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The Core of a Module over a Two-Dimensional Regular Local Ring
189, 1]22 Ž1997. JA966823
JOURNAL OF ALGEBRA ARTICLE NO.
The Core of a Module over a Two-Dimensional Regular Local Ring Radha Mohan* Department of Mathematics, Purdue Uni¨ ersity, West Lafayette, Indiana 47907 Communicated by Mel¨ in Hochster Received November 14, 1994
This paper explicitly determines the core of a torsion-free, integrally closed module over a two-dimensional regular local ring. It is analogous to a result of Huneke and Swanson which determines the core of an integrally closed ideal. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two-dimensional regular local ring is the product of the module and a certain Fitting ideal of the module. The technical tools used are quadratic transforms and Buchsbaum]Rim multiplicity. Q 1997 Academic Press
INTRODUCTION A recent result of Huneke and Swanson wHnkSwnx explicitly determines the core of an integrally closed ideal over a two-dimensional regular local ring. Here, the core of an ideal I is the intersection of all reductions of I. The concept of the core of an ideal was introduced by Sally in the late 1980s. It is first alluded to in wRsSllx. We recall that a subideal J of an ideal I is a reduction of I, if I is contained in the integral closure of J. Equivalently, J is a reduction of I if there exists a positive integer n such that I n s JI ny 1. Huneke and Swanson wHnkSwnx prove that if I is an integrally closed ideal over a two-dimensional regular local ring then the core of I is a product of I and a certain Fitting ideal of I. Furthermore, they show that this Fitting ideal is integrally closed and deduce that the core of I is integrally closed. *E-mail address: [email protected]. 1 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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RADHA MOHAN
This paper generalizes the main theorem of wHnkSwnx to finitely generated, torsion-free integrally closed modules. We obtain an explicit formula for the core of such a module and show that it is also an integrally closed module. The concept of reductions and integral closures for finitely generated, torsion-free modules over a Noetherian domain was introduced by Rees wRsx. ŽFor details see Section 1.. The study of the core of an ideal has been partly motivated by a theorem of Brianc¸on and Skoda Žsee wBrnSkd, LpmSth, LpmTss, Lpm, HchHnk, RsSll, Swn, AbrHnkx.. Likewise, the study of the core of a module, denoted coreŽ M . and defined to be the intersection of all reductions of M, is partly relevant due to a recent Brianc¸on]Skoda type theorem of Katz and Kodiyalam wKtzKdyx for finitely generated, torsionfree, integrally closed modules over two-dimensional regular local rings. One of the crucial ideas used by Huneke and Swanson wHnkSwnx in determining the core of a complete ideal I is the notion of a quadratic transform of I and the fact that the multiplicity of an ideal decreases on taking quadratic transforms. Recently, Kodiyalam wKdyx has introduced the ideas of transforms for finitely generated, torsion-free modules over twodimensional regular local rings. He also shows that the Buchsbaum]Rim multiplicity of a finitely generated, torsion-free module decreases on taking transforms. With these ideas in place we can generalize the theorem of Huneke and Swanson to finitely generated, torsion-free, integrally closed modules. The following theorem is the main result of this paper. THEOREM 2.10. Let Ž R, m, k . be a two-dimensional regular local ring with infinite residue field. Let M be a finitely generated, torsion-free, complete R-module of rank r. Let A be an n = n y r presenting matrix for M. Then, core Ž M . s Iny ry1 Ž A . M, where Iny ry1Ž A. is the ideal of R generated by the n y r y 1 minors of A. We prove this result in Section 2. The structure of the proof is patterned after the proof of Huneke and Swanson wHnkSwnx for the theorem for the core of an ideal. In Section 1 we discuss some preliminaries.
1. PRELIMINARIES Integral Closures and Reductions of Modules We review the notions of integral closures and reductions for torsion-free modules over arbitrary Noetherian domains as developed by Rees wRsx.
CORE OF A MODULE
3
Let R be a Noetherian domain and let K be its field of fractions. Let M be a finitely generated, torsion-free R module. By MK we denote the finite-dimensional K-vector space M mR K. If N is a submodule of M, then NK is naturally identified with a subspace of MK . We say a ring between R and its field of fractions K is a birational o¨ erring of R. For any birational overring S of R, we let MS denote the S-submodule of MK generated by M. There is a canonical R-module homomorphism from M mR S onto MS. Its kernel is the submodule of R-torsion elements. Hence, M mR S modulo S-torsion and MS are isomorphic as S modules. DEFINITION 1.1. With notation as above, an element ¨ g MK is said to be integral over M if ¨ g MV for every discrete valuation ring V of K containing R. The integral closure of M, denoted M, is the set of all elements of MK that are integral over M. The module M is said to be integrally closed or complete if M s M. It is clear that M is an R-submodule of MK which contains M. In fact, since any valuation domain of K which contains R also contains its integral closure R in K, we have that M is an R module. Rees shows in wRs, Lemma 1.1x that M is a finitely generated R-module exactly when R is a finitely generated R-module. He also shows that in this case, if Ž ] .q denotes the functor Hom R Ž ], R ., then Mqq can be naturally identified with a submodule of MK and that M : Mqq. Thus, if R is a Noetherian normal domain, in particular, if R is a two-dimensional regular local ring M : M** where Ž ] .* denotes the functor Hom R Ž ], R .. Before we state Theorem 1.2 we require some notations and results about symmetric algebras and exterior powers associated to a module. For a Noetherian domain R and a finitely generated, torsion-free R-module M, let SŽ M . denote the image of the symmetric algebra Sym R Ž M . in the Sym K Ž MK . under the canonical map. As an R-algebra, SŽ M . is Sym R Ž M . modulo its ideal of R-torsion elements. The symmetric algebra Sym R Ž M . is a non-negatively graded R-algebra whose nth graded piece is the nth symmetric power of M for each positive integer n. The canonical homomorphism of Sym R Ž M . onto SŽ M . is a graded R-module homomorphism, the kernel of this map being the ideal of R-torsion elements. Thus, SŽ M . is a graded R-algebra whose nth graded component is the nth torsion-free symmetric power, denoted S nŽ M . of M. If A s A n is a graded subalgebra of B s nG 0 Bn then the integral nG 0 closure of A in B is a graded subalgebra of B. Since Sym K Ž MK . is a polynomial ring over K and hence integrally closed, S Ž M . is contained in Sym K Ž MK . and is a graded domain. In particular, the homogeneous graded component of degree 1 of S Ž M . can be identified with a submodule of Sym 1K Ž MK . s MK .
[
[
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RADHA MOHAN
We denote by Hn Ž M . the nth exterior power of the module M and by EnŽ M . the nth exterior power modulo R-torsion. Then EnŽ M . is the image of Hn Ž M . in Hn Ž MK . via the canonical map. If rank R Ž M . s r, then Hr Ž M . is an R-module of rank 1 contained in K and hence is isomorphic to an ideal of R. If N is a submodule of M of the same rank, then Er Ž N . is contained in Er Ž M . and fixing an isomorphism of Er Ž M . with an ideal of R we can identify Er Ž N . as a subideal. The following fundamental theorem due to Rees wRs, Theorems 1.2 and 1.5x gives three different characterizations of when an element is integral over a module. THEOREM 1.2 ŽRees.. Let R be a Noetherian domain and let M be a finitely generated, torsion-free module of rank r. For an element ¨ g MK , the following are equi¨ alent: 1. The element ¨ is integral o¨ er M in the sense of Definition 1.1. 2. The element ¨ g MK s Sym 1K Ž MK . is integral o¨ er SŽ M .. 3. Under some isomorphism of Er Ž M q R¨ . with an ideal I of R, the subideal J corresponding to Er Ž M . is a reduction of the ideal I. We will next state the definition of reductions for modules and a fundamental theorem about minimal reductions. DEFINITION 1.3. Let R be a Noetherian domain and M a finitely generated, torsion-free R-module. A submodule N of M is said to be a reduction of M if M ; N. If in addition we assume that R is a local domain and let n Ž N . denote the number of minimal generators of M, N is said to be a minimal reduction of M if n Ž N . is minimal over the set of all possible reductions of M. The following result of Rees wRs, Lemma 2.2x generalizes to modules the theorem that any m-primary ideal of a d-dimensional, Noetherian local ring Ž R, m. with infinite residue field has a d-generated reduction, where d ) 0. THEOREM 1.4 ŽRees.. Let R be a d-dimensional, Noetherian local domain with infinite residue field and M a non-free, finitely generated, torsion-free R-module. Then M has a minimal reduction which is generated by at most rank R Ž M . q d y 1 elements. Further, a minimal basis of a minimal reduction of M forms part of a minimal basis for M. In particular, when d s 2, M has a rank R Ž M . q 1 generated minimal reduction. The next proposition shows that if a minimal reduction of M is generated by l elements then any general set of l elements of M generates a minimal reduction of M.
5
CORE OF A MODULE
PROPOSITION 1.5. Let Ž R, m. be a d-dimensional, Noetherian local domain with infinite residue field k, and M a non-free, finitely generated torsion-free R-module. Let f 1 , f 2 , . . . f l be minimal generators of a minimal reduction N of M. Then the set of elements g g M such that Ž f 1 , f 2 , . . . f ly1 , g . is a reduction of M generates M and its image is Zariski open in MrmM.1 Proof. Let h1 , h 2 , . . . h t be a fixed set of generators for M. Fix an embedding of M into a free module F where rank R Ž M . s rank R Ž F .. Fix a basis for F and express each h i in terms of this basis. Consider the torsion-free symmetric algebra of M, S Ž M ., which is contained in Sym R Ž F . s Rw T1 , T2 , . . . Tr x. Each h i corresponds to a linear form in Sym R Ž F ., which we denote as ˜ h i , and these linear forms are the generators of S1Ž M .. Let ‘‘ ’’ denote the image of ˜ h i in S1Ž M .rmS1Ž M .. Consider the fiber ring SŽ M . mS Ž M .
sk ˜ h1 , ˜ h2 , . . . ˜ ht
Noether
¢
Normalization
k f˜1 , f˜2 , . . . f˜l .
Fix f˜1 , f˜2 , . . . f˜ly1. By Noether Normalization we can choose ˜ g such that k f˜1 , f˜2 , . . . f˜ly1 , ˜ g ¨ k ˜ h1 , ˜ h2 , . . . ˜ ht finite
Ž ).
and ˜ g s Ýtjs1 u i j˜ h j; u i j g k, as k is infinite. We note that there exist infinitely many such ˜ g. Indeed, the set of ˜ g which are chosen as above generates S1Ž M .rmS1Ž M .. By the proof of normalization we can choose u i j g k such that they are in the complement of the solution set of a finite number of equations; hence, this set is Zariski open. Finally, Ž f˜1 , f˜2 , . . . f˜ly1 , ˜ g . is a reduction of M, for by the finiteness of the inclusion in Ž ) . there exists an integer n such that S n Ž M . : Ž f˜1 , f˜2 , . . . f˜ly1 , ˜ g . Sny1Ž M . q mSnŽ M .. By Nakayama’s lemma we have ˜ ˜ Ž . Ž Sn M s f 1 , f 2 , . . . f˜ly1 , ˜ g . Sny1Ž M .. By Theorem 1.5 of wRsx the result follows. Buchsbaum]Rim Multiplicity We will now review the definition of Buchsbaum]Rim multiplicity for modules. The concept of multiplicity for a module was introduced by Buchsbaum and Rim in 1964 wBchRmx Žsee also wKrbx.. 1
Here we are regarding MrmM as a finite-dimensional k-vector space and hence, as an affine space over the field k.
