Analytic spread and integral closure of integrally decomposable modules
aa r X i v : . [ m a t h . A C ] S e p ANALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLYDECOMPOSABLE MODULES
CARLES BIVI `A-AUSINA AND JONATHAN MONTA ˜NOA
BSTRACT . We relate the analytic spread of a module expressed as the direct sum of two sub-modules with the analytic spread of its components. We also study a class of submodules whoseintegral closure can be expressed in terms of the integral closure of its row ideals, and therefore canbe obtained by means of a simple computer algebra procedure. In particular, we analyze a class ofmodules, not necessarily of maximal rank, whose integral closure is determined by the family ofNewton polyhedra of their row ideals.
1. I
NTRODUCTION
Given an ideal I in a Noetherian local ring ( R , m ) , the notions of integral closure, reduction,analytic spread, and multiplicity of I are fundamental objects of study in commutative algebraand algebraic geometry (see for instance [27, 26, 46]). These notions have essential applicationsalso in singularity theory mainly due to the works of Lejeune and Teissier [35, 42, 43]. Theseapplications concern the study of the equisingularity of deformations of hypersurfaces in ( C n , ) with isolated singularity at the origin. The concept of integral closure of ideals was extendedby Rees to modules (see [39]). Moreover, the multiplicity of ideals was extended to modulesby Buchsbaum and Rim [11] (see also Kirby [31]), thus leading to what is commonly known asBuchsbaum-Rim multiplicity of a submodule of R p of finite colength.The integral closure and multiplicity of a submodule of a free module satisfy analogous proper-ties as those satisfied by ideals. For instance, they satisfy an analogous of the Rees’ multiplicitytheorem (see [30] or [46, Corollary 8.20]). Moreover, when the residual field is infinite, the an-alytic spread of a submodule (see Definition 2.6) also coincides with the minimum number ofelements needed to generate a reduction of the submodule (see [7, 27, 46]). We also remark that,by the results of Gaffney [18, 19] the notion of integral closure of modules and Buchsbaum-Rimmultiplicities have essential applications to the study of the equisingularity of deformations of iso-lated complete intersection singularities. We also refer to [20] for other applications in singularitytheory.In general, the computation of the analytic spread and the integral closure of a submodule isa non-trivial problem than can be approached from several points of view. Our objective in thiswork takes part of the general project of computing effectively the analytic spread and the integral Mathematics Subject Classification.
Primary 13B22; Secondary 13H15, 32S05.
Key words and phrases.
Integral closure of modules, analytic spread, Newton polyhedra, Rees algebra.The first author was partially supported by MICINN Grant PGC2018-094889-B-I00. closure for certain classes of modules. We relate the analytic spread of a module expressed asthe direct sum of two submodules with the analytic spread of its components (see Theorem 3.6and Corollary 3.8). Moreover, we analyze a class of submodules M ⊆ R p , that we call integrallydecomposable , for which a generating system of M can be obtained by means of an easy computeralgebra procedure once the integral closure of each row ideal M i is known (Theorem 4.9).In Section 2 we recall briefly some fundamental facts about the integral closure of modules,analytic spread, reductions, Buchsbaum-Rim multiplicity of submodules of a free module andRees algebras that will be used in subsequent sections. In particular, we highlight the connectionbetween the integral closure of a module M and the integral closure of the ideal generated by theminors of size rank ( M ) of M (Theorem 2.10 and Corollary 2.12).Section 3 is devoted to the study of the analytic spread of decomposable modules. The mainresult of this section is Theorem 3.6, where we relate ℓ ( M ⊕ N ) with ℓ ( M ) and ℓ ( N ) , and we derivea generalization of some results of [29] and [36] about the analytic spread of ideals (see also [27,8.4.4]). This result has required the study of multi-graded Rees algebras and their correspondingmulti-projective spectrum (see Subsections 3.1 and 3.2). As a corollary, given ideals I , . . . , I p of R , we prove that ℓ ( I ⊕ · · · ⊕ I p ) = ℓ ( I · · · I p ) + p − M ⊆ R p (Definition 4.1)and analyze their relation with the condition C ( M ) = M (see Theorem 4.9), where C ( M ) denotesthe submodule of R p generated by the elements h ∈ M ⊕ · · · ⊕ M p such that rank ( M ) = rank ( M , h ) .In general we have that M ⊆ C ( M ) . If equality holds, then we obtain a substantial simplificationof the computation of M , as can be seen in Examples 4.14 and 4.23.We also extend the notion of Newton non-degenerate submodule of O pn introduced in [2] to thecase where the rank of the module is not p . These modules constitute a wide class of integrallydecomposable submodules. We recall that O n denotes the local ring of analytic function germs ( C n , ) → C . As a consequence of our study we show in Example 4.28 an integrally closed andnon-decomposable submodule of O (see Definition 2.1) whose ideal of maximal minors can befactorized as the product of two proper integrally closed ideals.2. P RELIMINARIES : R
EES ALGEBRAS , ANALYTIC SPREAD , AND INTEGRAL CLOSURE
Throughout this paper R is a Noetherian ring and all R -modules are finitely generated. An R -module M has a rank if there exists e ∈ N such that M p ∼ = R e p for every p associated prime of R . Equivalently, if M ⊗ R Q ( R ) is a free Q ( R ) -module of rank e , where Q ( R ) is the total ring offractions of R . In this case we also say M has rank e (rank ( M ) = e ), and if e > M has positive rank . We note that an R -ideal I has positive rank if it contains non-zero divisors. If R is anintegral domain, then Q ( R ) is a field and hence every module over an integral domain has a rank.From now on, whenever M is a submodule of a free module R p , we identify M with a matrixof generators. In this case, we denote by I i ( M ) the ideal of R generated by the i × i minors of M .If i > p , then we set I i ( M ) = ( ) . We note that the ideals I i ( M ) are independent of the matrix ofgenerators chosen as they agree with the Fitting ideals of the module R p / M (see [16, Section 2.2]).If M has a rank, the maximum i such that I i ( M ) ⊗ R Q ( R ) = ( ) coincides with rank ( M ) . NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 3 If M ⊆ R p is a submodule, then for any L ⊆ { , . . . , p } , L = /0, we denote by M L the submoduleof R | L | obtained by projecting the components of M indexed by L , where | L | is the cardinal of L .In particular, we have M { i } = M i for all i = , . . . , p , where M i is the ideal of R generated by theelements of the i -th row of any matrix of generators of M . The ideals M , . . . , M p are called the row ideals of M . It is immediate to check that these ideals are independent of the chosen matrix ofgenerators of M . Definition 2.1.
