Bivariate Hilbert functions for the torsion functor

Transcription

1 Journal of Algebra ) Bvarate Hlbert functons for the torson functor Emanol Theodorescu Department of Mathematcs, Unversty of Mssour, Columba, MO 65211, USA Receved 27 February 2002 Communcated by Paul Roberts Abstract Let R, P ) be a commutatve, local Noetheran rng, I, J deals, M and N fntely generated R-modules. Suppose J +ann R M +ann R N s P -prmary. The man result of ths paper s Theorem 6, whch gves necessary and suffcent condtons for the length of Tor M/I n M,N/J m N),toagree wth a polynomal, for m, n 0. As a corollary, t s shown that the length of Tor M/I n M,N/I n N) always agrees wth a polynomal n n,forn 0, provded I + ann R M + ann R N s P -prmary Elsever Scence USA). All rghts reserved. 1. Introducton Throughout ths paper, unless otherwse stated, R, P ) s a commutatve, Noetheran local rng wth unt and I, J are proper) deals. Also, let M, N be fnte. R-modules, m, n be nonnegatve ntegers, and let λ denote length. We would lke to study the two-varable the Hlbert functon Hn,m):= λtor M/I n M,N/J m N)). On the one hand, we have n mnd extendng results on Hn,m) of the authors of [2,7] and [1], whle on the other hand we seek two varable analogues of recent results concernng the Hlbert functon Hn):= λtor M/I n M,N)). Prevous work on Hn) appears n [5,6] and [8]. In fact, n [8] t s shown that Hn)agrees wth a polynomal n n for n large, f we smply assume that the lengths λtor M/I n M,N)) are fnte. Here we seek to gve condtons under whch Hn,m) has polynomal growth for n and m suffcently large. In some specal cases, we gve a degree bound on the resultng polynomals n n and m. Determnng the exact degree of these polynomals seems to be a more dffcult task. In the one varable case, [5] and [8] gve upper bound estmates for the degree n general whle [4,8] and [6] determne the degree n some specal cases. E-mal address: theodore@math.mssour.edu /03/$ see front matter 2003 Elsever Scence USA). All rghts reserved. do: /s )00156-x

2 E. Theodorescu / Journal of Algebra ) In hs Doctoral Thess, Bruce Felds [2] nvestgates two-varable functons of the form λtor R/I n,r/j m )), where 0, under the assumpton that I + J s P -prmary. For 2, he proves that these lengths are eventually gven by polynomals n two varables. Actually, snce Tor R/I n,r/j m ) = Tor 1 I n,r/j m ) = Tor 2 I n,j m ) by applyng twce the shftng formula), hs proof essentally shows that m,n=0 Tor j I n M,J m N), j 0, s a fnte, bgraded module, over a sutable polynomal rng over R, wherem, N are two fnte R-modules. It s then well-known that, f the lengths of homogeneous peces of a fnte bgraded module over a sutable polynomal rng) are fnte, then they are eventually gven by a polynomal functon also see Notatons and conventons). For = 0and = 1, Felds only proves that polynomal growth holds under some rather restrctve condtons: he assumes that R s regular local, and that m,n=0 I n J m ) s a fnte bgraded module over some polynomal rng n two sets of varables. Ths s, n general, a very strong condton on two deals I, J. The functon λr/i n + J m )) has also been studed by Kshor Shah [7] and Wllam C. Brown [1], who gve suffcent condtons for t to be gven by a polynomal, for m, n 0. The present paper gves a characterzaton of those cases for whch the length of Tor M/I n M,N/J m N) has polynomal growth, provded the followng condton s satsfed: J +ann R M +ann R N s P -prmary see Theorem 6). It turns out that polynomal growth doesn t always hold, even n the case 2, as Felds work mght have suggested see Remark followng Corollary 8). On the other hand, Proposton 3 shows that, provded Tor I n M,N/J m N) has fnte length, for all large m, n, ts length s always gven by a polynomal, wthout any restrctve assumpton. As a corollary to the proof of Theorem 6, under the assumpton that I + ann R M + ann R N s P -prmary, we prove that λtor M/I n M,N/I n N)) has always polynomal growth. Corollary 8 shows that, under the hypothess that both I + ann R M + ann R N and J +ann R M +ann R N be P -prmary, the length of Tor M/I n M,N/J m N)has polynomal growth f and only f both Tor M, N) and Tor 1 M, N) have fnte length. Fnally, when M N has fnte length, Theorem 9 gves the formula λ Tor M/I n M,N/J m N )) = λ Tor M, N) ) + λ Tor 1 I n M,N )) + λ Tor 1 M,J m N )) + λ Tor 2 I n M,J m N )), whch works for all 0, by assumng that all Tor wth <0 are zero. The man result of ths paper shows that, at least when J + ann R M + ann R N s P -prmary, the nature of λtor M/I n M,N/J m N)) s controlled by modules of the form I n A J m B. Therefore, a study of modules of ths knd would deepen our understandng of λtor M/I n M,N/J m N)). 2. Notaton and conventons We wll be usng free) resolutons of modules over several dfferent rngs. There wll be resolutons of modules over R, graded resolutons of graded modules over the polynomal

