appears below in B. In this case Ta appears in N 9 and t ŽTa .qs 0. Hence TaTb s t Ž Ta . Tb y t Ž Ta .
yy
Tb y t Ž Ta .
qq
y
Tb y t Ž Ta . Tb .
121
ALGEBRAS OF REDUCTION NUMBER ONE
Note that all terms of t ŽTa .yy and t ŽTa .qq appear in a big block different from B. Hence the required representation for the second and third summand can be found according to Ž1.. Thus it suffices to consider the last term. If a s 1, the t ŽTa .ys 0 and we are done. Otherwise take Ta s t ŽTa .y and Tb s Tb . Then a s a y 1, and b s b . Apply Ž2.1. or Ž2.3. to TaTb . This is possible, since t ŽTa .y certainly appears above. Ž2.3. Tb only appears above in B. An argument similar to the one developed in Ž2.2. permits us to reduce the problem to the product Tat ŽTb .q. If b s s, then t ŽTb .qs 0 and we are done. Otherwise take Ta s Ta and Tb s t ŽTb .q. Then a s a , and b s b q 1. Apply Ž2.1. or Ž2.2. to TaTb . It is clear that the above algorithm stops after a finite number of steps, when either Ta is the leftmost entry of the upper row of B, or Tb is the rightmost entry of the lower row of B. Ž3. Ta and Tb appear in the same small block B of N . Ž3.1. Ta or Tb appears in N 9. Then TaTb s t Ž Ta . Tb y t Ž Ta .
yy
Tb y t Ž Ta .
qq
y
q
Tb y t Ž Ta . Tb y t Ž Ta . Tb .
The four last terms are sums of products TaX Tb , where TaX appears in a different small block with respect to Tb . Hence Ž1. or Ž2. can be applied to each of these products. Ž3.2. Ta and Tb do not appear in N 9. First suppose Ta s Tb . Note that Ta is not the leftmost element of the first row of B. Hence the block B has the following minor: 0s
TaX Ta
Ta s TaX TaY y Ta2 . TaY
Thus we may assume Ta / Tb . Then the block B is of one of the following forms: Bs
Ž a.
ž
... ...
TaX Ta
Ta Tb
Tb T bX
... ...
/
or
Ž b.
Bs
ž
Tc ...
... TaY
TaY Ta
Ta ...
... Tb
Tb T bY
T bY ...
... . Td
/
In case Ža. it holds TaTb s TaX TbX . If TaX is in the first column, then TaX appears in N 9, and we are back in Ž3.1.. The same is true if TbX is in the last column. Otherwise we are in the case Žb.. In this case TaTb s TaY TbY .
122
BARILE AND MORALES
Reapply this identity successively to move the first factor to the left, the second to the right. After a finite number of steps we end up with a product where the first factor is Tc , or the second is Td . Since Tc and Td both appear in N 9, we are back in Ž3.1.. COROLLARY 2.3. If d is the cardinality of L , then d s n y c q 1 and dim K w T xrJ s d. In particular, A is a Noether normalization of K w T xrJ. Proof. From Corollary 1.3 it follows that dim RrJ s n y c q 1, where n is the number of variables. Thus it suffices to show that d s n y c q 1. Now let V be the set of all variables appearing below in N : its cardinality is obviously equal to the number c of columns. Let V denote its complementary set. Let F be the set of the first terms of the sums in L . By construction it is clear that F s V j Tj1. But the cardinality of F is d. Hence d s n y c q 1.
3. MONOMIAL VARIETIES OF CODIMENSION 2 Theorem 2.2 can be applied to the study of the special fibre of a projective monomial variety of codimension 2: see w5x for a complete and detailed presentation of this subject. We first quote the basic notions and the main results from w6x. Let R be the polynomial ring in n q 2 indeterminates over the field K. Let I be an ideal minimally generated by t elements Fi 41 F iF t . The Rees algebra of I is defined to be the graded ring Rw It x s [nG 0 I n t n, and the M Rw It x, where M denotes the irrelevant special fibre is the quotient Rw It xrM maximal ideal of R. The dimension of the special fibre is called the analytic spread of I and is denoted by l Ž I .. We introduce t independent variables over R, say T s Ti 41 F iF t , and we consider the ideal I s ker Rw T x ª Rw It x ª 0, Ti ª Fi t 4 . We obtain a presentation Rw It x , I of the Rees algebra, from which we can deduce a presentation of Rw T xrI M Rw It x , K w T xrI I˜, where I˜ denotes the image of the special fibre: Rw It xrM w x I modulo M R T . Let us place ourselves in the case where I is the defining ideal of a toric variety admitting the parametrization x 1 s u1a1 ,
x 2 s u 2a 2 , . . . , x n s u na n ,
y s u1c1 u 2c 2 ??? u cnn ,
z s u1b 1 u 2b 2 ??? u nb n ,
ALGEBRAS OF REDUCTION NUMBER ONE
123
where, for all i, 1 F i F n, a i , bi , and c i are nonnegative integers such that a i / 0, Ž bi , c i . / Ž0, 0., and Ž b1 , . . . , bn . / Ž0, . . . , 0., Ž c1 , . . . , c n . / Ž0, . . . , 0.. We call this a monomial ¨ ariety of codimension 2. It is known that in general codimŽ I . F l Ž I ., and equality holds for I a prime ideal if and only if I is a complete intersection. In w6x the number l Ž I . was determined for all defining ideals I of a codimension 2 monomial variety, in spite of the face that no complete description of the ideal I˜ was known yet. The approach, indeed, is indirect: the computations are done on a subideal A˜: I˜, generated in degree two, called a reduced essential ideal. In w11x it is shown that I is generated by the 2 = 2-minors of a 2 = t-matrix. The ideal A˜ is generated by some of the Plucker relations. It ¨ A˜s 3 Žcf. w5, Corollary 5.26.1x.. This immediately turns out that dim Rw T xrA implies the following result: THEOREM 3.1 w6, Theorem 4.2x. The analytic spread of I is equal to two if I is a complete intersection and equal to three in all the remaining cases. 2.1.1 The ideal A˜ is always generated by monomials and binomials, with the exception of a very particular case, that is treated separatedly w5, Proposition 5.18.1x. In this case t s 4 and A˜s Ž T0 T1 y T22 y T32 . : K w T0 , T1 , T2 , T3 x . One easily sees that A˜ K w T0 , T1 , T2 x : K w T0 , T1 , T2 , T3 x rA is a Noether normalization and that the reduction number is 1. In the general case A˜ is the ideal associated to a barred matrix N whose big and small blocks coincide: see w5, 6, 11x for an explicit construction. We shall use our results to prove that in the general case A˜s I˜, i.e., the special fibre of I is entirely generated by the quadratic relations. We need the following preliminary LEMMA 3.2. Let R be a finitely generated K-algebra, J an ideal of R, and w : R ª RrJ the canonical epimorphism. Moreo¨ er let A : R be a Noether normalization of R. If dim R s dim RrJ, then A is a Noether normalization of RrJ with respect to the restriction w < A . Proof. We consider the A-module structure defined on RrJ by w < A . Since w is a finite A-homomorphism, the same is true for w < A . Thus it suffices to prove that w < A is an injection. Now K [ kerŽ w < A . s kerŽ w . l A, and the map w < A induces a finite monomorphism of rings K ª RrJ. ArK
