Powers of Ideals: Primary Decompositions, Artin-Rees Lemma and Regularity

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1 Powers of Ideas: Primary Decompositions, Artin-Rees Lemma and Reguarity Irena Swanson Department of Mathematica Sciences, New Mexico State University, Las Cruces, NM (e-mai: Mathematics Subject Cassification (1991): 13C99, 13H99, 14D25 Let R be a Noetherian ring and I an idea in R. Then there exists an integer k such that for a n 1 there exists a primary decomposition I n = q 1 q s such that for a i, q i nk q i. Aso, for each homogeneous idea I in a poynomia ring over a fied there exists an integer k such that the Castenuovo-Mumford reguarity of I n is bounded above by kn. The reguarity part foows from the primary decompositions part, so the heart of this paper is the anaysis of the primary decompositions. In [S], this was proved for the primary components of height at most one over the idea. This paper proves the existence of such a k but does not provide a formua for it. In the paper [SS], Karen E. Smith and mysef find expicit k for ordinary and Frobenius powers of monomia ideas in poynomia rings over fieds moduo a monomia idea and aso for Frobenius powers of a specia idea first studied by Katzman. Expicit k for the Castenuovo-Mumford reguarity for specia ideas is given in the papers by Chander [C] and Geramita, Gimigiano and Pitteoud [GGP]. Another method for proving the existence of k for primary decompositions of powers of an idea in Noetherian rings which are ocay formay equidimensiona and anayticay unramified is given in the paper by Heinzer and Swanson [HS]. The primary decomposition resut is not vaid for a primary decompositions. Here is an exampe: et I be the idea (X 2, XY ) in the poynomia ring k[x, Y ] in two variabes X and Y over a fied k. For each positive integer m, I = (X) (X 2, XY, Y m ) is an irredundant primary decomposition of I. However, for each integer k there exists an integer m, say m = k + 1, such that (X, Y ) k (X 2, XY, Y m ). Hence the resut can ony The author was partiay supported by the NSF. 1

2 hod for some primary decompositions. Section 1 contains the necessary background and some reductions towards the proof of the main theorem on the primary decompositions. The reductions make it necessary to prove a version of the inear uniform Artin-Rees emma in the spirit of Huneke s paper [Hu]. This is done in Section 2. Section 3 contains the main resuts concerning the primary decompositions and the reguarity of powers of an idea. Section 4 contains severa more versions of the uniform Artin-Rees emma reating to powers of ideas and products of powers of various ideas and some reated questions. 1. Background We first make the observation that if φ : R S is a ring homomorphism and I an idea in S, then any primary decomposition I = q 1 q s of I gives a possiby redundant primary decomposition φ 1 (I) = φ 1 (q 1 ) φ 1 (q s ) of the contraction of I to R. Namey, each φ 1 (q i ) is primary to φ 1 ( q i ). In a our appications of this, φ wi be the incusion homomorphism: by the observation then any primary decomposition of an idea I in S contracts to a primary decomposition of the contraction IS R. In the proof of the first theorem to come, we need the foowing we-known emma: Lemma 1.1: If a prime idea in a Noetherian ring R is associated to some idea generated by a non zerodivisor, then it is associated to every idea generated by a non zerodivisor that it contains. and 4: With this we prove a generaization of Ratiff s theorem which is needed in Sections 3 Theorem 1.2: (Ratiff [R]) Let I 1,..., I d be ideas in a Noetherian ring R. Then the set n1,...,n d Ass (R/I n 1 1 In d d ) of a associated prime ideas of R/In 1 1 In d d over non negative integers, is finite. Proof: as the n i vary First we reduce to the case when a the ideas I i are principa and generated by non zerodivisors. This we may do by first passing to the extended Rees ring S = R[I i t, t 1 ]: if we can prove the resut for ideas I 1 S,..., I i 1 S, t 1 S, I i+1 S,..., I d S, then as primary decompositions of ideas in S contract to a primary decomposition in R and as I n 1 1 In i 1 i 1 t n i I n i+1 i+1 In d d S R = In 1 1 In d d, we are done. Thus we may assume that a the ideas I i are principa and generated by non zerodivisors. 2

