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Length Approximations for Independently Generated Ideals
Journal of Algebra 237, 708᎐718 Ž2001. doi:10.1006rjabr.2000.8598, available online at http:rrwww.idealibrary.com on
Length Approximations for Independently Generated Ideals Douglas Hanes Department of Mathematics, Uni¨ ersity of Minnesota, Minneapolis, Minnesota 55455 E-mail: [email protected] Communicated by Craig Huneke Received March 28, 2000
We study ideals primary to the maximal ideal of a commutative Noetherian local ring. When such an ideal is generated by elements which are independent in the sense of C. Lech, we prove a lower bound on the length of the quotient ring in terms of the orders of the generators. As a corollary we obtain a substantial partial result on Lech’s conjecture on the multiplicities of a flat couple of local rings. 䊚 2001 Academic Press
Key Words: commutative algebra; matrix factorization.
1. INTRODUCTION In this paper all rings are assumed to be commutative, Noetherian, and to possess a multiplicative unit element. All ring homomorphisms are assumed to be unital. Let Ž R, m. be a local ring, and let I be an m-primary ideal, so that the homomorphic image RrI has finite length. Our primary object is to show that if the ideal I is generated by elements which are independent Žas defined by Lech in w6x., then one may give a lower estimate for the length of RrI in terms of the orders of the generators of I Žwhere by the ‘‘order’’ of an element x we mean the largest t such that x g m t .. Because of possible confusion caused by the much-used term ‘‘independent,’’ we will call the elements x 1 , . . . , x r of a commutative ring R Lech-independent, or independent in the sense of Lech, if for every relation of the form x 1 u1 q x 2 u 2 q ⭈⭈⭈ qx r u r s 0 708 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
LENGTHS OF INDEPENDENT IDEALS
709
in R, one may conclude that u i g Ž x 1 , . . . , x r . R for 1 F i F r Žsee w6x.. As Žimportant. examples, we note that the maximal ideal of any local ring is generated by Lech-independent elements, and that any regular sequence x 1 , . . . , x n is Lech-independent. Moreover, note that if I : R is generated by Lech-independent elements, and if S is any flat ring extension of R, then the extended ideal IS is still generated by Lech-independent elements. Lech’s main result w6, Theorem 2x on independent sequences is the following theorem. Note that if I s Ž x 1 , . . . , x m . is an ideal of R, then the sequence of elements x 1 , . . . , x m is Lech-independent if and only if IrI 2 is a free module Žof rank m. over RrI. In Lech’s presentation the latter of the equivalent hypotheses is used. THEOREM 1.1 ŽLech.. Let x 1 , . . . , x r be a sequence of independent elements in a local ring S of embedding dimension t, and assume that SrŽ x . S is equicharacteristic and has finite length. Then r F t. The theorem may be directly applied to flat couples of local rings, which are defined as follows: DEFINITION 1.1. A flat couple or flat local extension of local rings is an extension Ž R, m. : Ž S, n. of local rings with the properties that S is a flat R-module and that mS : n. By the embedding dimension edimŽ R . of a local ring Ž R, m. we mean the minimal number of generators Ž m. of its maximal ideal m. It follows from Theorem 1.1 that if Ž R, m. : Ž S, n. is a flat local extension of local rings of dimension d, then the embedding dimension edimŽ R . of R is less than or equal to that of S, since the minimal generators of m map to an independent set of elements of S. In fact, by employing a similar result with regard to localization, Lech shows in w6x that edim Ž R . F edim Ž S . for any flat local extension Ž R, m. : Ž S, n.. Our main result is the following theorem, which gives a lower bound on the length of SrI when I is an n-primary ideal of the local ring Ž S, n. generated by independent elements. THEOREM 1.2. Let x 1 , . . . , x r be a set of Lech-independent elements of a commutati¨ e local ring ŽT, mT . which generate an ideal primary to the maximal ideal. Suppose that for 1 F i F r, x i g mTt i . Then r
l Ž Tr Ž x 1 , . . . , x r . T . G
Ł ti .
is1
710
DOUGLAS HANES
Such an inequality is well known for any system of parameters x 1 , . . . , x d , and the theorem represents a generalization of this fact in the case that x 1 , . . . , x d is a regular sequence. We note that the statement of the theorem is implied by the results of w6x in the case that each x i is actually a t th power of some element of mT . Of course nothing like this can be assumed about a general element x i g mTt , but we will employ the following theorem of w1x in order to realize each x i as the t th power of some square matrix with entries in mT . THEOREM 1.3 ŽBackelin, Herzog, and Ulrich.. Let R be a commutati¨ e Noetherian ring, and let I be an ideal of R. If a g I t , then there exists, for some n ) 0, a matrix A g MnŽ R . such that: 1. At s a ⭈ In , where In is the n = n identity matrix. 2. The ideal generated by the entries of A is equal to I. The second statement impliesᎏand this will be the key point for our purposesᎏthat the entries of the matrix A are all in I. It turns out that for the considerations of this paper, these ‘‘matrix roots’’ of the elements are just as good as real roots within the ring. As for Lech, our applications, to be presented in the final section, will be to flat local ring extensions Ž R, m. : Ž S, n.. In fact, we are able to prove a very general partial result ŽTheorem 3.1. on Lech’s conjecture that there should be a general inequality eŽ R. F eŽ S . on the Hilbert᎐Samuel multiplicities of the base and flat extension rings with respect to their respective maximal ideals Žsee w5x..
2. THE MAIN RESULTS ON LENGTHS In order to prove our results, we need to first extend the definition of independence in order to include the notion of elements of a commutative ring R acting independently on an R-module M. DEFINITION 2.1. Let T be a commutative ring, and let M be any T-module. Then we say that the elements x 1 , . . . , x r of the ring T act Lech-independently on M if, for any relation of the form x 1 u1 q x 2 u 2 q ⭈⭈⭈ qx r u r , where each u i g M, one may conclude that each u i g Ž x 1 , . . . , x r . M. With this definition, one may prove the following lemma and corollary, which generalize similar results of w6x, and which are proved in a very similar fashion.