3 138 E. Theodorescu / Journal of Algebra ) rng n r varables, S 1 := R[X 1,...,X r ], as well as bgraded resolutons of bgraded modules over the polynomal rng n two sets of varables, S 2 := R[X 1,...,X r ; Y 1,...,Y s ]. Unless otherwse stated, the Tor s are over R. To further smplfy notaton, we denote M = n=0 M, whch s an nfntely generated) graded module over the Rees rng R I := n=0 I n.ifi s generated by x 1,...,x r,thenm s naturally an nfntely generated S 1 -graded module, va the canoncal rng homomorphsm S R I,gvenbyX x for all. The acton of S 1 on M s gven by X v k = x v k,wherev k denotes a homogeneous vector of degree k. Also, f we denote IM := n=0 I n M, then ths s a fntely generated graded module over R I, and hence over S 1, as before. It follows that M/IM = n=0 M/I n M) s a graded module over both R I and S 1. Smlarly, f we assume J = y 1,...,y s ), m,n=0 I n J m M s a bgraded module over the bgraded Rees rng R I,J := m,n=0 I n J m, and hence over the polynomal rng S 2,va a smlar map S 2 R I,J. Note that any graded free resoluton over S 1 or S 2 of some graded module, s also a free resoluton of that module over R. We wll be makng use of the fact that, n a b)graded resoluton of some S 1 -ors 2 -) graded module, say IM, by consderng just ts homogeneous part of degree k, we obtan a free resoluton, over R, of the module I k M,thekth homogeneous component of IM. We wll be makng repeated use of the fact that, f P := m,n=0 P m,n s a fnte bgraded S 2 -module, whose homogeneous peces have fnte length, then λp m,n ) s eventually gven by a polynomal. In partcular, λtor I n M,J m N)) s eventually gven by a polynomal. Indeed, we can take C a S 1 -graded free resoluton consstng of fnte free S 1 -modules) of n=0 I n M and, smlarly, D a S 1 -graded free resoluton of m=0 J m N, also consstng of fnte free S 1 -modules. Here, S 1 = R[Y 1,...,Y s ].) Then the modules n C R D have a natural structure of S 1 R S 1 = S 2 -modules. Actually, C R D s a complex of fnte, free, S 2 -modules, whose th homology s Tor R n=0 I n M, m=0 J m N). Of course, ths s a fntely generated bgraded S 2 -module. Snce the homogeneous components of ths are just Tor R I n M,J m N), t follows that, f ther lengths are fnte, then these lengths are eventually gven by a polynomal n m, n. 3. The man result In an attempt to study the length of Tor M/I n M,N/J m N) n as great generalty as possble, we frst nvestgate Tor I n M,N/J m N). It turns out that n ths case polynomal growth follows from the smplest assumpton that these Tor s have fnte length. The followng few results are essentally gven wthout proof, as ther proofs parallel those of correspondng one-varable statements see [8]). Proposton 1. Let R be a Noetheran rng not necessarly local), and J R an deal. Let S 1 be the polynomal rng over R n r varables, and let C : F 2 ψ φ F 1 F 0

