A Filtration of the Sally Module and the Associated Graded Ring of an Ideal

Transcription

1 Comm. Algebra to appear A Filtration of the Sally Module and the Associated Graded Ring of an Ideal Claudia Polini Department of Mathematics - Hope College - Holland, MI Introduction Let (R; m) be a Noetherian local ring and let I be an R-ideal. The associated graded ring of I, G = gr I (R), plays a significant role in the study of resolution of singularities. Its relevance lies upon the fact that it represents algebraically the exceptional fiber of the blowup of a variety along a subvariety. A commonly addressed issue is to find numerical conditions that imply lower bounds on the depth of G. In [7, 8] and [3], for instance, this depth has been measured by using the Hilbert coefficients of I. To better explain these results, let us introduce some notation: An ideal J I is called a reduction of I if I r+1 = JI r for some integer r. The least such r is called the reduction number of I with respect to J, and denoted r J (I). If R is Cohen Macaulay with infinite residue field and I is an m-primary ideal, then any minimal (with respect to inclusion) reduction of I is generated by a regular sequence. The Hilbert Samuel function of I is the numerical function H I (n) =λ(r=i n ) (where λ() denotes length) that measures the growth of the length of R=I n for all n 1. If d denotes the dimension of R, it is well-known that for n 0, H I (n) is a polynomial in n of degree d n + n + d, 2 P I (n) =e 0 (I), e 1 (I) + +(,1) d e d (I); d called the Hilbert or Hilbert Samuel polynomial of I, whose coefficients e 0 (I), e 1 (I), :::, e d (I) are uniquely determined by I and called the Hilbert coefficients of I. The author was partially supported by the NATO/CNR Advanced Fellowships Programme. polini@cs.hope.edu.

2 2 Claudia Polini A relation between e 1 (I) and the depth of G is proved in [7, 8] and [3]: More precisely, for any m-primary ideal I in a local Cohen Macaulay ring R with infinite residue field, e 1 (I) λ(i n+1 =JI n ) for any minimal reduction J of I. If equality holds, then depthg. This suggests the following conjecture Conjecture (see [9]): Let (R; m) be a Cohen Macaulay local ring of dimension d with infinite residue field and let I be an m-primary ideal of R. If for some minimal reduction J of I then λ(i n+1 =JI n ), e 1 (I) =t depthg d,t, 1: The t = 1 case has been recently settled in [9] (see also [2] for partial results when t = 2). In this paper we give a much easier proof of the result of [9]. The proof Some additional background and notation are needed first. The main tool is a filtration of the Sally module introduced by M. Vaz Pinto (see [7, 8]). LetJ be any minimal reduction of I; thesally module of I with respect to J, S = S J (I), isthe graded R[Jt]-module defined by the following short exact sequence 0! IR[Jt],! IR[It],! S =MI n+1 =J n I! 0; where R[It] and R[Jt] denote the Rees algebras of I and J respectively (see [5]). The importance of such a module is due to the fact that any information on its depth or its Hilbert coefficients can be passed onto G.Tobemoreprecise n1 depthg depths, 1 (see [7]) and if we let s i s denote the Hilbert coefficients of S then we have the following relations e 0 (I) =λ(r=j) e 1 (I) =λ(i=j)+s 0 e i (I) =s i,1 for i = 2;:::;d