LENGTHS OF INDEPENDENT IDEALS
711
LEMMA 2.1. Let x 1 , . . . , x r be elements of T which act Lech-independently on M, and suppose that x 1 s yz. Then the elements y, x 2 , . . . , x r also act Lech-independently on M, and
Ž x 1 , . . . , x r . M : M y s Ž z, x 2 , . . . , x r . M. Proof. Suppose that one has a relation yu1 q x 2 u 2 q ⭈⭈⭈ qx r u r s 0, where the u i g M. By multiplying with z, we get a new relation: x 1 u1 q x 2 zu 2 q ⭈⭈⭈ qx r zu r s 0. By hypothesis, this implies that u1 g Ž x 1 , . . . , x r . M. So if u1 s x 1¨ 1 q x 2¨ 2 q ⭈⭈⭈ x r¨ r , we may rewrite the first relation as x 1 y¨ 1 q x 2 Ž u 2 q y¨ 2 . q ⭈⭈⭈ qx r Ž u r q y¨ r . s 0. The independence of the x’s now implies that u i q y¨ i g Ž x 1 , . . . , x r . M for i G 2, which shows that u i g Ž y, x 2 , . . . , x r . M for all i. For the second statement, suppose that for ¨ g M, one has y¨ g Ž x 1 , . . . , x r . M. Then there exist elements w 1 , . . . , wr of M such that y¨ q x 1w 1 q x 2 w 2 q ⭈⭈⭈ qx r wr s 0. Multiplying by z then gives x 1 Ž ¨ q zw1 . q x 2 zw 2 q ⭈⭈⭈ qx r zwr s 0. The independence of the x’s implies that ¨ q zw1 g Ž x 1 , . . . , x r . M, so that
¨ g Ž z, x 2 , . . . , x r . M, as claimed.
COROLLARY 2.1. Suppose that the elements x 1 , . . . , x r of the ring T act Lech-independently on the module M, and that there exist elements y 1 , . . . , yr of T and positi¨ e integers si such that x i s yis i for 1 F i F r. If we set I s Ž x 1 , . . . , x r . and J s Ž y 1 , . . . , yr ., then MrIM has a filtration by Ł 1r si copies of MrJM. In particular, if ŽT, m. is local and I is primary to m, then r
l Ž MrIM . s
si ⭈ l Ž MrJM . .
žŁ / 1
Proof. We proceed by induction on Ý1r si ; the conclusion is immediate if this sum is equal to r. Thus, we may assume without loss of generality that s1 G 2, so that x 1 s y 1 ⭈ y 1s1y1 . Note that if A is any ideal of T, and z is an element not in A, then one has an exact sequence: AM ª Mr Ž A q z . M ª 0. O ª Mr Ž AM : M z . ª MrA
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DOUGLAS HANES
By Lemma 2.2, we know that IM: M y 1 s Ž y 1s1y1 , x 2 , . . . , x r . M. Thus, applying the exact sequence given above with A s I and z s y 1 gives the short exact sequence Oª
M
Ž
y 1s1y1 ,
x2 , . . . , xr . M
ª
M IM
ª
M
Ž y1 , x 2 , . . . , x r . M
ª 0.
Now, by the induction hypothesis, the modules on the outside of the exact sequence may be filtered by Ž s1 y 1. ⭈ Ł 2r si and Ł 2r si copies of MrJM, respectively. It follows that MrIM may be filtered by Ł 1r si copies of MrJM, which completes the proof of the corollary. With the corollary as well as the theory of matrix factorization in hand, we are now able to give a proof of the main theorem. The central idea is that, if an element x in the t th power of the maximal ideal of a local ring Ž S, n. is allowed to act upon a suitable free module S Ž N ., then it follows from Theorem 1.3 that this action can be realized as the t th power of a certain matrix action on S Ž N ., where the matrix in question has entries in the maximal ideal n. Proof of Theorem 1.2. It follows from Theorem 1.3 that, for each i, there exists n i and a matrix A i g Mn iŽT . such that Atii s x i ⭈ In i , and such that all entries of A i are contained in the maximal ideal mT . For each i, denote the free T-module of rank n i acted upon by A i by Vi . Now replace each matrix A i by the matrix V1 mT ⭈⭈⭈ Viy1 mT A i mT Viq1 mT ⭈⭈⭈ mT Vr g MN Ž T . , where N s Ł 1r n i . The new matrices A i still have entries in mT , and still satisfy Atii s x i ⭈ IN . But now the A i all act on the same free module T Ž N ., and also commute. Thus, the ring T X s T w A1 , . . . , A r x : MN ŽT . is commutative local Žeach A i has a power in mT . and module-finite over T ŽT embeds into T X as scalar matrices.. Moreover, the free T-module T Ž N . has a natural T X-module structure which extends the usual T-module structure. We next note that the elements x 1 , . . . , x r of T X act independently on the module T Ž N . Žsince the x i are in T, this really has nothing to do with the extension ring T X .. So by the corollary to the lemma, we see that r
l Ž T Ž N .r Ž x 1 , . . . , x r . T Ž N . . s
Ł ti ⭈ l
is1
ž
T ŽN. Ý1r A i T Ž N .
/
.
Ž 1.
ŽTechnically, these lengths are taken over T X , but since T and T X have the same residue field, they are equal to the lengths taken over T.. Since the
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LENGTHS OF INDEPENDENT IDEALS
matrices A i all have entries in mT , it is clear that l
ž
T ŽN. Ý1r A i T Ž N .
/
G N.
Ž 2.