4 E. Theodorescu / Journal of Algebra ) be a graded complex of graded S 1 -modules, graded by total degree. Assume that F 1, F 0 are fntely generated S 1 -modules. Then, there s l 0, such that, for all m l H 1 C R ) J m = U + J m l V Z + J m l W, where Z U and W V are fnte, graded S 1 -modules. Proof. It essentally goes as n Proposton 3 n [8]. Proposton 2. Let R, S 1, J be as n Proposton 1. LetT be a graded S 1 -module, and U, V, W, Z be fnte graded S 1 -submodules of T. Assume that Z U, and that W V, and denote L m := U + J m V Z + J m W. Then, f L m ) n,thenth degree homogeneous component of L m, has fnte length for all large values of m and n, λl m ) n ) s eventually gven by a polynomal n m and n. Proof. It follows the same path as Lemma 2b) n [8]. Proposton 3. Let R be a Noetheran rng, I,J R deals, M, N be fnte R-modules, and 0. IfTor I n M,N/J m N) has fnte length for all m, n 0, then ths length s eventually gven by a polynomal n m, n. Proof. Take an S 1 -graded resoluton by fnte free S 1 -modules of the fnte graded S 1 -module n=0 I n M. Tensor t wth N/J m N, n two steps, frst wth N call the resultng S 1 -complex C), then wth R/J m. The part gvng Tor R m=0 I n M,N/J m N), looks just lke the stuaton descrbed n Proposton 1. Therefore, by Proposton 1, we see that Tor R n=0 ) I n M,N/J m N = U + J m l V Z + J m l W for some l, allm l, whereu, V, Z and W are all fnte graded S 1 -modules. It follows that Tor R I n M,N/J m N ) = U n + J m l V n Z n + J m l W n, by lookng at homogeneous peces of degree n n the prevous Tor formula. Thus, the concluson follows from Proposton 2. Lemma 4. Let R, P ) be Noetheran, local, I,J R deals, 0. Then, for two fnte R-modules M, N, we have

5 140 E. Theodorescu / Journal of Algebra ) a) The mage of the nduced map Tor I n M,N ) Hf ) Tor M, N) s of the form I n k A for some k 0 and n k, wherea s the mage of the map Tor I k M,N) Hf ) Tor M, N). b) The mage of the nduced map Tor M, N) Hg ) Tor M,N/J m N ) has the form Tor M, N) + J m B J m B for some module B, such that Tor M, N) B. Proof. a) Let R β +1 R β R β 1 1) be a free resoluton of N. Then we have the followng commutatve dagram I n M β +1 M β +1 ψ n ψ I n M β φ n f I n Mβ 1 M β φ M β 1. Let K = ker φ and L = m ψ, sotor M, N) = K/L. Wealsohavethatkerφ n = K I n M β and m ψ n = I n L, and thus Tor I n M,N) = K I n M β )/I n L. It follows that m Hf ) ) = K I n M β + L L = I n k K I k M β ) + L L for some k and all n k. Note that ths s of the form I n k A,whereA s the mage of the map Tor I k M,N) Hf ) Tor M, N), as stated. b) Now assume that 1) gves a free resoluton of M, and tensor t wth N/J m N.We get N β +1 ψ N β φ g Nβ 1 N β +1/J m N β +1 ψ m N β /J m N β φ m N β 1 /J m N β 1.

6 E. Theodorescu / Journal of Algebra ) Agan, f we denote K = ker φ and L = m ψ, then Tor M, N) = K/L and, moreover, we obtan that for some l and m l. We also get ker φ m = K + J m l φ 1 J l N β 1)) J m N β so m ψ m = L + J m N β J m N β, Tor M,N/J m N ) = K + J m l φ 1 J l N β 1)) L + J m N β. It follows that m Hg ) = K + J m N β L + J m N β = Tor M, N) + J m B J m, B where B = N β /L. Of course, Tor M, N) B. The next proposton s an extended verson of the followng well-known result: Let R, P ) be Noetheran, local, and I R an deal. If L, M are fntely generated modules, L of fnte length, then, for any 0, the natural map Tor I n M,L) Tor M, L) s zero for n 0 see [3]). Proposton 5. Let R, P ) be a Noetheran, local rng. Let I R be an deal, M, N two fnte R-modules and 0, fxed. Then the followng are equvalent: a) I radann R Tor M, N)). b) I radann R Tor I k M,N)) for some k 0. c) I radann R Tor I n M,N)) for all n 0. d) I radann R m Tor I n M,N) Tor M, N))) for all n 0. e) mtor I n M,N) Tor M, N)) = 0 for all n 0. Proof. Clearly, c) mples a) and b). Conversely, consder the long exact sequence Tor +1 M/I n M,N ) Tor I n M,N ) α Tor M, N) β Tor M/I n M,N ). a) mples b), c) follows by consderng α and,sncei radann R Tor j M/I n M,N)) for all n 0. b) mples a) follows from c) mples a). a) mples d) and d) mples a) are mmedate, consderng α.