124
BARILE AND MORALES
K .. But the left-hand side is equal to dim R s Thus dimŽ RrJ . s dimŽ ArK dim A. Thus the preceding equality is only possible if K s 0. Let us fix a monomial variety of codimension 2, with defining ideal I, which is not a complete intersection and does not belog to the particular case. We shall refer to the notation just introduced. Proposition 2.3 permits us to determine a Noether normalization A of the quotient Rw T x A˜. Consider the composition of maps A˜ª R w T x rI I˜, A ¨ R w T x rA where the right map is the canonical epimorphism. The equality of dimensions and Lemma 3.2 imply that A is a Noether normalization of I˜ with respect to the composed map. Since Rw T xrA A˜ is generated by Rw T xrI I˜. linear forms as an A-module, the same is true for its quotient Rw T xrI For the rest of this section we consider the local ring R M . The notions of Rees algebra, special fibre, and reduction naturally extend to this local ring. The above results apply to R M and to the localized ideals and subrings. An ideal J is called a reduction of I if JI n s I nq 1 for some nonnegative integer n. Let r J Ž I . denote the least number n such that the above equality holds. Then the Ž absolute. reduction number of I is defined to be the minimum of r J Ž I . taken over all possible reductions J of I. The following result is due to Vasconcelos: PROPOSITION 3.3 w15, Proposition 5.1.3x. Let Ž R, M , K . be a local Noetherian ring with infinite residue field and I an ideal of R of analytic spread l. Suppose A s K w z1 , . . . , z l x ¨ F Ž I . is a Noether normalization of the special fibre F Ž I ., and assume that all the z i are of degree 1. Furthermore let b1 , . . . , bs 4 be a minimal set of homogeneous generators of F Ž I . as an A-module. For all i, . . . , l, let a i g I be a lift of z i . Then J s Ž a1 , . . . , a l . is a Ž minimal . reduction of I and r J Ž I . s max deg bi < i s 1, . . . , s 4 . Putting the above results together one immediately obtains the following COROLLARY 3.4. With respect to the notation introduced in Subsection 2.1.1 it holds that r Ž I M . s 1. Next consider the following result by Cortadellas and Zarzuela: PROPOSITION 3.5 w1, Theorem 4.2 and Corollary 5.7x. Let Ž R, M . be a local Noetherian ring and let I be an ideal of R. Suppose I is generically a
ALGEBRAS OF REDUCTION NUMBER ONE
125
complete intersection, and the analytic spread is l s heightŽ I . q 1. If r Ž I . s 1, then F Ž I . is Cohen]Macaulay. Moreo¨ er the Hilbert function of F Ž I . is HF Ž I . Ž t . s
1 q Žt y l . t
Ž1 y t .
l
,
where t denotes the minimal number of generators of I. More results on the Hilbert functions of Cohen]Macaulay special fibres are contained in the work by D’Cruz, Raghavan, and Verma w3x. Now we are able to conclude: PROPOSITION 3.6. Let I : K w x 1 , . . . , x n , y, z x be the defining ideal of a monomial ¨ ariety of codimension 2. Suppose I is not a complete intersection. Then the reduced essential ideal A˜ coincides with the presentation ideal I˜ of the special fibre F Ž I .. A˜ is Proof. In w5, p. 117x it is proven that the Hilbert function of K w T xrA H A˜Ž t . s
1 q Ž t y 3. t
Ž1 y t .
3
.
A˜ Since l Ž I . s 3, it follows from Proposition 3.5 that F Ž I . and K w T xrA A˜. have the same Hilbert function. But F Ž I . is a quotient of K w T xrA Note that Theorem 2.2 and Proposition 3.3 yield an explicit construction for a minimal reduction of any ideal I M . We perform such a construction in the next examples, where I is throughout an ideal of K w x, y, z, w x defining a projective monomial curve in P 3. EXAMPLES 3.7. Ža. Suppose I is minimally generated by 4 elements. Coudurier and Morales w2x call this a monomial curve of type I. A system of generators is given by X
F2 s y pqn w ryl y x ky r z sqm
F1 s x k w l y y n z m , X
F3 s y pq2 n y x 2 kyr z s w 2 lyr ,
X
F4 s x r y p w r y z sq2 m ,
where all the exponents are supposed to be nonnegative. The ideal I˜ is the ideal associated to the barred matrix Ns
ž
T3 T2
T2 . T4
/
IM is A M s K w t 1 , t 2 , t 3 q t 4 x, Thus a Noether normalization of K w T x MrI where t i denotes the image of the variable Ti via the localization at M . Let J s Ž F1 , F2 , F3 q F4 . .
126
BARILE AND MORALES
According to Proposition 3.3 the ideal J M is a minimal reduction of I M . Moreover note that F12 s F1 Ž F1 q F3 . y F1 F3 , F32 s F3 Ž F1 q F3 . y F1 F3 , and X
F1 F3 s x r w 2 lyr F22 y y p z s F02 . Hence J is a minimal reduction of I note only locally at M , but also globally. In particular we have that r Ž I . s 1. The next example shows that this direct passage from a local to a global reduction is not always possible: in general the generators of the subalgebra A do not yield a global reduction. Žb. Consider the ideal I generated by the following 6 binomials: F1 s y 6 z 38 y x 13 w 31 ,
F2 s y 18 z 25 y x 15 w 28
F3 s y 30 z 12 y x 17 w 25 ,
F4 s y 42 y zx 19 w 22
F5 s z 51 y y 6 x 11 w 34 ,
F6 s y 12 w 3 y x 2 z 13 .