Combining the formulas Ž1. and Ž2. now implies that r
l Ž Tr Ž x 1 , . . . , x r . T . G
Ł ti ,
is1
as claimed. Before moving on to the applications, we should make a few remarks about the proof. Remark 2.1. If we just let T U be the free T-algebra TU s
T w z1 , . . . , z r x
Ž z it y x i . i
,
then the existence of commuting matrix roots for the x i allow us to give some free T-module T Ž N . a T U-module structure in such a way that mT U ⭈ T Ž N . s mT ⭈ T Ž N . . The proof could now be carried out by thinking of T Ž N . as a module over T U with this module structure, thus removing from view the more strange-looking ring T X . r Remark 2.2. Of course, T U is itself a free T-module of rank Ł is1 ti , U and the elements x i act on T as the powers of the z i . However, if one writes down the matrix for the action of z i on the free T-module T U , one sees that the matrix involves many unit entries. This prevents us from drawing the conclusion that
l Ž T Ur Ž z1 , . . . , z r . T U . G rank T Ž T U . . This is why we cannot use the much more familiar matrix action of the free extension T U on itself in the proof. Remark 2.3. What the proof of the theorem really shows is that r
l Ž Tr Ž x 1 , . . . , x r . T . s Ž 1rN .
ž / ž Ł ti ⭈ l
is1
It was shown above that l
ž
T ŽN. Ý1r A i T Ž N .
/
G N,
T ŽN. Ý1r A i T Ž N .
/
.
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DOUGLAS HANES
but in fact, if s represents the embedding dimension of T, then we know by Lech’s result ŽTheorem 1.1. that r F s, and it is easy to see that we must have l
ž
T ŽN. Ý1r A i T Ž N .
/
G N ⭈ Ž1 q s y r . .
This provides the improved estimate: r
l Ž Tr Ž x 1 , . . . , x r . T . G
ž / Ł ti
⭈ Ž 1 q edim Ž T . y r . .
is1
3. APPLICATIONS TO FLAT LOCAL EXTENSIONS We believe that Theorem 1.2 will have many interesting applications to the theory of Hilbert functions and multiplicities. In this section, we will consider two applications to flat couples of local Žor graded. rings. First we apply the results on length to Lech’s conjecture that if Ž R, m. : Ž S, n. is a flat local extension of local rings, then eŽ R . F eŽ S .. This conjecture, which first appeared in the 1960 paper w5x, naturally arises out of Nagata’s work on multiplicities in w7x. The conjecture remains open in most cases, as long as the base ring R has dimension at least 3. For the best partial results, the reader is referred to Lech’s original papers w5, 6x, as well as to w2x by Herzog. The result to be presented here is most satisfying in the case that Ž R, m. : Ž S, n. is obtained by localization from a flat extension of graded rings, but we begin with the most general conclusion, which only requires reference to the associated graded ring of the extension ring S. Recall that an extension Ž R, m. : Ž S, n. of local rings is called flat local if S is a flat R-module and m embeds into n. Also note that if x is an element of R, then by the initial form of x in G s grnŽ S . we mean the initial form inŽ x . of the image of x in S Ž not the image in G of the initial form of x in grmŽ R ... COROLLARY 3.1. Let Ž R, m. : Ž S, n. be a flat local extension of d-dimensional local rings. Assume that there exists a minimal reduction I s Ž x 1 , . . . , x d . of m with the property that the initial forms inŽ x 1 ., . . . , inŽ x d . in the associated graded ring G s grnŽ S . of S form a system of parameters for G. Then eŽ S . G eŽ R .. In fact, if mS : n t , then e Ž S . G Ž 1 q Ž n . y Ž m . . Ž t Ž m.yd . e Ž R . . Proof. First, we may extend the system of parameters x 1 , . . . , x d to a minimal generating set x 1 , . . . , x r of m. For each i, we fix t i such that x i maps into n t i _ n t iq1 .
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LENGTHS OF INDEPENDENT IDEALS
We conclude from the flatness condition and Theorem 1.2 Žor Remark 2.3. that r
e Ž m; S . s e Ž R . ⭈ l Ž SrmS . G Ž 1 q Ž n . y Ž m . .
ž / Ł ti
⭈ e Ž R . . Ž 3.
1
On the other hand, we may note that for any n-primary ideal J of S, if inŽ J . represents the ideal of initial forms of J in G, then one has e Ž J ; S . F e Ž in Ž J . ; G . . This follows from the fact that, for any t ) 0, ŽinŽ J .. t : inŽ J t .. Applying this to the ideal mS, we see that e Ž m; S . F e Ž in Ž mS . ; G . F e Ž in Ž x 1 . , . . . , in Ž x d . ; G . . Finally, since the elements inŽ x 1 ., . . . , inŽ x d . form a homogeneous system of parameters of G of degrees t 1 , . . . , t d , respectively, we have that d
e Ž in Ž x 1 . , . . . , in Ž x d . ; G . s
d
Ł ti ⭈ e Ž G . s
ž / 1
ti ⭈ e Ž S . .
žŁ / 1
Thus, we may conclude that d
e Ž m, S . F
ž / Ł ti
⭈ eŽ S .
Ž 4.
1
Žactually, we could prove equality.. Now combining the inequalities Ž3. and Ž4. implies that r
e Ž S . G Ž 1 q Ž n. y Ž m. .
ž / Ł ti
⭈ eŽ R. .
dq1
This certainly implies all of the conclusions of the corollary. Remark 3.1. We remark that the hypothesis of Corollary 3.1 on the flat couple Ž R, m. : Ž S, n. is considerably weaker than that of tangential flatness. The extension is called tangentially flat if the associated homomorphism grmŽ R . ª grnŽ S . which maps m trm tq1 to n trn tq1 , for each t G 0, is flat. This condition implies, among other things, that the initial forms in the graded ring of S of a minimal reduction of m will constitute a system of parameters of 1-forms in the graded ring of S. The condition of tangential flatness, and its relation to the various Lech᎐Hironaka conjectures, has been considered by Herzog in, e.g., w2, 3x Žfor Hironaka’s statements see w4x..