7 142 E. Theodorescu / Journal of Algebra ) e) mples a): f α = 0, then β s an njecton, so the concluson follows. a) mples e) follows from Lemma 4a). Here s the man result of ths paper. Theorem 6. Let R, P ) be Noetheran, local, I,J R two deals, M, N fntely generated R-modules, 0. Assume that ann R M + ann R N + J s P -prmary. Then λ Tor M/I n M,N/J m N )) s eventually gven by a polynomal n m and n f and only f I radann R Tor j M, N)) for j { 1,}. Proof. Consder the long exact sequence Tor I n M,N/J m N ) αm,n Tor M,N/J m N ) Tor M/I n M,N/J m N ) Tor 1 I n M,N/J m N ) αm,n 1 Tor 1 M,N/J m N ). We already know that the lengths of the modules above, save the one n the mddle, are eventually) gven by polynomals n one or two varables see Proposton 3). Thus, we have λ Tor M/I n M,N/J m N )) = [ λ Tor M,N/J m N )) λ m α m,n )] + λ ker α m,n) 1 = [ λ Tor M,N/J m N )) λ m α m,n )] + [ λ Tor 1 I n M,N/J m N )) λ m α 1)] m,n. 2) Therefore, we need to examne λm α m,n j ) for j { 1,}. Consder the followng commutatve dagram Tor I n M,N) ψ m,n Tor I n M,N/J m N) φ m,n Tor 1 I n M,J m N) σ m,n α m,n τ m,n Tor M, N) θ m,n Tor M, N/J m N) Tor 1 M, J m N) π m,n 3) Tor M, N/J m N)/α m,n L m,n ) 0

8 E. Theodorescu / Journal of Algebra ) where L m,n = m ψ m,n = ker φ m,n. Note that the commutatve dagram 3) s a homogeneous pece of the dagram 3 ) below. That s because Tor R s addtve, and the natural maps n 3) commute wth the acton of I and J on the modules occurrng n ths dagram. It follows that the dagram 3 ) s a commutatve dagram of bgraded S 2 -modules and maps. Tor R IM, N ) ψ Tor R φ IM, N /JN) Tor R 1 IM, JN) σ α τ Tor R M, N ) θ Tor R M, N /JN) Tor R 1 M, JN) π 3 ) Tor R M, N /JN)/α L) 0 where L = m,n=0 L m,n. Observe now that π α factors through the mage of φ, whch s a fntely generated, bgraded S 2 -module snce Tor R 1 IM, JN) s so), hence mπ α ) s a fnte, bgraded S 2 -module. Then λmπ α ) m,n ) s eventually gven by a polynomal, by classcal theory. Note that λ m α m,n ) = λ mπ α ) m,n) + λ α m,n L m,n ) ), and a smlar equalty holds for 1 n place of. From 2) and what we have just seen, t follows that λtor M/I n M,N/J m N)) s eventually gven by a polynomal, f and only f the same s true of λα m,n 1 L m,n)) + λα m,n L m,n )). We now examne λα m,n L m,n )). From 3), we fnd that α m,n L m,n ) = α m,n From Lemma 4a) and b), we get that ψ m,n Tor I n M,N ))) = θ σ) m,n Tor I n M,N )). 4) θ σ) m,n Tor I n M,N )) = I n k A + J m B J m B = I n k A I n k A J m B 5) for some k 0andn k,wherea = mtor I k M,N) Tor M, N)). We now clam that λi n k A/I n k A J m B) s dentcally zero for m, n 0fand only f t s polynomal for m, n 0, f and only f I radann R Tor M, N)). To prove ths clam, assume I radann R Tor M, N)). ThenI n k A = 0forlargen, and so λi n k A/I n k A J m B) = 0, hence polynomal, for n 0andallm. It remans to check that, f I radann R Tor M, N)), thenλi n k A/I n k A J m B) s nonzero and