The barred matrix associated to I is Ns
ž
T4 T6
T3 T2
T2 T1
T1 . T5
/
Then N9 s
ž
T4 T6
T3 T1
T1 , T5
/
whence one obtains A M s K w t 1 q t 4 , t 3 q t5 , t6 x as a Noether normalizaI˜M . Let J s Ž F1 q F4 , F3 q F5 , F6 .. Then J M is a reduction tion of K w T x MrI of I M , but J is not a reduction of I. One can show that it even holds RadŽ J . / I. A reduction of I is given by J X s Ž Ž 1 q y 6 . F1 q F4 , F3 q F5 , F6 . . We are not able to give the general form of three elements generating a minimal reduction of the ideal I defining a projective monomial curve. The problem was solved by Morales and Simis Žcf. w13, Proposition 3.1.2x. for projective monomial curves lying on a quadric surface.
ALGEBRAS OF REDUCTION NUMBER ONE
127
It was conjectured in w5x that the ideal of presentation of the Rees algebra Rw It x is generated by forms of degree two at most. According to Micali w12x this is certainly true if I is a complete intersection. Now we are able to answer the question in the general case. The crucial result is due to Huckaba and Huneke. THEOREM 3.8 w10, Theorems 2.9 and 4.5x. Let R be a Cohen]Macaulay local ring and I an ideal ha¨ ing height d G 1 and analytic spread l Ž I . s d q 1. Assume that the minimal primes of RrI all ha¨ e the same height, and the associated primes of RrI ha¨ e height at most d q 1. Assume also that I is generically a complete intersection and there exists a minimal reduction J of I such that r J Ž IQ . F 1 for e¨ ery prime ideal Q = I with codimŽ QrI . s 1. Finally assume that depthŽ RrI . G dimŽ RrI . y 1. Then the presentation ideal of the Rees algebra Rw It x of I is defined by elements of degree two at most. Moreover Peeva and Sturmfels Žcf. w14, Theorem 2.3x. showed that for any toric ideal I it holds that projdimŽ RrI . F 2 codimŽ I . y 1. In particular, if I is the defining ideal of a monomial variety of codimension 2, by Auslander and Buchsbaum it holds that depth Ž RrI . s dim Ž R . y projdimŽ RrI . G dim Ž R . y 3 s dim Ž RrI . y 1. Suppose I is not a complete intersection. In view of Theorem 3.1, the preceding inequality, and Corollary 3.4 the ideal I fulfills all the assumptions of Theorem 3.8. This proves: THEOREM 3.9. Let I : R be the defining ideal of a monomial ¨ ariety of codimension 2. Then the presentation ideal of the Rees algebra Rw It x of I is generated by forms of degree two at most. Note that Gimenez Žcf. w5, Theorem 6.3.1x. already showed Proposition 3.6 under the hypothesis that Theorem 3.9 be true.
ACKNOWLEDGMENTS The authors express their gratitude to Santiago Zarzuela for his helpful comments. The first author is indebted to the Institut Fourier of the University of Grenoble for hospitality and support during the preparation of this paper.
128
BARILE AND MORALES
REFERENCES 1. T. Cortadellas and S. Zarzuela, On the depth of the fiber cone of filtrations, preprint. 2. L. Coudurier and M. Morales, Classification des courbes toriques dans l’espace projectif, module de Rao et liaison, preprint. 3. C. D’Cruz, K. N. Raghavan, and J. K. Verma, Cohen]Macaulay fiber cones, preprint. 4. D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 Ž1984., 89]133. ´ 5. Ph. Gimenez, ‘‘Etude de la fibre speciale de l’eclatement d’une variete ´ ´ ´ ´ monomiale en codimension deux,’’ Dissertation, Univ. of Grenoble, 1993. 6. Ph. Gimenez, M. Morales, and A. Simis, L’analytic spread de l’ideal d’une ´ de definition ´ variete ou ´ egal ` a trois, C. R. ´ ´ monomiale de codimension deux est toujours inferieur ´ Acad. Sci. Paris 319 Ž1994., 703]706. 7. Ph. Gimenez, M. Morales, and A. Simis, The analytic spread of certain toric varieties of codimension two, preprint. 8. J. Harris, ‘‘Algebraic Geometry,’’ Springer-Verlag, New York, 1992. 9. R. Hartshorne, Complete intersections and connectedness, Amer. J. Math. 96 Ž1974., 602]639. 10. S. Huckaba and C. Huneke, Powers of ideals having small analytic deviation, Amer. J. Math. 144 Ž1992., 367]403. ´ 11. M. Morales, Equations des varietes ´ ´ monomiales en codimension deux, J. Algebra 175 Ž1995., 1082]1095. 12. A. Micali, Sur les algebres universelles, Ann. Inst. Fourier 14 Ž1964., 33]88. ` 13. M. Morales and A. Simis, Symbolic powers of monomial curves in P 3 lying on a quadric surface, Comm. Algebra 20 Ž1992., 1109]1121. 14. I. Peeva and B. Sturmfels, Syzygies of codimension 2 lattice ideals, preprint. 15. W. Vasconcelos, ‘‘The Arithmetic of Blowup Algebras,’’ London Math. Soc., Vol. 195, Cambridge Univ. Press, Cambridge, UK, 1994. 16. W. Vasconcelos, The reduction number of an algebra, Compositio Math. 104 Ž1996., 189]197.
JOURNAL OF ALGEBRA ARTICLE NO.
On Certain Algebras of Reduction Number One Margherita BarileU Dipartimento di Matematica, Uni¨ ersita ` degli Studi di Bari, Via Orabona 4, 70125, Bari, Italy
and Marcel Morales Uni¨ ersite´ de Grenoble I, Institut Fourier, Laboratoire de Mathematiques associe´ au CNRS, ´ URA 188, B.P. 74, 38402 Saint-Martin D’Heres ` Cedex, and IUFM de Lyon, 5 rue Anselme, 69317 Lyon Cedex, France Communicated by Craig Huneke Received April 1, 1997
INTRODUCTION In this paper we consider a large class of coordinate rings of certain unions of projective scrolls. This class appears in several recent works studying properties of the special fibre F Ž I . of an ideal I in a local ring w5]7x. We prove that the reduction number of these algebras is always equal to one w15x. We also prove that these reduced algebras are Cohen]Macaulay of minimal degree, so that the above assertion also follows from a theorem by w4x. Our methods, however, are constructive, and we can explicitly describe a Noether subalgebra. This can be used to find explicit reductions for the ideal I Žsee Examples 3.7.. As an application we describe the special fibre F Ž I . when I is the defining ideal of a projective monomial variety of codimension 2, and prove a conjecture contained in w5, 6x: we show that the Rees algebra of I is defined by relations of degree two at most. * Supported by an Institutional Fellowship from the Commission of the European Communities, Contract ERBCHBG CT 94-0540. 113 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
114
BARILE AND MORALES
1. THE IDEAL ASSOCIATED TO A BARRED MATRIX Let K be an algebraically closed field. Let T s Ta <1 F a F n4 be a set of variables over K. We consider the barred matrix N s Ž B1 < B2 < . . . < Bs1 5 Bs1q1 < . . . < Bs 2 5 . . . 5 Bs ry 1q1 < . . . < Bs r . , where for all n s 1, . . . , sr , Bn s
ž
Tinq1
Tinq2
???