Let M be a submodule of R p . We say that M is decomposable when M = M ⊕· · · ⊕ M p .2.1. Rees algebras and the analytic spread.
In this subsection we include the definition andsome of the properties of Rees algebras of modules. We also define the analytic spread of modules.For more details see [17] and [41].Henceforth, we denote by Sym R ( M ) the symmetric algebra of the R -module M , or simplySym ( M ) when the base ring is clear. We also denote by τ R ( M ) the R-torsion of M , i.e., τ R ( M ) = { x ∈ M | ( R x ) contains non-zero divisors of R } . Definition 2.2. If M has a rank, the Rees algebra of M is defined as R ( M ) : = Sym ( M ) / τ R ( Sym ( M )) . The above definition coincides with the usual one for ideals, i.e., R ( I ) = R [ It ] = ⊕ n ∈ N I n t n ,although we note that the latter does not require the rank assumption. Remark 2.3.
Assume M has a rank, then the natural map R ( M ) → R ( M / τ R ( M )) is an isomor-phism, i.e., Sym ( M ) / τ R ( Sym ( M )) ∼ = Sym ( M / τ R ( M )) / τ R (cid:0) Sym ( M / τ R ( M )) (cid:1) . To see this, we note that since Sym ( M / τ R ( M )) / τ R (cid:0) Sym ( M / τ R ( M )) (cid:1) is torsion-free, the kernel ofthe natural map ϕ : Sym ( M ) → Sym ( M / τ R ( M )) / τ R (cid:0) Sym ( M / τ R ( M )) (cid:1) contains τ R ( Sym ( M )) . Onthe other hand, since M has a rank, M ⊗ R Q ( R ) is free and then ϕ ⊗ R Q ( R ) is an isomorphism.Thus, ker ( ϕ ) has rank zero which is equivalent to being contained in τ R ( Sym ( M )) .We also note that M / τ R ( M ) is a torsion-free module with a rank, then it is contained in a free R -module. The latter implies that when dealing with the Rees algebra of a module with a rank, onemay always assume it is contained in a free module. Remark 2.4.
Assume M has a rank and M / τ R ( M ) ⊆ F for a free R -module F ∼ = R r , then R ( M ) isisomorphic to the image of the map Sym ( M ) α −→ Sym ( F ) ∼ = R [ t , . . . , t r ] .In the following proposition we recall some facts about the dimension and associated primes ofRees algebras. Following the notation from Remark 2.4, let T : = R [ t , . . . , t r ] . For any I ∈ Spec R we denote by I ′ the R -ideal I T ∩ R ( M ) . Proposition 2.5.
Let M be an R-module that has a rank. Then ( ) Min ( R ( M )) = { P ′ | P ∈ Min ( R ) } and Ass ( R ( M )) = { P ′ | P ∈ Ass ( R ) } . CARLES BIVI `A-AUSINA AND JONATHAN MONTA ˜NO ( ) dim R ( M ) = dim R + rank ( M ) .Proof. See [37, Section 15.4] and [41, 2.2]. (cid:3)
We are now ready to define the analytic spread.
Definition 2.6.
Assume ( R , m , k ) is local and M is an R -module having a rank. The fiber cone of M is defined as F ( M ) : = R ( M ) ⊗ R k . The analytic spread of M is then ℓ ( M ) : = dim F ( M ) . The following proposition will be needed in several of our arguments.
Proposition 2.7 ([41, 2.3]) . Let M be an R-module having a rank. Then rank ( M ) ℓ ( M ) dim R + rank ( M ) − . Integral closure of modules.
In this subsection we include the definition of integral closureof modules and some basic properties of it. We restrict ourselves to the case of torsion-free moduleswith a rank. For more details see [27, Chapter 16] and [46, Chapter 8].
Definition 2.8 (Rees [39]) . Let R be a Noetherian ring and let M be a submodule of R p . ( ) The element h ∈ R p is integral over M if for every minimal prime p of R and every discretevaluation ring (DVR) or field V between R / p and R p / p R p , the image hV of h in V p is inthe image MV of the composition of R -maps M ֒ → R p → V p (see [27, 16.4.9]). ( ) The integral closure of M in R p is defined as M : = { h ∈ R p : h is integral over M } , whichis a submodule of R p . If M = M , we say M is integrally closed . We note that if M ⊆ R isan ideal, then the integral closure of M as a module coincides with that as an ideal (see [27,6.8.3]). ( ) Assume M has a rank. A submodule U ⊆ M having a rank is a reduction of M if M ⊆ U .As shown in [39] (see also [27, 16.2.3]), this is equivalent to R ( M ) being integral overthe subalgebra generated by the image of U . The latter condition is in turn equivalent to [ R ( M )] n + = U [ R ( M )] n for n ≫
0, where U is identified with its image in [ R ( M )] . Areduction is minimal if it does not properly contain any other reduction of M . Remark 2.9.
Let M ⊂ R p be a submodule having a rank, then ( ) M = [ R ( M )] , where R ( M ) is the integral closure of R ( M ) in Sym ( R p ) (cf. [27, 5.2.1]). ( ) If R is local, then for every reduction U of M we have µ ( U ) > ℓ ( M ) , where µ ( − ) denotesthe minimal number of generators. Moreover, if R has infinite residue field then everyminimal reduction is generated by exactly ℓ ( M ) elements. ( ) It is clear from the definition that free modules R q ⊆ R p are integrally closed. Moreover, if U ⊆ M is a reduction, then rank ( U ) = rank ( M ) (see [46, p. 416]). In particular, rank ( M ) = rank ( M ) .The integral closure of modules admits several characterizations. The following theorem relatesthe integral closure of modules with the integral closure of ideals. As far as the authors are aware,this result had not appeared in the literature in this generality (see [18, 1.7], [27, 16.3.2], [39, 1.2],[46, 8.66] for related statements). NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 5
Theorem 2.10.
Let R be a Noetherian ring and M ⊆ R p a submodule having a rank. Let h ∈ R p be such that M + Rh also has a rank and rank ( M ) = rank ( M + Rh ) . Then the following conditionsare equivalent. ( ) h ∈ M. ( ) I i ( M ) = I i ( M + Rh ) , for all i > . ( ) I r ( M ) = I r ( M + Rh ) , for r = rank ( M ) . For the proof of the theorem we need the following lemma whose proof is essentially the sameas [18, 1.6]. We include here the details for completeness.
Lemma 2.11.
