Minimal reductions of monomial ideals

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1 ISSN: Minimal reductions of monomial ideals Veronica Crispin Quiñonez Research Reports in Mathematics Number 10, 2004 Department of Mathematics Stocholm Universit

2 Electronic versions of this document are available at Date of publication: November 23, Mathematics Subject Classification: Primar 13C05. Kewords: reduction, monomial ideal, integrall closed ideal. Postal address: Department of Mathematics Stocholm Universit S Stocholm Sweden Electronic addresses:

3 Minimal reductions of monomial ideals Veronica Crispin Quiñonez Department of Mathematics Stocholm Universit S Stocholm Sweden 1 Introduction For an ideal I in a ring R a reduction is defined as an ideal J I such that JI l = I l+1 for some integer l. An ideal which is minimal with respect to this propert is called minimal reduction, and the least integer l for these ideals is called the reduction number of I. Further, J is a reduction of I if and onl if I is integral over J. To a monomial ideal I in [x 1,..., x n ] or [[x 1,... x n ]] we associate the set log I which consists of the exponents of all monomials belonging to I, that is, log I = {(a 1,... a n ) x a1 1 xan n I}. It is proved ([2], [3], [6]) that the integral closure of I is the integrall closed monomial ideal, Ī, generated b all the monomials with exponents in the convex hull of log I in N n. In this paper we determine minimal reductions of an monomial ideal in the ring [[x, ]] using the relation between reduction and integral closure. 2 Monomial ideals Consider the two-dimensional local ring R = [[x, ]] and an ideal of dimension two in it. According to the general theor for minimal reductions, a minimal reduction of the ideal will be generated b two elements in the ring, if the residue field is infinite. However, the results that will follow are valid for an field in [[x, ]]. An monomial ideal I in R is of the form I = mi, where m is a monomial and I an m-primar monomial ideal. An ideal mj is a reduction of mi if and onl if J is a reduction of I. Thus, we ma assume that I is m-primar, that is, that x a and b belong to I for some a and b. For a moment we ma let the ring R be the polnomial ring also. Let I = x Ai Bi 0 i s [x, ] or [[x, ]] where A i < A i+1, B i > B i+1 and A 0 = B s = 0. We will construct a monomial ideal, which we call I lmr, having the same integral closure as I, and show that it is the unique smallest one. In other words, conv(log I lmr ) = log Ī. 1

4 Definition 2.1. Let I = x Ai Bi. Define I lmr = x Ai j B ij as follows: i 0 = 0, i 1 be the greatest i such that the minimal value of the expression obtained, A i A ij 1 for j 2 let i j = max { i > i j 1 ; B ij 1 B i is minimal }. A i B 0 B i Graphicall we define the generators of I lmr in N 2 recursivel b starting with (0, B 0 ) and choosing the greatest index i such that (A i, B i ) gives the steepest slope between the two points. Taing this exponent as our new starting point we repeat the procedure. Example 2.2. Let I = 12, x 11, x 2 7, x 5 4, x 8. Then the integral closure is Ī = 12, x 10, x 2 7, x 3 6, x 4 4, x 5 3, x 6, x 8 and the least monomial reduction of both ideals is I lmr = 12, x 2 7, x 6, x 8, the generators of which are mared b empt circles. is I Ī Ī lmr = I lmr x x It is clear that among all monomial ideals with integral closure Ī the ideal I lmr is the least one. Equivalentl, among all monomial ideals which are reductions of Ī, the ideal I lmr is least. We call it least monomial reduction. Moreover, I lmr is the least monomial reduction of an ideal ling between itself and Ī. Remar 2.3. An integrall closed monomial ideal is a product of blocs (Theorem 3.8 in [2]), where we b an (a, b)-bloc mean the unique simple integrall closed monomial ideal containing the elements x a and b in its minimal generating set, that is, x a, b where a and b are relativel prime. (We recall that an ideal is called simple if it cannot be written as a product of two proper ideals.) The product of s number of (a i, b i )-blocs satisfing the condition ai b i the integrall closed ideal s x ai, bi = i=1 s i=1 x ai+1 b i+1 (x i 1 i =1 a i s i =i+1 b i ) x ai, bi. (2.1) In N 2 we illustrate this product as the vertices (a 1 + +a i, b i b s ) = (a i, b i ), 0 i s, and the diagonal lines between two consecutive vertices. Then is 2