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RADHA MOHAN
Let R be a Noetherian local ring of dimension d. Let P be an R-module of finite length with a free presentation G ª F ª P ª 0. Let SŽ G . denote the image of Sym R Ž G . in Sym R Ž F . s SŽ F .. Then SŽ G . is a graded subring of SŽ F . whose homogeneous components are denoted SnŽ G .. Buchsbaum and Rim wBchRm, Theorem 3.1x showed that if R is a Noetherian local ring of dimension d and P a finite length, non-zero R-module, then l R ŽSym Rn Ž F .rSŽ G .. is asymptotically given by a polynomial function, pŽ n., of n of degree rank R Ž F . q d y 1 and that the normalized leading coefficient of this polynomial is independent of the presentation chosen. DEFINITION 1.6. With notation as above, the normalized leading coefficient of pŽ n., denoted eŽ P ., is an invariant of P and is called the Buchsbaum]Rim multiplicity of P. The Buchsbaum]Rim multiplicity of the zero module is defined to be zero. We note here that if I is an m-primary ideal of a Noetherian local ring, Ž R, m., then the Buchsbaum]Rim multiplicity of RrI is the usual multiplicity of the ideal I. Contracted Modules and Module Transforms Throughout, Ž R, m, k . is a two-dimensional regular local ring with infinite residue field. We next discuss the structure of a finitely generated, torsion-free R-module and review the notions of contracted modules and module transforms as developed by Kodiyalam wKdyx. Let M be a non-free, finitely generated, torsion-free R-module. M embeds canonically in its double dual M** and it is well known that depthŽ M**. G 2. From the Auslander]Buchsbaum formula it follows that the projective dimension of M** is 0 and hence, M** is free. Let M** s F. Since M is free on the punctured spectrum of R, FrM has finite length. We have the following exact sequence G
M
F ª FrM ª 0,
6
A
6
0ªK
where K and G are free R-modules. Further, if F9 is any free module containing M such that F9rM is of finite length then F9 is isomorphic to F up to unique isomorphism wKdy, Proposition 2.1x. Let rank R Ž M . denote the rank of M and let n R Ž M . denote the minimal number of generators of M. We have rank R Ž M . s rank R Ž F . since FrM is of finite length. Choose a basis for F and a minimal generating set for M
CORE OF A MODULE
7
and consider the matrix expressing this set of generators in terms of the chosen basis of F. Considering the elements of F as column vectors we get a rank R Ž M . = n R Ž M . representing matrix for M. By an abuse of notation we let M denote the representing matrix for M. The ideal of maximal minors, i.e., the minors of size rank R Ž M ., is denoted I Ž M .. It is easy to see that I Ž M . is independent of choices made and is an invariant of the module. ŽIn the statement of the main theorem we consider a presenting matrix A for M. We note here that this matrix is different from the representing matrix for M mentioned above.. We also note that if M is a free module then I Ž M . s R and if M is non-free then it follows from the above exact sequence and the fact that FrM is finite length that I Ž M . is m-primary. DEFINITION 1.7. Let M be a finitely generated, torsion-free R-module. Let S s Rw mrx x be a birational overring of R where x is a minimal generator of the maximal ideal m. We call MS the transform of M in S. The module M is said to be contracted from S if M s MS l F regarded as submodules of FS. For an m-primary ideal I of R, the proper transform of I in S, denoted by I S , is defined to be the ideal xyr IS where r s ordŽ I . as in wHnk, p. 326x. While by the above definition the transform of an ideal would merely be its extension to IS. However, both of these definitions give isomorphic S-modules which is all that is required in this context. Now we recall, briefly, the notion of quadratic transforms of a two-dimensional regular local ring. A first quadratic transform of R is a ring obtained by localizing a ring of the form S s Rw mrx x where x g m y m2 at a maximal ideal P containing mS. Such a ring is a two-dimensional regular local ring and we define the nth quadratic transform of R as the first quadratic transform of an Ž n y 1.st quadratic transform of R. In general, a quadratic transform of R is an nth quadratic transform of R for some n. By convention, we regard R as a quadratic transform of R with n s 0. By well-known results on quadratic transforms wZrsSml, p. 392x, if T is a quadratic transform of R, there is a unique sequence of quadratic transforms, R s T0 ; T1 ; ??? Tny1 ; Tn s T, where each Tiq1 is a first quadratic transform of Ti for i s 0, 1, . . . , n y 1. For an m-primary ideal I of R, the proper transform of the I in T is denoted I T and is defined to be xyr IT where r s ordŽ I .. If T is a localization of S, then it is clear that I T s I S T. For an iterated quadratic transform T of R with the associated sequence R s T0 ; T1 ; ??? Tny1 ; Tn s T, the proper transform of I in T is defined inductively as Ž I T ny 1 .T .
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RADHA MOHAN
We are now in a position to state several results of Kodiyalam wKdyx which we will need. THEOREM 1.8 ŽKodiyalam.. Let M be a finitely generated, torsion-free R-module and F s M**. Then the following conditions are equi¨ alent: 1. There exists x g m y m2 so that M is contracted from S s Rw mrx x. 2. ord R Ž I Ž M .. s n R Ž M . y rank R Ž M .. THEOREM 1.9 ŽKodiyalam.. Let M be a finitely generated, torsion-free complete R-module and T a quadratic transform of R. Then the transform MT is a complete T-module. The next two results discuss respectively the structure of the ideal I Ž M . when M is contracted from S s Rw mrx x for some x g m y m2 wKdy, Theorem 2.8x and the structure of the ideal of maximal minors of the transform, MT of M wKdy, Proposition 4.7x. THEOREM 1.10 ŽKodiyalam.. Let M be a finitely generated, torsion-free R-module that is contracted from S s Rw mrx x for some x g m y m2 . Then for e¨ ery n G 0, the module EnŽ M . is contracted from S. In particular, when n s rank R Ž M . then the ideal I Ž M . s Er Ž M . is contracted from S. THEOREM 1.11 ŽKodiyalam.. Let T be a quadratic transform of R and let M be a finitely generated, torsion-free R module. Then I Ž MT . s I Ž M .T . Finally, we state the theorem of Kodiyalam which says that the Buchsbaum]Rim multiplicity of a module decreases on taking quadratic transforms wKdy, Theorem 4.8x. THEOREM 1.12 ŽKodiyalam.. Let T be a quadratic transform of R and let M be a finitely generated, torsion-free R-module. Then eŽŽ MT .**rMT . eŽ M**rM ..
2. THE CORE OF A MODULE Let Ž R, m, k . be a two-dimensional regular local ring with maximal ideal m and infinite residue field k. 2.1. Some Notation Let M be a finitely generated, torsion-free R-module of rank R Ž M . s r. Choose a set of minimal generators for M so that the first r q 1 of these generators generate a minimal reduction N of M. We can do this by Theorem 1.4.
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CORE OF A MODULE
Let A be an n = n y r presenting matrix for M with the above choice of generators of M. Then the sequence G
M
F ª FrM ª 0
Ž ).
6
A
6
0ªK
is exact. Define B to be the n y r y 1 = n y r matrix obtained by deleting the first r q 1 rows of A. Let B t denote the transpose of B. Let r i denote the ith row of A. Let J be the ideal generated by detŽr i , B ., i s 1, 2 ??? r q 1. In analogy with Lemma 3.1 of wHnkSwnx, we have the following lemma. LEMMA 2.2. With notation as in 2.1, B t presents the ideal Ž N : M .. R Furthermore, Iny ry1Ž B . s Ž N : M .. R
Proof. Let L be the complex G
M
F ª FrM ª 0,
6
A
6
L: 0 ª K
where the free module F has rank r and the free module G has rank n. Let K be the complex N
F ª FrN ª 0,
6
K: 0 ª R ª G9
where the free module G9 has rank r q 1. This last fact follows from Theorem 1.4. The map from FrN to FrM induces a map from K ª L. Consider the standard construct of the mapping cone M of the dual map L* ª K* Žsee wRtm, p. 175x.. It follows from the fact that FrN is finite length that K* is an acyclic complex and further, H 0 ŽK*. s Ext 2R Ž FrN, R .. Similarly, H 0 ŽL*. s Ext 2R Ž FrM, R .. Consider the exact sequence 0 ª MrN ª FrN ª FrM ª 0. On dualizing this sequence, we get the exact sequence 0 ª Ext 2R Ž FrM, R . ª Ext 2R Ž FrN, R . ª Ext 2R Ž MrN, R . ª 0. It then follows, by a standard result of mapping cones wPskSpz, Proposition 2.6x, that M is acyclic. Also, after splitting off free copies of R from M we get that a presenting matrix of H 0 ŽM. is B t. It remains to show that Nt
H 0 ŽM. s Ž N : M .. To see this, consider the complex K*: 0 ª F ª G9 ª R R ª 0 and note that H 0 ŽK*. s RrI Ž N . by the Hilbert]Burch Theorem wBrnHrz, Theorem 1.4.16x. This shows that Ext 2R Ž FrN, R . s RrI Ž N ..2 It 2
This implies that I Ž N . multiplies F into N. For, by local duality, Ext 2R Ž FrN, R . ks s FrN and I Ž N . annihilates this module.
Hm0 Ž FrN .
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RADHA MOHAN
follows that Ext 2R Ž MrN, R . s RrJ where J s annŽExt 2R Ž MrN, R ... It is clear that ann R Ž MrN . s Ž N : M . : J. In fact, equality holds in this R inclusion. To see this we note that by local duality, Ext 2R Ž MrN, R . ks Hm0 Ž MrN . s MrN Žsee wBrnHrz, Corollary 3.5.9x.. Hence, J : ann R Ž MrN . s Ž N : M .. Therefore, H0 ŽM. s Ž N : M .. R
R
LEMMA 2.3. With notation as in 2.1, I Ž N . s Ž det Ž r 1 , B . , det Ž r 2 , B . ??? det Ž rrq1 , B . . . Proof. By the Buchsbaum]Eisenbud Structure Theorem wHch, Theorem 7.1x and the theory of multipliers Žsee wHch, pp. 44 and 45x. there exists an element c s arb, b / 0, of the fraction field of R such that cI Ž N . is equal to the ideal ŽdetŽr 1 , B ., detŽr 2 , B . ??? detŽrrq1 , B ... This then implies that aI Ž N . s bŽdetŽr 1 , B ., detŽr 2 , B . ??? detŽrrq1 , B ... By localizing the sequence Ž). on the punctured spectrum of R we get a split exact sequence since FrM is of finite length. This shows that ŽdetŽr 1 , B ., detŽr 2 , B . ??? detŽrrq1 , B .. is m-primary. Also, I Ž N . is mprimary. Hence, we can conclude by a standard argument that I Ž N . s ŽdetŽr 1 , B ., detŽr 2 , B . ??? detŽrrq1 , B ... PROPOSITION 2.4. Let M be a contracted module satisfying the hypothesis of 2.1. Then, the ideal Iny ry1Ž B . is also contracted. Proof. Take a minimal resolution of FrM satisfying the conditions Ži. the sequence G
M
F ª FrM ª 0
6
A
6
0ªK
is exact, where the free module F has rank r and the free module G has rank n. Žii. the first r q 1 columns of M generate a minimal reduction N of M. By minimality of the resolution we know that all entries of A and hence those of B are in the maximal ideal. Hence, m Ž Iny ry1Ž B .. s n y r. It is enough to show that ordŽ Iny ry1Ž B .. s n y r y 1 wHnk, Proposition 2.3x. By way of contradiction, assume that ordŽ Iny ry1Ž B .. G n y r. This implies that Iny ry1Ž B . is contained in m ny r. Consequently, I Ž N . ; mIny ry1Ž B . ; m ny rq1. By Theorem 1.1, I Ž N . is a reduction of I Ž M .; hence ordŽ I Ž N .. s ordŽ I Ž M ... By Theorem 1.10 we know that since M is contracted from S, I Ž M . is contracted and it follows from Theorem 1.8 that ordŽ I Ž M .. s n y r. This completes the proof.
CORE OF A MODULE
11
Remarks. 1. With notation as in 2.1 if some entry of A is a unit of R, then there exists an entry of B which is a unit of R. 2. We first note that if Ž y 1 , y 2 . generate a minimal reduction of I Ž N . then y 1 and y 2 form part of a minimal generating set for I Ž N . and so by Lemma 2.2 we can write rq1
y1 s
Ý l i det Ž r i , B . is1 rq1
y2 s
Ý t i det Ž r i , B . , is1
where l i , t j are units for some i, j. Also, Ž y 1 , y 2 . generate a minimal reduction of I Ž M . and so y 1 and y 2 form part of a minimal generating set for I Ž M .. Proof of Remark 1. Assume that some entry of A is a unit. Further, we may assume that this entry is of the form a i1 after interchanging columns of A. If i g r q 2, r q 3 ??? n4 then there is nothing to prove. So, assume after interchanging rows of A that a11 is a unit. We can, further, assume that a11 s 1. After performing elementary row operations consider the n = n y r matrix 1 0 A9 s .. .
??? ??? ???
) ) .. .
0 0
???