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DOUGLAS HANES
The condition on the initial forms of the parameters x i of R need not hold in general. However, it is clear in the case that S is a standard graded algebra over a field and the elements x i map to homogeneous elements of S. Notice that one may easily extend the notion of a flat couple of rings to the case of a ring extension R : S of N-graded rings with homogeneous maximal ideals m and n, respectively. In this case, one says that Ž R, m. : Ž S, n. is a flat couple of N-graded rings if the homogeneous maximal ideal m of R is mapped into the homogeneous maximal ideal n of S, and if S is a flat R-module. It is then easy to see that Ž R, m. : Ž S, n. is a flat couple of N-graded rings if and only if the extension R m : Sn of local rings is flat local. In the graded case we obtain the following: COROLLARY 3.2. Let Ž R, m. : Ž S, n. be a flat extension of standard graded algebras of dimension d. If m has a minimal reduction I s Ž x 1 , . . . , x d . with the property that each x i maps to a homogeneous element of positi¨ e degree in S, then eŽ m; R . F eŽ n; S .. Finally, in the case of a flat extension of graded rings, we may obtain a similar inequality on the a-invariants of the respective rings. We recall the definition, which may be found in w8x. DEFINITION 3.1. Let R be a finitely generated N-graded K-algebra of dimension d, where R 0 s K is a field. If m is the homogeneous maximal ideal of R, then the a-invariant aŽ R . of R is defined to be the greatest integer i for which the ith graded piece of the dth local cohomology module Hmd Ž R . is nonzero. If R is Cohen᎐Macaulay, and if x 1 , . . . , x d is a homogeneous system of parameters of degrees t 1 , . . . , t d , respectively, then the a-invariant may be computed as d
a Ž R . s deg G y
Ý ti , 1
where G is a form of maximal degree not in Ž x 1 , . . . , x d .. It follows that if R is a standard graded algebra Ži.e., generated by 1-forms over a field., then aŽ R . q d is equal to the reduction number of R. DEFINITION 3.2. Let R be a standard graded algebra over a field K. For any system of parameters of 1-forms x 1 , . . . , x d , let r x denote the maximal degree of a form of R not in the ideal Ž x 1 , . . . , x d .. Then the reduction number r Ž R . of R is defined to be the minimum of all such r x . If Ž R, m. is a Noetherian local ring, then we define the reduction number of R to be r Ž R . s r Žgrm R .. Since the a-invariant Žor reduction number. is yet another measure of the complexity of a graded ring, the following corollary provides added
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LENGTHS OF INDEPENDENT IDEALS
evidence for the general notion that a flat extension ring must be at least as ‘‘complex’’ as the base ring. COROLLARY 3.3. Let Ž R, m. : Ž S, n. be a flat extension of positi¨ ely graded rings, generated by 1-forms o¨ er a field K. Assume that R and S are Cohen᎐Macaulay, and that R possesses a system of parameters x 1 , . . . , x d such that each x i maps to a homogeneous element of positi¨ e degree in S. Then aŽ R . F aŽ S .. Proof. Assume as before that m is generated by x 1 , . . . , x d , where each x i maps into n t i _ n t iq1 . The proof of Lemma 2.2 implies that A1t 1y1 At22y1 ⭈⭈⭈ Atrry1 S Ž N . mS Ž N . , where the A i are commuting matrix roots of the generators x i of I Žas in the proof of Theorem 1.2.. Since the matrices all have entries in n, this implies that nŽÝt i .yr mS. We know that the highest degree form x of R which is not in I has degree aŽ R . q d. The image of x in S may not be homogeneous, but all components have degree at least aŽ R . q d in S. Moreover, it follows from the flatness that nŽÝt i .yr x f IS. Putting all this together shows that there exists a form y g S _ IS of degree at least ŽÝ1r t i . y r q aŽ R . q d. Finally, since S is Cohen᎐Macaulay and x 1 , . . . , x d is a homogeneous system of parameters, this shows that r
aŽ S . G
ž / Ý ti 1
d
y r q aŽ R . q d y
r
s
ž / ž / Ý ti 1
Ý ti
y Ž r y d . q aŽ R . ,
dq1
and thus aŽ S . G aŽ R . simply because the t i are all positive. As in the case of Lech’s conjecture, this provides a proof in the graded Cohen᎐Macaulay case about a statement which would more naturally be made of local rings. However, I think the results presented here are persuasive enough in order to venture the general conjecture on reduction numbers. CONJECTURE 3.1. Let Ž R, m. : Ž S, n. be a flat local extension of local rings. Then the reduction number of R is less than or equal to that of S. A proof of the conjecture would be of interest in its own right, but would also contribute to continuing efforts to prove Lech’s proposed inequality on the Hilbert᎐Samuel multiplicities, since rough estimates of one invariant can often be made in terms of the other.
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REFERENCES 1. J. Backelin, J. Herzog, and B. Ulrich, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 Ž1991., 187᎐202. 2. B. Herzog, Lech᎐Hironaka inequalities for flat couples of local rings, Manuscripta Math. 68 Ž1990., 351᎐371. 3. B. Herzog, Local singularities such that all deformations are tangentially flat, Trans. Amer. Math. Soc. 324 Ž1991., 555᎐601. 4. H. Hironaka, Certain numerical characters of singularities, J. Math. Kyoto Uni¨ . 10 Ž1970., 151᎐187. 5. C. Lech, Note on multiplicities of ideals, Ark. Math. 4 Ž1960., 63᎐86. 6. C. Lech, Inequalities related to certain couples of local rings, Acta Math. 112 Ž1964., 69᎐89. 7. M. Nagata, The theory of multiplicity in general local rings, in ‘‘Proceedings of the International Symposium on Algebraic Number Theory. Tokyo-Nikko, 1955,’’ pp. 191᎐226, Science Council of Japan, Tokyo, 1956. 8. K. I. Watanabe, Rational singularities with kU action, in ‘‘Commutative Algebra ŽProc. of the Trento Conference.,’’ Lecture Notes in Pure and Applied Mathematics, Vol. 84, pp. 339᎐351, Dekker, New YorkrBasel, 1983.