3 A filtration of the Sally module and the associated graded ring of an ideal 3 (see [5]). To determine some lower bounds on the depth of S we filter S itself with other modules, called reduction modules, endowed with a good structure, namely they are factor of polynomial rings. Furthermore, we will prove that under our assumptions the reduction modules are polynomial modules except for one which, however, turns out to be Cohen Macaulay. The filtration of S is defined as follows (see [7, 8]). LetC n be the R[Jt]-module M C n = in I i+1 =J i,n+1 I n : Note that C 1 = S. Thereduction modules L n of I with respect to J are defined to be the R[Jt]-modules generated by the first component, I n+1 =JI n,ofc n, namely L n = I n+1 =JI n R[Jt]: The L n s and the C n s fit into the short exact sequence of R[Jt]-modules 0! L n,! C n,! C n+1! 0: If r = r J (I) is the reduction number of I with respect to J then C r = 0andL ' C. In conclusion, we have then the following set of short exact sequences of R[Jt]-modules 0! L 1,! S,! C 2! 0 0! L 2,! C 2,! C 3! ! L r,2,! C r,2,! L! 0 (1) which give a filtration of S. To determine a lower bound on the depth of S and hence on the depth of G, we need to study the structure of the L n s. In doing so, we use the so called virtual reduction modules D n of I with respect to J. To define the D n s, let A n = Ann R (I n+1 =JI n ) and let B n = R=A n [T 1 ;:::;T d ]. Note that B n is a d-dimensional Cohen Macaulay ring with Ass R[Jt] (B n )=fmr[jt]g, also B n ' R[Jt]=A n R[Jt]. Now D n = B n R I n+1 =JI n ' I n+1 =JI n [T 1 ;:::;T d ] is a maximal Cohen Macaulay B n -module with Ass R[Jt] (D n )=fmr[jt]g. Then the D n s and the L n s fit into the following short exact sequence of R[Jt]-modules 0! K n,! D n [,n] ϕ n,! L n! 0; (2) where ϕ n is the epimorphism of R[Jt]-modules that is the identity on I n+1 =JI n and sends each T i to a generator of R[Jt],andK n = Kerϕ n.

4 4 Claudia Polini We are now ready to give a simplified proof of the main result of [9]. Theorem: Let (R; m) be a Cohen Macaulay local ring of dimension d with infinite residue field. Let I be an m-primary ideal of R. If e 1 (I) = λ(i n+1 =JI n ), 1 for some minimal reduction J of I, then depth G = d, 2. Proof. Let J beaminimalreductionof I for which e 1 (I) = λ(i n+1 =JI n ), 1: Claim 1: All the K n defined in (2) are zero except for one, K N. For i 0, one has λ(s i ) = s 0 i + = λ(i n+1 =JI n ), 1 because s 0 = e 1, λ(i=j), by [5]. On the other hand from (1) and (2) we get, λ(s i )= i + d, 2, s 1 + +(,1) d,1 s d,1 d, 2! i + i + d, 2, s 1 + +(,1) d,1 s d,1 ; d, 2 λ((l n )i) = (λ((d n [,n])i), λ((k n )i)) ; for all i 0. The contribution of λ((d n [,n])i) to s 0 is exactly λ(i n+1 =JI n ) hence λ((k n )i) =1 i +, f 1 i + d, 2 d, 2 + +(,1) d,1 f d,1 : But by (2) Ass(K n )=fmr[jt]g, hence K n is either (0) or has dimension d. This implies that all K n must be (0) except for one, say K N. Claim 2: K N is a rank 1 torsionfree B = R=m[T 1 ;:::;T d ]-module. From the proof of Claim 1, one has i + λ((k N )i) =1 + lower terms; for all i 0. This implies, together with Ass R[Jt] (K N )=fmr[jt]g, that λ((k N ) mr[jt] )=1: (3)