)
which has as its rows r 1 , r 2 y a21 r 1 , . . . rn y a n1 r 1. Now, detŽr i y a i1 r 1 , rrq2 y aŽ rq2.1 r 1 , . . . , rn y a n1 r 1 . s 0 where i s 2, 3, . . . r q 1. This says that det Ž r i , B . s a i1 det Ž r 1 , rrq2 , . . . rn . q aŽ rq2.1 det Ž r i , r 1 , . . . rn . q ??? qan1 det Ž r i , rrq2 ??? r 1 . . Assume that each a i1 is in the maximal ideal m of R for i s r q 2, . . . n. Ž . Ž . Then y 1 s Ý rq1 is1 a i1 l i det r 1 , r rq2 , . . . r n q g where g g mI M . This forces rq1 Ý rq1 a l to be a unit. Similarly, Ý a t is also a unit. This implies that is1 i1 i is1 i1 i y 1 y ky 2 g mI Ž M . where k is a unit of R. But this contradicts the fact that y 1 y ky 2 , y 2 also generates a minimal reduction of IŽM.. Hence, one of the a i1 for i s r q 2, . . . n is a unit of R. In analogy with Proposition 3.3 of wHnkSwnx we have the following proposition.
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RADHA MOHAN
PROPOSITION 2.5. Let M be a non-free, finitely generated, torsion-free complete R-module. Let A be an n = n y r presenting matrix for M satisfying the conditions of 2.1. Let B be the submatrix of A obtained after deleting the first r q 1 rows of A. Then, the ideals Iny ry1Ž B . and Inyry1Ž A. are equal and are integrally closed ideals. Proof. Suppose that the resolution where A is the presenting matrix of M is not minimal. This means that one of the entries of A is a unit of R and by Remark 1, one of the entries of B is a unit of R. Without loss of generality we may assume that some entry a i j of B is a unit and so all other entries in the ith row and jth column are 0. This is because elementary column transformations leave Ii Ž A. and Ii Ž B . unchanged for all integers i. Then the matrix A9 obtained by deleting the ith row and jth column is still a presenting matrix for M leaving out the redundant generator of M. We note that for all integers i, Ii Ž A. s Iiy1Ž A9. and Ii Ž B . s Iiy1Ž B9., where B9 is the submatrix of A9 obtained by deleting the first r q 1 rows of A9. So, we may assume that n s m Ž M .. Since M is complete it is contracted wKdy, Proposition 4.3x. It then follows by Proposition 2.4 that Iny ry1Ž B . is contracted from some Rw mrx x, x g m y m2 . By Theorem 1.10, I Ž M . is contracted. As the residue field is infinite we can choose x g m y m2 such that both Iny ry1Ž B . and I Ž M . are contracted from S s Rw mrx x. Let P be a height 2 maximal ideal of S containing mS and let T be the two-dimensional regular local ring SP . Let Arx denote the matrix with entries in S obtained by dividing each entry of A by x. We claim that matrix Ž Arx .P presents the finitely generated torsion-free module MT. By wKdy, Proposition 2.2x, I Ž A. s I Ž M .. It is then clear that I Ž Arx . s I Ž A. Srx ny r s I Ž M . Srx ny r s I Ž M . S . The last equality follows since ord R Ž I Ž M .. s n y r by Theorem 1.8. Let G1 G2 Ms . ..
0 Gr
be the r = n matrix representing a set of generators of the module M in terms of a chosen basis for F s M**. Let G1rx n1 G 2rx n 2 M9 s , .. .
0 Grrx n r
CORE OF A MODULE
13
where the order of the ideal generated by Gi , the ith row of the matrix M, is n i . Consider the complex GT
Ž M 9.P
MT** ª 0
6
Ž Arx .P
6
0 ª KT
which is acyclic by the Buchsbaum]Eisenbud exactness criterion wBchEnb2x. As noted above, I ŽŽ Arx .P . s Ž I Ž M . S .P s I Ž M .T . By Theorem 1.11 I Ž M .T s I Ž MT .; hence heightŽ I ŽŽ Arx .P .. G 2. The above complex is a resolution of the module Ž MT .**rMT and once again, by Remark 1 we may assume that the resolution is minimal. By Theorem 1.12, eŽ MT**rMT . - eŽ M**rM . and by Theorem 1.9 MT is a complete module, so by induction on the Buchsbaum]Rim multiplicity of M we have, Inyry1 Ž Ž Arx . P . s Inyry1 Ž Ž Brx . P . and both these ideals are integrally closed. So, Inyry1 Ž Arx . R w mrx x P s Inyry1 Ž Brx . R w mrx x P and is integrally closed for each height 2 prime ideal P of Rw mrx x. Hence, Iny ry1Ž Arx . s Inyry1Ž Brx . and is integrally closed in S. Finally, Iny ry1 Ž A . : x ny ry1 Inyry1 Ž Arx . l R s x ny ry1 Inyry1 Ž Brx . l R s Iny ry1 Ž B . : Inyry1 Ž A . and equality holds throughout. Thus, Iny ry1Ž A. is contracted from S and complete. In analogy with Corollary 3.6 of wHnkSwnx, we have the following corollary. COROLLARY 2.6. Let M be a non-free, finitely generated, torsion-free, complete R-module. Let N be any minimal reduction of M. Let A be an n = n y r presenting matrix of M. Then, Iny ry1 Ž A . s N : M .
ž
R
/
Proof. Choose a presenting matrix A9 and B9 satisfying the conditions of 2.1. By Proposition 2.5 and Lemma 2.2 we can conclude that Iny ry1Ž A9. s Iny ry1Ž B9. s Ž N : M .. By the invariance of Fitting ideals Inyry1Ž A. s R Iny ry1Ž A9. s Ž N : M .. R
14
RADHA MOHAN
COROLLARY 2.7. Let M be a non-free, finitely generated, torsion-free, complete R-module. Let N be a minimal reduction of M. Then with A and B as abo¨ e we ha¨ e that Iny ry1 Ž A . M s Inyry1 Ž A . N s Inyry1 Ž B . M s Inyry1 Ž B . N. In particular, Iny ry1 Ž A . M ; core Ž M . . Proof. We will show that Iny ry1Ž A. N s Inyry1Ž B . M; the other equalities follow from Proposition 2.5. In particular, Iny ry1Ž A. N is contained in Iny ry1Ž B . M. We want to show that Iny ry1Ž B . M ; Inyry1Ž A. N. As before let G1 G2 Ms . ..
0 Gr
be the r = n matrix representing a set of generators of the module M in terms of a chosen basis for F s M**. Consider the matrix equation MA s 0. Summing the rows of this equation times suitable n y r y 2 minors of B gives Iny ry1 Ž B . M ; Inyry1 Ž A . N. Finally, any reduction of M contains a minimal reduction of M, hence, Iny ry1 Ž A . M s Inyry1 Ž B . M s N : M M ; N ; core Ž M . .
ž
R
/
The next lemma and proposition together help us prove the equality coreŽ M . s Iny ry1Ž A. M. LEMMA 2.8. Let Ž R, m, k . be a two-dimensional regular local ring. Let M be a non-free, finitely generated, torsion-free R-module of rank r and let N be a minimal reduction of M. N is minimally generated by r q 1 elements, say Ž f 1 , f 2 , . . . f rq1 .. If a1 f 1 q a2 f 2 q ??? qa rq1 f rq1 s 0, where the a i g R, then a i g I Ž N .. Furthermore, if M is a complete R-module then, for each i, a i g Inyry1Ž A. where A is an n = n y r presenting matrix for M.
CORE OF A MODULE
15
Proof. Choose a minimal set of generators Ž f 1 , f 2 , . . . f rq1 , . . . f n . for the module M. Let M be the r = n matrix representing a set of generators of the module M in terms of a chosen basis for F s M**. Let N be the r = r q 1 matrix Ž C1 , C2 , . . . Crq1 . where C1 , . . . Crq1 are the first r q 1 columns of the matrix M. Let I Ž N . be the ideal of maximal minors of the matrix N; i.e., I Ž N . s Ž D 1 , D 2 , . . . D rq1 . where D i is the minor computed after leaving out the column Ci . Then, D 1 f 1 q D 2 f 2 q ??? qD rq1 f rq1 s det Ž G1 , N . q det Ž G 2 , N . q ??? qdet Ž Gr , N . s 0, where Gi is the ith row of the matrix N. Now, if a1 f 1 q a2 f 2 q ??? qa rq1 f rq1 s 0, then the r q 1 tuple Ž a1 , a2 , . . . a rq1 . is a multiple of the tuple Ž D 1 , D 2 , . . . D rq1 . over the fraction field of R. Hence, there exist non-zero elements c and d of R such that cŽ a1 , a2 , . . . a rq1 . s dŽ D 1 , D 2 , . . . D rq1 .. Let J be the ideal generated by a1 , a2 , . . . a rq1. Then cJ s dI Ž N .. This says that I Ž N . ; Ž c : d .. R Since I Ž N . is m-primary, we have that J s bI Ž N ., for some non-zero element b of R and it follows that each a i g I Ž N .. The last part follows from Lemma 2.2, Proposition 2.5, and footnote 2. In order to state Proposition 2.9 we need some notation. Let M be a non-free, finitely generated, torsion-free Ž R, m, k . module. We say that f 1 , f 2 , . . . f r are part of a minimal reduction of M if there exists f rq1 g M Žactually, infinitely many. such that f 1 , f 2 , . . . f r , f rq1 is a minimal reduction of M. PROPOSITION 2.9. Let M be a non-free, finitely generated, torsion-free, complete R-module o¨ er a two-dimensional regular local ring Ž R, m, k . with infinite residue field. Let f 1 , f 2 , . . . f r be part of a minimal reduction of M. Then,
F Ž f 1 , f 2 , . . . f r , f rq1 . s Ž f 1 , f 2 , . . . f r . q Inyry1Ž A . M f rq1
for any n = n y r presenting matrix A of M where f rq1 ¨ aries o¨ er all elements such that Ž f 1 , f 2 , . . . f r , f rq1 . is a reduction of M. Proof. The right-hand side is contained in the left-hand side by Corollary 2.7. We want to show that the left-hand side is contained in the right-hand side. We note that if N and M are R-modules such that N : M and MrN has finite length, then N s M if and only if N s M l Ž0 : m.. N
16
RADHA MOHAN
Let
ag
F Ž f 1 , f 2 , . . . f r , f rq1 . . f rq1
Fix one f rq1 and write
a s a1 f 1 q a2 f 2 q ??? qar f r q a rq1 f rq1 . It is enough to show that a rq1 g Inyry1Ž A.. Assume, by way of contradiction, that a rq1 f Inyry1Ž A.. As Inyry1Ž A. is m-primary there exists a least integer t such that m ty1 a rq1 f Inyry1Ž A. and m t a rq1 g Inyry1Ž A.. By multiplying a by an element of m ty1, we may then assume that a rq1 f Inyry1Ž A. but that xarq1 g Inyry1Ž A. for each x g m. Now, a rq1 f Inyry1Ž A. s Ž N : M .. Hence, there exists an element g g R M such that a rq1 g f Ž f 1 , f 2 , . . . f r , f rq1 .; i.e., a rq1 f Ž f 1 , f 2 , . . . f r , f rq1 . : g. R
Further, we may choose g such that Ž f 1 , f 2 , . . . f r , g . is a reduction of M. This follows from the fact that the set of all elements g g M such that Ž f 1 , f 2 , . . . f r , g . is a reduction of M, generates M and is Zariski open in MrmM by Proposition 1.5. It is also true that Ž f 1 , f 2 , . . . f r , f rq1 q ug . is a reduction of M for infinitely many units u of R. For each such unit u,
a s a1 u f 1 q a2 u f 2 q ??? qa r u f r q a rq1 u Ž f rq1 q ug . . Now for any x g m, xarq1 f rq1 g Inyry1 Ž A . M s Inyry1 Ž A . Ž f 1 , f 2 , . . . f r , f rq1 q ug . by Corollary 2.7. Therefore, there exist t 1 , t 2 , . . . t rq1 g Inyry1Ž A. such that xarq1 f rq1 s t 1 f 1 q t 2 f 2 q ??? qt rq1 Ž f rq1 q ug . , i.e., r
xa y
Ý ai f i s t1 f 1 q t 2 f 2 q ??? qtrq1Ž f rq1 y ug . is1 r
x Ž a1 u f 1 q a2 u f 2 q ??? qa r u f r q a rq1 u Ž f rq1 q ug . . y
Ý ai fi is1
s t 1 f 1 q t 2 f 2 q ??? qt rq1 Ž f rq1 y ug . . By Lemma 2.8, the coefficient of f rq1 is in I Ž N ..