???
Tinq1
Tinq2
Tinq3
???
Tinq1
Tjn
/
.
We call Bn the n th small block of N . For all i s 1, . . . , r, the submatrix Bi s Ž Bs iy 1q1 < Bs iy 1q2 < . . . < Bs i . will be called the ith big block of N . We suppose that different indices correspond to different variables, and that the entires of each big block are pairwise distinct. Moreover we assume that the indices jn are pairwise distinct and for all n s 1, . . . , sr , the index jn verifies one of the following two conditions: Ži. Tj does not appear anywhere else in the matrix N , or n Žii. there exists a unique m , n - m F sr , such that Bn and Bm belong to different big blocks and Tjn s Tim q1. In other words, all the entries of N appear one time only, except for the last entry of the second row of each small block, which can appear a second time as the first entry of the first row of a small block belonging to one of the following big blocks. Let J be the ideal of K w T x generated by the 2 = 2-minors of every big block of N and by every product TaTb , where Ta is in the first row of Bi , and Tb is in the second row of Bj for some indices i and j, i - j. The latter will be called trans¨ ersal products. We shall say that J is the ideal associated to the barred matrix N . We introduce the following notation. For all i let Mi be the set of 2 = 2-minors of the big block Bi . Let Pi be the set of entries of the first rows of the big blocks B1 , . . . , Biy1 and Di the set of entries of the second rows of the big blocks Biq1 , . . . , Br . Set P1 s B and Dr s B. The following result generalizes Proposition 5.1 in w7x. The technique used in the proof is similar. PROPOSITION 1.1.
For all i, 1 F i F r, let Ji s Ž Mi , Pi , Di .. Then r
Js
F Ji is1
is a primary decomposition of the ideal J.
ALGEBRAS OF REDUCTION NUMBER ONE
115
Proof. The inclusion : is trivial. We prove the other inclusion. We use induction on i to show that every r
fg
F Ji is1
can be written in the form f s fi q di ,
where f i g J , d i g Ž Di . ,
for all i, 1 F i F r. Taking i s r then yields the required result. Let i s 1. Since f g J1 , we have that f s f 1 q d1 ,
for some f 1 g Ž M1 . : J , d1 g Ž D 1 . .
Now let i G 2, and suppose that f s f iy1 q d iy1
for some f iy1 g J , d iy1 g Ž Diy1 . .
Ž 1.
Since f g Ji , and f iy1 g Ji , one has that d iy1 g Ji , whence d iy1 s m i q
Ý
lT T q
TgPi
Ý
gT T ,
TgD i
for some m i g Ž Mi . , l T , g T g K w T x . lXT
lYT ,
lXT
Ž 2.
lYT
One can write l T s q where g Ž Diy1 ., where does not contain any of the variables of Diy1. Since Mi : Ž Diy1 ., Di : Diy1 , Eq. Ž2. implies that ÝT g P i lYT T g Ž Diy1 .. But by property Žb. one has that Pi l Di s B, so that ÝT g P i lYT T s 0. On the other hand by definition
Ý
lXT T g J.
TgPi
Thus f i s m i q ÝT g P i lXT T and d i s ÝT g D i g T T provide the required decomposition of f. For all i let c i be the number of columns of Bi . One has that codim Ž Mi . s c i y 1. For the generalities on rational normal scrolls see w4x or w8x. Remark 1.2. Note that the ideal Ž Mi . defines a scroll not in P n, but in its space of immersion. The latter is the linear subspace of P n defined by the vanishing of the variables in Pi j Di . We are going to show that J has minimal degree. We now give some immediate properties of the sets Pi and Di , which will be useful in the proofs of the next results.
116
BARILE AND MORALES
Ža. One has that B s P1 : P2 : ??? : Pr , and D 1 = D 2 = ??? = Dr s B. Žb. Pi l Di s B for all i s 1, . . . , r. Žc. < Piq1 < s Ýijs1 c j , and < Di < s Ý rjsiq1 c i , for all i s 1, . . . , r y 1. Žd. Mi : Ž Diy1 ., and Miy1 : Ž Pi . for all i s 2, . . . , r. COROLLARY 1.3.
Let c be the number of columns of N. Then codim Ji s c y 1
for all i s 1, . . . , c. In particular J is of pure codimension c y 1. Proof. By properties Žb. and Žc. one has the Pi j Di is a set of c y c i pairwise distinct variables. By construction of N none of these appears in the minors lying in Mi . Thus codim Ji s c i y 1 q c y c i s c y 1 for all i s 1, . . . , r. PROPOSITION 1.4.
Let J be the ideal defined abo¨ e. Then deg J s codim J q 1.
Proof. The degree of a rational normal scroll X is equal to the number of columns of the associated matrix. The degree is the same for all cylinders over X. Hence deg Ji s c i for all i s 1, . . . , r, so that r
deg J s
r
Ý deg Ji s
Ý c i s c.
is1
is1
The claim then follows from Corollary 1.3. Next we prove that J is connected in codimension 1. According to the definition given by Hartshorne w9x, for an equidimensional ideal this property follows from the condition ky1
codim Jk q
ž
F Ji is1
/
s codim Jk q 1
for k s 1, . . . , r .
Ž ).
PROPOSITION 1.5. The prime decomposition of J ¨ erifies condition Ž).. Proof. Let k g 1, . . . , r 4 . We show that ky1
Jk q
F Ji s Ž Pk , Dky1 . . is1
ALGEBRAS OF REDUCTION NUMBER ONE
117
Let
˜ky 1 s Ž B1 5 . . . 5 Bky1 . . B for all i s 1, . . . , k y 1 let J˜i s Ž Mi , Pi , D iq1 , . . . , D ky1 . , i.e., the intersection of Ji with the subring generated by the entries of ˜ky 1. Note that B Ji s J˜i q Ž D ky1 .
for i s 1, . . . , k y 1.