Length Approximations for Independently Generated Ideals Douglas Hanes Department of Mathematics, Uni¨ ersity of Minnesota, Minneapolis, Minnesota 55455 E-mail: [email protected] Communicated by Craig Huneke Received March 28, 2000
We study ideals primary to the maximal ideal of a commutative Noetherian local ring. When such an ideal is generated by elements which are independent in the sense of C. Lech, we prove a lower bound on the length of the quotient ring in terms of the orders of the generators. As a corollary we obtain a substantial partial result on Lech’s conjecture on the multiplicities of a flat couple of local rings. 䊚 2001 Academic Press
Key Words: commutative algebra; matrix factorization.
1. INTRODUCTION In this paper all rings are assumed to be commutative, Noetherian, and to possess a multiplicative unit element. All ring homomorphisms are assumed to be unital. Let Ž R, m. be a local ring, and let I be an m-primary ideal, so that the homomorphic image RrI has finite length. Our primary object is to show that if the ideal I is generated by elements which are independent Žas defined by Lech in w6x., then one may give a lower estimate for the length of RrI in terms of the orders of the generators of I Žwhere by the ‘‘order’’ of an element x we mean the largest t such that x g m t .. Because of possible confusion caused by the much-used term ‘‘independent,’’ we will call the elements x 1 , . . . , x r of a commutative ring R Lech-independent, or independent in the sense of Lech, if for every relation of the form x 1 u1 q x 2 u 2 q ⭈⭈⭈ qx r u r s 0 708 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
LENGTHS OF INDEPENDENT IDEALS
709
in R, one may conclude that u i g Ž x 1 , . . . , x r . R for 1 F i F r Žsee w6x.. As Žimportant. examples, we note that the maximal ideal of any local ring is generated by Lech-independent elements, and that any regular sequence x 1 , . . . , x n is Lech-independent. Moreover, note that if I : R is generated by Lech-independent elements, and if S is any flat ring extension of R, then the extended ideal IS is still generated by Lech-independent elements. Lech’s main result w6, Theorem 2x on independent sequences is the following theorem. Note that if I s Ž x 1 , . . . , x m . is an ideal of R, then the sequence of elements x 1 , . . . , x m is Lech-independent if and only if IrI 2 is a free module Žof rank m. over RrI. In Lech’s presentation the latter of the equivalent hypotheses is used. THEOREM 1.1 ŽLech.. Let x 1 , . . . , x r be a sequence of independent elements in a local ring S of embedding dimension t, and assume that SrŽ x . S is equicharacteristic and has finite length. Then r F t. The theorem may be directly applied to flat couples of local rings, which are defined as follows: DEFINITION 1.1. A flat couple or flat local extension of local rings is an extension Ž R, m. : Ž S, n. of local rings with the properties that S is a flat R-module and that mS : n. By the embedding dimension edimŽ R . of a local ring Ž R, m. we mean the minimal number of generators Ž m. of its maximal ideal m. It follows from Theorem 1.1 that if Ž R, m. : Ž S, n. is a flat local extension of local rings of dimension d, then the embedding dimension edimŽ R . of R is less than or equal to that of S, since the minimal generators of m map to an independent set of elements of S. In fact, by employing a similar result with regard to localization, Lech shows in w6x that edim Ž R . F edim Ž S . for any flat local extension Ž R, m. : Ž S, n.. Our main result is the following theorem, which gives a lower bound on the length of SrI when I is an n-primary ideal of the local ring Ž S, n. generated by independent elements. THEOREM 1.2. Let x 1 , . . . , x r be a set of Lech-independent elements of a commutati¨ e local ring ŽT, mT . which generate an ideal primary to the maximal ideal. Suppose that for 1 F i F r, x i g mTt i . Then r
l Ž Tr Ž x 1 , . . . , x r . T . G
Ł ti .
is1
710
DOUGLAS HANES
Such an inequality is well known for any system of parameters x 1 , . . . , x d , and the theorem represents a generalization of this fact in the case that x 1 , . . . , x d is a regular sequence. We note that the statement of the theorem is implied by the results of w6x in the case that each x i is actually a t th power of some element of mT . Of course nothing like this can be assumed about a general element x i g mTt , but we will employ the following theorem of w1x in order to realize each x i as the t th power of some square matrix with entries in mT . THEOREM 1.3 ŽBackelin, Herzog, and Ulrich.. Let R be a commutati¨ e Noetherian ring, and let I be an ideal of R. If a g I t , then there exists, for some n ) 0, a matrix A g MnŽ R . such that: 1. At s a ⭈ In , where In is the n = n identity matrix. 2. The ideal generated by the entries of A is equal to I. The second statement impliesᎏand this will be the key point for our purposesᎏthat the entries of the matrix A are all in I. It turns out that for the considerations of this paper, these ‘‘matrix roots’’ of the elements are just as good as real roots within the ring. As for Lech, our applications, to be presented in the final section, will be to flat local ring extensions Ž R, m. : Ž S, n.. In fact, we are able to prove a very general partial result ŽTheorem 3.1. on Lech’s conjecture that there should be a general inequality eŽ R. F eŽ S . on the Hilbert᎐Samuel multiplicities of the base and flat extension rings with respect to their respective maximal ideals Žsee w5x..
2. THE MAIN RESULTS ON LENGTHS In order to prove our results, we need to first extend the definition of independence in order to include the notion of elements of a commutative ring R acting independently on an R-module M. DEFINITION 2.1. Let T be a commutative ring, and let M be any T-module. Then we say that the elements x 1 , . . . , x r of the ring T act Lech-independently on M if, for any relation of the form x 1 u1 q x 2 u 2 q ⭈⭈⭈ qx r u r , where each u i g M, one may conclude that each u i g Ž x 1 , . . . , x r . M. With this definition, one may prove the following lemma and corollary, which generalize similar results of w6x, and which are proved in a very similar fashion.