CORE OF A MODULE
17
So, we conclude that xarq1 u y t rq1 g I Ž N . ; Ž N : M . s Inyry1Ž A.. R Hence, xarq1 u g Inyry1Ž A.. Since this holds for all x g m and all allowable units u, the set of all a rq1 u generate an Rrm vector space in RrIny ry1Ž A.. Let u1 , u 2 , . . . u c be units of R such that the elements a rq1 u1, a rq1 u 2 , . . . a rq1 u c are minimal generators of this vector space. Choose u different from all the u i ; this is possible since the residue field k is infinite. We claim that a rq1 u g Inyry1Ž A.. Now, there exist elements l i in R such that c
uarq1 u y
Ý l i u i arq1 u g Inyry1Ž A . . i
is1
Consider the following cases: Case i. c
a rq1 u y
Ý l i arq1 u g Inyry1Ž A . . i
is1
Then going mod Iny ry1Ž A. we get two distinct linear combinations of a rq1 u in terms of the basis vectors. Consequently, the l i are zero mod Iny ry1Ž A.. Hence, a rq1 u g Inyry1Ž A.. But
a s a1 u f 1 q a2 u f 2 q ??? qa r u f r q a rq1 u Ž f rq1 q ug . . This means that a g Ž f 1 , f 2 , . . . f r . q Inyry1Ž A. M and hence the proposition. Case ii. c
a rq1 u y
Ý l i arq1 u f Inyry1Ž A . . i
is1
Let l s Ýcis1 l i y 1. Then, c
la s
Ý li a y a is1 c
s
Ý l i Ž a1 u
i
f 1 q a2 u i f 2 q ??? qa rq1 u iŽ f rq1 q ug . .
is1
y Ž a1 u f 1 q a2 u f 2 q ??? qarq1 u Ž f rq1 q ug . . .
18
RADHA MOHAN
Therefore, c
la s
Ý l i Ž a1 u
i
f 1 q a2 u i f 2 q ??? qa rq1 u i f rq1 .
is1 rq1
y
c
Ý ai u f i q Ý l i u i arq1 u y uarq1 u
ž
is1
i
is1
/
g.
But, c
ž
uarq1 u y
Ý l i u i arq1 u is1
i
/
g g Inyry1 Ž A . M s Iny ry1 Ž A . N.
So, there exist elements t 1 , t 2 , . . . t rq1 g Inyry1Ž A. such that, c
ž
Ý l i u i arq1 u y uarq1 u i
is1
/
g s t 1 f 1 q t 2 f 2 q ??? qt rq1 f rq1 .
Thus, rq1
la s
Ý ai f i is1 c
s
Ý l i Ž a1 u
i
f 1 q a2 u i f 2 q ??? qar u i f r q a rq1 u i f rq1 .
is1 rq1
y
rq1
Ý aiu f i q Ý t i f i . is1
is1
It follows that, la rq1 y Ýcis1 l i a rq1 u i q a rq1 u y t rq1 g Inyry1Ž A.. Consequently, la rq1 y Ýcis1 l i a rq1 u i q a rq1 u g Inyry1Ž A.. If l is a unit then a rq1 g Inyry1Ž A . q Ž a rq1 u , a rq1 u i 4.. But, a rq1 u g, a rq1 u i , g g Ž f 1 , f 2 , . . . f r , f rq1 .. This implies that a rq1 g Inyry1Ž A. q Ž f 1 , f 2 , . . . f r , f rq1 . : g. Hence, a rq1 g Ž f 1 , f 2 , . . . f r , f rq1 . : g. But this R
R
contradicts our choice of g. If l is not a unit then la rq1 g Inyry1Ž A. which implies that a rq1 u y Ýcis1 l i a rq1 u i g Inyry1Ž A.. This contradicts our basic assumption for Case ii. The proof is now complete. We can now prove the main theorem.
CORE OF A MODULE
19
THEOREM 2.10. Let Ž R, m, k . be a two-dimensional regular local ring with infinite residue field. Let M be a finitely generated, torsion-free, complete R-module of rank r. Let A be an n = n y r presenting matrix A for M. Then, core Ž M . s Iny ry1 Ž A . M, where Iny ry1Ž A. is the ideal generated by the n y r y 1 minors of A. Proof of Theorem 2.10. Case i. We first consider the case when M is a non-free, finitely generated, torsion-free, complete R module. We have already seen by Corollary 2.7 that Iny ry1Ž A. M ; coreŽ M .. It remains to prove the reverse inclusion. Let a g coreŽ M . and let N be a minimal reduction of M that is minimally generated by the elements f 1 , f 2 , . . . f rq1. Then, by Proposition 2.9, we have that
a s a11 f 1 q a21 f 2 q ??? qa r1 f r q b 1 s a22 f 2 q ??? qar 2 f r q a rq12 f rq1 q b 2 .. . s a1 rq1 f 1 q a2 rq1 f 2 q ??? qa rq1 rq1 f rq1 q brq1 , where the bi ’s are in Inyry1Ž A. M. Subtracting each of the equations pairwise and using Lemma 2.8 we conclude that each a i j g Inyry1Ž A.. Thus, coreŽ M . ; Inyry1Ž A. M. Case ii. Let M be a free module of rank r. Let A be any presenting matrix for M. The ideal of n y r y 1 minors of A is by definition the r q 1st Fitting ideal of A which by definition is R. Hence, coreŽ M . s M s Iny ry1Ž A. M. This completes the proof of the theorem. As an application of Theorem 2.10 we have a formula relating the Buchsbaum]Rim multiplicity of a complete module M with the length of M**rcoreŽ M . and the length of RrIny ry1Ž A.. COROLLARY 2.11. With notation as in Theorem 2.10, e Ž M**rM . s l Ž Frcore Ž M . . y Ž r q 1 . l Ž RrIny ry1 Ž A . . . Proof. Let N be a minimal reduction of M. Then eŽ M**rM . s lŽ FrN . wBchRm, Corollary 4.5; Kdy, Proposition 3.8x. Consider the exact sequence, 0 ª NrIny ry1Ž A. N ª FrcoreŽ M . ª FrN. It follows that eŽ M**rM . s lŽ FrcoreŽ M .. y lŽ NrIny ry1Ž A. N .. We claim that Nr Iny ry1Ž A. N is isomorphic to Ž RrInyry1Ž A.. rq1. Let f 1 , f 2 , . . . f rq1 be a set of minimal generators for N. Define a homomorphism F: R rq1 ª
20
RADHA MOHAN
NrIny ry1Ž A. N given by F Ž e i . s f i q Inyry1Ž A. N, where e i is a standard basis vector of R rq1. F is a surjective homomorphism. It is clear that Iny ry1Ž A. R rq1 is contained in the kernel of F. If a1 e1 q a2 e2 q ??? qarq1 e rq1 is in the kernel of F then a1 f 1 q a2 f 2 q ??? a rq1 f rq1 g Iny ry1Ž A. N. Thus, a1 f 1 q a2 f 2 q ??? a rq1 f rq1 s t 1 f 1 q ??? t rq1 f rq1 and by Lemma 2.8 we can conclude that each a i g Inyry1Ž A.. Hence, the kernel of F is Iny ry1Ž A. R rq1. This proves the claim. The corollary follows. EXAMPLE 1. Let R s k w x, y xŽ x, y . . Let M be the rank 2 R-module generated by the columns of the matrix Ms
ž
x 0
y
0
0
x
2
xy
y
/
.
Let N be the minimal reduction of M given by Ns
ž
x
y
0
0
x
y2
/
.
M is a complete, indecomposable module and the first three generators of M generate a minimal reduction of M Žsatisfying notation 2.1.. We compute coreŽ M .. A presenting matrix of M is given by yy 2 xy As 0 yx
0 0 . x yy
0
The submatrix B obtained from A by deleting the first 2 q 1 s 3 rows is Žyx yy .. Thus, the ideal of 1 minors of B and of A is m. Hence, coreŽ M . s mM. EXAMPLE 2. Let R s k w x, y xŽ x, y . . Let M be the rank 2 R-module generated by the columns of the matrix Ms
ž
x
y2
2
3
y
x
x2
0
0
xy
0
3
2
0
x
x y
Let N be the minimal reduction of M given by Ns
ž
x
y2
x2
y2
x3
0
/
.
/
.
21
CORE OF A MODULE
M is a complete module and the first three generators of M generate a minimal reduction of M Žsatisfying notation 2.1.. We compute coreŽ M .. A presenting matrix of M is given by 0 yy 0 As 0 0 x
0 0 0 yy x 0
x2 yx 0 0 yy 0
0 0 yx . x 0 y
0
The submatrix B obtained from A by deleting the first 2 q 1 s 3 rows is 0 Bs 0 x
yy x 0
0 yy 0
x 0 . y
0
I3 Ž B . s I3 Ž A. s m3. Thus, coreŽ M . s m3 M. I Ž M . s m3 and I Ž N . s Ž x 4 y y 4 , x 2 y 2 , x 5 .. We note that m2 [ m3 ; M ; m [ m2 . Hence, 4 lŽ FrM . - 9. We use Corollary 2.11 to compute the length of FrM. We have e Ž M**rM . s l Ž FrN . s l Ž Frcore Ž M . . y Ž r q 1 . l Ž RrIny ry1 Ž A . . . By the proof of Lemma 2.2 Ext 2R Ž FrN, R . s RrI Ž N .. By local duality, Ext 2R Ž FrN, R . ks Hm0 Ž FrN . s FrN Žsee wBrnHrz, Corollary 3.5.9x.. Hence, lŽ FrN . s lŽ RrI Ž N .. ks lŽ RrI Ž N ... Therefore, eŽ M**rM . s lŽ FrN . s lŽ RrI Ž N .. s 14. Hence, lŽ FrcoreŽ M .. s 14 q 3Ž6. s 32. Now consider the exact sequence, 0 ª m2 Mrm3 M ª Frm3 M ª Frm2 M ª 0. The length of m2 Mrm3 M s n Ž m2 M . and since m2 M is a complete module it follows by Theorem 1.8 that n Ž m2 M . s rank R Ž m2 M . q ordŽ m2 M . s 2 q Ž2Ž2. q 4. s 10. Hence, lŽ Frm2 M . s 22. Similarly, by considering the exact sequence 0 ª mMrm2 M ª Frm2 M ª FrmM ª 0, we get that the length of FrmM is 14. Then, finally, considering the sequence 0 ª MrmM ª FrmM ª FrM ª 0 we have lŽ FrM . s 14 y 6 s 8.
ACKNOWLEDGMENTS I am deeply grateful to my thesis advisor Professor Bill Heinzer for his encouragement and many useful conversations regarding this work. I thank Vijay Kodiyalam for several discussions regarding complete modules. For a careful reading of the paper and comments I thank Irena Swanson.