˜ky 1 one deduces that the intersection Applying Proposition 1.1 to B ky 1 ˜ J˜s F is1 J1 is generated by M1 , . . . , Mky1 and the transversal products of ˜ky 1. In particular J˜: Ž Pk .. It follows that B ky1
Jk q
F Ji s Ž Mk , Pk , Dk . q J˜q Ž Dky1 . : Ž Pk , Dky1 . , is1
where we used properties Ža. and Žd.. Since the opposite inclusion is obvious, this suffices to conclude. The following definition is due to Vasconcelos: DEFINITION 1.6 w16, Definition 1x. Let R be a finitely generated standard algebra over K. Let A s K w z1 , . . . , z l x ¨ R be a Noether normalization of R, and assume that all the z i are of degree 1. Let b1 , b 2 , . . . , bs 4 be a minimal set of homogeneous generators of R as an A-module. The number rA Ž R . s max deg bi < i s 1, . . . , s 4 is called the reduction number of R with respect to A. The minimum of rAŽ R . taken over all possible Noether normalizations A of R is called the Ž absolute. reduction number of R. From Propositions 1.4, 1.5, and w4, Theorem 4.2x it follows PROPOSITION 1.7. Let N be a barred matrix whose set of entries is a system T of ¨ ariables o¨ er the field K. Let J be the ideal of K w T x associated to N . Then K w T x is Cohen]Macaulay and r Ž K w T x rJ . s 1.
118
BARILE AND MORALES
2. THE NOETHER SUBALGEBRA In the sequel we present an explicit construction of a Noether normalization of K w T xrJ. First of all we introduce the barred matrix N 9 obtained by replacing the n th small block of N with the column Tinq1
ž / Tjn
.
We consider all sequences of the form Tin
1
Ž 1 F n 1 - n 2 - ??? - ns F r .
q1 , Tjn 1 , Tjn 2 , . . . , Tjn s
which verify the following conditions: Ži. Tj s Ti , for all k s 1, . . . , s, nk n kq 1 y 1 q1 Žii. the sequence is maximal with respect to Ži.. Note that Tj1 is the only entry of N that possibly does not appear in any of the above sequences. All the others occur one time exactly. We form the sum of the elements of each sequence: ns
t s Tin
q1 1
q
Ý
is n 1
Tj i .
Let L be the set whose elements are all these sums, Tj1 if it does not appear in any of these sums, and all variables not appearing in N . EXAMPLE 2.1. In the ring K w T1 , . . . , T11 x consider the barred matrix Ns
ž
T1 T2
T2 T3
T3 T4
T5 T6
T6 T7
T7 T8
T7 T8
T9 . T10
T9 . T10
/
Then N9 s
ž
T1 T4
T5 T7
/
In this case the elements of L are T1 q T7 q T10 , T5 q T8 , T4 , T9 , T11 . Remarks and Notations. Ž1. For every variable Ta appearing in N 9 we shall denote: }by t ŽTa . the Žunique. element of L containing Ta ; }by t ŽTa .y the sum of all terms of t ŽTa . that precede Ta and appear in the same big block of N ;
ALGEBRAS OF REDUCTION NUMBER ONE
119
}by t ŽTa .yy the sum of the remaining terms of t ŽTa . preceding Ta . In a similar way we define t ŽTa .q and t ŽTa .qq. Ž2. If a variable Ta only appears above Žbelow. in N , then it also appears in N 9, and it is the first Žlast. term of t ŽTa .. This follows from condition Žii.. Each of the remaining terms of t ŽTa . appears one time below and one time above. The variables not appearing in N 9 are those appearing one time below and one time above in the same small block of N . Ž3. Each of t ŽTa .y and t ŽTa .q contains one summand at most. More precisely we have that } t ŽTa .y/ 0 if and only if Ta s Tjn for some n , and the n th small block is not the first left in a big block; } t ŽTa .q/ 0 if and only if Ta s Tinq1 for some n , and the n th small block is not the last right in a big block. We are now ready to prove our main result: THEOREM 2.2. Let A be the sub-K-algebra of K w T xrJ generated by the images mod J of the elements of L . Then the image mod J of the set of ¨ ariables T generates K w T xrJ as an A-module. Proof. For the sake of simplicity we shall keep the notation Ta for the image of Ta mod J. This will not cause any confusion, since in this proof all equalities will be written in K w T xrJ. It suffices to show that, for all entries Ta and Tb of N , the product TaTb is an element of Ý i ATi . We shall throughout suppose that there exist two indices m , n such that Ta appears in the m th column, Tb appears in the n th column of N , and m F n . We distinguish between several cases. For the sake of clearness, we structure our proof according to the top-down numbering. Ž1. Ta and Tb belong to different big blocks. In this case we shall say that Tb is right to Ta or that Ta is left to Tb . Ž1.1. Ta only appears below in N . Then Ta also appears in N 9 and t ŽTa .qs t ŽTa .qqs 0. Ž1.1.1. Tb appears below in N . Since every term of t ŽTa .yy and y t ŽTa . appears above in N 9, and is left to Tb , we have that t ŽTa .yy Tb s t ŽTa .y Tb s 0. Hence TaTb s t Ž Ta . Tb . Ž1.1.2. Tb only appears above in N . Then Tb certainly appears in N 9, and t ŽTb .yys t ŽTb .ys 0. Thus TaTb s t Ž Tb . Ta y t Ž Tb .
q
Ta y t Ž Tb . Ta .
120
BARILE AND MORALES
Since every term in t ŽTb .qqq and t ŽTb .q appears below, and is right to Ta , we can apply the result in Ž1.1.1. to the last two summands. Ž1.2. Ta appears above in N . Ž1.2.1. Tb appears below in N . Then TaTb s 0. Ž1.2.2. Tb only appears above in N . Then Tb certainly appears in N 9, and t ŽTb .yys t ŽTb .ys 0. Since all terms of t ŽTb .q and t ŽTb .qq appear below, and are right to Ta , one has that Tat ŽTb .qs Tat ŽTb .qqs 0, hence TaTb s Tat Ž Tb . . Ž2. Ta and Tb belong to the same big block B, but to different small blocks. The subcases Ž2.1., Ž2.2., and Ž2.3. correspond to three different steps of an algorithm. At each step we replace Ta and Tb by new entries Ta and Tb , respectively, which also verify the assumption Ž2.. Let s be the number of small blocks in B. Suppose Ta and Tb appear in the a th and in the b th small block of B, respectively Ž a - b .. Let a and b be the indices of the small blocks of B occupied by Ta and Tb , respectively. Ž2.1. Ta appears above and Tb appears below in B. Consider the following submatrix of B:
ž
. . . Ta TaX TaY . . . TaX TaY . . .