5 A filtration of the Sally module and the associated graded ring of an ideal 5 To prove that mr[jt] AnnK N, we use the fact that Ass R[Jt] (AnnK N )=fmr[jt]g together with (3), and we have that K N is a rank 1 B-module. Furthermore, K N is torsionfree, since Ass B (K N )=f0g. Claim 3: depthk N = d. By Claim 2, K N is isomorphic to a B-ideal L. We will prove that L is unmixed of height 1, hence it is a principal B-ideal since B is a UFD. Let p 2 SpecB with ht p 2. We want to show that depth(l) p 2. Indeed, this yields depth(b=l) p 1 and therefore p 62 Ass B (L). Since depthl p depth p L, it is enough to show that depth p L 2. Being B = R[Jt]=mR[Jt] we can find a q 2 SpecR[Jt] such that p = q=mr[jt]. Now depth q S 1, because Ass R[Jt] (S) = fmr[jt]g ( q, and depth q L i 2fori < N, sincel i ' D i [,i] and the D i [,i] s are maximal Cohen Macaulay B i -modules. Depth counting (see [1, Proposition 1.2.9]) on the short exact sequences of (1) implies that depth q C i 1 for all i N. Hence depth q L N 1. Finally, depth counting on 0! K N,! D N [,N] ϕ N,! L N! 0 yields depth q K N 2, which gives our result. Conclusion: From Claim 3 one has depth R[Jt] L N, while from Claim 1 depth R[Jt] L i = d for all i 6= N. Hence depth counting on (1) yields depths, which yields depthg d, 2 by [7, Proposition ]. Actually, depthg = d, 2 by [3, Theorem 3.1]. From the proof of the theorem we can deduce the following corollary. Corollary: Let (R; m) be a Cohen Macaulay ring of dimension d with infinite residue field. Let I be an m-primary ideal of R. If e 1 (I) = λ(i n+1 =JI n ), 1 for some minimal reduction J of I then (a) for j = 1;:::; the Hilbert coefficients of S are s j = n= j (b) the Hilbert Poincaré series of S is HP(S;t) = n λ(i n+1 =JI n ), j s ; j λ(i n+1 =JI n )t n,t s (1,t) d ;

6 6 Claudia Polini (c) for j = 1;:::;d the Hilbert coefficients of G are e j = n= j,1 n j, 1 (d) the Hilbert PoincaréseriesofG is HP(G;t) = where s is a positive integer.. r λ(r=i)+ Proof. By the previous proof λ(i n+1 =JI n ), s j, 1 λ(i n =JI n,1 ), λ(i n+1 =JI n ) t n +t s (1,t) L n ' I n+1 =JI n [T 1 ;:::;T d ][,n] (1,t) d ; for all n 6= N. Hence the Hilbert polynomial of L n is given by (i, n)+ P(L n ;i) =λ(i n+1 =JI n ) and its Hilbert Poincaréseriesis HP(L n ;t) = λ(in+1 =JI n )t n (1,t) d : For n = N, the structure of L N can be recovered from the short exact sequence 0! K N,! D N [,N],! L N! 0; where K N =(f ), f 2 R=m[T 1 ;:::;T d ],deg( f )=s, andd N = I N+1 =JI N [T 1 ;:::;T d ]. Thus, the Hilbert polynomial of L N is given by (i, N)+ (i, s)+ P(L N ;i) =λ(i N+1 =JI N ), and its Hilbert Poincaréseriesis HP(L N ;t) = λ(in+1 =JI N )t n,t s (1,t) d : By (1), we only have to add up these expressions from n = 1ton = to conclude that the Hilbert coefficients of S and its Hilbert Poincaré series are given by (a) and (b). Now, using the formula HP(G;t) = λ(r=j) λ(i=j), (1,t) (1,t) d (1,t) d + HP(S;t) (see [6, 7]) relating the Hilbert Poincaré series of G and S we obtain (c) and (d). ;

7 A filtration of the Sally module and the associated graded ring of an ideal 7 References [1] W. Bruns and J. Herzog, Cohen Macaulay Rings, Cambridge University Press, Cambridge, [2] A. Guerrieri and M. Rossi, Estimates on the depth of the associated graded ring, J. Algebra (to appear). [3] S. Huckaba, A d-dimensional extension of a lemma of Huneke s and formulas for the Hilbert coefficients, Proc. Amer. Math. Soc. 124 (1996), [4] D.G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Camb. Phil. Soc. 50 (1954), [5] W.V. Vasconcelos, Hilbert functions, analytic spread and Koszul homology, Contemporary Mathematics 159 (1994), [6] W.V. Vasconcelos, The degrees of graded modules, in Six Lectures on Commutative Algebra, (Bellaterra, 1996), , Progr. Math., 166, Birkhäuser, Basel, [7] M.T.R. Vaz Pinto, Structure of Sally modules and Hilbert functions, Ph.D. Thesis, Rutgers University, [8] M.T.R. Vaz Pinto, Hilbert functions and Sally modules, J. Algebra 192 (1997), [9] H.-J. Wang, Hilbert coefficients and the associated graded rings, Proc. Amer. Math. Soc. (to appear).