Let R be a Noetherian integral domain and M ⊆ R p a submodule. Let h ∈ R p be anarbitrary element and set r = rank ( M + Rh ) , then I r ( M ) h ⊆ I r ( M + Rh ) M.Proof.
If rank ( M ) < r , then I r ( M ) = ( M ) = r . We identify M with a matrix of generators and M + Rh with the matrix [ M | h ] .Let M ′ be a r × r submatrix of M such that d = det ( M ′ ) = L ⊆ { , . . . , p } be the rowsof M corresponding to the rows of M ′ . By Cramer’s rule there exists x , . . . , x r ∈ I r ( M ′ | h L ) ⊆ I r ( M | h ) such that Mx = dh L , where x = [ x . . . x r ] T ∈ R r . Let N be the p × r submatrix of M corresponding to the columns of M ′ and consider the vector g = dh − Nx . By construction, wehave g ∈ M + Rh and g L =
0. Let i ∈ { , . . . , p } \ L , then the ( r + ) × ( r + ) minor of [ N | g ] corresponding to the rows L ∪ { i } is ± g i d and it must vanish since rank ( M + Rh ) = r . Therefore, g i d = g i =
0. Thus g = dh = Nx ⊆ I r ( M + Rh ) M . Since M ′ was chosenarbitrarily the proof is complete. (cid:3) We are now ready to prove the theorem.
Proof of Theorem 2.10.
We begin with (1) ⇒ (2). Let p be a minimal prime of R and V a DVR ora field between R / p and R p / p R p . Since hV ∈ MV , for every i > I i ( M + Rh ) V = I i ( MV + R ( hV )) ⊆ I i ( MV ) = I i ( M ) V . Thus I i ( M + Rh ) ⊆ I i ( M ) and (2) follows.Since (2) ⇒ (3) is clear, it suffices to show (3) ⇒ (1). Let p be a minimal prime of R andfor a submodule N ⊆ R q let N ( R / p ) its image in ( R / p ) q . By assumption we have that M ( R / p ) and ( M + Rh )( R / p ) both have rank r . In particular, I r ( M )( R / p ) = I r ( M ( R / p )) =
0, and likewise I r ( M + Rh )( R / p ) =
0. Let V a DVR or a field between R / p and R p / p R p . Then by the assumptionand Lemma 2.11, applied to R / p , we have ( I r ( M ) V ) hV = ( I r ( M ) h ) V ⊆ ( I r ( M + Rh ) M ) V = ( I r ( M + Rh ) V ) MV = ( I r ( M ) V ) MV . Thus hV ∈ MV . We conclude h ∈ M , as desired. (cid:3) As an immediate consequence of Theorem 2.10 we have the following result.
CARLES BIVI `A-AUSINA AND JONATHAN MONTA ˜NO
Corollary 2.12.
Let R be a Noetherian ring and let M ⊆ R p be a submodule having a rank. Letr = rank ( M ) . ThenM = n h ∈ R p : rank ( M + Rh ) = r and I r ( M + Rh ) ⊆ I r ( M ) o . Assume R is local of dimension d and let λ ( − ) denote the length function of R -modules. If λ ( R p / M ) < ∞ , we say M has finite colength and in this case the limit e ( M ) = ( d + p − ) ! lim n → ∞ λ ([ Sym ( R p )] n / [ R ( M )] n ) n d + p − is called the Buchsbaum-Rim multiplicity of M . It is known that if R is Cohen Macaulay and M is generated by d + p − e ( M ) = λ ( R p / M ) = λ ( R / I p ( M )) (see for instance [19,p. 214]).We recall the following numerical characterization of integral closures due to Rees [38] in thecase of ideals and Katz [30] for modules. Theorem 2.13 ([27, p. 317],[30]) . Let R be a formally equidimensional Noetherian local ring ofdimension d > . Let N ⊆ M ⊆ R p be submodules such that λ ( R p / N ) < ∞ . Then M = N if andonly if e ( N ) = e ( M ) . Remark 2.14.
Let M ⊆ R p be a submodule. In general we have(2.1) M ⊆ M ⊕ · · · ⊕ M p = M ⊕ · · · ⊕ M p . However, the first inclusion in (2.1) might be strict. For instance, consider the submodule of O generated by the columns of the matrix " x + y x y x y x . It is clear that x ∈ M , x ∈ M . Let h = [ x x ] T , we can see that h / ∈ M . By Theorem 2.10 wehave that h ∈ M ⇐⇒ I ( M + O h ) ⊆ I ( M ) ⇐⇒ e ( I ( M )) = e ( I ( M + O h )) , where the last equivalence follows from Theorem 2.13. However e ( I ( M )) = e ( I ( M + O h )) =
6, as can be computed using Singular [14]. Hence h / ∈ M .Another argument leading to the conclusion that h / ∈ M is the following. We have that e ( M ) = e ( M + O h ) =
5, computed again using Singular. Since these multiplicities are different, itfollows that h / ∈ M , by Theorem 2.13. Moreover, by using Macaulay2 (see Remark 4.25) it ispossible to prove that M is generated by the columns of the matrix " x + y x y x y x y x x + y . That is, M = M + O [ x y x + y ] T . NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 7
Given an analytic map ϕ : ( C m , ) → ( C n , ) , we denote by ϕ ∗ the morphism O n → O m given by ϕ ∗ ( h ) = h ◦ ϕ , for all h ∈ O n . For submodules of O pn we have the following alternative definitionof integral closure. Theorem 2.15 (Gaffney [18, p. 303]) . Let M ⊆ O pn be a submodule and let h ∈ O pn . Then h isintegral over M if and only if ϕ ∗ ( h ) ∈ O ϕ ∗ ( M ) , for any analytic curve ϕ : ( C , ) → ( C n , ) . Example 2.16.
It is also possible to check that h / ∈ M in the example from Remark 2.14 by con-sidering the arc ϕ : ( C , ) → ( C , ) given by ϕ ( t ) = ( − t + t , t ) , for all t ∈ C . We have that ϕ ∗ ( h ) = [( − t + t ) − t + t ] T and that ϕ ∗ ( M ) is generated by the columns of the matrix " t ( − t + t ) t − t + t t − t + t . We note that the first and third columns of the previous matrix coincide. If ϕ ∗ ( h ) ∈ ϕ ∗ ( M ) , thenwe would have(2.2) I " t ( − t + t ) − t + t t = I " t ( − t + t ) ( − t + t ) − t + t t − t + t . The ideal on the left of (2.2) is equal to ( t ) and the ideal on the right of (2.2) is equal to ( t ) .Hence ϕ ∗ ( h ) / ∈ ϕ ∗ ( M ) and by Theorem 2.15 it follows that h M .We finish this section with the following relation between integral closures and projections. Proposition 2.17.