3 To finish the proof of the theorem it now suffices to prove that when a the n i 1, Ass (R/I n 1 1 In d ) is contained in Ass (R/I 1 I d ). But this foows by the emma. d The main resuts of this paper are about inear properties of powers of an idea. The foowing resut of Katz and McAdam is the first step in this direction: Theorem 1.3: (Katz-McAdam [KM, (1.5)]) Let R be a Noetherian ring. For any idea I there exists an integer such that I n : J n = I n : J n+1 for a ideas J and a n 0. Actuay, Katz and McAdam prove the existence of such an integer depending on I and J, but their argument can be easiy extended to show that there is an independent of J. A proof of this generaization appeared in [S], but I incude it here for competeness: Proof: Let S = R[It, t 1 ], where t is an indeterminate over R. It is cear that I n : R J = (t n S : S J) R for a n and a ideas J of R. Thus it suffices to show that there exists an integer such that t n S : S J n = t n S : S J n+1 for a n and a ideas J of S. So by repacing R by S we may assume without oss of generaity that I is a principa idea generated by a reguar eement a. Now et (a) = q 1 q s be a primary decomposition of I = (a). Let be such that ( q i ) q i for a i = 1,..., s. We prove by induction on n that this works for a ideas J. First assume that n = 1. If J q i then by the choice of we have J q i. If, however, J q i, then q i : J = q i : J +1 = q i as q i is primary. Thus I : J = (q 1 : J ) (q s : J ) = (q 1 : J +1 ) (q s : J +1 ) = I : J +1, so we are done. Now et n 1. It is enough to show that I n : J n+1 I n : J n. Let x be an eement of I n : J n+1. Then x I n 1 : J n+1 and by induction assumption x ies in I n 1 : J (n 1). Let y be an eement of J (n 1) and z an eement of J +1. Then xy ies in I n 1, so xy = ba n 1 for some b. As yz is in J n+1, we get that xyz ies in I n, so xyz = ca n for some c. These two equations say that ca n = bza n 1, hence that ca = bz. Since z is an arbitrary eement of J +1, this says that b (a) : J +1. So by induction assumption for n = 1 we get that b (a) : J. Now et w be an eement of J. Then xyw = ba n 1 w (a n ). This hods for arbitrary y J (n 1) and arbitrary w J, so x I n : J n. In particuar, if R is a oca ring with maxima idea m and I is a prime idea of dimension one, then m kn I (n) I n, where I (n) denotes the n th symboic power of I. This answers a question of Herzog s in [He]. With this, we are ready for some reductions towards the main theorem: 3

4 Main Theorem: (see Theorem 3.4) For every idea I in a Noetherian ring R there exists an integer k such that for a n 1 there exists an irredundant primary decomposition I n = q 1 q s such that q i nk q i for a i. We start outining the proof here, but we compete it ony in Section 3. Let P be a prime idea associated to some R/I n. We want to prove that we can find an integer k, independent of n but possiby dependent on P, for which there exists a primary decomposition I n = q 1 q s such that whenever P = q i, then P nk q i. If we can find such a k for each P, then as there are ony finitey many such P (see Theorem 1.2), we have proved the theorem. Our strategy is to first ocaize at P : if we can prove that P nk R P q i R P, then certainy P nk q i R P R = q i. Thus we may assume that R is a oca ring and that P is the maxima idea of R. We may aso assume that the resut is aready known for Q-primary components, Q propery contained in P, as those components do not affect any possibe P -primary components. Let be the Katz-McAdam s constant for I, i.e., has the property that for a integers n and a ideas J, I n : J n = I n : J n+1. Thus in particuar if P is the maxima idea in the ring, I n : P n = I n : P n+1, and this is the intersection of a the primary components of I n not primary to P. To simpify notation we denote this idea as I <n>. Note that I n = I <n> (I n + P m ) for a arge integers m and that this intersection decomposes into a primary decomposition of I n. In order to prove the main theorem we have to bound the east possibe m above by a inear function in n. One way of accompishing this is to find some sort of inear Artin-Rees emma: namey, we want to find an integer k such that P m I <n> P m kn I <n> ( ) for a m kn and a n. If we can do that, then by the choice of for a m kn + n, P m I <n> P m kn n I n. Hence I n = I <n> (I n + P (k+)n ) gives a desired primary decomposition of I n. We prove the inear Artin-Rees statement (*) for specia I in Section 3 (see Theorem 3.3) where we then aso finish the proof of the main theorem for a I (see Theorem 3.4). 4