5 conv( (a i, b i )+N 2 ) constitutes the log-set of the product. If we eep onl those vertices (a j, b j ) that satisf the condition ai b i < ai+1 b i+1, then the convex hull of (a i, b i ) + N 2 will also constitute the desired log-set. Thus, the generators of the minimal monomial reduction are illustrated b such vertices (a j, b j ) that < aj+1 aj b j b j+1. a j a j 1 b j 1 b j Example 2.4. The ideal Ī in the previous example is a product of the blocs x 2, 5, x 2, 3, x 2, 3, x 2,. It is depicted b the vertices (12,0), (2,7), (4,4), (6,1) and (8,0), where (4,4) is omitted to give the least monomial reduction, since 12, x 2 7, x 6, x 8 = Ī. The reduction number of an integrall closed ideal is two (Theorem 5.1 [4]). Hence, I lmr Ī = Ī2. Proposition 2.5. An power of a simple and m-primar integrall closed monomial ideal in [[x, ]] (or [x, ])) has the ideal b, x a as its minimal reduction, where b denotes the highest power of and a the highest power of x in the minimal set of generators. Proof. There are two tpes of such ideals [2]. Let I = x j Bj 0 j s where B j = s j s B 0. Theorem 3.8 in [2] states that I 2 = x j B0+Bj, x s+j Bj and it is obvious that I B0, x s = I 2. Let I = x Ai r i 0 i r where A i = i Ar r. Then I2 = x Ai 2r i, x Ar+Ai r i and the result follows similarl. Assume that the integral closure of a monomial ideal is some power of a bloc. Then the least monomial reduction is generated b two monomials and is, of course, a minimal reduction itself. Suppose that the least monomial reduction of an ideal is generated b more than two monomials. Then the generators satisf a certain condition which we at first will demonstrate b an example. Example 2.6. Let I lmr = 12, x 2 7, x 6, x 8 = m j 0 j 3 as previousl. m 0 m 0 m 1 I lmr m 1 I lmr m 2 m 2 m 3 x m 3 x 3

6 It is clearl seen that moving m 1 either verticall or horisontall it will intersect the line between m 0 and m 2.The same is valid for m 2 and the line between m 1 and m 3. The pictures correspond to certain relations between an three consecutive generators. We have m 3 1 m 2 0m 2 and m 3 2 m 1 m 2 3. (2.2) In (2.2) we could as well have chosen x instead of and get: m 5 1x m 3 0m 2 2 and m 4 2x m 1 m 3 3. (2.3) An I lmr is constructed in such a wa that there are relations similar to (2.2) between the generators. Let I lmr = x Aj Bj = m j 0 j r. Let further c j = A j+1 A j and l j = A j+1 A j 1. Then we can easil deduce that { lj A j = c j A j 1 + (l j c j )A j+1 l j B j + 1 c j B j 1 + (l j c j )B j+1, that is, m lj j mcj j 1 mlj cj j+1. (2.4) Remar 2.7. The results that will follow will depend on the ring being local. Henceforth we will consider onl the formal power series ring. Often we can choose smaller l j, compared to Example 2.6 where c 1 = 4 and l 1 = 6 to start with. In the local ring with the maximal ideal m = x, the expression (1.4) sas m cj j 1 mlj cj j+1 m lj j m. (2.5) B taing l = lcm(l j ), we can assume that all l j are equal. The relation between three consecutive generators can be extended to three arbitrar generators. For example, for an 0 < j < r 2 we have: m cjl j 1 (mcj+1 j m l cj+1 j+2 ) l cj (m cj j 1 ml cj j+1 )l m m l2 j m. The power products m j are nonzerodivisors, hence m c j 1m l c j+2 ml j m. This small result is worth generalizing in a lemma. Lemma 2.8. Let (R, m) be a local integral domain and I = m j 0 j r an ideal in R. Suppose that the generators fulfil (2.5). Then, for each triple of indices i < i < j there are positive integers c and l, c < l, such that m c i m l c j m l i m. (2.6) There is an alternative wa to express (2.6). For an pair of indices i and j, j i 2, there are positive integers c and l, c < l, such that m c i ml c j I l m. Multipling b a proper power of some of the two generators we can formulate the following result. 4

7 Proposition 2.9. Let I lmr = m i be a least monomial reduction of some ideal I in [[x, ]]. Assume that its generators are ordered b descending powers of (or x). Then there is an integer l such that m l i ml j I2l m for an two indices i and j such that j i 2. In the next section we determine minimal reductions for a class of ideals in an local commutative ring and will show that the least monomial reductions we have defined belong to this class. In that wa we will be able to determine a minimal reduction J of an monomial ideal I in [[x, ]], because J I lmr I Ī where I lmr is integral over J. Hence, J is a minimal reduction of an ideal between J and Ī. 3 Minimal reductions Let (R, m) be a local ring and I = m i 0 i r an ideal in it. Suppose that the generators are ordered in such wa that the satisf the condition m l im l j I 2l m (3.1) for some integer l if j i 2. A reduction of an ideal, the generators of which satisf (3.1), can be expressed in a quite convenient wa. Before we prove our theorem we need two lemmas. The first one is found on p.147 in [6]. Lemma 3.1. Let J I be ideals. Then J is a reduction of I if and onl if J + Im is a reduction of I. Proof. If JI l = I l+1, then (J + Im)I l = JI l + I l+1 m = I l+1. If (J + Im)I l = I l+1, then we use Naaama s lemma on m(i l+1 /JI l ) = (I l+1 m + JI l )/JI l = I l+1 /JI l which gives us I l+1 /JI l = 0 and, hence, I l+1 = JI l. The proof is complete. Lemma 3.2. Let I = m i 0 i r be an ideal. Assume further that m l i JIl 1 + I l m for all i and some integer l. Then J is a reduction of I. Proof. Let l = (l 1)r, then I l +1 = r i=0 mli i r i=0 l i = l + 1. For ever generator (product) there is some index such that l l, according to the pigeon hole principle. Then m l (JI l 1 + I l m)i l l and, hence, r i=0 mli i = m l ( ) i mli i JI l + I l +1 m. Thus, I l +1 JI l + I l +1 m and we are done due to Lemma 3.1. Theorem 3.3. Let I = m i 0 i r be an ideal in (R, m). Assume that there is a partition {0,... r} = 0 α s S α, where s r, such that if i, j S α, i j, then m l i ml j I2l m for some integer l. Let J = i S α m i 0 α s, then J is a reduction of I. 5