Let R be a Noetherian ring and M ⊆ R p a submodule, then for every non-empty L ⊆ { , . . . , p } we have ( M ) L ⊆ M L .Proof. Fix h ∈ M . For every a minimal prime p of R and every DVR or field V between R / p and R p / p R p we have h L V = ( hV ) L ∈ ( MV ) L = M L V . Thus h L ∈ M L . The result follows. (cid:3)
3. T
HE ANALYTIC SPREAD OF DECOMPOSABLE MODULES
In this section we study the analytic spread of decomposable modules and its relation with theanalytic spread of their components. Our main results are Theorem 3.6 and its corollaries. Webegin with some necessary background information.3.1.
Multi-graded algebras and multi-projective spectrum.
In this subsection we recall severalfacts about multi-graded algebras and their multi-homogeneous spectrum, we refer the reader to[28] for more information. We start by setting up some notation.Let p ∈ Z > . We denote by n the vector ( n , . . . , n p ) ∈ N p . For convenience we also set =( , . . . , ) and = ( , . . . , ) where each of these vectors belongs to N p . We call the sum n + · · · + n p the total degree of n and denote it by | n | .Let R be a Noetherian ring and A = ⊕ n ∈ N p A n a Noetherian N p -graded algebra with A = R andgenerated by the elements of total degree one ( standard graded ). We denote by A ∆ the diagonal CARLES BIVI `A-AUSINA AND JONATHAN MONTA ˜NO subalgebra of A , i.e., A ∆ = ⊕ n ∈ N A n . For every 1 i p we write A ( i ) = ⊕ n i = A n . We also con-sider the following N p -homogeneous A -ideals A + i = ⊕ n i > A n for 1 i p and A + = ⊕ n ,..., n p > A n . We write Proj p A = { P ∈ Spec A | P is N p -homogeneous, and A + P } . The dimension of Proj p A is one minus the maximal length of an increasing chain of elements of Proj p A , P ( P ( · · · ( P d .The relation between the dimensions of Proj p A and A is explained in the following lemma. Lemma 3.1 ([28, 1.2]) . Let Z = Proj p A and assume Z = /0 , then ( ) dim Z = max { dim A / P | P ∈ Z } − p dim A − p. ( ) If dim A ( i ) < dim A for every i p, then dim Z = dim A − p. It is possible to give Proj p A a structure of scheme and to show that it is isomorphic to Proj A ∆ (see [22, Part II, Exercise 5.11] and also [24, Lemma 3.2] and [33, Lemma 7.1]). For the reader’sconvenience, we provide a proof of the following particular result which suffices for our applica-tions. Proposition 3.2.
Let ι : A ∆ → A be the natural inclusion. Then ι ∗ : Proj p A → Proj A ∆ is a bijec-tion.Proof. Clearly Proj p A = /0 if and only if Proj A ∆ = /0 if and only if A n = n ≫
0, then we mayassume these two sets are both non-empty. For every 1 i p , let e i = ( , . . . , , , , . . . , ) ∈ N p where the 1 is in the i th-position. Fix 0 = f i ∈ A e i for 1 i p and let f = f · · · f p . Since everyelement in the localization A f is a unit times an element of A ∆ f , one can easily see that ι ∗ f is bijective.We first show ι ∗ is injective. Let P , P ∈ Proj p A and assume ι ∗ ( P ) = ι ∗ ( P ) . If f is as aboveand such that f P (thus f P ), then by assumption ι ∗ f ( P A f ) = ι ∗ f ( P A f ) . Hence P A f = P A f ,which implies P = P .We now show ι ∗ is surjective. Let P ∈ Proj A ∆ and f P as above. Then there exists Q ∈ A such that ι ∗ f ( QA f ) = PA ∆ f , which implies ι ∗ ( Q ) = P , finishing the proof. (cid:3) We end this subsection with the following lemma that will be used in the proofs of our mainresults.
Lemma 3.3.
Let A = ⊕ n ∈ N p A n be a Noetherian standard N p -graded algebra and p ∈ Proj p − A ( p ) (if p = , Proj A is simply Spec A ). Fix e ∈ N , then the following statements are equivalent. ( ) There exists a chain of elements in
Proj p A, P ( P ( · · · ( P e − such that p = P i ∩ A ( p ) forevery i e − . ( ) dim Q ( A ( p ) / p ) ⊗ A ( p ) A > e.Proof. Set W = Q ( A ( p ) / p ) ⊗ A ( p ) A . If (1) holds, then P W ( · · · ( P e − W ( ( p + A + p ) W = W + is achain of prime ideals in W . Thus, dim W > e and (2) follows.Conversely, if (2) holds then dim ( A / p A ) p A + A + p > e . Since associated primes of N p -graded ringsare N p -homogeneous ([27, A.3.1]), a direct adaptation of [9, 1.5.8(a)] to N p -graded rings showsthat there exist N p -homogeneous A -ideals p A ⊆ P ( · · · ( P e − ( ( p A + A + p ) whose images inthe ring ( A / p A ) p A + A + p are all different. Since p = P i ∩ A ( p ) for every 0 i e −
1, the resultfollows. (cid:3)
NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 9
Multi-graded Rees algebras.
In this subsection we describe a standard multi-graded struc-ture for the Rees algebras of direct sums of modules.
Definition 3.4.
Let M , . . . , M p be R -modules having a rank. We define a natural standard N p -graded structure on R ( M ⊕ · · · ⊕ M p ) . By [16, A2.2.c] we haveSym ( M ⊕ · · · ⊕ M p ) ∼ = p O i = Sym ( M i ) , and since each of the algebras Sym ( M i ) has a standard N -grading, we can combine these to an N p -grading of R ( M ⊕ · · · ⊕ M p ) by setting [ N pi = Sym ( M i )] n = N pi = Sym ( M i ) n i . Proposition 3.5.
Let M , . . . , M p be R-modules having a rank and set R ′ = R ( M ⊕ · · · ⊕ M p − ) .Then there is a natural graded R ′ -isomorphism R ( M ⊕ · · · ⊕ M p ) ∼ = R ( M p ⊗ R R ′ ) . Proof.