5 2. Artin-Rees Lemma (foowing [Hu]) The goa of this section is the foowing generaized Artin-Rees Lemma aong the ines of Huneke s Uniform bounds in Noetherian rings : (See Theorem 2.7.) Let R be a compete Noetherian oca ring with infinite residue fied. Let N M be finitey generated R modues. Then there exists an integer k such that for a a satisfying a technica condition to be described ater, for a proper ideas J in R and for a integers m k, aj m M N aj m k N. The case a = 1 was proved with ess restrictive hypotheses in [Hu, Theorem 4.12]. The case we need for the main theorem of this paper is when a = 1, M = R and N equas I <n> for some n. The integer k in the statement of the goa certainy depends on n. By using the statement for a arge set of eements a, we prove in Theorem 3.3 that for specia I we can bound k(n) above by a inear function in n. The connection between the statement and the promised inear bound for k(n) may not be immediatey obvious and we expain it in Section 3. This section is somewhat technica and the reader may skip the rest of it. Lemma 2.1: (Compare with [Hu, Proposition 2.2].) Let N K M be R modues and et a be an eement of R. Assume that there exist integers h and k such that for a ideas J in R, aj m M K aj m h K for a m h and aj m K N aj m k N for a m k. Then aj m M N aj m h k N for a m h + k. The proof is straightforward. An important consequence is Proposition 2.2: (Compare with [Hu, Proposition 2.2].) Let R be a Noetherian ring, a an eement of R and N M two finitey generated R modues. Assume that 0 = K 0 K 1 K t = M/N is a fitration of M/N such that K i /K i 1 = R/Pi, where P i is an idea of R. Let k i be an integer satisfying aj m P i aj m k i P i for a ideas J and a m k i. Then for a J and a m k k t, Proof: aj m M N aj m k 1 k t N. It suffices to prove that if M/N = R/P for some idea P in R, then aj m P aj m k P impies aj m M N aj m k N. By assumption M = N + xr for some x M such that N : R x = P. Then aj m M N = aj m N + aj m x N. Let y aj m x N. Write y = arx for some r J m. Then ar is an eement of aj m (N : R x) = aj m P aj m k P. Thus y = arx ies in 5

6 aj m k P x aj m k N. The foowing is a technica emma which wi aow us to concude the main resut for a ideas primary to the maxima ideas once we know the resut for any minima reduction: Lemma 2.3: (Compare with [Hu, Lemma 3.1].) Let J, L, and P be ideas in a ring R and et a be an eement of R. Suppose that (i) al m P aj m k P for a m k, (ii) L m P J m k P for a m k, and (iii) J m L m h + P for a m h + 1. Then aj m P aj m k h 1 P for a m k + h + 1. Proof: We first prove J m L m h + J m k h 1 P for a m h + k + 1. (#) If m = h + k + 1, this is just (iii). Now assume (#) for some m h + k + 1. Mutipy through by J and use (iii): J m+1 (JL m h + J m k h P ) (L m+1 h + P ). Let x J m+1. Write x = u + v = y + z for some u JL m h, v J m k h P, y L m+1 h and z P. Then u y = z v (JL m h + L m+1 h ) P L m h P J m h k P by (ii). Thus x = (u y) + y + v J m h k P + L m+1 h, which finishes the induction. Hence aj m P a(l m h + J m k h 1 P ) P al m h P + aj m k h 1 P aj m h k P + aj m k h 1 P. In the proof of the main resut of this section we proceed by induction on the dimension of M/N: important ingredients in the induction step are eements which may be thought of as Cohen-Macauayfiers. We denote their set as CM(R): Definition 2.4: (See [Hu, 2.11].) Let F : 0 F t f t Ft 1 F 1 f 1 F0 be a compex of finitey generated free R modues. We say that F satisfies the standard condition on rank if rank (f t ) = rank F t and for 1 i < t, rank (f i )+rank (f i+1 ) = rank F i. Let I(f i ) denote the idea generated by the rank-size minors of f i. We say that F satisfies the standard condition on height if ht (I(f i )) i for a i. 6

7 With this, we define CM(R) to be the set of a eements x R such that for a compexes F satisfying the standard rank and height conditions, xh i (F) = 0 for a i 1. Lemma 2.5: (Compare with [Hu, Lemma 3.3].) Let R be a Noetherian ring and P a prime idea in R. Let a and c be eements of R not contained in P such that the image of c in R/P ies in CM(R/P ). Let L be any idea in R generated by eements a 1,..., a d such that ht (a 1,..., a d )R/P = d. Then if there exists an integer k such that al m (P + cr) al m k (P + cr) for a m k, then aso for a m k. al m P al m k P Note that this emma does the inductive step: assuming the main resut of this section (Theorem 2.7) for a (N, M) such that dim(m/n) < dim(r/p ), we can prove it aso for the pair (P, R), but ony for the specia ideas L. However, Lemma 2.3, together with some more technicaities, takes care of the rest of the ideas. Proof of Lemma 2.5: First note that al m P = al m (P +cr) P al m k (P +cr) P = al m k P + al m k cr P. Thus it suffices to prove that al m k cr P ies in al m k P. First note that al m k cr P equas c(al m k P ). Thus by the choice of c it suffices to prove that alm k P al m k P ies in some H i(f) for some F satisfying standard rank and height conditions and for some i 1. First et G be a minima homogeneous free resoution of (X 1,..., X d ) m k in the poynomia ring T = Z[X 1,..., X d ]. Let the maps in the compex be caed g i. It is weknown that I(g i ) = (X 1,..., X d ) for a i (cf. the proof of Lemma 3.3 in [Hu]). Let G = G T R, where the map T R takes X i to a i. Let F be the same as G except that the ast map is composed with mutipication by a. Namey, F equas g t 1 F : 0 G t G t 1 G 1 = R (m k+d 1 m k 1 ) af 1 G0 = R 0, where the entries of (the matrix) f 1 are a the eements of a generating set of L m k. Note that H 0 (F ) = R/aL m k. We compare F with a free resoution H of R/aL m k : F : G 2 R (m k+d 1 m k 1 ) af 1 H : R 2 R (m k+d 1 m k 1 ) af 1 R 0 R 0 where the eftmost vertica arrow exists because G 2 is free and the second compex is exact. Now tensor both compexes with R R/P and set F to be F R R/P. We get a 7

