On the depth of graded rings associated to lex-segment ideals in K[x,y]
aa r X i v : . [ m a t h . A C ] J u l ON THE DEPTH OF GRADED RINGS ASSOCIATED TOLEX-SEGMENT IDEALS IN K [ x, y ] A. V. JAYANTHAN
Abstract.
In this article, we show that the depths of the associated gradedring and fiber cone of a lex-segment ideal in K [ x, y ] are equal. Introduction
Let K be a field of characteristic zero and R = K [ x , . . . , x n ] be the poly-nomial ring in n variables over K . Let R i denote the K -vector subspace of allmonomials of degree i . We fix the ordering of variables as x > x > · · · > x n .For monomials u = x a · · · x a n n and v = x b · · · x b n n , we say that u < Lex v ifdeg u ≤ deg v or deg u = deg v and b i − a i > d is the set of all monomials of theform { m ∈ R d : m ≥ u } , where u ∈ R d . A graded ideal I is said to bea lex-segment ideal if I d is generated by initial lex-segments for each d with I d = 0. Lex-segment ideals are important due to many reasons. It is wellknown that among ideals with a given Hilbert function, the lex-segment idealhas the largest number of generators. A. M. Bigatti [1] and H. A. Hulett [8] incharacteristic zero and K. Pardue [11] in positive characteristic generalized thisto all Betti numbers. They proved that the lex-segment ideals have the largestBetti numbers among all ideals with a given Hilbert function. Lex-segmentideals are of interest also due to classical reasons. O. Zariski used the theoryof contracted ideals to study complete ideals in 2-dimensional regular localrings ( R, m ). In the graded setting, when K is algebraically closed, Zariski’sfactorization theorem for homogeneous contracted ideals asserts that any ho-mogeneous contracted ideal I can be written as I = m c L · · · L t , where each Key words and phrases.
Lex-segment ideals, associated graded ring, fiber cone, Reesalgebra, Cohen-Macaulay. L i is a lex-segment ideal with respect to an appropriate system of coordinates x i , y i which depends on i . [14, Theorem 1, Appendix 5], [4, Theorem 3.8].In this article, we study the blowup algebras, namely, the associated gradedring and the fiber cone of lex-segment ideals in a two dimensional polyno-mial ring. Let R be a ring, I any ideal of R and m a maximal ideal. Thenthe associated graded ring and the fiber cone of I are respectively defined asgr I ( R ) = ⊕ n ≥ I n /I n +1 and F ( I ) = ⊕ n ≥ I n / m I n . In [7], Huckaba and Marleyshowed that in a regular local ring ( R, m ), depth gr I ( R ) = depth R ( I ) − m -primary ideal I , where R ( I ) = ⊕ n ≥ I n t n denotes the Rees algebra of I .It is interesting to ask if there is a similar relation between the depths of thefiber cone and the associated graded ring. It is well known that this is not thecase in general (cf. Example 11, Example 12). In this article, we prove thatthe depths of these algebras are equal for lex-segment ideals in K [ x, y ], where K is a field of characteristic zero. Acknowledgements:
The author would like to thank Aldo Conca, M. E.Rossi, G. Valla, J. K. Verma and S. Goto for useful discussions regarding thecontents of the paper. 2.
Equality of depths
Let R = K [ x, y ], where K is a field of characteristic zero and M = ( x, y ).In this case, the lex-segment ideals are easy to describe. If I is a lex-segmentideal in K [ x, y ], then I = ( x d , x d − y a , . . . , x d − k y a k ) for some 0 ≤ k ≤ d and1 ≤ a < a < · · · < a k . Note that if I is a lex-segment ideal, then I n is also alex-segment ideal for all n ≥ Remark 1.