We claim that for any R -module M with a rank we have τ R ′ ( M ⊗ R R ′ ) is equal to the image T of τ R ( M ) ⊗ R R ′ in M ⊗ R R ′ . First observe that by Proposition 2.5(1), M ⊗ R R ′ has a rank as R ′ -module and it is equal to rank ( M ) . Now, consider a short exact sequence0 → τ R ( M ) → M → F ∼ = R r → . By tensoring with R ′ it follows that T contains τ R ′ ( M ⊗ R R ′ ) . On the other hand, T has rank zeroas R ′ -module (since rank ( τ R ( M )) = R ′ -torsion. The claim follows. We obtainthe following natural maps S : = p O i = Sym R ( M i ) [16, A2.2.c] −−−−−−→ R ′ ⊗ R Sym R ( M p ) [16, A2.2.b] ∼ = Sym R ′ ( M p ⊗ R R ′ ) onto −−→ Sym R ′ ( M p ⊗ R R ′ ) / τ R ( Sym R ′ ( M p ⊗ R R ′ )) claim = R ( M p ⊗ R R ′ ) . Clearly the kernel of the composition of these maps contains τ R ( S ) and, since tensoring by Q ( R ) leads to an monomorphism, this kernel must be equal to τ R ( S ) . The result follows. (cid:3) Main results about the analytic spread of modules.
This subsection contains the mainresults of this section. We assume ( R , m , k ) is a Noetherian local ring.The following is the main theorem of this section. This result, in particular, allows us to recover,and extend, the results in [29, Lemma 4.7], [36, 5.5], and [41, 2.3]. Theorem 3.6.
Let M and N be R-modules having a rank. Then max { ℓ ( M ) + rank ( N ) , ℓ ( N ) + rank ( M ) } ℓ ( M ⊕ N ) ℓ ( M ) + ℓ ( N ) . Proof.
We may assume M and N are torsion-free and hence contained in free R -modules (Remark2.3). If either M or N has rank zero, then it has to be the zero module. Then we may assume theyboth have positive rank. Consider the following natural surjective maps R ( M ) ⊗ R R ( N ) α −→ R ( M ) ⊗ R R ( N ) / τ R ( R ( M ) ⊗ R R ( N )) β ←− Sym ( M ) ⊗ R Sym ( N ) . Since Q ( R ) ⊗ R β is an isomorphism and the image of β is torsion-free, it follows that ker β ⊆ τ R ( Sym ( M ) ⊗ R Sym ( N )) ⊆ ker β . Then we obtain a surjective map R ( M ) ⊗ R R ( N ) onto −−→ Sym ( M ) ⊗ R Sym ( N ) / τ R ( Sym ( M ) ⊗ R Sym ( N )) [16, A2.2.c] ∼ = Sym ( M ⊕ N ) / τ R ( Sym ( M ⊕ N ))= R ( M ⊕ N ) . By tensoring this map by k we observe that F ( M ⊕ N ) is a quotient of F ( M ) ⊗ k F ( N ) , andsince the latter is a tensor product of affine algebras, it has dimension dim F ( M ) + dim F ( N ) = ℓ ( M ) + ℓ ( N ) . The right-hand inequality follows.We now show the left-hand inequality. Set R = R ( M ⊕ N ) . Following the multi-grading inDefinition 3.4 we have R ( ) = R ( M ) . We also observe that R ∼ = R ( N ′ ) , where N ′ = N ⊗ R R ( ) (Proposition 3.5). Fix p ∈ Proj R ( ) such that p ∩ R = m and dim R ( ) / p = ℓ ( M ) , which existsby Lemma 3.3 and the fact that ℓ ( M ) > ( N ′ ) p is an R ( ) p -module with the same rank as N ; let e be this rank. Then,(3.1) dim Q ( R ( ) / p ) ⊗ R ( ) R = ℓ (( N ′ ) p ) > e (Proposition 2.7) . Therefore, by Lemma 3.3 there exist P ( · · · ( P e − in Proj R with P i ∩ R ( ) = p for every0 i e −
1. We have an inclusion of domains A = R ( ) / p ֒ → B : = R / P and Lemma 3.3 impliesdim Q ( A ) ⊗ A B > e . Hence, dim Q ( A / p ′ ) ⊗ A B > e for every p ′ ∈ Proj A ([16, 14.8(b)]). Choosea p ′ that avoids a general element of A (cf. [16, 14.5]) and that dim A / p ′ =
1; such p ′ exists byHilbert’s Nullstellensatz. Additionally, choose P ′ e − ∈ Proj B such that P ′ e − ∩ A = p ′ and its imagein Q ( A / p ′ ) ⊗ A B has height > e − A = dim A p ′ = dim B P ′ e − − dim B P ′ e − / p ′ B P ′ e − dim Proj B − e + . Thus, ℓ ( M ⊕ N ) = dim R ⊗ R k > dim B > dim Proj B + > dim Proj A + e + = ℓ ( M ) + e . Where the second inequality follows from Lemma 3.1(1). Likewise, ℓ ( M ⊕ N ) > ℓ ( N ) + rank ( M ) ,finishing the proof. (cid:3) Remark 3.7.
Under the conditions of Theorem 3.6, let us assume R has infinite residue field. Then ℓ ( M ) = ℓ ( M ) . Hence Theorem 3.6 applies when M is decomposable.In the following corollary we observe that if a module satisfies equality in one of the inequalitiesin Proposition 2.7, then we obtain a closed formula for the analytic spread of its direct sum withany other module. NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 11
Corollary 3.8.
Let M and N be R-modules having a rank. ( ) If ℓ ( N ) = rank ( N ) , then ℓ ( M ⊕ N ) = ℓ ( M ) + rank ( N ) . ( ) If ℓ ( N ) = dim R + rank ( N ) − , then ℓ ( M ⊕ N ) = dim R + rank ( M ) + rank ( N ) − .Proof. The conclusion follows from Theorem 3.6, Proposition 2.7, and the fact that rank ( M ⊕ N ) = rank ( M ) + rank ( N ) . (cid:3) Remark 3.9.
We note that the equality ℓ ( N ) = dim R + rank ( N ) − N is torsion-free and F / N has finite length for some free R -module F ([46, 8.4]); if N is an ideal module (i.e., N is torsion-free and Hom R ( N , R ) is free), and such that N p is free for any p ∈ Spec ( R ) \ { m } ([41, 5.2]); and if R is a two-dimensional local normal domainwith infinite residue field and N is not free ([46, page 418]).The equality ℓ ( N ) = rank ( N ) trivially holds for any free R -module.In the following corollary we relate the analytic spread of direct sums and products of ideals andmodules. We remark that the estimates for the analytic spread in [5, 6.5 6.8] and [6, 5.9] followfrom our next result. Corollary 3.10.