22
RADHA MOHAN
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I. Aberbach and C. Huneke, An improved Brianc¸on-Skoda theorem with applications to the Cohen-Macaulayness of Rees algebras, Math. Ann. 297 Ž1993., 343]369. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge studies in ad¨ anced mathematics, vol. 39, Cambridge University Press. H. Skoda and J. Brianc¸on, Sur la cloture integrale d’un ideal ˆ ´ ´ de germes de fonctions holomorphes en un point de C n , C. R. Acad. Sc. Paris Serie ´ A 278 Ž1974., 949]951. D. A. Buchsbaum and D. Eisenbud, Some structure theorems for finite free resolutions, Ad¨ ances in Mathematics 12 Ž1974., 84]139. D. A. Buchsbaum and D. Eisenbud, What makes a complex exact?, J. of Algebra 25 Ž1973., 259]268. D. A. Buchsbaum and D. S. Rim, A generalized Koszul complex II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 Ž1964., 197]224. M. Hochster, Topics in the homological theory of modules over commutative rings, C.B.M.S., vol. 24, Amer. Math. Soc. M. Hochster and C. Huneke, Tight closure, invariant theory, and the Brianc¸onSkoda Theorem, J. Amer. Math. Soc. 3 Ž1990., 31]116. C. Huneke, Complete ideals in two dimensional regular local rings, Commutati¨ e Algebra, Proceedings of a Microprogram, MSRI publication no. 15, SpringerVerlag, 1989, pp. 417]436. C. Huneke and I. Swanson, Cores of ideals in two dimensional regular local rings, Michigan Math. J. 42 Ž1995., 193]208. D. Katz and V. Kodiyalam, Symmetric powers of complete modules over a two dimensional regular local ring, Trans. of Amer. Math. Soc. Žto appear.. D. Kirby, On the Buchsbaum-Rim multiplicity associated with a matrix, J. of London Math. Soc. (2) 32 Ž1985., 57]61. V. Kodiyalam, Integrally closed modules over two dimensional regular local rings, Trans. of Amer. Math. Soc. Ž1995.. J. Lipman, Adjoints of ideals in regular local rings, Mathematical Research Letters 1 Ž1994., 739]755. J. Lipman, Adjoints and polars of simple complete ideals in two dimensional regular local rings, Bull. Soc. Math. Belgique 45 Ž1993., 223]244. J. Lipman and A. Sathaye, Jacobian ideals and a theorem of Brianc¸on-Skoda, Michigan Math. J. 28 Ž1981., 199]222. J. Lipman and B. Tessier, Pseudo-local rational rings and a theorem of Brianc¸on-Skoda, Michigan Math. J. 28 Ž1981., 97]112. D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 Ž1954., 145]158. C. Peskine and L. Szpiro, Liason des varieties algebriques, In¨ ent. Math. 26 ´ ´ Ž1974., 271]302. D. Rees, Reduction of modules, Proc. Cambridge Philos. Soc. 101 Ž1987., 431]449. D. Rees and J. Sally, General elements and joint reductions, Michigan Math. J. 35 Ž1988., 241]254. J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979. I. Swanson, Joint reductions, tight closure and the Brianc¸on-Skoda theorem, J. of Algebra 147 Ž1992., 128]136. O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Van Nostrand Reinhold, New York, N.Y., 1960.
JOURNAL OF ALGEBRA ARTICLE NO.
The Core of a Module over a Two-Dimensional Regular Local Ring Radha Mohan* Department of Mathematics, Purdue Uni¨ ersity, West Lafayette, Indiana 47907 Communicated by Mel¨ in Hochster Received November 14, 1994
This paper explicitly determines the core of a torsion-free, integrally closed module over a two-dimensional regular local ring. It is analogous to a result of Huneke and Swanson which determines the core of an integrally closed ideal. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two-dimensional regular local ring is the product of the module and a certain Fitting ideal of the module. The technical tools used are quadratic transforms and Buchsbaum]Rim multiplicity. Q 1997 Academic Press
INTRODUCTION A recent result of Huneke and Swanson wHnkSwnx explicitly determines the core of an integrally closed ideal over a two-dimensional regular local ring. Here, the core of an ideal I is the intersection of all reductions of I. The concept of the core of an ideal was introduced by Sally in the late 1980s. It is first alluded to in wRsSllx. We recall that a subideal J of an ideal I is a reduction of I, if I is contained in the integral closure of J. Equivalently, J is a reduction of I if there exists a positive integer n such that I n s JI ny 1. Huneke and Swanson wHnkSwnx prove that if I is an integrally closed ideal over a two-dimensional regular local ring then the core of I is a product of I and a certain Fitting ideal of I. Furthermore, they show that this Fitting ideal is integrally closed and deduce that the core of I is integrally closed. *E-mail address: [email protected]. 1 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
2
RADHA MOHAN
This paper generalizes the main theorem of wHnkSwnx to finitely generated, torsion-free integrally closed modules. We obtain an explicit formula for the core of such a module and show that it is also an integrally closed module. The concept of reductions and integral closures for finitely generated, torsion-free modules over a Noetherian domain was introduced by Rees wRsx. ŽFor details see Section 1.. The study of the core of an ideal has been partly motivated by a theorem of Brianc¸on and Skoda Žsee wBrnSkd, LpmSth, LpmTss, Lpm, HchHnk, RsSll, Swn, AbrHnkx.. Likewise, the study of the core of a module, denoted coreŽ M . and defined to be the intersection of all reductions of M, is partly relevant due to a recent Brianc¸on]Skoda type theorem of Katz and Kodiyalam wKtzKdyx for finitely generated, torsionfree, integrally closed modules over two-dimensional regular local rings. One of the crucial ideas used by Huneke and Swanson wHnkSwnx in determining the core of a complete ideal I is the notion of a quadratic transform of I and the fact that the multiplicity of an ideal decreases on taking quadratic transforms. Recently, Kodiyalam wKdyx has introduced the ideas of transforms for finitely generated, torsion-free modules over twodimensional regular local rings. He also shows that the Buchsbaum]Rim multiplicity of a finitely generated, torsion-free module decreases on taking transforms. With these ideas in place we can generalize the theorem of Huneke and Swanson to finitely generated, torsion-free, integrally closed modules. The following theorem is the main result of this paper. THEOREM 2.10. Let Ž R, m, k . be a two-dimensional regular local ring with infinite residue field. Let M be a finitely generated, torsion-free, complete R-module of rank r. Let A be an n = n y r presenting matrix for M. Then, core Ž M . s Iny ry1 Ž A . M, where Iny ry1Ž A. is the ideal of R generated by the n y r y 1 minors of A. We prove this result in Section 2. The structure of the proof is patterned after the proof of Huneke and Swanson wHnkSwnx for the theorem for the core of an ideal. In Section 1 we discuss some preliminaries.
1. PRELIMINARIES Integral Closures and Reductions of Modules We review the notions of integral closures and reductions for torsion-free modules over arbitrary Noetherian domains as developed by Rees wRsx.
CORE OF A MODULE
3
Let R be a Noetherian domain and let K be its field of fractions. Let M be a finitely generated, torsion-free R module. By MK we denote the finite-dimensional K-vector space M mR K. If N is a submodule of M, then NK is naturally identified with a subspace of MK . We say a ring between R and its field of fractions K is a birational o¨ erring of R. For any birational overring S of R, we let MS denote the S-submodule of MK generated by M. There is a canonical R-module homomorphism from M mR S onto MS. Its kernel is the submodule of R-torsion elements. Hence, M mR S modulo S-torsion and MS are isomorphic as S modules. DEFINITION 1.1. With notation as above, an element ¨ g MK is said to be integral over M if ¨ g MV for every discrete valuation ring V of K containing R. The integral closure of M, denoted M, is the set of all elements of MK that are integral over M. The module M is said to be integrally closed or complete if M s M. It is clear that M is an R-submodule of MK which contains M. In fact, since any valuation domain of K which contains R also contains its integral closure R in K, we have that M is an R module. Rees shows in wRs, Lemma 1.1x that M is a finitely generated R-module exactly when R is a finitely generated R-module. He also shows that in this case, if Ž ] .q denotes the functor Hom R Ž ], R ., then Mqq can be naturally identified with a submodule of MK and that M : Mqq. Thus, if R is a Noetherian normal domain, in particular, if R is a two-dimensional regular local ring M : M** where Ž ] .* denotes the functor Hom R Ž ], R .. Before we state Theorem 1.2 we require some notations and results about symmetric algebras and exterior powers associated to a module. For a Noetherian domain R and a finitely generated, torsion-free R-module M, let SŽ M . denote the image of the symmetric algebra Sym R Ž M . in the Sym K Ž MK . under the canonical map. As an R-algebra, SŽ M . is Sym R Ž M . modulo its ideal of R-torsion elements. The symmetric algebra Sym R Ž M . is a non-negatively graded R-algebra whose nth graded piece is the nth symmetric power of M for each positive integer n. The canonical homomorphism of Sym R Ž M . onto SŽ M . is a graded R-module homomorphism, the kernel of this map being the ideal of R-torsion elements. Thus, SŽ M . is a graded R-algebra whose nth graded component is the nth torsion-free symmetric power, denoted S nŽ M . of M. If A s A n is a graded subalgebra of B s nG 0 Bn then the integral nG 0 closure of A in B is a graded subalgebra of B. Since Sym K Ž MK . is a polynomial ring over K and hence integrally closed, S Ž M . is contained in Sym K Ž MK . and is a graded domain. In particular, the homogeneous graded component of degree 1 of S Ž M . can be identified with a submodule of Sym 1K Ž MK . s MK .
[
[
4
RADHA MOHAN
We denote by Hn Ž M . the nth exterior power of the module M and by EnŽ M . the nth exterior power modulo R-torsion. Then EnŽ M . is the image of Hn Ž M . in Hn Ž MK . via the canonical map. If rank R Ž M . s r, then Hr Ž M . is an R-module of rank 1 contained in K and hence is isomorphic to an ideal of R. If N is a submodule of M of the same rank, then Er Ž N . is contained in Er Ž M . and fixing an isomorphism of Er Ž M . with an ideal of R we can identify Er Ž N . as a subideal. The following fundamental theorem due to Rees wRs, Theorems 1.2 and 1.5x gives three different characterizations of when an element is integral over a module. THEOREM 1.2 ŽRees.. Let R be a Noetherian domain and let M be a finitely generated, torsion-free module of rank r. For an element ¨ g MK , the following are equi¨ alent: 1. The element ¨ is integral o¨ er M in the sense of Definition 1.1. 2. The element ¨ g MK s Sym 1K Ž MK . is integral o¨ er SŽ M .. 3. Under some isomorphism of Er Ž M q R¨ . with an ideal I of R, the subideal J corresponding to Er Ž M . is a reduction of the ideal I. We will next state the definition of reductions for modules and a fundamental theorem about minimal reductions. DEFINITION 1.3. Let R be a Noetherian domain and M a finitely generated, torsion-free R-module. A submodule N of M is said to be a reduction of M if M ; N. If in addition we assume that R is a local domain and let n Ž N . denote the number of minimal generators of M, N is said to be a minimal reduction of M if n Ž N . is minimal over the set of all possible reductions of M. The following result of Rees wRs, Lemma 2.2x generalizes to modules the theorem that any m-primary ideal of a d-dimensional, Noetherian local ring Ž R, m. with infinite residue field has a d-generated reduction, where d ) 0. THEOREM 1.4 ŽRees.. Let R be a d-dimensional, Noetherian local domain with infinite residue field and M a non-free, finitely generated, torsion-free R-module. Then M has a minimal reduction which is generated by at most rank R Ž M . q d y 1 elements. Further, a minimal basis of a minimal reduction of M forms part of a minimal basis for M. In particular, when d s 2, M has a rank R Ž M . q 1 generated minimal reduction. The next proposition shows that if a minimal reduction of M is generated by l elements then any general set of l elements of M generates a minimal reduction of M.
5
CORE OF A MODULE
PROPOSITION 1.5. Let Ž R, m. be a d-dimensional, Noetherian local domain with infinite residue field k, and M a non-free, finitely generated torsion-free R-module. Let f 1 , f 2 , . . . f l be minimal generators of a minimal reduction N of M. Then the set of elements g g M such that Ž f 1 , f 2 , . . . f ly1 , g . is a reduction of M generates M and its image is Zariski open in MrmM.1 Proof. Let h1 , h 2 , . . . h t be a fixed set of generators for M. Fix an embedding of M into a free module F where rank R Ž M . s rank R Ž F .. Fix a basis for F and express each h i in terms of this basis. Consider the torsion-free symmetric algebra of M, S Ž M ., which is contained in Sym R Ž F . s Rw T1 , T2 , . . . Tr x. Each h i corresponds to a linear form in Sym R Ž F ., which we denote as ˜ h i , and these linear forms are the generators of S1Ž M .. Let ‘‘ ’’ denote the image of ˜ h i in S1Ž M .rmS1Ž M .. Consider the fiber ring SŽ M . mS Ž M .
sk ˜ h1 , ˜ h2 , . . . ˜ ht
Noether
¢
Normalization
k f˜1 , f˜2 , . . . f˜l .
Fix f˜1 , f˜2 , . . . f˜ly1. By Noether Normalization we can choose ˜ g such that k f˜1 , f˜2 , . . . f˜ly1 , ˜ g ¨ k ˜ h1 , ˜ h2 , . . . ˜ ht finite
Ž ).
and ˜ g s Ýtjs1 u i j˜ h j; u i j g k, as k is infinite. We note that there exist infinitely many such ˜ g. Indeed, the set of ˜ g which are chosen as above generates S1Ž M .rmS1Ž M .. By the proof of normalization we can choose u i j g k such that they are in the complement of the solution set of a finite number of equations; hence, this set is Zariski open. Finally, Ž f˜1 , f˜2 , . . . f˜ly1 , ˜ g . is a reduction of M, for by the finiteness of the inclusion in Ž ) . there exists an integer n such that S n Ž M . : Ž f˜1 , f˜2 , . . . f˜ly1 , ˜ g . Sny1Ž M . q mSnŽ M .. By Nakayama’s lemma we have ˜ ˜ Ž . Ž Sn M s f 1 , f 2 , . . . f˜ly1 , ˜ g . Sny1Ž M .. By Theorem 1.5 of wRsx the result follows. Buchsbaum]Rim Multiplicity We will now review the definition of Buchsbaum]Rim multiplicity for modules. The concept of multiplicity for a module was introduced by Buchsbaum and Rim in 1964 wBchRmx Žsee also wKrbx.. 1
Here we are regarding MrmM as a finite-dimensional k-vector space and hence, as an affine space over the field k.