9 144 E. Theodorescu / Journal of Algebra ) not gven by a polynomal, for all m, n 0. Indeed, by Proposton 5, 1) 3), we know that I radann R m Tor I n M,N) Tor M, N))) for all n,soi n k A 0foralln k. Now, snce ann R M + ann R N + J s P -prmary, there s a l 0, such that I l ann R M + ann R N + J. It follows that, for n lm + k,wehave so I n k J m + ann R M + ann R N, I n k A J m A J m B, snce we know that A B. Thus, for n lm + k, l and k fxed, λi n k A/I n k A J m B) vanshes. On the other hand, note that, for every n k, I n k A/I n k A J m B 0forallm 0. Ths s so snce, for every n k, n fxed, I n k A J m B I n k A for all large m, by Krull s Intersecton Theorem. Hence λi n k A/I n k A J m B) 0foreveryn k and m 0. Ths proves the clam, snce we proved that, above the lne d : n = lm + k n the m, n)- plane, λi n k A/I n k A J m B) always vanshes, for large m and n, whle below ths lne, the length n queston s nonzero, n case I radann R Tor M, N)). Fnally, note that both terms of the form λi n k A/I n k A J m B) occurrng n the formula 2) of λtor M/I n M,N/J m N)) also see 4) and 5)), actually occur wth the same sgn. By the clam, t follows that the sum of these two terms vanshes for all large m and n, fi radann R Tor M, N)) radann R Tor 1 M, N)). On the other hand, f I radann R Tor M, N)) radann R Tor 1 M, N)), then the sum n queston vanshes above both lnes d : n = lm+ k, d : n = l m + k one lne for each term), but t s nonzero below both these lnes, d and d. Ths means that λtor M/I n M,N/J m N)) can only then be eventually) polynomal, when both terms of the form λi n k A/I n k A J m B) vansh. And ths happens f and only f I radann R Tor j M, N)) for j { 1,}, as stated. The proof of Theorem 6 yelds the followng nterestng corollary. Corollary 7. Let R, P ) be Noetheran, local, I an deal M, N two fnte R-modules and 0. Assume that I + ann R M + ann R N s P -prmary. Then λ Tor M/I n M,N/I n N )) s gven by a polynomal, for n 0. Proof. Note that, by the proof of Theorem 6, we only have to look at each of the two smlar) terms n λtor M/I n M,N/J m N)), that turned out not to be polynomal, n general. If n each of them we set, J = I and m = n, we get two terms, each of whch looks lke λ I n k A I n k A I n B ).

10 E. Theodorescu / Journal of Algebra ) It s mmedate, by the Artn Rees Lemma, that n=0 I n k A/I n k A I n B s a fnte graded module over the Rees rng R I = n=0 I n, hence the concluson. Corollary 8. Assume that both I + ann R M + ann R N and J + ann R M + ann R N are P -prmary, n the statement of Theorem 6. ThenλTor M/I n M,N/J m N)) s eventually gven by a polynomal f and only f Tor j M, N) has fnte length for both j =, j = 1. Proof. λtor M/I n M,N/J m N)) s eventually gven by a polynomal f and only f I radann R Tor j M, N)) for j { 1,}, f and only f I + ann R M + ann R N radann R Tor j M, N)) for j { 1,}, f and only f Tor j M, N) has fnte length for both j = 1andj =. Remark. From ths corollary alone we could construct numerous examples n whch λtor M/I n M,N/J m N)) s not eventually polynomal. It suffces to take I and J to be P -prmary deals and M, N two fnte R-modules wth at least one of the two modules Tor M, N) and Tor 1 M, N) not havng fnte length. Let us gve two such examples of Tor M/I n M,N/J m N) that have non-polynomal length, the second of whch works for any value of. Frst, assume that R has postve depth and dmenson at least two. Take x 1,x 2,...,x t, t 1, to be a regular sequence, such that the deal generated by these elements s not P -prmary. Take M = R/x 1,...,x t ) s and N = R/x 1,...,x t ) r for some s r 1. Then Tor 1 M, N) = x 1,...,x t ) s /x 1,...,x t ) s+r has fnte length f and only f R/x 1,...,x t ) has fnte length. Ths s so because, by Rees theorem, x 1,...,x t ) j /x 1,...,x t ) j+1 s a free R/x 1,...,x t )-module for all j 0. Therefore Tor 1 M, N) can not have fnte length by the choce of the regular sequence. Now take I and J any two P -prmary deals: by Corollary 8, the length of Tor M/I n M,N/J m N) s not gven by a polynomal, for {1, 2}. Secondly, assume that R s nether regular, nor an solated sngularty. Then R Q s not regular for some non-maxmal prme Q. TakeM and N to be any two fnte R-modules, such that ther annhlator s Q. Note that both M Q and N Q are drect sums of copes of the resdue feld of R Q. Then Tor M, N) cannot have fnte length for any. For 1ths would mply that the localzaton at Q of Tor M, N) vanshes, gvng that R Q s regular, contrary to the choce of R.) Now, Corollary 8 says that for any choce of two prmary deals I and J, the length of Tor M/I n M,N/J m N) s not polynomal for all 0. Theorem 9. Let R, P ) be Noetheran local, I,J R deals, M, N fnte R-modules and 0. Assume that M N has fnte length. Then λ Tor M/I n M,N/J m N )) s gven by a polynomal, for m, n 0. Moreover, λ Tor M/I n M,N/J m N )) = λ Tor M, N) ) + λ Tor 1 I n M,N )) + λ Tor 1 M,J m N )) + λ Tor 2 I n M,J m N )).