Let I , . . . , I p − be R-ideals for some p > and let M be an R-module, all ofpositive rank. Then ℓ ( I · · · I p − M ) + p − = ℓ ( I ⊕ · · · ⊕ I p − ⊕ M ) > max i p − { ℓ ( I i ) + rank ( M ) − , ℓ ( M ) } + p − . Proof.
As the inequality follows directly from Theorem 3.6, it suffices to show the equality.We may assume M is torsion-free (Remark 2.3). We proceed by induction on p >
1, the case p = p > A = F ( I ⊕ · · · ⊕ I p − ⊕ M ) . Notice that A hasa natural N p -graded structure (Definition 3.4). Moreover, A ∆ = F ( I · · · I p − M ) , A ( i ) = F ( I ⊕· · · ⊕ I i − ⊕ I i + ⊕ · · · ⊕ I p − ⊕ M ) for every 1 i p −
1, and A ( p ) = F ( I ⊕ · · · ⊕ I p − ) . Hence,dim A ( i ) < dim A for every 1 i p (Theorem 3.6). Therefore, by Lemmas 3.1(2) and 3.2 we havedim A = dim Proj p A + p = dim Proj A ∆ + p = dim A ∆ + p − , and the result follows. (cid:3) The following example extends [46, 8.6]. Here we are able to provide a formula for the analyticspread of a certain class of modules.
Example 3.11.
Let A , . . . , A p be standard graded k -algebras and for each i = , . . . , p let I i be an R i -ideal of positive rank and generated by elements of degree δ i . Consider A = A ⊗ k · · · ⊗ k A p and identify each I i with its image in A . Then I · · · I p is generated in degree δ + · · · + δ p and itsminimal number of generators is the dimension of the k -vector space [ I ] δ ⊗ k · · · ⊗ k [ I p ] δ p , i.e., ∏ pi = dim k [ I i ] δ i = ∏ pi = µ ( I i ) , where µ ( − ) denotes minimal number of generators. Likewise, for every n ∈ N , we have µ (( I · · · I p ) n ) = ∏ pi = µ ( I ni ) . Hence, ℓ ( I · · · I p ) − = ( ℓ ( I ) − ) + · · · +( ℓ ( I p ) − ) ([9, 4.1.3]). From Corollary 3.10 we conclude that ℓ ( I ⊕ · · · ⊕ I p ) = ℓ ( I ) + · · · + ℓ ( I p ) . In the following corollary we recover, and slightly extend, the results in [36, 5.5] (see also[27, 8.4.4]) and [29, Lemma 4.7]). We recall that the analytic spread of an ideal is defined as ℓ ( I ) = dim R ( I ) ⊗ R k regardless of any rank assumption. Corollary 3.12.
Let I and J be R-ideals (not necessarily with a rank). Then ( ) If I or J is not nilpotent, then ℓ ( I ) + ℓ ( J ) > ℓ ( IJ ) . ( ) If IJ has positive height, or √ I = √ J, then ℓ ( IJ ) > max { ℓ ( I ) , ℓ ( J ) } .Proof. For (1), assume I is not nilpotent. If J is nilpotent, i.e., ℓ ( J ) =
0, the inequality clearlyholds. Otherwise, for any p minimal prime of R that does not contain IJ we have ℓ ( I ) + ℓ ( J ) > ℓ ( I ( R / p )) + ℓ ( J ( R / p )) > ℓ ( IJ ( R / p )) where the first inequality follows from [27, 5.1.7] and the second one from Theorem 3.6 andCorollary 3.10. The result then follows from [27, 5.1.7]. Similarly, for (2), let p be a minimalprime of R such that ℓ ( I ) = ℓ ( I ( R / p )) ([27, 5.1.7] ), then ℓ ( IJ ) > ℓ ( IJ ( R / p )) > ℓ ( I ( R / p )) = ℓ ( I ) , where the second inequality follows from Corollary 3.10. Likewise, ℓ ( IJ ) > ℓ ( J ) , and the resultfollows. (cid:3) Our results allow us to build a minimal reduction of a direct sum of multiple copies of an ideal I as we show in the next corollary. This result extends [46, 8.67] to arbitrary ideals. Moreover, thecomputation of integral closure in [34, 3.5] follows from this result.Given elements a , . . . , a s ∈ R and an integer p >
1, we define the matrix A p ( a , . . . , a s ) : = a a a · · · a a · · · · · · a s − a s
00 0 0 · · · a s − a s − a s . Corollary 3.13.
Let I be an R-ideal of positive rank and let s be its analytic spread. Fix p ∈ Z > and consider the R-module M = I ⊕ · · · ⊕ I | {z } p times . Then, ℓ ( M ) = s + p − and given any (minimal) reduction ( a , . . . , a s ) ⊆ I, the R-submodule of R p generated by the columns of the matrix A p ( a , . . . , a s ) is a minimal reduction of M. NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 13
Proof.
Let U be the module generated by the columns of this matrix and notice that U ⊆ M . Wefirst show that U is a reduction of M . For this, note that by [10, page 15], I p ( U ) = I p , and the latteris clearly also equal to I p ( M ) . By Theorem 2.10, it follows that U is a reduction of M .It remains to show ℓ ( M ) = s + p −
1, but this follows from Corollary 3.10 since ℓ ( M ) = ℓ ( I p ) + p − = s + p − , finishing the proof. (cid:3) Example 3.14.
Let I be a monomial ideal of O . Let Γ + ( I ) denote the Newton polyhedron of I (seethe definition of this notion before Example 4.14) and let { ( a , b ) , ( a , b ) , . . . , ( a n , b n ) } ⊂ N bethe set of vertices of Γ + ( I ) , with n > a < a < · · · < a n and b > b > · · · > b n . Considerthe polynomials of C [ x , y ] given by g = ∑ i is odd x a i y b i and g = ∑ i is even x a i y b i . By [4] (see also [13, 3.6] or [12, 3.7]), the ideal ( g , g ) is a reduction of I . Thus, by Corollary3.13, the module generated by the columns of A p ( g , g ) is a minimal reduction of the module M = I ⊕ · · · ⊕ I ⊂ O p .4. I NTEGRALLY DECOMPOSABLE MODULES , N
EWTON NON - DEGENERACY , AND THECOMPUTATION OF THE INTEGRAL CLOSURE
In this section we address the task of computing the integral closure of modules. In general,this is a difficult and involved process as it requires the computation of the normalization of Reesalgebras. In our main results we focus on a wide family of modules, that we call integrally decom-posable , for which an important example are the
Newton non-degenerate modules (see Definitions4.1 and 4.15). In our main results, we express the integral closure of these modules in terms ofthe integral closure of its component ideals (see Theorem 4.9 and Corollary 4.21). Therefore, wetranslate the problem of computing integral closures of modules to integral closures of ideals, forwhich several algorithms are available in the literature (see for instance [45, Chapter 6]).Throughout this section R is a Noetherian ring.4.1. Integrally decomposable modules.