LENGTHS OF INDEPENDENT IDEALS
711
LEMMA 2.1. Let x 1 , . . . , x r be elements of T which act Lech-independently on M, and suppose that x 1 s yz. Then the elements y, x 2 , . . . , x r also act Lech-independently on M, and
Ž x 1 , . . . , x r . M : M y s Ž z, x 2 , . . . , x r . M. Proof. Suppose that one has a relation yu1 q x 2 u 2 q ⭈⭈⭈ qx r u r s 0, where the u i g M. By multiplying with z, we get a new relation: x 1 u1 q x 2 zu 2 q ⭈⭈⭈ qx r zu r s 0. By hypothesis, this implies that u1 g Ž x 1 , . . . , x r . M. So if u1 s x 1¨ 1 q x 2¨ 2 q ⭈⭈⭈ x r¨ r , we may rewrite the first relation as x 1 y¨ 1 q x 2 Ž u 2 q y¨ 2 . q ⭈⭈⭈ qx r Ž u r q y¨ r . s 0. The independence of the x’s now implies that u i q y¨ i g Ž x 1 , . . . , x r . M for i G 2, which shows that u i g Ž y, x 2 , . . . , x r . M for all i. For the second statement, suppose that for ¨ g M, one has y¨ g Ž x 1 , . . . , x r . M. Then there exist elements w 1 , . . . , wr of M such that y¨ q x 1w 1 q x 2 w 2 q ⭈⭈⭈ qx r wr s 0. Multiplying by z then gives x 1 Ž ¨ q zw1 . q x 2 zw 2 q ⭈⭈⭈ qx r zwr s 0. The independence of the x’s implies that ¨ q zw1 g Ž x 1 , . . . , x r . M, so that
¨ g Ž z, x 2 , . . . , x r . M, as claimed.
COROLLARY 2.1. Suppose that the elements x 1 , . . . , x r of the ring T act Lech-independently on the module M, and that there exist elements y 1 , . . . , yr of T and positi¨ e integers si such that x i s yis i for 1 F i F r. If we set I s Ž x 1 , . . . , x r . and J s Ž y 1 , . . . , yr ., then MrIM has a filtration by Ł 1r si copies of MrJM. In particular, if ŽT, m. is local and I is primary to m, then r
l Ž MrIM . s
si ⭈ l Ž MrJM . .
žŁ / 1
Proof. We proceed by induction on Ý1r si ; the conclusion is immediate if this sum is equal to r. Thus, we may assume without loss of generality that s1 G 2, so that x 1 s y 1 ⭈ y 1s1y1 . Note that if A is any ideal of T, and z is an element not in A, then one has an exact sequence: AM ª Mr Ž A q z . M ª 0. O ª Mr Ž AM : M z . ª MrA
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DOUGLAS HANES
By Lemma 2.2, we know that IM: M y 1 s Ž y 1s1y1 , x 2 , . . . , x r . M. Thus, applying the exact sequence given above with A s I and z s y 1 gives the short exact sequence Oª
M
Ž
y 1s1y1 ,
x2 , . . . , xr . M
ª
M IM
ª
M
Ž y1 , x 2 , . . . , x r . M
ª 0.
Now, by the induction hypothesis, the modules on the outside of the exact sequence may be filtered by Ž s1 y 1. ⭈ Ł 2r si and Ł 2r si copies of MrJM, respectively. It follows that MrIM may be filtered by Ł 1r si copies of MrJM, which completes the proof of the corollary. With the corollary as well as the theory of matrix factorization in hand, we are now able to give a proof of the main theorem. The central idea is that, if an element x in the t th power of the maximal ideal of a local ring Ž S, n. is allowed to act upon a suitable free module S Ž N ., then it follows from Theorem 1.3 that this action can be realized as the t th power of a certain matrix action on S Ž N ., where the matrix in question has entries in the maximal ideal n. Proof of Theorem 1.2. It follows from Theorem 1.3 that, for each i, there exists n i and a matrix A i g Mn iŽT . such that Atii s x i ⭈ In i , and such that all entries of A i are contained in the maximal ideal mT . For each i, denote the free T-module of rank n i acted upon by A i by Vi . Now replace each matrix A i by the matrix V1 mT ⭈⭈⭈ Viy1 mT A i mT Viq1 mT ⭈⭈⭈ mT Vr g MN Ž T . , where N s Ł 1r n i . The new matrices A i still have entries in mT , and still satisfy Atii s x i ⭈ IN . But now the A i all act on the same free module T Ž N ., and also commute. Thus, the ring T X s T w A1 , . . . , A r x : MN ŽT . is commutative local Žeach A i has a power in mT . and module-finite over T ŽT embeds into T X as scalar matrices.. Moreover, the free T-module T Ž N . has a natural T X-module structure which extends the usual T-module structure. We next note that the elements x 1 , . . . , x r of T X act independently on the module T Ž N . Žsince the x i are in T, this really has nothing to do with the extension ring T X .. So by the corollary to the lemma, we see that r
l Ž T Ž N .r Ž x 1 , . . . , x r . T Ž N . . s
Ł ti ⭈ l
is1
ž
T ŽN. Ý1r A i T Ž N .
/
.
Ž 1.
ŽTechnically, these lengths are taken over T X , but since T and T X have the same residue field, they are equal to the lengths taken over T.. Since the
713
LENGTHS OF INDEPENDENT IDEALS
matrices A i all have entries in mT , it is clear that l
ž
T ŽN. Ý1r A i T Ž N .
/
G N.
Ž 2.