8 surjective map from H 1 (F) = H 1 (F R R/P ) to H 1 (H R R/P ) = Tor R 1 (R/aL m k, R/P ) = al m k P al m k P. As by assumption F satisfies the standard conditions on height and rank, then ch 1 (F) = 0, so the emma is proved. Note that if P is the ony maxima idea of the ring, the conusion of the emma is triviay true with k = 1 for a a in the ring. However, the concusion of this emma is fase if a is an eement of P and P is not the maxima idea. For this reason we need to put restrictions on a in the main theorem (Theorem 2.7) of this section: Definition 2.6: Let R be a compete Noetherian oca ring and et N M be finitey generated R modues. We define the condition (C) inductivey on dimension of M/N: if dim(m/n) = 0, we say that a the eements of R satisfy the condition (C) with respect to the pair (N, M). Now suppose that M/N has positive dimension. We say that an eement a satisfies the condition (C) with respect to (N, M) if there exists a prime fitration 0 = K 0 K 1 K t = M/N of M/N (i.e., for each i, K i /K i 1 = R/Pi for some prime idea P i in R), such that a satisfies the condition (C) with respect to each (P i, R). It remains to define the condition (C) with respect to (P, R) where P is a nonmaxima prime idea in R. By [Hu, Proposition 4.5 (i)] and the Cohen Structure Theorem, CM(R/P ) is nonzero. We say that a satisfies the condition (C) with respect to (P, R) if a is not contained in P and if there exists an eement c of R whose image in R/P is nonzero in CM(R/P ) such that a satisfies the condition (C) with respect to (P + cr, R). Note that an eement satisfies the condition (C) with respect to (N, M) if and ony if it avoids a finite set of nonmaxima primes. Thus 1 satisfies (C) for a (N, M), and an eement a satisfies (C) with respect to (N, M) if and ony if a of its powers satisfy (C) with respect to (N, M). Note that by the Prime Avoidance Theorem there are aways eements satisfying (C) with respect to some (N, M) even in the maxima idea. With this we are ready to prove the main resut of this section: Theorem 2.7: (Compare with [Hu, Theorem 4.12].) Let R be a compete Noetherian oca ring with infinite residue fied. Let N M be finitey generated R modues. Then there exists an integer k such that for a proper ideas J in R, for any eement a in R satisfying condition (C) with respect to (N, M), and for a m k, aj m M N aj m k N. Proof: First note that it suffices to prove the theorem for ideas primary to the maxima idea m ony, because then for an arbitrary J we have: aj m M N a(j+m ) m M N a(j + m ) m k N (aj m k + m )N = aj m k N. Consider a prime fitration of M/N as in the definition of the condition (C). Note that 8