Let M be a submodule of R p and let r = rank ( M ) . Weidentify M with any matrix of generators and denote by Λ M the set of vectors ( i , . . . , i r ) ∈ Z r > such that 1 i < · · · < i r p and there exists some non-zero minor of M formed from rows i , . . . , i r . Definition 4.1.
Let M be submodule of R p and let r = rank ( M ) . We say that M is integrallydecomposable when M L is decomposable, for all L ∈ Λ M .We remark that, under the conditions of the above definition, if L ∈ Λ M and we write L =( i , . . . , i r ) , where 1 i < · · · < i r p , then M L is decomposable if and only if M L = ( M L ) i ⊕ · · · ⊕ ( M L ) i r . In particular, we observe that Definition 4.1 constitutes a void condition when rank ( M ) = Lemma 4.2.
Let M be submodule of R p . Then ( M ) i = M i , for all i = , . . . , p. Proof.
Fix an index i ∈ { , . . . , p } . The inclusion M ⊆ M implies that M i ⊆ ( M ) i . Thus M i ⊆ ( M ) i .From Proposition 2.17 we deduce that ( M ) i ⊆ M i . Therefore, ( M ) i ⊆ M i , and hence the resultfollows. (cid:3) Proposition 4.3.
Let M be submodule of R p and let r = rank ( M ) . Then M is integrally decompos-able if and only if (4.1) M L = M i ⊕ · · · ⊕ M i r . for all L = ( i , . . . , i r ) ∈ Λ M , where i < · · · < i r p.Proof. Since M L is a submodule of R r of rank r , for all L ∈ Λ M , it suffices to show the result in thecase r = p . So let us assume that rank ( M ) = p . In general we have the following inclusions: M ⊆ ( M ) ⊕ · · · ⊕ ( M ) p ⊆ ( M ) ⊕ · · · ⊕ ( M ) p = M ⊕ · · · ⊕ M p where the last equality is an application of Lemma 4.2. This shows that if relation (4.1) holds, then M is decomposable.Conversely, if M is decomposable, then M = ( M ) ⊕ · · · ⊕ ( M ) p . Taking integral closures in thisequality it follows that M = M = ( M ) ⊕ · · · ⊕ ( M ) p = ( M ) ⊕ · · · ⊕ ( M ) p = M ⊕ · · · ⊕ M p again by Lemma 4.2, and thus equality (4.1) follows. (cid:3) In the following proposition we characterize integrally decomposable modules in terms of theirideals of minors.
Proposition 4.4.
Let M be a submodule of R p and let r = rank ( M ) . Then the following conditionsare equivalent. ( ) M is integrally decomposable. ( ) I r ( M L ) = ∏ i ∈ L M i , for all L ∈ Λ M .Proof. Fix L = ( i , . . . , i r ) ∈ Λ M and let N = M i ⊕ · · · ⊕ M i r . Then M L ⊆ N = M i ⊕ · · · ⊕ M i r , where the last equality holds by Remark 2.14. Therefore, by Theorem 2.10, M L = N if and only if I r ( M L ) = I r ( N ) = M i · · · M i r . Then the result follows as a direct application of Proposition 4.3. (cid:3) Let R be a Noetherian local ring of dimension d and let I , . . . , I d be a family of ideals of R offinite colength. We denote by e ( I , . . . , I d ) the mixed multiplicity of the family of ideals I , . . . , I d (see [27, p. 339]). We recall that when the ideals I , . . . , I d coincide with a given ideal I of finitecolength, then e ( I , . . . , I d ) = e ( I ) , where e ( I ) is the multiplicity of I , in the usual sense.Let ( i , . . . , i p ) ∈ Z p > , for some p d , such that i + · · · + i p = d . We denote by e i ,..., i p ( I , . . . , I p ) the mixed multiplicity e ( I , . . . , I , . . . , I p , . . . , I p ) where I j is repeated i j times, for all j = , . . . , p .Let M be a submodule of R p of finite colength. Following [2, p. 418], we define δ ( M ) = ∑ i + ··· + i p = di ,..., i p > e i ,..., i p ( M , . . . , M p ) . NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 15
We remark that the condition that M has finite colength in R p implies that M i has finite colength in R , for all i = , . . . , p .By a result of Kirby and Rees in [32, p. 444] (see also [2, p. 417]), we have that e ( I ⊕ · · · ⊕ I p ) = δ ( I ⊕ · · · ⊕ I p ) , for any family of ideals I , . . . , I p of R of finite colength. Therefore δ ( M ) = e ( M ⊕ · · · ⊕ M p ) . Proposition 4.5.
Let R be a formally equidimensional Noetherian local ring of dimension d > .Let M be a submodule of R p . Let r = rank ( M ) . Assume M L has finite colength, as a submoduleof R r , for all L ∈ Λ M . Then M is integrally decomposable if and only if e ( M L ) = δ ( M L ) , for all L ∈ Λ M .Proof. Let us fix any L = ( i , . . . , i r ) ∈ Λ M . By Proposition 4.3, the submodule M L ⊆ R r is decom-posable if and only if M L = M i ⊕ · · · ⊕ M i r . Let us recall that M i ⊕ · · · ⊕ M i r = M i ⊕ · · · ⊕ M i r . Thus M L is integrally decomposable if and only if M L is a reduction of M i ⊕ · · · ⊕ M i r , which is tosay that e ( M L ) = e ( M i ⊕ · · · ⊕ M i r ) , by Theorem 2.13. But e ( M i ⊕ · · · ⊕ M i r ) = δ ( M L ) , thus theresult follows. (cid:3) For a submodule of R p , we introduce the following objects. Definition 4.6.
Let M ⊆ R p be a submodule of rank r . We define the ideal J M = ∑ ( i ,..., i r ) ∈ Λ M M i · · · M i r and the following modules Z ( M ) = { h ∈ R p : rank ( M ) = rank ( M + Rh ) } C ( M ) = Z ( M ) ∩ (cid:0) M ⊕ · · · ⊕ M p (cid:1) . Remark 4.7.
In the previous definition, if r = p then Z ( M ) = R p and thus C ( M ) = M ⊕ · · · ⊕ M p .From Remarks 2.9 and 2.14 it follows that M is always contained in C ( M ) but this containmentcan be strict. We ask the following question. Question 4.8.