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RADHA MOHAN
Let R be a Noetherian local ring of dimension d. Let P be an R-module of finite length with a free presentation G ª F ª P ª 0. Let SŽ G . denote the image of Sym R Ž G . in Sym R Ž F . s SŽ F .. Then SŽ G . is a graded subring of SŽ F . whose homogeneous components are denoted SnŽ G .. Buchsbaum and Rim wBchRm, Theorem 3.1x showed that if R is a Noetherian local ring of dimension d and P a finite length, non-zero R-module, then l R ŽSym Rn Ž F .rSŽ G .. is asymptotically given by a polynomial function, pŽ n., of n of degree rank R Ž F . q d y 1 and that the normalized leading coefficient of this polynomial is independent of the presentation chosen. DEFINITION 1.6. With notation as above, the normalized leading coefficient of pŽ n., denoted eŽ P ., is an invariant of P and is called the Buchsbaum]Rim multiplicity of P. The Buchsbaum]Rim multiplicity of the zero module is defined to be zero. We note here that if I is an m-primary ideal of a Noetherian local ring, Ž R, m., then the Buchsbaum]Rim multiplicity of RrI is the usual multiplicity of the ideal I. Contracted Modules and Module Transforms Throughout, Ž R, m, k . is a two-dimensional regular local ring with infinite residue field. We next discuss the structure of a finitely generated, torsion-free R-module and review the notions of contracted modules and module transforms as developed by Kodiyalam wKdyx. Let M be a non-free, finitely generated, torsion-free R-module. M embeds canonically in its double dual M** and it is well known that depthŽ M**. G 2. From the Auslander]Buchsbaum formula it follows that the projective dimension of M** is 0 and hence, M** is free. Let M** s F. Since M is free on the punctured spectrum of R, FrM has finite length. We have the following exact sequence G
M
F ª FrM ª 0,
6
A
6
0ªK
where K and G are free R-modules. Further, if F9 is any free module containing M such that F9rM is of finite length then F9 is isomorphic to F up to unique isomorphism wKdy, Proposition 2.1x. Let rank R Ž M . denote the rank of M and let n R Ž M . denote the minimal number of generators of M. We have rank R Ž M . s rank R Ž F . since FrM is of finite length. Choose a basis for F and a minimal generating set for M
CORE OF A MODULE
7
and consider the matrix expressing this set of generators in terms of the chosen basis of F. Considering the elements of F as column vectors we get a rank R Ž M . = n R Ž M . representing matrix for M. By an abuse of notation we let M denote the representing matrix for M. The ideal of maximal minors, i.e., the minors of size rank R Ž M ., is denoted I Ž M .. It is easy to see that I Ž M . is independent of choices made and is an invariant of the module. ŽIn the statement of the main theorem we consider a presenting matrix A for M. We note here that this matrix is different from the representing matrix for M mentioned above.. We also note that if M is a free module then I Ž M . s R and if M is non-free then it follows from the above exact sequence and the fact that FrM is finite length that I Ž M . is m-primary. DEFINITION 1.7. Let M be a finitely generated, torsion-free R-module. Let S s Rw mrx x be a birational overring of R where x is a minimal generator of the maximal ideal m. We call MS the transform of M in S. The module M is said to be contracted from S if M s MS l F regarded as submodules of FS. For an m-primary ideal I of R, the proper transform of I in S, denoted by I S , is defined to be the ideal xyr IS where r s ordŽ I . as in wHnk, p. 326x. While by the above definition the transform of an ideal would merely be its extension to IS. However, both of these definitions give isomorphic S-modules which is all that is required in this context. Now we recall, briefly, the notion of quadratic transforms of a two-dimensional regular local ring. A first quadratic transform of R is a ring obtained by localizing a ring of the form S s Rw mrx x where x g m y m2 at a maximal ideal P containing mS. Such a ring is a two-dimensional regular local ring and we define the nth quadratic transform of R as the first quadratic transform of an Ž n y 1.st quadratic transform of R. In general, a quadratic transform of R is an nth quadratic transform of R for some n. By convention, we regard R as a quadratic transform of R with n s 0. By well-known results on quadratic transforms wZrsSml, p. 392x, if T is a quadratic transform of R, there is a unique sequence of quadratic transforms, R s T0 ; T1 ; ??? Tny1 ; Tn s T, where each Tiq1 is a first quadratic transform of Ti for i s 0, 1, . . . , n y 1. For an m-primary ideal I of R, the proper transform of the I in T is denoted I T and is defined to be xyr IT where r s ordŽ I .. If T is a localization of S, then it is clear that I T s I S T. For an iterated quadratic transform T of R with the associated sequence R s T0 ; T1 ; ??? Tny1 ; Tn s T, the proper transform of I in T is defined inductively as Ž I T ny 1 .T .
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RADHA MOHAN
We are now in a position to state several results of Kodiyalam wKdyx which we will need. THEOREM 1.8 ŽKodiyalam.. Let M be a finitely generated, torsion-free R-module and F s M**. Then the following conditions are equi¨ alent: 1. There exists x g m y m2 so that M is contracted from S s Rw mrx x. 2. ord R Ž I Ž M .. s n R Ž M . y rank R Ž M .. THEOREM 1.9 ŽKodiyalam.. Let M be a finitely generated, torsion-free complete R-module and T a quadratic transform of R. Then the transform MT is a complete T-module. The next two results discuss respectively the structure of the ideal I Ž M . when M is contracted from S s Rw mrx x for some x g m y m2 wKdy, Theorem 2.8x and the structure of the ideal of maximal minors of the transform, MT of M wKdy, Proposition 4.7x. THEOREM 1.10 ŽKodiyalam.. Let M be a finitely generated, torsion-free R-module that is contracted from S s Rw mrx x for some x g m y m2 . Then for e¨ ery n G 0, the module EnŽ M . is contracted from S. In particular, when n s rank R Ž M . then the ideal I Ž M . s Er Ž M . is contracted from S. THEOREM 1.11 ŽKodiyalam.. Let T be a quadratic transform of R and let M be a finitely generated, torsion-free R module. Then I Ž MT . s I Ž M .T . Finally, we state the theorem of Kodiyalam which says that the Buchsbaum]Rim multiplicity of a module decreases on taking quadratic transforms wKdy, Theorem 4.8x. THEOREM 1.12 ŽKodiyalam.. Let T be a quadratic transform of R and let M be a finitely generated, torsion-free R-module. Then eŽŽ MT .**rMT . eŽ M**rM ..
2. THE CORE OF A MODULE Let Ž R, m, k . be a two-dimensional regular local ring with maximal ideal m and infinite residue field k. 2.1. Some Notation Let M be a finitely generated, torsion-free R-module of rank R Ž M . s r. Choose a set of minimal generators for M so that the first r q 1 of these generators generate a minimal reduction N of M. We can do this by Theorem 1.4.
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CORE OF A MODULE
Let A be an n = n y r presenting matrix for M with the above choice of generators of M. Then the sequence G
M
F ª FrM ª 0
Ž ).
6
A
6
0ªK
is exact. Define B to be the n y r y 1 = n y r matrix obtained by deleting the first r q 1 rows of A. Let B t denote the transpose of B. Let r i denote the ith row of A. Let J be the ideal generated by detŽr i , B ., i s 1, 2 ??? r q 1. In analogy with Lemma 3.1 of wHnkSwnx, we have the following lemma. LEMMA 2.2. With notation as in 2.1, B t presents the ideal Ž N : M .. R Furthermore, Iny ry1Ž B . s Ž N : M .. R
Proof. Let L be the complex G
M
F ª FrM ª 0,
6
A
6
L: 0 ª K
where the free module F has rank r and the free module G has rank n. Let K be the complex N
F ª FrN ª 0,
6
K: 0 ª R ª G9
where the free module G9 has rank r q 1. This last fact follows from Theorem 1.4. The map from FrN to FrM induces a map from K ª L. Consider the standard construct of the mapping cone M of the dual map L* ª K* Žsee wRtm, p. 175x.. It follows from the fact that FrN is finite length that K* is an acyclic complex and further, H 0 ŽK*. s Ext 2R Ž FrN, R .. Similarly, H 0 ŽL*. s Ext 2R Ž FrM, R .. Consider the exact sequence 0 ª MrN ª FrN ª FrM ª 0. On dualizing this sequence, we get the exact sequence 0 ª Ext 2R Ž FrM, R . ª Ext 2R Ž FrN, R . ª Ext 2R Ž MrN, R . ª 0. It then follows, by a standard result of mapping cones wPskSpz, Proposition 2.6x, that M is acyclic. Also, after splitting off free copies of R from M we get that a presenting matrix of H 0 ŽM. is B t. It remains to show that Nt
H 0 ŽM. s Ž N : M .. To see this, consider the complex K*: 0 ª F ª G9 ª R R ª 0 and note that H 0 ŽK*. s RrI Ž N . by the Hilbert]Burch Theorem wBrnHrz, Theorem 1.4.16x. This shows that Ext 2R Ž FrN, R . s RrI Ž N ..2 It 2
This implies that I Ž N . multiplies F into N. For, by local duality, Ext 2R Ž FrN, R . ks s FrN and I Ž N . annihilates this module.
Hm0 Ž FrN .
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RADHA MOHAN
follows that Ext 2R Ž MrN, R . s RrJ where J s annŽExt 2R Ž MrN, R ... It is clear that ann R Ž MrN . s Ž N : M . : J. In fact, equality holds in this R inclusion. To see this we note that by local duality, Ext 2R Ž MrN, R . ks Hm0 Ž MrN . s MrN Žsee wBrnHrz, Corollary 3.5.9x.. Hence, J : ann R Ž MrN . s Ž N : M .. Therefore, H0 ŽM. s Ž N : M .. R
R
LEMMA 2.3. With notation as in 2.1, I Ž N . s Ž det Ž r 1 , B . , det Ž r 2 , B . ??? det Ž rrq1 , B . . . Proof. By the Buchsbaum]Eisenbud Structure Theorem wHch, Theorem 7.1x and the theory of multipliers Žsee wHch, pp. 44 and 45x. there exists an element c s arb, b / 0, of the fraction field of R such that cI Ž N . is equal to the ideal ŽdetŽr 1 , B ., detŽr 2 , B . ??? detŽrrq1 , B ... This then implies that aI Ž N . s bŽdetŽr 1 , B ., detŽr 2 , B . ??? detŽrrq1 , B ... By localizing the sequence Ž). on the punctured spectrum of R we get a split exact sequence since FrM is of finite length. This shows that ŽdetŽr 1 , B ., detŽr 2 , B . ??? detŽrrq1 , B .. is m-primary. Also, I Ž N . is mprimary. Hence, we can conclude by a standard argument that I Ž N . s ŽdetŽr 1 , B ., detŽr 2 , B . ??? detŽrrq1 , B ... PROPOSITION 2.4. Let M be a contracted module satisfying the hypothesis of 2.1. Then, the ideal Iny ry1Ž B . is also contracted. Proof. Take a minimal resolution of FrM satisfying the conditions Ži. the sequence G
M
F ª FrM ª 0
6
A
6
0ªK
is exact, where the free module F has rank r and the free module G has rank n. Žii. the first r q 1 columns of M generate a minimal reduction N of M. By minimality of the resolution we know that all entries of A and hence those of B are in the maximal ideal. Hence, m Ž Iny ry1Ž B .. s n y r. It is enough to show that ordŽ Iny ry1Ž B .. s n y r y 1 wHnk, Proposition 2.3x. By way of contradiction, assume that ordŽ Iny ry1Ž B .. G n y r. This implies that Iny ry1Ž B . is contained in m ny r. Consequently, I Ž N . ; mIny ry1Ž B . ; m ny rq1. By Theorem 1.1, I Ž N . is a reduction of I Ž M .; hence ordŽ I Ž N .. s ordŽ I Ž M ... By Theorem 1.10 we know that since M is contracted from S, I Ž M . is contracted and it follows from Theorem 1.8 that ordŽ I Ž M .. s n y r. This completes the proof.