Combining the formulas Ž1. and Ž2. now implies that r
l Ž Tr Ž x 1 , . . . , x r . T . G
Ł ti ,
is1
as claimed. Before moving on to the applications, we should make a few remarks about the proof. Remark 2.1. If we just let T U be the free T-algebra TU s
T w z1 , . . . , z r x
Ž z it y x i . i
,
then the existence of commuting matrix roots for the x i allow us to give some free T-module T Ž N . a T U-module structure in such a way that mT U ⭈ T Ž N . s mT ⭈ T Ž N . . The proof could now be carried out by thinking of T Ž N . as a module over T U with this module structure, thus removing from view the more strange-looking ring T X . r Remark 2.2. Of course, T U is itself a free T-module of rank Ł is1 ti , U and the elements x i act on T as the powers of the z i . However, if one writes down the matrix for the action of z i on the free T-module T U , one sees that the matrix involves many unit entries. This prevents us from drawing the conclusion that
l Ž T Ur Ž z1 , . . . , z r . T U . G rank T Ž T U . . This is why we cannot use the much more familiar matrix action of the free extension T U on itself in the proof. Remark 2.3. What the proof of the theorem really shows is that r
l Ž Tr Ž x 1 , . . . , x r . T . s Ž 1rN .
ž / ž Ł ti ⭈ l
is1
It was shown above that l
ž
T ŽN. Ý1r A i T Ž N .
/
G N,
T ŽN. Ý1r A i T Ž N .
/
.
714
DOUGLAS HANES
but in fact, if s represents the embedding dimension of T, then we know by Lech’s result ŽTheorem 1.1. that r F s, and it is easy to see that we must have l
ž
T ŽN. Ý1r A i T Ž N .
/
G N ⭈ Ž1 q s y r . .
This provides the improved estimate: r
l Ž Tr Ž x 1 , . . . , x r . T . G
ž / Ł ti
⭈ Ž 1 q edim Ž T . y r . .
is1
3. APPLICATIONS TO FLAT LOCAL EXTENSIONS We believe that Theorem 1.2 will have many interesting applications to the theory of Hilbert functions and multiplicities. In this section, we will consider two applications to flat couples of local Žor graded. rings. First we apply the results on length to Lech’s conjecture that if Ž R, m. : Ž S, n. is a flat local extension of local rings, then eŽ R . F eŽ S .. This conjecture, which first appeared in the 1960 paper w5x, naturally arises out of Nagata’s work on multiplicities in w7x. The conjecture remains open in most cases, as long as the base ring R has dimension at least 3. For the best partial results, the reader is referred to Lech’s original papers w5, 6x, as well as to w2x by Herzog. The result to be presented here is most satisfying in the case that Ž R, m. : Ž S, n. is obtained by localization from a flat extension of graded rings, but we begin with the most general conclusion, which only requires reference to the associated graded ring of the extension ring S. Recall that an extension Ž R, m. : Ž S, n. of local rings is called flat local if S is a flat R-module and m embeds into n. Also note that if x is an element of R, then by the initial form of x in G s grnŽ S . we mean the initial form inŽ x . of the image of x in S Ž not the image in G of the initial form of x in grmŽ R ... COROLLARY 3.1. Let Ž R, m. : Ž S, n. be a flat local extension of d-dimensional local rings. Assume that there exists a minimal reduction I s Ž x 1 , . . . , x d . of m with the property that the initial forms inŽ x 1 ., . . . , inŽ x d . in the associated graded ring G s grnŽ S . of S form a system of parameters for G. Then eŽ S . G eŽ R .. In fact, if mS : n t , then e Ž S . G Ž 1 q Ž n . y Ž m . . Ž t Ž m.yd . e Ž R . . Proof. First, we may extend the system of parameters x 1 , . . . , x d to a minimal generating set x 1 , . . . , x r of m. For each i, we fix t i such that x i maps into n t i _ n t iq1 .
715
LENGTHS OF INDEPENDENT IDEALS
We conclude from the flatness condition and Theorem 1.2 Žor Remark 2.3. that r
e Ž m; S . s e Ž R . ⭈ l Ž SrmS . G Ž 1 q Ž n . y Ž m . .
ž / Ł ti
⭈ e Ž R . . Ž 3.
1
On the other hand, we may note that for any n-primary ideal J of S, if inŽ J . represents the ideal of initial forms of J in G, then one has e Ž J ; S . F e Ž in Ž J . ; G . . This follows from the fact that, for any t ) 0, ŽinŽ J .. t : inŽ J t .. Applying this to the ideal mS, we see that e Ž m; S . F e Ž in Ž mS . ; G . F e Ž in Ž x 1 . , . . . , in Ž x d . ; G . . Finally, since the elements inŽ x 1 ., . . . , inŽ x d . form a homogeneous system of parameters of G of degrees t 1 , . . . , t d , respectively, we have that d
e Ž in Ž x 1 . , . . . , in Ž x d . ; G . s
d
Ł ti ⭈ e Ž G . s
ž / 1
ti ⭈ e Ž S . .
žŁ / 1
Thus, we may conclude that d
e Ž m, S . F
ž / Ł ti
⭈ eŽ S .
Ž 4.
1
Žactually, we could prove equality.. Now combining the inequalities Ž3. and Ž4. implies that r
e Ž S . G Ž 1 q Ž n. y Ž m. .
ž / Ł ti
⭈ eŽ R. .
dq1
This certainly implies all of the conclusions of the corollary. Remark 3.1. We remark that the hypothesis of Corollary 3.1 on the flat couple Ž R, m. : Ž S, n. is considerably weaker than that of tangential flatness. The extension is called tangentially flat if the associated homomorphism grmŽ R . ª grnŽ S . which maps m trm tq1 to n trn tq1 , for each t G 0, is flat. This condition implies, among other things, that the initial forms in the graded ring of S of a minimal reduction of m will constitute a system of parameters of 1-forms in the graded ring of S. The condition of tangential flatness, and its relation to the various Lech᎐Hironaka conjectures, has been considered by Herzog in, e.g., w2, 3x Žfor Hironaka’s statements see w4x..