9 by the choice of P i, a satisfies the condition (C) with respect to a these (P i, R). Aso, for each of these P i m there exists some c i whose image is nonzero in CM(R/P i ), such that a aso satisfies the condition (C) with respect to (P i + c i R, R). Now we proceed as in [Hu, Theorem 3.4] using induction on dim(m/n). By using Proposition 2.2 and the given prime fitration of M/N, it suffices to prove that for each P i there exists an integer k such that for a m-primary ideas J and for a m k, aj m P i aj m k P i. If dim(r/p i ) = 0, then P i is the maxima idea m of R, and the theorem foows easiy with k = 1 for a a. Now assume that d = dim(r/p i ) > 0. We rename P i as P and c i as c. By induction on dimension there exists an integer k c such that for a a satisfying the condition (C) with respect to (N, M), for a m-primary ideas J and for a m k c, aj m (P + cr) aj m k c (P + cr). Aso, as in the proof of [Hu, Theorem 4.12], there exists an integer s and an eement t m \ P such that for a ideas J in R/P, t mutipies the integra cosure of J n R/P into J n s R/P. (In the notation of [Hu], t T s (R/P ). In fact, in this case T s (R/P ) = R/P by the Uniform Briançon-Skoda Theorem [Hu, Theorem 4.13].) We may assume either by induction with a = 1 or by using Huneke s [Hu, Theorem 4.12] that there exists an integer k t such that for a ideas J and for a m k t, J m (P + tr) J m k t (P + tr). We caim now that aj m P aj m k t k c s 1 P for a m k t + k c + s + 1 and for a m-primary ideas J. As R/P has an infinite residue fied, we may choose eements a 1,..., a d in J whose images in R/P generate a reduction of the image of J in R/P (see [NR, Section 5]). As we may assume that J is m-primary, ht (a 1,..., a d )R/P = d. Let L = (a 1,..., a d )R. Then by Lemma 2.5 al m P al m k c P aj m k c P (i) and (ii) for a a R satisfying condition (C) with respect to (N, M) and for a m k c. Now note that the integra cosures of JR/P and LR/P are the same, so by the choice of t, t(jr/p ) m (LR/P ) m s. Thus tj m (L m s + P ) (P + tr) = L m s (P + tr) + P L m s k t (P + tr) + P = L m s k t tr + P 9

10 Moduo P then tj m R/P L m s k t tr/p, so as t is not in P, J m R/P L m s k t R/P. Thus J m L m s k t + P (iii) for a m s + k t. Now we appy Lemma 2.3 to the dispayed equations (i), (ii) and (iii) to obtain that aj m P aj m k t k c s 1 P. Note that the assumptions in this theorem are overy restrictive. However, they hep keep the notation simpe and the statement is enough for the main theorem of the paper which is to be proved in the next section. 3. Primary decompositions and reguarity of powers of ideas We first appy the main theorem of the previous section (Theorem 2.7) to the case when M = R and N varies over powers of a specia idea: Theorem 3.1: Let R be a compete Noetherian oca ring with infinite residue fied and x an eement of R which is not a zerodivisor. Then there exists an integer k such that for any proper idea J of R, for a integers n, a m kn and any eement a in R which satisfies the condition (C) with respect to the pair ((x), R), Proof: aj m (x) n aj m kn (x) n. Let 0 = K 0 K 1 K t = R/(x) be a prime fitration of R/(x), i.e., each K i /K i 1 is isomorphic to R/P i for some prime idea P i in R. The P i may not be distinct, but to simpify notation we think of them as distinct. Note that for every integer n, R/(x) n has a prime fitration in which each P i occurs n times and no other prime ideas occur in the fitration. This foows by induction, using the fact that (x n )/(x n+1 ) = R/(x) and using the short exact sequences 0 (x) n /(x) n+1 R/(x) n+1 R/(x) n 0. By Theorem 2.7 there exists an integer k i such that for a proper ideas J in R and a integers m k i, aj m P i J m k i P i. Let k = i k i. Then by Proposition 2.2, for a proper ideas J, a given eements a in R, a n and a m kn, aj m (x) n aj m kn (x) n. This proof aso shows: 10

11 Proposition 3.2: Let x be a non zerodivisor in a compete Noetherian ring R. Then an eement a of R which satisfies the condition (C) with respect to the pair ((x), R) aso satisfies the condition (C) with respect to a ((x n ), R). A coroary is the promised uniform inear Artin-Rees emma for symboic powers: Theorem 3.3: Let I be a principa idea generated by a non zerodivisor in a Noetherian oca ring (R, P ). Let I <n> denote the intersection of a the primary components of I n which are not primary to P. Then there exists an integer k such that for a n and a m kn, P m I <n> P m kn I n. Proof: Let X be an indeterminate over R and S the competion of R[X] P R[X] in the P R[X]-adic topoogy. As S is faithfuy fat over R, I <n> S is the intersection of a primary components of I n S which are not primary to P S. Suppose that there exists an integer k such that P m S I <n> S P m kn I n S. Then P m I <n> P m S I <n> S R P m kn I n S R = P m kn I n, so we are done. Thus we may repace R by S and assume that R is a compete oca ring with infinite residue fied. Let a be an eement of R such that a satisfies the condition (C) with respect to (I, R), a ies in P but not in any other prime idea associated to any power of I, and a is a non zerodivisor. As a of these conditions on a just mean that a has to avoid finitey many prime ideas propery contained in P, then by the Prime Avoidance Theorem a exists. Thus by Theorem 3.1 there exists an integer k such that a P m I n a P m kn I n for a n,, and a m kn. As a is not contained in contained in any associated prime idea of I <n>, then I <n> = I n : a for a sufficienty arge. Thus as a is not a zerodivisor, the dispayed equation says that P m I <n> = P m (I n : a )(a P m I n ) : a P m kn I n. Now we can prove the main theorem: Theorem 3.4: For every idea I in a Noetherian ring R there exists an integer k such that for a n 1 there exists an irredundant primary decomposition I n = q 1 q s such that q i nk q i for a i. Proof: Suppose we can prove the theorem for the idea (t 1 ) in R[It, t 1 ]. Then as t n R[It, t 1 ] R = I n and as every primary decomposition of t n R[It, t 1 ] contracts to a primary decomposition of I n, we have aso proved the theorem for the idea I in R. Thus 11