... ...
... Tc
... ...
Td ...
. . . T b Y T bX T b Y T bX T b
... ...
... ...
/
.
One has that 0s
Ta TaX
T bX s TaTb y TaX TbX , Tb
and 0s
TaX TaY
T bY s TaX TbX y TaY TbY . T bX
Finally TaTb s TaY TbY . Hence the problem is reduced to finding the required presentation for TaY TbY . Take Ta s TaY , and Tb s TbY . If TaY s Tc apply Ž2.2.; if TbY s Td , apply Ž2.3.. Otherwise apply Ž2.1. once again. Ž2.2. Ta only appears below in B. In this case Ta appears in N 9 and t ŽTa .qs 0. Hence TaTb s t Ž Ta . Tb y t Ž Ta .
yy
Tb y t Ž Ta .
y
Tb y t Ž Ta . Tb .
121
ALGEBRAS OF REDUCTION NUMBER ONE
Note that all terms of t ŽTa .yy and t ŽTa .qq appear in a big block different from B. Hence the required representation for the second and third summand can be found according to Ž1.. Thus it suffices to consider the last term. If a s 1, the t ŽTa .ys 0 and we are done. Otherwise take Ta s t ŽTa .y and Tb s Tb . Then a s a y 1, and b s b . Apply Ž2.1. or Ž2.3. to TaTb . This is possible, since t ŽTa .y certainly appears above. Ž2.3. Tb only appears above in B. An argument similar to the one developed in Ž2.2. permits us to reduce the problem to the product Tat ŽTb .q. If b s s, then t ŽTb .qs 0 and we are done. Otherwise take Ta s Ta and Tb s t ŽTb .q. Then a s a , and b s b q 1. Apply Ž2.1. or Ž2.2. to TaTb . It is clear that the above algorithm stops after a finite number of steps, when either Ta is the leftmost entry of the upper row of B, or Tb is the rightmost entry of the lower row of B. Ž3. Ta and Tb appear in the same small block B of N . Ž3.1. Ta or Tb appears in N 9. Then TaTb s t Ž Ta . Tb y t Ž Ta .
yy
Tb y t Ž Ta .
y
q
Tb y t Ž Ta . Tb y t Ž Ta . Tb .
The four last terms are sums of products TaX Tb , where TaX appears in a different small block with respect to Tb . Hence Ž1. or Ž2. can be applied to each of these products. Ž3.2. Ta and Tb do not appear in N 9. First suppose Ta s Tb . Note that Ta is not the leftmost element of the first row of B. Hence the block B has the following minor: 0s
TaX Ta
Ta s TaX TaY y Ta2 . TaY
Thus we may assume Ta / Tb . Then the block B is of one of the following forms: Bs
Ž a.
ž
... ...
TaX Ta
Ta Tb
Tb T bX
... ...
/
or
Ž b.
Bs
ž
Tc ...
... TaY
TaY Ta
Ta ...
... Tb
Tb T bY
T bY ...
... . Td
/
In case Ža. it holds TaTb s TaX TbX . If TaX is in the first column, then TaX appears in N 9, and we are back in Ž3.1.. The same is true if TbX is in the last column. Otherwise we are in the case Žb.. In this case TaTb s TaY TbY .
122
BARILE AND MORALES
Reapply this identity successively to move the first factor to the left, the second to the right. After a finite number of steps we end up with a product where the first factor is Tc , or the second is Td . Since Tc and Td both appear in N 9, we are back in Ž3.1.. COROLLARY 2.3. If d is the cardinality of L , then d s n y c q 1 and dim K w T xrJ s d. In particular, A is a Noether normalization of K w T xrJ. Proof. From Corollary 1.3 it follows that dim RrJ s n y c q 1, where n is the number of variables. Thus it suffices to show that d s n y c q 1. Now let V be the set of all variables appearing below in N : its cardinality is obviously equal to the number c of columns. Let V denote its complementary set. Let F be the set of the first terms of the sums in L . By construction it is clear that F s V j Tj1. But the cardinality of F is d. Hence d s n y c q 1.