716
DOUGLAS HANES
The condition on the initial forms of the parameters x i of R need not hold in general. However, it is clear in the case that S is a standard graded algebra over a field and the elements x i map to homogeneous elements of S. Notice that one may easily extend the notion of a flat couple of rings to the case of a ring extension R : S of N-graded rings with homogeneous maximal ideals m and n, respectively. In this case, one says that Ž R, m. : Ž S, n. is a flat couple of N-graded rings if the homogeneous maximal ideal m of R is mapped into the homogeneous maximal ideal n of S, and if S is a flat R-module. It is then easy to see that Ž R, m. : Ž S, n. is a flat couple of N-graded rings if and only if the extension R m : Sn of local rings is flat local. In the graded case we obtain the following: COROLLARY 3.2. Let Ž R, m. : Ž S, n. be a flat extension of standard graded algebras of dimension d. If m has a minimal reduction I s Ž x 1 , . . . , x d . with the property that each x i maps to a homogeneous element of positi¨ e degree in S, then eŽ m; R . F eŽ n; S .. Finally, in the case of a flat extension of graded rings, we may obtain a similar inequality on the a-invariants of the respective rings. We recall the definition, which may be found in w8x. DEFINITION 3.1. Let R be a finitely generated N-graded K-algebra of dimension d, where R 0 s K is a field. If m is the homogeneous maximal ideal of R, then the a-invariant aŽ R . of R is defined to be the greatest integer i for which the ith graded piece of the dth local cohomology module Hmd Ž R . is nonzero. If R is Cohen᎐Macaulay, and if x 1 , . . . , x d is a homogeneous system of parameters of degrees t 1 , . . . , t d , respectively, then the a-invariant may be computed as d
a Ž R . s deg G y
Ý ti , 1
where G is a form of maximal degree not in Ž x 1 , . . . , x d .. It follows that if R is a standard graded algebra Ži.e., generated by 1-forms over a field., then aŽ R . q d is equal to the reduction number of R. DEFINITION 3.2. Let R be a standard graded algebra over a field K. For any system of parameters of 1-forms x 1 , . . . , x d , let r x denote the maximal degree of a form of R not in the ideal Ž x 1 , . . . , x d .. Then the reduction number r Ž R . of R is defined to be the minimum of all such r x . If Ž R, m. is a Noetherian local ring, then we define the reduction number of R to be r Ž R . s r Žgrm R .. Since the a-invariant Žor reduction number. is yet another measure of the complexity of a graded ring, the following corollary provides added
717
LENGTHS OF INDEPENDENT IDEALS
evidence for the general notion that a flat extension ring must be at least as ‘‘complex’’ as the base ring. COROLLARY 3.3. Let Ž R, m. : Ž S, n. be a flat extension of positi¨ ely graded rings, generated by 1-forms o¨ er a field K. Assume that R and S are Cohen᎐Macaulay, and that R possesses a system of parameters x 1 , . . . , x d such that each x i maps to a homogeneous element of positi¨ e degree in S. Then aŽ R . F aŽ S .. Proof. Assume as before that m is generated by x 1 , . . . , x d , where each x i maps into n t i _ n t iq1 . The proof of Lemma 2.2 implies that A1t 1y1 At22y1 ⭈⭈⭈ Atrry1 S Ž N . mS Ž N . , where the A i are commuting matrix roots of the generators x i of I Žas in the proof of Theorem 1.2.. Since the matrices all have entries in n, this implies that nŽÝt i .yr mS. We know that the highest degree form x of R which is not in I has degree aŽ R . q d. The image of x in S may not be homogeneous, but all components have degree at least aŽ R . q d in S. Moreover, it follows from the flatness that nŽÝt i .yr x f IS. Putting all this together shows that there exists a form y g S _ IS of degree at least ŽÝ1r t i . y r q aŽ R . q d. Finally, since S is Cohen᎐Macaulay and x 1 , . . . , x d is a homogeneous system of parameters, this shows that r
aŽ S . G
ž / Ý ti 1
d
y r q aŽ R . q d y
r
s
ž / ž / Ý ti 1
Ý ti
y Ž r y d . q aŽ R . ,
dq1
and thus aŽ S . G aŽ R . simply because the t i are all positive. As in the case of Lech’s conjecture, this provides a proof in the graded Cohen᎐Macaulay case about a statement which would more naturally be made of local rings. However, I think the results presented here are persuasive enough in order to venture the general conjecture on reduction numbers. CONJECTURE 3.1. Let Ž R, m. : Ž S, n. be a flat local extension of local rings. Then the reduction number of R is less than or equal to that of S. A proof of the conjecture would be of interest in its own right, but would also contribute to continuing efforts to prove Lech’s proposed inequality on the Hilbert᎐Samuel multiplicities, since rough estimates of one invariant can often be made in terms of the other.
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REFERENCES 1. J. Backelin, J. Herzog, and B. Ulrich, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 Ž1991., 187᎐202. 2. B. Herzog, Lech᎐Hironaka inequalities for flat couples of local rings, Manuscripta Math. 68 Ž1990., 351᎐371. 3. B. Herzog, Local singularities such that all deformations are tangentially flat, Trans. Amer. Math. Soc. 324 Ž1991., 555᎐601. 4. H. Hironaka, Certain numerical characters of singularities, J. Math. Kyoto Uni¨ . 10 Ž1970., 151᎐187. 5. C. Lech, Note on multiplicities of ideals, Ark. Math. 4 Ž1960., 63᎐86. 6. C. Lech, Inequalities related to certain couples of local rings, Acta Math. 112 Ž1964., 69᎐89. 7. M. Nagata, The theory of multiplicity in general local rings, in ‘‘Proceedings of the International Symposium on Algebraic Number Theory. Tokyo-Nikko, 1955,’’ pp. 191᎐226, Science Council of Japan, Tokyo, 1956. 8. K. I. Watanabe, Rational singularities with kU action, in ‘‘Commutative Algebra ŽProc. of the Trento Conference.,’’ Lecture Notes in Pure and Applied Mathematics, Vol. 84, pp. 339᎐351, Dekker, New YorkrBasel, 1983.