12 without oss of generaity we may assume that I is a principa idea generated by a non zerodivisor x. As in the reductions at the end of Section 1, we may ocaize and assume that R is a Noetherian oca ring with maxima idea P and it then suffices to find an integer k and primary decompositions I n = q 1 q s such that whenever q i = P, then P nk q i. Set I <n> to be the intersection of those primary components of I n which are not primary to P. Then by Theorem 3.3 there exists an integer k such that for a n and for a m kn, P m I <n> P m kn I n. This means that I n equas I <n> (I n + P kn ), which gives a desired primary decomposition of I n. For genera ideas, it is very difficut to find the integer k which satisfies the theorem. For one thing, computing the embedded components of primary decompositions of arbitrary ideas is very difficut (see [EHV]). However, for monomia ideas in poynomia rings, computing primary decompositions is quite fast, see for exampe [STV]. It turns out that one can obtain k for monomia ideas even without any primary decomposition agorithms: in [SS] Smith and I prove that if I and J are monomia ideas in the poynomia ring K[x 1,..., x d ] over a fied K and if is the maximum of the degrees of eements in a minima monomia generating set, then for every integer n there exists a primary decomposition I n + J = q 1 q s such that q 2dn i q i for a i. Thus in this case, k = 2d, which is easiy computabe. For a more precise statement and for the anaogous resut for Frobenius powers of I moduo J, refer to [SS]. A consequence of the existence of the integer k in Theorem 3.4 is that we can bound the Castenuovo-Mumford reguarity of powers of ideas ineary in the powers. Namey, for every homogeneous idea I in a poynomia ring R = K[x 1,..., x d ] over a fied K there exists an integer k such that the reguarity of I n is bounded above by kn (see Theorem 3.6 beow). When dim(r/i) 1, Chander [C] and Geramita, Gimigiano and Pitteoud [GGV] found expicit formuas for upper bounds of the reguarity of I n. In higher dimensions, the obstruction to finding expicit bounds is precisey the ack of understanding in which degrees I n and the saturation of I n start agreeing (in our notation the saturation of I n is I <n> = m I n : (x 1,..., x d ) m ). We say that I is m-saturated if I and its saturation agree in degrees m and higher. Definition 3.5: Let I be a homogeneous idea in the poynomia ring R = K[x 1,..., x d ] over a fied K. Let 0 F d F 1 F 0 I 0 12

13 be a minima graded free resoution of I. Thus each F i = R( b ij ) for some integers b ij. We say that I is m-reguar if b ij i m for a i and a j. The (Castenuovo-Mumford) reguarity reg (I) of I is then defined to be the east integer m for which I is m-reguar. Theorem 3.6: Let I be a homogeneous idea in the poynomia ring R = K[x 1,..., x d ] of d variabes over a fied K. Then there exists an integer k such that reg (I n ) kn for a n. Proof: We proceed by induction on dim(r/i). If dim(r/i) = 0 (or 1 aso) this is true by [C] (or Theorem 1.1 in [GGP]). Thus we may assume that dim(r/i) > 0. Note that the hypotheses or the concusion of the theorem are unaffected if we repace K by an infinite fied extension. Thus we may assume that K is infinite to start with. Hence there exists a homogeneous eement h of degree 1 in R which avoids a the finitey many prime ideas different from (x 1,..., x d ) which are associated to the powers of I. By induction on the dimension then there exists an integer k 1 such that reg (I n +hr) k 1 n for a n. Aso, by Theorem 3.4 there exists an integer k 2 such that I n and the saturation I <n> of I n agree in degrees k 2 n and arger. Now et k = max{k 1, k 2 }. Then by assumptions I n is kn saturated and I n + h is kn-reguar. Now we use a emma of Bayer and Stiman ([BS, (1.8) Lemma]) which says that if h is a form of degree one in R which is not a zerodivisor on R moduo the saturation of J such that J is m-saturated and J + hr is m-reguar, then J is m-reguar. By appying that we get that I n is kn-reguar. 4. More uniform Artin-Rees emmas for powers of ideas Simiar techniques as in the proof of Theorem 3.4 show: Theorem 4.1: Let I be an idea in a Noetherian ring R. Then there exists an integer k such that for a ideas J in R, for a integers n and for a m kn, J m I n J m kn I n. Proof: As in the proof of Theorem 3.4 we may assume that I is principa and generated by a non zerodivisor. To prove that J m I n J m kn I n, it suffices to prove the incusion after ocaization at every prime idea associated to J m kn I n. By Theorem 1.2 there are ony finitey many such prime ideas, thus if we find a k for each one of them, we are done. Hence we may ocaize and assume that R is a Noetherian oca ring with, say, maxima idea P. As in the proofs in Section 3 we may pass to the competion of R[X] P R[X] in the P R[X]-adic topoogy, whence we may appy Theorem 3.1 to compete the proof. 13