8 Proof. In the case when ever S α = 1, the ideal J = I is triviall a reduction. Suppose that S α 2 for some α. For that α define p α = i S α m i and fix a S α. B assumption m l ml i I2l m for all i S α. Let l = S α (l 1). Then for an t, 0 t l + l 1 we have m l+l t (p α m ) t+1 + m l+l t 1 (p α m ) t+2 = = m l+l t 1 (p α m ) t+1 p α I l+l J. Using this we can rewrite the zero element in the quotient ring R/JI l+l (3.2) as: [m l+l p α ] = [m l+l +1 + m l+l (p α m )] (1.8) = [m l+l = [m l+l +1 ± m l (p α m ) l +1 ] = [m l+l +1 ± m l m l+l 1 (p α m ) 2 ] =... ( ( )) β... ], l i= l +1 l j i m i, i S α (3.3) where the β... s are the multinomial coefficients. According to the pigeon hole principle there is some in each product m li i such that l l. For that we have m l ( ) m li i I 2l mi l +1 l = I l+l +1 m. (3.4) i, l i=l +1 From (3.3) and (3.4) we can deduce that m l+l +1 JI l+l + I l+l +1 m. Since the index was chosen arbitraril it follows easil that there is an integer L such that m L i JI L 1 + I L m for all i. Lemma 3.2 completes the proof. Example 3.4. Let I = x 3 z 2, x 2 2 z, x 3 z, 4 z 2 = m i 0 i 3 R = [[x,, z]]. The partition of the indices is {0, 2} {1, 3}, since m 0 m 2 m 2 1m and m 1 m 3 m 2 2 m. Appling Theorem 3.3 gives us a minimal reduction of I which is x3 z 2 + x 3 z, x 2 2 z + 4 z 2. Example 3.5. Let I = x 4, x 2, 4, 2 z, z 4, z 2 x = m i 0 i 5 R = [[x,, z]]. This is an m-primar ideal, so the number of the generators of a reduction of it must be at least three. We see that (x 2 ) 2 z (x 4 )( 2 z) or, equivalentl, m 0 m 3 (x 2 2 )m. B smmetr there are similar relations for the other generators. Hence, we get a partition {0, 3} {2, 5} {1, 4} and a minimal reduction x z, 4 + z 2 x, z 4 + x 2. We return to the monomial ideals in [[x, ]]. The minimal reductions of an m-primar ideal is generated b two elements in this ring. Corollar 3.6. Let I = m j 0 j s be a monomial ideal in [[x, ]] and I lmr = m i 0 i r its monomial reduction where the generators are ordered in such wa that the powers of (or x) are descending. Then J = even i m i, odd i m i is a minimal reduction of I. Moreover, if J is a minimal reduction of an ideal between J and Ī. 6

9 Proof. We have shown that the relation (3.1) is valid for all the generators of I lmr except for an two consecutive. Since two consecutive generators must lie in different subsets of the partition, a split into even and odd indices is the onl one. The rest follows from the theorem. Example 3.7. Consider Example 2.2. The ideal J = 12 + x 6, x x 8 is a minimal reduction of all ideals ling between J and Ī. References [1] M. F. Atiah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesle, [2] V. Crispin Quiñonez, Integrall Closed Monomial Ideals and Powers of Ideals, Research Reports in Mathematics 7, Department of Mathematics, Stochom Universit, [3] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometr, Springer Verlag, [4] C. Hunee, Complete Ideals in Two-Dimensional Regular Local Rings, Commutative Algebra: Proceedings from a microprogram held June 15 - Jul 2, 1987 (M. Hochster, C. Hunee, J. Sall, eds), , Springer-Verlag, [5] D. G. Northcott, D. Rees, Reductions of Ideals in Local Rings, Proc. Cambridge Philos. Soc. 50 (1954), [6] R. Villarreal, Monomial Algebras, Marcel Deer, [7] O. Zarisi, P. Samuel, Commutative Algebra. Vol. 2, Van Nostrand,