3. MONOMIAL VARIETIES OF CODIMENSION 2 Theorem 2.2 can be applied to the study of the special fibre of a projective monomial variety of codimension 2: see w5x for a complete and detailed presentation of this subject. We first quote the basic notions and the main results from w6x. Let R be the polynomial ring in n q 2 indeterminates over the field K. Let I be an ideal minimally generated by t elements Fi 41 F iF t . The Rees algebra of I is defined to be the graded ring Rw It x s [nG 0 I n t n, and the M Rw It x, where M denotes the irrelevant special fibre is the quotient Rw It xrM maximal ideal of R. The dimension of the special fibre is called the analytic spread of I and is denoted by l Ž I .. We introduce t independent variables over R, say T s Ti 41 F iF t , and we consider the ideal I s ker Rw T x ª Rw It x ª 0, Ti ª Fi t 4 . We obtain a presentation Rw It x , I of the Rees algebra, from which we can deduce a presentation of Rw T xrI M Rw It x , K w T xrI I˜, where I˜ denotes the image of the special fibre: Rw It xrM w x I modulo M R T . Let us place ourselves in the case where I is the defining ideal of a toric variety admitting the parametrization x 1 s u1a1 ,
x 2 s u 2a 2 , . . . , x n s u na n ,
y s u1c1 u 2c 2 ??? u cnn ,
z s u1b 1 u 2b 2 ??? u nb n ,
ALGEBRAS OF REDUCTION NUMBER ONE
123
where, for all i, 1 F i F n, a i , bi , and c i are nonnegative integers such that a i / 0, Ž bi , c i . / Ž0, 0., and Ž b1 , . . . , bn . / Ž0, . . . , 0., Ž c1 , . . . , c n . / Ž0, . . . , 0.. We call this a monomial ¨ ariety of codimension 2. It is known that in general codimŽ I . F l Ž I ., and equality holds for I a prime ideal if and only if I is a complete intersection. In w6x the number l Ž I . was determined for all defining ideals I of a codimension 2 monomial variety, in spite of the face that no complete description of the ideal I˜ was known yet. The approach, indeed, is indirect: the computations are done on a subideal A˜: I˜, generated in degree two, called a reduced essential ideal. In w11x it is shown that I is generated by the 2 = 2-minors of a 2 = t-matrix. The ideal A˜ is generated by some of the Plucker relations. It ¨ A˜s 3 Žcf. w5, Corollary 5.26.1x.. This immediately turns out that dim Rw T xrA implies the following result: THEOREM 3.1 w6, Theorem 4.2x. The analytic spread of I is equal to two if I is a complete intersection and equal to three in all the remaining cases. 2.1.1 The ideal A˜ is always generated by monomials and binomials, with the exception of a very particular case, that is treated separatedly w5, Proposition 5.18.1x. In this case t s 4 and A˜s Ž T0 T1 y T22 y T32 . : K w T0 , T1 , T2 , T3 x . One easily sees that A˜ K w T0 , T1 , T2 x : K w T0 , T1 , T2 , T3 x rA is a Noether normalization and that the reduction number is 1. In the general case A˜ is the ideal associated to a barred matrix N whose big and small blocks coincide: see w5, 6, 11x for an explicit construction. We shall use our results to prove that in the general case A˜s I˜, i.e., the special fibre of I is entirely generated by the quadratic relations. We need the following preliminary LEMMA 3.2. Let R be a finitely generated K-algebra, J an ideal of R, and w : R ª RrJ the canonical epimorphism. Moreo¨ er let A : R be a Noether normalization of R. If dim R s dim RrJ, then A is a Noether normalization of RrJ with respect to the restriction w < A . Proof. We consider the A-module structure defined on RrJ by w < A . Since w is a finite A-homomorphism, the same is true for w < A . Thus it suffices to prove that w < A is an injection. Now K [ kerŽ w < A . s kerŽ w . l A, and the map w < A induces a finite monomorphism of rings K ª RrJ. ArK
124
BARILE AND MORALES
K .. But the left-hand side is equal to dim R s Thus dimŽ RrJ . s dimŽ ArK dim A. Thus the preceding equality is only possible if K s 0. Let us fix a monomial variety of codimension 2, with defining ideal I, which is not a complete intersection and does not belog to the particular case. We shall refer to the notation just introduced. Proposition 2.3 permits us to determine a Noether normalization A of the quotient Rw T x A˜. Consider the composition of maps A˜ª R w T x rI I˜, A ¨ R w T x rA where the right map is the canonical epimorphism. The equality of dimensions and Lemma 3.2 imply that A is a Noether normalization of I˜ with respect to the composed map. Since Rw T xrA A˜ is generated by Rw T xrI I˜. linear forms as an A-module, the same is true for its quotient Rw T xrI For the rest of this section we consider the local ring R M . The notions of Rees algebra, special fibre, and reduction naturally extend to this local ring. The above results apply to R M and to the localized ideals and subrings. An ideal J is called a reduction of I if JI n s I nq 1 for some nonnegative integer n. Let r J Ž I . denote the least number n such that the above equality holds. Then the Ž absolute. reduction number of I is defined to be the minimum of r J Ž I . taken over all possible reductions J of I. The following result is due to Vasconcelos: PROPOSITION 3.3 w15, Proposition 5.1.3x. Let Ž R, M , K . be a local Noetherian ring with infinite residue field and I an ideal of R of analytic spread l. Suppose A s K w z1 , . . . , z l x ¨ F Ž I . is a Noether normalization of the special fibre F Ž I ., and assume that all the z i are of degree 1. Furthermore let b1 , . . . , bs 4 be a minimal set of homogeneous generators of F Ž I . as an A-module. For all i, . . . , l, let a i g I be a lift of z i . Then J s Ž a1 , . . . , a l . is a Ž minimal . reduction of I and r J Ž I . s max deg bi < i s 1, . . . , s 4 . Putting the above results together one immediately obtains the following COROLLARY 3.4. With respect to the notation introduced in Subsection 2.1.1 it holds that r Ž I M . s 1. Next consider the following result by Cortadellas and Zarzuela: PROPOSITION 3.5 w1, Theorem 4.2 and Corollary 5.7x. Let Ž R, M . be a local Noetherian ring and let I be an ideal of R. Suppose I is generically a
ALGEBRAS OF REDUCTION NUMBER ONE
125
complete intersection, and the analytic spread is l s heightŽ I . q 1. If r Ž I . s 1, then F Ž I . is Cohen]Macaulay. Moreo¨ er the Hilbert function of F Ž I . is HF Ž I . Ž t . s
1 q Žt y l . t
Ž1 y t .
l
,
where t denotes the minimal number of generators of I. More results on the Hilbert functions of Cohen]Macaulay special fibres are contained in the work by D’Cruz, Raghavan, and Verma w3x. Now we are able to conclude: PROPOSITION 3.6. Let I : K w x 1 , . . . , x n , y, z x be the defining ideal of a monomial ¨ ariety of codimension 2. Suppose I is not a complete intersection. Then the reduced essential ideal A˜ coincides with the presentation ideal I˜ of the special fibre F Ž I .. A˜ is Proof. In w5, p. 117x it is proven that the Hilbert function of K w T xrA H A˜Ž t . s
1 q Ž t y 3. t
Ž1 y t .
3
.
A˜ Since l Ž I . s 3, it follows from Proposition 3.5 that F Ž I . and K w T xrA A˜. have the same Hilbert function. But F Ž I . is a quotient of K w T xrA Note that Theorem 2.2 and Proposition 3.3 yield an explicit construction for a minimal reduction of any ideal I M . We perform such a construction in the next examples, where I is throughout an ideal of K w x, y, z, w x defining a projective monomial curve in P 3. EXAMPLES 3.7. Ža. Suppose I is minimally generated by 4 elements. Coudurier and Morales w2x call this a monomial curve of type I. A system of generators is given by X
F2 s y pqn w ryl y x ky r z sqm
F1 s x k w l y y n z m , X
F3 s y pq2 n y x 2 kyr z s w 2 lyr ,
X
F4 s x r y p w r y z sq2 m ,
where all the exponents are supposed to be nonnegative. The ideal I˜ is the ideal associated to the barred matrix Ns
ž
T3 T2
T2 . T4
/
IM is A M s K w t 1 , t 2 , t 3 q t 4 x, Thus a Noether normalization of K w T x MrI where t i denotes the image of the variable Ti via the localization at M . Let J s Ž F1 , F2 , F3 q F4 . .