CORE OF A MODULE
11
Remarks. 1. With notation as in 2.1 if some entry of A is a unit of R, then there exists an entry of B which is a unit of R. 2. We first note that if Ž y 1 , y 2 . generate a minimal reduction of I Ž N . then y 1 and y 2 form part of a minimal generating set for I Ž N . and so by Lemma 2.2 we can write rq1
y1 s
Ý l i det Ž r i , B . is1 rq1
y2 s
Ý t i det Ž r i , B . , is1
where l i , t j are units for some i, j. Also, Ž y 1 , y 2 . generate a minimal reduction of I Ž M . and so y 1 and y 2 form part of a minimal generating set for I Ž M .. Proof of Remark 1. Assume that some entry of A is a unit. Further, we may assume that this entry is of the form a i1 after interchanging columns of A. If i g r q 2, r q 3 ??? n4 then there is nothing to prove. So, assume after interchanging rows of A that a11 is a unit. We can, further, assume that a11 s 1. After performing elementary row operations consider the n = n y r matrix 1 0 A9 s .. .
??? ??? ???
) ) .. .
0 0
???
)
which has as its rows r 1 , r 2 y a21 r 1 , . . . rn y a n1 r 1. Now, detŽr i y a i1 r 1 , rrq2 y aŽ rq2.1 r 1 , . . . , rn y a n1 r 1 . s 0 where i s 2, 3, . . . r q 1. This says that det Ž r i , B . s a i1 det Ž r 1 , rrq2 , . . . rn . q aŽ rq2.1 det Ž r i , r 1 , . . . rn . q ??? qan1 det Ž r i , rrq2 ??? r 1 . . Assume that each a i1 is in the maximal ideal m of R for i s r q 2, . . . n. Ž . Ž . Then y 1 s Ý rq1 is1 a i1 l i det r 1 , r rq2 , . . . r n q g where g g mI M . This forces rq1 Ý rq1 a l to be a unit. Similarly, Ý a t is also a unit. This implies that is1 i1 i is1 i1 i y 1 y ky 2 g mI Ž M . where k is a unit of R. But this contradicts the fact that y 1 y ky 2 , y 2 also generates a minimal reduction of IŽM.. Hence, one of the a i1 for i s r q 2, . . . n is a unit of R. In analogy with Proposition 3.3 of wHnkSwnx we have the following proposition.
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RADHA MOHAN
PROPOSITION 2.5. Let M be a non-free, finitely generated, torsion-free complete R-module. Let A be an n = n y r presenting matrix for M satisfying the conditions of 2.1. Let B be the submatrix of A obtained after deleting the first r q 1 rows of A. Then, the ideals Iny ry1Ž B . and Inyry1Ž A. are equal and are integrally closed ideals. Proof. Suppose that the resolution where A is the presenting matrix of M is not minimal. This means that one of the entries of A is a unit of R and by Remark 1, one of the entries of B is a unit of R. Without loss of generality we may assume that some entry a i j of B is a unit and so all other entries in the ith row and jth column are 0. This is because elementary column transformations leave Ii Ž A. and Ii Ž B . unchanged for all integers i. Then the matrix A9 obtained by deleting the ith row and jth column is still a presenting matrix for M leaving out the redundant generator of M. We note that for all integers i, Ii Ž A. s Iiy1Ž A9. and Ii Ž B . s Iiy1Ž B9., where B9 is the submatrix of A9 obtained by deleting the first r q 1 rows of A9. So, we may assume that n s m Ž M .. Since M is complete it is contracted wKdy, Proposition 4.3x. It then follows by Proposition 2.4 that Iny ry1Ž B . is contracted from some Rw mrx x, x g m y m2 . By Theorem 1.10, I Ž M . is contracted. As the residue field is infinite we can choose x g m y m2 such that both Iny ry1Ž B . and I Ž M . are contracted from S s Rw mrx x. Let P be a height 2 maximal ideal of S containing mS and let T be the two-dimensional regular local ring SP . Let Arx denote the matrix with entries in S obtained by dividing each entry of A by x. We claim that matrix Ž Arx .P presents the finitely generated torsion-free module MT. By wKdy, Proposition 2.2x, I Ž A. s I Ž M .. It is then clear that I Ž Arx . s I Ž A. Srx ny r s I Ž M . Srx ny r s I Ž M . S . The last equality follows since ord R Ž I Ž M .. s n y r by Theorem 1.8. Let G1 G2 Ms . ..
0 Gr
be the r = n matrix representing a set of generators of the module M in terms of a chosen basis for F s M**. Let G1rx n1 G 2rx n 2 M9 s , .. .
0 Grrx n r
CORE OF A MODULE
13
where the order of the ideal generated by Gi , the ith row of the matrix M, is n i . Consider the complex GT
Ž M 9.P
MT** ª 0
6
Ž Arx .P
6
0 ª KT
which is acyclic by the Buchsbaum]Eisenbud exactness criterion wBchEnb2x. As noted above, I ŽŽ Arx .P . s Ž I Ž M . S .P s I Ž M .T . By Theorem 1.11 I Ž M .T s I Ž MT .; hence heightŽ I ŽŽ Arx .P .. G 2. The above complex is a resolution of the module Ž MT .**rMT and once again, by Remark 1 we may assume that the resolution is minimal. By Theorem 1.12, eŽ MT**rMT . - eŽ M**rM . and by Theorem 1.9 MT is a complete module, so by induction on the Buchsbaum]Rim multiplicity of M we have, Inyry1 Ž Ž Arx . P . s Inyry1 Ž Ž Brx . P . and both these ideals are integrally closed. So, Inyry1 Ž Arx . R w mrx x P s Inyry1 Ž Brx . R w mrx x P and is integrally closed for each height 2 prime ideal P of Rw mrx x. Hence, Iny ry1Ž Arx . s Inyry1Ž Brx . and is integrally closed in S. Finally, Iny ry1 Ž A . : x ny ry1 Inyry1 Ž Arx . l R s x ny ry1 Inyry1 Ž Brx . l R s Iny ry1 Ž B . : Inyry1 Ž A . and equality holds throughout. Thus, Iny ry1Ž A. is contracted from S and complete. In analogy with Corollary 3.6 of wHnkSwnx, we have the following corollary. COROLLARY 2.6. Let M be a non-free, finitely generated, torsion-free, complete R-module. Let N be any minimal reduction of M. Let A be an n = n y r presenting matrix of M. Then, Iny ry1 Ž A . s N : M .
ž
R
/
Proof. Choose a presenting matrix A9 and B9 satisfying the conditions of 2.1. By Proposition 2.5 and Lemma 2.2 we can conclude that Iny ry1Ž A9. s Iny ry1Ž B9. s Ž N : M .. By the invariance of Fitting ideals Inyry1Ž A. s R Iny ry1Ž A9. s Ž N : M .. R
14
RADHA MOHAN
COROLLARY 2.7. Let M be a non-free, finitely generated, torsion-free, complete R-module. Let N be a minimal reduction of M. Then with A and B as abo¨ e we ha¨ e that Iny ry1 Ž A . M s Inyry1 Ž A . N s Inyry1 Ž B . M s Inyry1 Ž B . N. In particular, Iny ry1 Ž A . M ; core Ž M . . Proof. We will show that Iny ry1Ž A. N s Inyry1Ž B . M; the other equalities follow from Proposition 2.5. In particular, Iny ry1Ž A. N is contained in Iny ry1Ž B . M. We want to show that Iny ry1Ž B . M ; Inyry1Ž A. N. As before let G1 G2 Ms . ..
0 Gr
be the r = n matrix representing a set of generators of the module M in terms of a chosen basis for F s M**. Consider the matrix equation MA s 0. Summing the rows of this equation times suitable n y r y 2 minors of B gives Iny ry1 Ž B . M ; Inyry1 Ž A . N. Finally, any reduction of M contains a minimal reduction of M, hence, Iny ry1 Ž A . M s Inyry1 Ž B . M s N : M M ; N ; core Ž M . .
ž
R
/
The next lemma and proposition together help us prove the equality coreŽ M . s Iny ry1Ž A. M. LEMMA 2.8. Let Ž R, m, k . be a two-dimensional regular local ring. Let M be a non-free, finitely generated, torsion-free R-module of rank r and let N be a minimal reduction of M. N is minimally generated by r q 1 elements, say Ž f 1 , f 2 , . . . f rq1 .. If a1 f 1 q a2 f 2 q ??? qa rq1 f rq1 s 0, where the a i g R, then a i g I Ž N .. Furthermore, if M is a complete R-module then, for each i, a i g Inyry1Ž A. where A is an n = n y r presenting matrix for M.
CORE OF A MODULE
15
Proof. Choose a minimal set of generators Ž f 1 , f 2 , . . . f rq1 , . . . f n . for the module M. Let M be the r = n matrix representing a set of generators of the module M in terms of a chosen basis for F s M**. Let N be the r = r q 1 matrix Ž C1 , C2 , . . . Crq1 . where C1 , . . . Crq1 are the first r q 1 columns of the matrix M. Let I Ž N . be the ideal of maximal minors of the matrix N; i.e., I Ž N . s Ž D 1 , D 2 , . . . D rq1 . where D i is the minor computed after leaving out the column Ci . Then, D 1 f 1 q D 2 f 2 q ??? qD rq1 f rq1 s det Ž G1 , N . q det Ž G 2 , N . q ??? qdet Ž Gr , N . s 0, where Gi is the ith row of the matrix N. Now, if a1 f 1 q a2 f 2 q ??? qa rq1 f rq1 s 0, then the r q 1 tuple Ž a1 , a2 , . . . a rq1 . is a multiple of the tuple Ž D 1 , D 2 , . . . D rq1 . over the fraction field of R. Hence, there exist non-zero elements c and d of R such that cŽ a1 , a2 , . . . a rq1 . s dŽ D 1 , D 2 , . . . D rq1 .. Let J be the ideal generated by a1 , a2 , . . . a rq1. Then cJ s dI Ž N .. This says that I Ž N . ; Ž c : d .. R Since I Ž N . is m-primary, we have that J s bI Ž N ., for some non-zero element b of R and it follows that each a i g I Ž N .. The last part follows from Lemma 2.2, Proposition 2.5, and footnote 2. In order to state Proposition 2.9 we need some notation. Let M be a non-free, finitely generated, torsion-free Ž R, m, k . module. We say that f 1 , f 2 , . . . f r are part of a minimal reduction of M if there exists f rq1 g M Žactually, infinitely many. such that f 1 , f 2 , . . . f r , f rq1 is a minimal reduction of M. PROPOSITION 2.9. Let M be a non-free, finitely generated, torsion-free, complete R-module o¨ er a two-dimensional regular local ring Ž R, m, k . with infinite residue field. Let f 1 , f 2 , . . . f r be part of a minimal reduction of M. Then,
F Ž f 1 , f 2 , . . . f r , f rq1 . s Ž f 1 , f 2 , . . . f r . q Inyry1Ž A . M f rq1
for any n = n y r presenting matrix A of M where f rq1 ¨ aries o¨ er all elements such that Ž f 1 , f 2 , . . . f r , f rq1 . is a reduction of M. Proof. The right-hand side is contained in the left-hand side by Corollary 2.7. We want to show that the left-hand side is contained in the right-hand side. We note that if N and M are R-modules such that N : M and MrN has finite length, then N s M if and only if N s M l Ž0 : m.. N
16
RADHA MOHAN
Let
ag
F Ž f 1 , f 2 , . . . f r , f rq1 . . f rq1
Fix one f rq1 and write
a s a1 f 1 q a2 f 2 q ??? qar f r q a rq1 f rq1 . It is enough to show that a rq1 g Inyry1Ž A.. Assume, by way of contradiction, that a rq1 f Inyry1Ž A.. As Inyry1Ž A. is m-primary there exists a least integer t such that m ty1 a rq1 f Inyry1Ž A. and m t a rq1 g Inyry1Ž A.. By multiplying a by an element of m ty1, we may then assume that a rq1 f Inyry1Ž A. but that xarq1 g Inyry1Ž A. for each x g m. Now, a rq1 f Inyry1Ž A. s Ž N : M .. Hence, there exists an element g g R M such that a rq1 g f Ž f 1 , f 2 , . . . f r , f rq1 .; i.e., a rq1 f Ž f 1 , f 2 , . . . f r , f rq1 . : g. R
Further, we may choose g such that Ž f 1 , f 2 , . . . f r , g . is a reduction of M. This follows from the fact that the set of all elements g g M such that Ž f 1 , f 2 , . . . f r , g . is a reduction of M, generates M and is Zariski open in MrmM by Proposition 1.5. It is also true that Ž f 1 , f 2 , . . . f r , f rq1 q ug . is a reduction of M for infinitely many units u of R. For each such unit u,
a s a1 u f 1 q a2 u f 2 q ??? qa r u f r q a rq1 u Ž f rq1 q ug . . Now for any x g m, xarq1 f rq1 g Inyry1 Ž A . M s Inyry1 Ž A . Ž f 1 , f 2 , . . . f r , f rq1 q ug . by Corollary 2.7. Therefore, there exist t 1 , t 2 , . . . t rq1 g Inyry1Ž A. such that xarq1 f rq1 s t 1 f 1 q t 2 f 2 q ??? qt rq1 Ž f rq1 q ug . , i.e., r
xa y
Ý ai f i s t1 f 1 q t 2 f 2 q ??? qtrq1Ž f rq1 y ug . is1 r
x Ž a1 u f 1 q a2 u f 2 q ??? qa r u f r q a rq1 u Ž f rq1 q ug . . y
Ý ai fi is1
s t 1 f 1 q t 2 f 2 q ??? qt rq1 Ž f rq1 y ug . . By Lemma 2.8, the coefficient of f rq1 is in I Ž N ..