11 146 E. Theodorescu / Journal of Algebra ) Proof. The frst statement follows mmedately from Theorem 6, snce, trvally, ts hypotheses are met. For the last statement, let s observe that, there s a k 0, such that, for all m 0, and n k, σ m,n n 3) s the zero map, by Proposton 5. It follows that α m,n m ψ m,n ) = α m,n ker φ m,n ) = 0, hence α m,n factors through m φ m,n, and thus as before) λm α m,n ) s eventually gven by a polynomal n m, n. Fnally, by Proposton 5 agan, we see that for each fxed m, mα m,n ) vanshes for n 0. Therefore, mα m,n ) s dentcally zero for all large m and n. We also have the long exact sequence Tor I n M,N/J m N ) αm,n Tor M,N/J m N ) Tor M/I n M,N/J m N ) Tor 1 I n M,N/J m N ) αm,n 1 Tor 1 M,N/J m N ), and we now know that α m,n = α m,n 1 = 0form, n 0. Then, λ Tor M/I n M,N/J m N )) = λ Tor M,N/J m N )) + λ Tor 1 I n M,N/J m N )). 6) We apply ths trck two more tmes. We have Tor M,J m N ) 0 Tor M, N) Tor M,N/J m N ) Tor 1 M,J m N ) 0 Tor 1 M, N), 7) where the maps marked as 0 are so by Proposton 5. We get that λ Tor M,N/J m N )) = λ Tor M, N) ) + λ Tor 1 M,J m N )). 8) Replacng M by I n M n 7) and usng the fact that m,n=0 Tor I n M,J m N) s a fnte bgraded S 2 -module, we see that the maps marked as 0 wll reman so, for every n and large m, agan by Proposton 5. We then get that λ Tor 1 I n M,N/J m N )) = λ Tor 1 I n M,N )) + λ Tor 2 I n M,J m N )). 9) Puttng together 6), 8) and 9), we obtan λ Tor M/I n M,N/J m N )) = λ Tor M, N) ) + λ Tor 1 I n M,N )) + λ Tor 1 M,J m N )) +λ Tor 2 I n M,J m N )), as stated. Note that ths also yelds a drect proof of the frst statement of ths theorem, snce the four terms on the rght-hand sde of the equalty above are eventually gven by polynomals, by classcal theory of fnte b)graded modules.

12 E. Theodorescu / Journal of Algebra ) Fnally, we gve an upper bound for the degree of the polynomal that arses n Corollary 8. Note that ths estmate also apples to the case of Theorem 9. Proposton 10. Assume the hypotheses n Corollary 8 and suppose that the length of Tor M/I n M,J m N) s gven by a polynomal, for m, n 0.Then deg λ Tor M/I n M,J m N )) l M I) + l N J ) 2. Proof. Ths s a rather crude estmate, based on the one-varable case. We smply apply Corollary 4 n [8], separately, for fxed, large enough values of m and n, then add. For the exact degree n some specal cases n one varable, though), see [6]. Acknowledgments The work n ths paper forms part of my dssertaton wrtten at the Unversty of Kansas. I would partcularly lke to thank my advsor, Prof. D. Katz, for hs constant patence and steady encouragements. I would also lke to thank both D. Katz and A. Vracu for ther comments on ths paper. Specal thanks also go to the referee for a number of valuable suggestons. References [1] W.C. Brown, Hlbert functons for two deals, J. Algebra ) [2] J.B. Felds, Length functons determned by kllng powers of several deals n a local rng, Ph.D. Dssertaton, Unversty of Mchgan 2000). [3] G. Levn, Poncare seres of modules over local rngs, Proc. Amer. Math. Soc. 72 1) 1978) [4] D. Krby, Hlbert functons and the extenson functor, Math. Proc. Cambrdge Phl. Soc ) 1989) [5] V. Kodyalam, Homologcal nvarants of powers of an deal, Proc. Amer. Math. Soc ) 1993) [6] T. Marley, Hlbert functons n Cohen Macaulay rngs, Ph.D. Dssertaton, Purdue Unversty 1989). [7] K. Shah, On equmultple deals, Math. Z ) [8] C. Theodorescu, Derved functors and Hlbert polynomals, Proc. Cambrdge Phl. Soc )