126
BARILE AND MORALES
According to Proposition 3.3 the ideal J M is a minimal reduction of I M . Moreover note that F12 s F1 Ž F1 q F3 . y F1 F3 , F32 s F3 Ž F1 q F3 . y F1 F3 , and X
F1 F3 s x r w 2 lyr F22 y y p z s F02 . Hence J is a minimal reduction of I note only locally at M , but also globally. In particular we have that r Ž I . s 1. The next example shows that this direct passage from a local to a global reduction is not always possible: in general the generators of the subalgebra A do not yield a global reduction. Žb. Consider the ideal I generated by the following 6 binomials: F1 s y 6 z 38 y x 13 w 31 ,
F2 s y 18 z 25 y x 15 w 28
F3 s y 30 z 12 y x 17 w 25 ,
F4 s y 42 y zx 19 w 22
F5 s z 51 y y 6 x 11 w 34 ,
F6 s y 12 w 3 y x 2 z 13 .
The barred matrix associated to I is Ns
ž
T4 T6
T3 T2
T2 T1
T1 . T5
/
Then N9 s
ž
T4 T6
T3 T1
T1 , T5
/
whence one obtains A M s K w t 1 q t 4 , t 3 q t5 , t6 x as a Noether normalizaI˜M . Let J s Ž F1 q F4 , F3 q F5 , F6 .. Then J M is a reduction tion of K w T x MrI of I M , but J is not a reduction of I. One can show that it even holds RadŽ J . / I. A reduction of I is given by J X s Ž Ž 1 q y 6 . F1 q F4 , F3 q F5 , F6 . . We are not able to give the general form of three elements generating a minimal reduction of the ideal I defining a projective monomial curve. The problem was solved by Morales and Simis Žcf. w13, Proposition 3.1.2x. for projective monomial curves lying on a quadric surface.
ALGEBRAS OF REDUCTION NUMBER ONE
127
It was conjectured in w5x that the ideal of presentation of the Rees algebra Rw It x is generated by forms of degree two at most. According to Micali w12x this is certainly true if I is a complete intersection. Now we are able to answer the question in the general case. The crucial result is due to Huckaba and Huneke. THEOREM 3.8 w10, Theorems 2.9 and 4.5x. Let R be a Cohen]Macaulay local ring and I an ideal ha¨ ing height d G 1 and analytic spread l Ž I . s d q 1. Assume that the minimal primes of RrI all ha¨ e the same height, and the associated primes of RrI ha¨ e height at most d q 1. Assume also that I is generically a complete intersection and there exists a minimal reduction J of I such that r J Ž IQ . F 1 for e¨ ery prime ideal Q = I with codimŽ QrI . s 1. Finally assume that depthŽ RrI . G dimŽ RrI . y 1. Then the presentation ideal of the Rees algebra Rw It x of I is defined by elements of degree two at most. Moreover Peeva and Sturmfels Žcf. w14, Theorem 2.3x. showed that for any toric ideal I it holds that projdimŽ RrI . F 2 codimŽ I . y 1. In particular, if I is the defining ideal of a monomial variety of codimension 2, by Auslander and Buchsbaum it holds that depth Ž RrI . s dim Ž R . y projdimŽ RrI . G dim Ž R . y 3 s dim Ž RrI . y 1. Suppose I is not a complete intersection. In view of Theorem 3.1, the preceding inequality, and Corollary 3.4 the ideal I fulfills all the assumptions of Theorem 3.8. This proves: THEOREM 3.9. Let I : R be the defining ideal of a monomial ¨ ariety of codimension 2. Then the presentation ideal of the Rees algebra Rw It x of I is generated by forms of degree two at most. Note that Gimenez Žcf. w5, Theorem 6.3.1x. already showed Proposition 3.6 under the hypothesis that Theorem 3.9 be true.
ACKNOWLEDGMENTS The authors express their gratitude to Santiago Zarzuela for his helpful comments. The first author is indebted to the Institut Fourier of the University of Grenoble for hospitality and support during the preparation of this paper.
128
BARILE AND MORALES
REFERENCES 1. T. Cortadellas and S. Zarzuela, On the depth of the fiber cone of filtrations, preprint. 2. L. Coudurier and M. Morales, Classification des courbes toriques dans l’espace projectif, module de Rao et liaison, preprint. 3. C. D’Cruz, K. N. Raghavan, and J. K. Verma, Cohen]Macaulay fiber cones, preprint. 4. D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 Ž1984., 89]133. ´ 5. Ph. Gimenez, ‘‘Etude de la fibre speciale de l’eclatement d’une variete ´ ´ ´ ´ monomiale en codimension deux,’’ Dissertation, Univ. of Grenoble, 1993. 6. Ph. Gimenez, M. Morales, and A. Simis, L’analytic spread de l’ideal d’une ´ de definition ´ variete ou ´ egal ` a trois, C. R. ´ ´ monomiale de codimension deux est toujours inferieur ´ Acad. Sci. Paris 319 Ž1994., 703]706. 7. Ph. Gimenez, M. Morales, and A. Simis, The analytic spread of certain toric varieties of codimension two, preprint. 8. J. Harris, ‘‘Algebraic Geometry,’’ Springer-Verlag, New York, 1992. 9. R. Hartshorne, Complete intersections and connectedness, Amer. J. Math. 96 Ž1974., 602]639. 10. S. Huckaba and C. Huneke, Powers of ideals having small analytic deviation, Amer. J. Math. 144 Ž1992., 367]403. ´ 11. M. Morales, Equations des varietes ´ ´ monomiales en codimension deux, J. Algebra 175 Ž1995., 1082]1095. 12. A. Micali, Sur les algebres universelles, Ann. Inst. Fourier 14 Ž1964., 33]88. ` 13. M. Morales and A. Simis, Symbolic powers of monomial curves in P 3 lying on a quadric surface, Comm. Algebra 20 Ž1992., 1109]1121. 14. I. Peeva and B. Sturmfels, Syzygies of codimension 2 lattice ideals, preprint. 15. W. Vasconcelos, ‘‘The Arithmetic of Blowup Algebras,’’ London Math. Soc., Vol. 195, Cambridge Univ. Press, Cambridge, UK, 1994. 16. W. Vasconcelos, The reduction number of an algebra, Compositio Math. 104 Ž1996., 189]197.