CORE OF A MODULE
17
So, we conclude that xarq1 u y t rq1 g I Ž N . ; Ž N : M . s Inyry1Ž A.. R Hence, xarq1 u g Inyry1Ž A.. Since this holds for all x g m and all allowable units u, the set of all a rq1 u generate an Rrm vector space in RrIny ry1Ž A.. Let u1 , u 2 , . . . u c be units of R such that the elements a rq1 u1, a rq1 u 2 , . . . a rq1 u c are minimal generators of this vector space. Choose u different from all the u i ; this is possible since the residue field k is infinite. We claim that a rq1 u g Inyry1Ž A.. Now, there exist elements l i in R such that c
uarq1 u y
Ý l i u i arq1 u g Inyry1Ž A . . i
is1
Consider the following cases: Case i. c
a rq1 u y
Ý l i arq1 u g Inyry1Ž A . . i
is1
Then going mod Iny ry1Ž A. we get two distinct linear combinations of a rq1 u in terms of the basis vectors. Consequently, the l i are zero mod Iny ry1Ž A.. Hence, a rq1 u g Inyry1Ž A.. But
a s a1 u f 1 q a2 u f 2 q ??? qa r u f r q a rq1 u Ž f rq1 q ug . . This means that a g Ž f 1 , f 2 , . . . f r . q Inyry1Ž A. M and hence the proposition. Case ii. c
a rq1 u y
Ý l i arq1 u f Inyry1Ž A . . i
is1
Let l s Ýcis1 l i y 1. Then, c
la s
Ý li a y a is1 c
s
Ý l i Ž a1 u
i
f 1 q a2 u i f 2 q ??? qa rq1 u iŽ f rq1 q ug . .
is1
y Ž a1 u f 1 q a2 u f 2 q ??? qarq1 u Ž f rq1 q ug . . .
18
RADHA MOHAN
Therefore, c
la s
Ý l i Ž a1 u
i
f 1 q a2 u i f 2 q ??? qa rq1 u i f rq1 .
is1 rq1
y
c
Ý ai u f i q Ý l i u i arq1 u y uarq1 u
ž
is1
i
is1
/
g.
But, c
ž
uarq1 u y
Ý l i u i arq1 u is1
i
/
g g Inyry1 Ž A . M s Iny ry1 Ž A . N.
So, there exist elements t 1 , t 2 , . . . t rq1 g Inyry1Ž A. such that, c
ž
Ý l i u i arq1 u y uarq1 u i
is1
/
g s t 1 f 1 q t 2 f 2 q ??? qt rq1 f rq1 .
Thus, rq1
la s
Ý ai f i is1 c
s
Ý l i Ž a1 u
i
f 1 q a2 u i f 2 q ??? qar u i f r q a rq1 u i f rq1 .
is1 rq1
y
rq1
Ý aiu f i q Ý t i f i . is1
is1
It follows that, la rq1 y Ýcis1 l i a rq1 u i q a rq1 u y t rq1 g Inyry1Ž A.. Consequently, la rq1 y Ýcis1 l i a rq1 u i q a rq1 u g Inyry1Ž A.. If l is a unit then a rq1 g Inyry1Ž A . q Ž a rq1 u , a rq1 u i 4.. But, a rq1 u g, a rq1 u i , g g Ž f 1 , f 2 , . . . f r , f rq1 .. This implies that a rq1 g Inyry1Ž A. q Ž f 1 , f 2 , . . . f r , f rq1 . : g. Hence, a rq1 g Ž f 1 , f 2 , . . . f r , f rq1 . : g. But this R
R
contradicts our choice of g. If l is not a unit then la rq1 g Inyry1Ž A. which implies that a rq1 u y Ýcis1 l i a rq1 u i g Inyry1Ž A.. This contradicts our basic assumption for Case ii. The proof is now complete. We can now prove the main theorem.
CORE OF A MODULE
19
THEOREM 2.10. Let Ž R, m, k . be a two-dimensional regular local ring with infinite residue field. Let M be a finitely generated, torsion-free, complete R-module of rank r. Let A be an n = n y r presenting matrix A for M. Then, core Ž M . s Iny ry1 Ž A . M, where Iny ry1Ž A. is the ideal generated by the n y r y 1 minors of A. Proof of Theorem 2.10. Case i. We first consider the case when M is a non-free, finitely generated, torsion-free, complete R module. We have already seen by Corollary 2.7 that Iny ry1Ž A. M ; coreŽ M .. It remains to prove the reverse inclusion. Let a g coreŽ M . and let N be a minimal reduction of M that is minimally generated by the elements f 1 , f 2 , . . . f rq1. Then, by Proposition 2.9, we have that
a s a11 f 1 q a21 f 2 q ??? qa r1 f r q b 1 s a22 f 2 q ??? qar 2 f r q a rq12 f rq1 q b 2 .. . s a1 rq1 f 1 q a2 rq1 f 2 q ??? qa rq1 rq1 f rq1 q brq1 , where the bi ’s are in Inyry1Ž A. M. Subtracting each of the equations pairwise and using Lemma 2.8 we conclude that each a i j g Inyry1Ž A.. Thus, coreŽ M . ; Inyry1Ž A. M. Case ii. Let M be a free module of rank r. Let A be any presenting matrix for M. The ideal of n y r y 1 minors of A is by definition the r q 1st Fitting ideal of A which by definition is R. Hence, coreŽ M . s M s Iny ry1Ž A. M. This completes the proof of the theorem. As an application of Theorem 2.10 we have a formula relating the Buchsbaum]Rim multiplicity of a complete module M with the length of M**rcoreŽ M . and the length of RrIny ry1Ž A.. COROLLARY 2.11. With notation as in Theorem 2.10, e Ž M**rM . s l Ž Frcore Ž M . . y Ž r q 1 . l Ž RrIny ry1 Ž A . . . Proof. Let N be a minimal reduction of M. Then eŽ M**rM . s lŽ FrN . wBchRm, Corollary 4.5; Kdy, Proposition 3.8x. Consider the exact sequence, 0 ª NrIny ry1Ž A. N ª FrcoreŽ M . ª FrN. It follows that eŽ M**rM . s lŽ FrcoreŽ M .. y lŽ NrIny ry1Ž A. N .. We claim that Nr Iny ry1Ž A. N is isomorphic to Ž RrInyry1Ž A.. rq1. Let f 1 , f 2 , . . . f rq1 be a set of minimal generators for N. Define a homomorphism F: R rq1 ª
20
RADHA MOHAN
NrIny ry1Ž A. N given by F Ž e i . s f i q Inyry1Ž A. N, where e i is a standard basis vector of R rq1. F is a surjective homomorphism. It is clear that Iny ry1Ž A. R rq1 is contained in the kernel of F. If a1 e1 q a2 e2 q ??? qarq1 e rq1 is in the kernel of F then a1 f 1 q a2 f 2 q ??? a rq1 f rq1 g Iny ry1Ž A. N. Thus, a1 f 1 q a2 f 2 q ??? a rq1 f rq1 s t 1 f 1 q ??? t rq1 f rq1 and by Lemma 2.8 we can conclude that each a i g Inyry1Ž A.. Hence, the kernel of F is Iny ry1Ž A. R rq1. This proves the claim. The corollary follows. EXAMPLE 1. Let R s k w x, y xŽ x, y . . Let M be the rank 2 R-module generated by the columns of the matrix Ms
ž
x 0
y
0
0
x
2
xy
y
/
.
Let N be the minimal reduction of M given by Ns
ž
x
y
0
0
x
y2
/
.
M is a complete, indecomposable module and the first three generators of M generate a minimal reduction of M Žsatisfying notation 2.1.. We compute coreŽ M .. A presenting matrix of M is given by yy 2 xy As 0 yx
0 0 . x yy
0
The submatrix B obtained from A by deleting the first 2 q 1 s 3 rows is Žyx yy .. Thus, the ideal of 1 minors of B and of A is m. Hence, coreŽ M . s mM. EXAMPLE 2. Let R s k w x, y xŽ x, y . . Let M be the rank 2 R-module generated by the columns of the matrix Ms
ž
x
y2
2
3
y
x
x2
0
0
xy
0
3
2
0
x
x y
Let N be the minimal reduction of M given by Ns
ž
x
y2
x2
y2
x3
0
/
.
/
.
21
CORE OF A MODULE
M is a complete module and the first three generators of M generate a minimal reduction of M Žsatisfying notation 2.1.. We compute coreŽ M .. A presenting matrix of M is given by 0 yy 0 As 0 0 x
0 0 0 yy x 0
x2 yx 0 0 yy 0
0 0 yx . x 0 y
0
The submatrix B obtained from A by deleting the first 2 q 1 s 3 rows is 0 Bs 0 x
yy x 0
0 yy 0
x 0 . y
0
I3 Ž B . s I3 Ž A. s m3. Thus, coreŽ M . s m3 M. I Ž M . s m3 and I Ž N . s Ž x 4 y y 4 , x 2 y 2 , x 5 .. We note that m2 [ m3 ; M ; m [ m2 . Hence, 4 lŽ FrM . - 9. We use Corollary 2.11 to compute the length of FrM. We have e Ž M**rM . s l Ž FrN . s l Ž Frcore Ž M . . y Ž r q 1 . l Ž RrIny ry1 Ž A . . . By the proof of Lemma 2.2 Ext 2R Ž FrN, R . s RrI Ž N .. By local duality, Ext 2R Ž FrN, R . ks Hm0 Ž FrN . s FrN Žsee wBrnHrz, Corollary 3.5.9x.. Hence, lŽ FrN . s lŽ RrI Ž N .. ks lŽ RrI Ž N ... Therefore, eŽ M**rM . s lŽ FrN . s lŽ RrI Ž N .. s 14. Hence, lŽ FrcoreŽ M .. s 14 q 3Ž6. s 32. Now consider the exact sequence, 0 ª m2 Mrm3 M ª Frm3 M ª Frm2 M ª 0. The length of m2 Mrm3 M s n Ž m2 M . and since m2 M is a complete module it follows by Theorem 1.8 that n Ž m2 M . s rank R Ž m2 M . q ordŽ m2 M . s 2 q Ž2Ž2. q 4. s 10. Hence, lŽ Frm2 M . s 22. Similarly, by considering the exact sequence 0 ª mMrm2 M ª Frm2 M ª FrmM ª 0, we get that the length of FrmM is 14. Then, finally, considering the sequence 0 ª MrmM ª FrmM ª FrM ª 0 we have lŽ FrM . s 14 y 6 s 8.
ACKNOWLEDGMENTS I am deeply grateful to my thesis advisor Professor Bill Heinzer for his encouragement and many useful conversations regarding this work. I thank Vijay Kodiyalam for several discussions regarding complete modules. For a careful reading of the paper and comments I thank Irena Swanson.
22
RADHA MOHAN
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