14 Moreover, a simiar but messier proof shows that given ideas I 1,..., I in a Noetherian ring R, there exist integers k i, i = 1,...,, such that for a ideas J, a integers n i 1 and a m k 1 n k n, J m I n 1 1 In J m k 1n 1 k n I n 1 1 In. The ony new observation we need for this is that if x 1,..., x are non zerodivisors in R, then R/x n 1 1 xn R has a prime fitration in which each of the prime ideas occurring in a prime fitration of R/x i R occurs exacty n i times. And this can be easiy generaized to the foowing: Let I 1,..., I be ideas in a Noetherian ring R. Then there exist integers k i, i = 1,...,, such that I n 1 1 I n I n 1 1 I n 2 2 I n 1 1 I n 3 3 In 1 k n 1 I n In 2 k 1 (n 1 k n ) k n 2 I n 1 k n 1 I n This brings us to the muti-idea version of the uniform Artin-Rees emma: etc. Theorem 4.2: (Compare with [Hu, Theorem 4.12].) Let R be a Noetherian ring satisfying at east one of the foowing conditions: (i) R is essentiay of finite type over a Noetherian oca ring. (ii) R is a ring of positive prime characteristic p and is modue-finite over R p (R p is the subring generated by pth powers of eements of R). (iii) R is essentiay of finite type over Z. Then given any finitey generated R modues N M there exists an integer k such that for a integers, for a ideas J 1,..., J in R and a integers n i k, i = 1,...,, J n 1 1 J n M N J n 1 k 1 J n k N. Moreover, if R is a compete oca Noetherian ring with infinite residue fied and a satisfies the condition (C) with respect to (N, M), then aso (for a possiby different k) aj n 1 1 J n M N aj n 1 k 1 J n k N. In Section 2 we proved the specia case of this when = 1 and the ring is compete and oca. The proof proceeds as in Section 2 or as in [Hu], with some modifications: 1) As in Proposition 2.2, we may assume that M = R and N is a prime idea P in R. 2) As in the proof of Theorem 2.7 we may assume that each J i is primary to some maxima idea in R. 3) We define T (R) = {c R : t such that cj n 1 1 J n 14 J n 1 t 1 J n t for a ideas J i, a n i }

15 (Here, the overine above an idea stands for the integra cosure.) Then the natura generaizations of [Hu, Proposition 4.7] and [Hu, Theorem 4.10] (by using a muti-idea version of the Briançon-Skoda Theorem for Proposition 4.7 and a muti-rees ring for Theorem 4.10) show that for any R satisfying (i), (ii) or (iii), T (R/P ) is nonzero for a prime ideas P in R and, moreover, that T (R/P ) is nonzero. 4) As in the proof of [Hu, Theorem 4.12], CM(R/P ) is nonzero for every prime idea P in R. 5) (Compare with Lemma 2.3.) Let L 1,..., L, J 1,..., J, and P be ideas in R. For any -tupe (n 1,..., n ), J n stands for the idea J n 1 1 J n, and for any integer n, n stands for the -tupe (n,..., n). Let h and k be integers. Assume that (i) al m P al m k P for a m i k, (ii) L m P L m k P for a m i k, and (iii) J m L m h + P for a m i h + 1. Then aj m P aj m k h 1 P for a m i k + h ) A muti-idea version of Lemma 2.5 hods: Let eements a, c not ie in a prime idea P and assume that the image of c in R/P ies in CM(R/P ). Let L 1,..., L be any d- generated ideas in R such that for each i, i = 1,...,, ht (L i R/P ) = d. Suppose that for a given integer there exists an integer k such that for a integers m 1,..., m k, al m (P + cr) al m k (P + cr). Then for a m 1,..., m k, al m P al m k P. Now the proof proceeds as in Theorem 2.7, except that if the ring is not oca, we aso need to use some methods of the proof of [Hu, Theorem 3.4]. The coroary of this is that for every idea I in a Noetherian ring R there exists an integer k such that for a ideas J 1,..., J and a m 1,..., m k, J m I n J m kn I n. A natura foow-up, which was actuay my main motivation for studying these asymptotic properties of primary decompositions, is: Open Question: Given eements x 1,..., x in a Noetherian ring R of positive prime characteristic p, does there exist an integer k such that for a integers n there exists a primary decomposition (x pn 1,..., xpn ) = q 1 q s such that q i kµ( q i )p n q i? Here, µ( ) stands for the minima number of generators. In anaogy with the proof of Theorem 2.7, one may want to start with something ike 15