Let S = K [[ x, y ]] and m = ( x, y ) . Then for any ideal I ⊂ m , S/IS ∼ = R/I and S/ m S ∼ = R/ m . Therefore gr IS ( S ) ∼ = gr I ( R ) and F ( IS ) ∼ = F ( I ) [4, Lemma 2.1] . Hence, we may use the local techniques to prove theresults for gr I S ( S ) and F ( IS ) and derive the same for gr I ( R ) and F ( I ) . We first show that the Cohen-Macaulay property of the associated gradedring and the fiber cone are equivalent. The dimension of the fiber cone, denotedby s ( I ), is called the analytic spread. It is well known that h ( I ) ≤ s ( I ), where h ( I ) denote the height of the ideal I . The difference, s ( I ) − h ( I ) , is called N THE DEPTH OF GRADED RINGS ASSOCIATED TO LEX-SEGMENT IDEALS 3 the analytic deviation. Let I = ( x d , x d − y a , . . . , x d − k y a k ). If k = d , then I is an M -primary homogeneous contracted ideal. Because of Remark 1, wecan use local theory of M -primary contracted ideals in 2-dimensional regularlocal rings to study the blowup algebras. If 0 < k < d , then I is a non- M -primary ideal of analytic deviation one. Here we note that if 0 < k < d , then I = x d − k ( x k , x k − y a , . . . , xy a k − , y a k ), which is of the form I = zL , where z isan R -regular element and L an M -primary homogeneous contracted ideal. Weshow that the depth of gr I ( R ) is at most the depth of gr L ( R ). In particular,when gr I ( R ) is Cohen-Macaulay, so is gr L ( R ). For an element a ∈ I , let a ∗ denote its initial form in gr I ( R ) and a o denote its initial form in F ( I ).Let I be an ideal of a ring R . An ideal J ⊆ I is said to be reduction of I if I n +1 = J I n for some n ≥
0. A reduction which is minimal with respect toinclusion is called a minimal reduction. For a reduction J of I , the number r J ( I ) = min { n | I n +1 = J I n } , is called the reduction number of I with respectto J . Proposition 2.
Let ( R, m ) be a Noetherian local ring and L an m -primaryideal of R . Let x be a regular element in R and I = xL . Then depth gr I ( R ) ≤ depth gr L ( R ) . In particular, if gr I ( R ) is Cohen-Macaulay, then so is gr L ( R ) .Proof. Let depth gr I ( R ) = t . Let a , . . . , a t ∈ L \ L and b i = xa i be suchthat b ∗ , . . . , b ∗ t ∈ gr I ( R ) is a regular sequence. Then by Valabrega-Valla [13],( b , . . . , b t ) ∩ I n = ( b , . . . , b t ) I n − for all n ≥
1. We show that ( a , . . . , a t ) ∩ L n =( a , . . . , a t ) L n − for all n ≥ p ∈ ( a , . . . , a t ) ∩ L n for some n ≥
1. Then x n p ∈ ( b , . . . , b t ) ∩ I n = ( b , . . . , b t ) I n − = x n ( a , . . . , a t ) L n − . Therefore x n p = x n q for some q ∈ ( a , . . . , a t ) L n − . Since x is regular in R , p = q which implies that p ∈ ( a , . . . , a t ) L n − . Therefore, by Valabrega-Valla condition, a ∗ , . . . , a ∗ t ∈ gr L ( R )is a regular sequence. (cid:3) Remark 3.
In the above Proposition, we have shown that if b ∗ , . . . , b ∗ t is aregular sequence in gr I ( R ) , then a ∗ , . . . , a ∗ t is a regular sequence in gr L ( R ) .The following example shows that the converse is not true in general. Example 4.
Let R = K [ x, y ] . Let L = M = ( x, y ) and I = ( x , x y ) . Then x ∗ , y ∗ is a regular sequence in gr L ( R ) . It can be easily seen that I : x = A. V. JAYANTHAN ( x , x y, xy ) = I . Therefore ( x ) ∗ ∈ gr I ( R ) is not regular. However, this doesnot imply that the depth gr I ( R ) < . In fact, in this case, it can be seen (usingany of the computational commutative algebra packages) that gr I ( R ) is indeedCohen-Macaulay. The following result follows directly from Theorem 2.1 of [5].
Proposition 5.
Let ( R, m ) be a Cohen-Macaulay local ring and I be an idealof R with s ( I ) = r and H ( F ( I ) , t ) = a + bt (1 − t ) r . If F ( I ) is Cohen-Macaulay, then r J ( I ) ≤ for any minimal reduction J of I . We show that the Cohen-Macaulay property of the associated graded ringand the fiber cone are equivalent:
Theorem 6.