16 Theorem 1.2 for ideas (x pn 1,..., xpn ) as n varies. However, Katzman gave a counterexampe in [K]: if R is K[X, Y, t]/(xy (X + Y )(X + ty )), where K is a fied of characteristic p and X, Y, t are indeterminates, then the set n Ass (R/(X pn, Y pn )) is infinite. Nevertheess, Smith and I verified that the stated open question and the anaog of the Katz-McAdam s Theorem 1.3 do hod in this exampe (see [SS]). We aso proved that the stated open question has a positive answer when the x i are monomias and R is a poynomia ring moduo a monomia idea (see [SS] for precise statements). In genera, even the anaog of the Katz-McAdam theorem is an open question for ideas of the form (x pn 1,..., xpn ) as n varies. The foowing specia case is known: if the x i form a reguar sequence, a simiar argument as in Theorem 1.3, together with an argument from [HH, Theorem 4.5], gives: Proposition 4.3: Let R be a Noetherian ring and x 1,..., x s a reguar sequence in R. Let be as in Katz-McAdam s theorem for I = (x 1,..., x s ). Then for any n 1,..., n s 1 and any idea J, (x n 1 1,..., xn s s ) : J N = (x n 1 1,..., xn s s ) : J N+1, where N = ( n i ) s + 1. Acknowedgement: I thank Karen Smith for a the discussions regarding this materia and for a the suggestions which heped improve the carity of this presentation. 16

17 Bibiography [BS] D. Bayer and M. Stiman, A criterion for detecting m-reguarity, Invent. Math. 87 (1987), [C] K. A. Chander, Reguarity of the powers of an idea 1, preprint. [EHV] D. Eisenbud, C. Huneke and W. Vasconceos, Direct methods for primary decomposition, Invent. math. 110 (1992), [GGP] A. V. Geramita, A. Gimigiano and Y. Pitteoud, Graded Betti numbers of some embedded rationa n-fods, Math. Ann. 301 (1995), [HS] W. Heinzer and I. Swanson, Ideas contracted from 1-dimensiona overrings with an appication to the primary decomposition of ideas, preprint. [He] J. Herzog, A homoogica approach to symboic powers, in Commutative Agebra, Proc. of a Workshop hed in Savador, Brazi, 1988, Lecture Notes in Mathematics 1430, Springer-Verag, Berin, 1990, [HH] M. Hochster and C. Huneke, F-reguarity, test eements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), [Hu] C. Huneke, Uniform bounds in Noetherian rings, Invent. Math. 107 (1992), [KM] D. Katz and S. McAdam, Two asymptotic functions, Comm. in Ag. 17 (1989), [K] M. Katzman, Finiteness of e Ass F e (M) and its connections to tight cosure, preprint. [NR] D. G. Northcott and D. Rees, Reductions of ideas in oca rings, Proc. Cambridge Phi. Soc. 50 (1954), [R] L. J. Ratiff, Jr., On prime divisors of I n, n arge, Michigan Math. J. 23 (1976), [SS] Karen E. Smith and I. Swanson, Linear bounds on growth of associated primes for monomia ideas, preprint. [STV] B. Sturmfes, N. V. Trung and W. Voge, Bounds on degrees of projective schemes, Math. Annaen 302 (1995), [S] I. Swanson, Primary decompositions of powers of ideas, Commutative Agebra: Syzygies, Mutipicities, and Birationa Agebra: Proceedings of a summer research conference on commutative agebra hed Juy 4-10, 1992 (W. Heinzer, C. Huneke, J. D. Say, ed.), Contemporary Mathematics, Voume 159, Amer. Math. Soc., Providence, 1994, pp