Let M be a submodule of R p , when do we have M = C ( M ) ?The following is the main theorem of this section, here we provide a partial answer to Question4.8 by showing that integrally decomposable modules satisfy this equality. Theorem 4.9.
Let M be a submodule of R p and let r = rank ( M ) . Consider the following conditions. ( ) M is integrally decomposable. ( ) I r ( M ) = J M . ( ) M = C ( M ) .Then (1) ⇒ (2) ⇒ (3) . Moreover, if r = p, then these implications become equivalences. We remark that (3) ; (2). In particular (3) ; (1) in general. This is shown in Example 4.26. In awide variety of examples of modules M ⊆ R with rank ( M ) = M is integrallydecomposable when I r ( M ) = J M . However we have not yet found a proof or a counterexample ofthe implication (2) ⇒ (1); we conjecture that this implication holds in general.We present the proof of Theorem 4.9 after the following remark and lemma. Remark 4.10.
We observe that I r ( M ) ⊆ J M . In general, this inclusion might be strict. For instance,consider the submodule M ⊆ O generated by the columns of the following matrix x xy x y y y x + y y x + y . Notice that M = ( x , xy ) , M = ( y ) and M = ( x , y ) . We see that rank ( M ) = J M = M M + M M + M M = ( x , y ) . However, I ( M ) = ( x y , xy , y ) . Therefore I r ( M ) is strictly contained in J M .We need one more lemma prior presenting the proof of the theorem. Lemma 4.11.
Let M ⊆ R p be a submodule and let h ∈ R p . If rank ( M ) = rank ( M + Rh ) , then rank ( M L ) = rank ( M L + Rh L ) , for any L ⊆ { , . . . , p } , L = /0 .Proof. Let us identify M with a given matrix of generators. Let Q ( R ) denote the total ring of frac-tions of R . We note that rank ( M ) = rank ( M + Rh ) if and only if h is equal to a linear combinationof the columns of M with coefficients in Q ( R ) . By projecting this linear combination onto the rowscorresponding to L we obtain that h L is equal to a linear combination of the columns of M L , whichmeans rank ( M L ) = rank ( M L + Rh L ) , as desired. (cid:3) We are now ready to present the proof of Theorem 4.9.
Proof of Theorem 4.9.
We begin with (1) ⇒ (2). From I r ( M ) = ∑ L ∈ Λ M I r ( M L ) and Proposition 4.4we obtain I r ( M ) = ∑ L ∈ Λ M I r ( M L ) = ∑ L ∈ Λ M I r ( M L ) = ∑ L ∈ Λ M ∏ i ∈ L M i = J M . We continue with (2) ⇒ (3). The inclusion M ⊆ C ( M ) follows immediately from Remarks 2.9and 2.14, then we need to show the reverse inclusion. Let h ∈ C ( M ) , we claim that I r ( M + Rh ) ⊆ I r ( M ) . We note that if the claim holds then h is integral over M , by Theorem 2.10, finishing theproof.Now we prove the claim. Identify M with a matrix of generators and let g be a non-zero minorof size r of the matrix [ M | h ] with row set L = { i , . . . , i r } . By Lemma 4.11, we have rank ( M L ) = rank ( M L | h L ) . In particular, the matrix M L has some non-zero minor of order r . This implies that L ∈ Λ M . Since h ∈ M ⊕ · · · ⊕ M p , we have g ∈ ∏ i ∈ L M i ⊆ ∏ i ∈ L M i ⊆ J M ([27, 1.3.2]). Therefore, I r ( M + Rh ) ⊆ J M = I r ( M ) , and the claim follows. NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 17
Let us suppose that r = p . In this case C ( M ) = M ⊕ · · · M p and therefore the equivalence of theconditions follows as a direct consequence of Propositions 4.3. (cid:3) The following result shows a procedure to compute the module Z ( M ) with the aid of Singular[14] or other computational algebra programs. If N is a submodule of R p , then we denote by N T the transpose of any matrix whose columns generate N . Lemma 4.12.
Let R be an integral domain and let M be an p × m matrix with entries in R. Then { h ∈ R p : rank ( M ) = rank ([ M | h ]) } = ker (cid:0) ( ker ( M T )) T (cid:1) . Proof.
Let Q ( R ) be the field of fractions of R and let ker Q ( R ) ( − ) the kernel of matrices computedover Q ( R ) .Clearly the rank of a matrix over R is equal to the rank as a matrix over Q ( R ) . Let h ∈ R p , thenby the dimension theorem for matrices we haverank ( M ) = rank ( M T ) = p − dim ker Q ( R ) ( M T ) , and rank ([ M | h ]) = p − dim ker Q ( R ) ([ M | h ] T ) . Since we always have ker Q ( R ) ([ M | h ] T ) ⊆ ker Q ( R ) ( M T ) , it follows thatrank ( M ) = rank ([ M | h ]) ⇐⇒ ker Q ( R ) ([ M | h ] T ) = ker Q ( R ) ( M T ) ⇐⇒ h T v = v ∈ ker Q ( R ) ( M T ) ⇐⇒ h T v = v ∈ ker ( M T ) ⇐⇒ h ∈ ker (cid:0) ( ker ( M T )) T (cid:1) . This finishes the proof. (cid:3)
Remark 4.13.
Given a submodule M of R p , the computation of Z ( M ) can be done with Singular[14] as follows. Denoting also by M a matrix whose columns generate this module, then Z ( M ) isgenerated by the columns of the matrix obtained as syz(transpose(syz(transpose(M)))) .In the next example we show an application of Theorem 4.9 in order to compute the integralclosure of a module. First, we introduce some concepts.Let us fix coordinates x , . . . , x n for C n . If n =
2, we simply write x , y instead of x , x . If k = ( k , . . . , k n ) ∈ N n , then we denote the monomial x k · · · x k n n by x k . If f ∈ O n and f = ∑ k a k x k is the Taylor expansion of f around the origin, then the support of f , denoted by supp ( f ) , is theset { k ∈ N n : a k = } . The support of a non-zero ideal I of O n is the union of the supports of theelements of I . We denote this set by supp ( I ) .Given a subset A ⊆ R n > , the Newton polyhedron determined by A , denoted by Γ + ( A ) , is theconvex hull of the set { k + v : k ∈ A , v ∈ R n > } . The Newton polyhedron of f is defined as Γ + ( f ) = Γ + ( supp ( f )) . For an ideal I of O n , the Newton polyhedron of I is defined as Γ + ( I ) = Γ + ( supp ( I )) .It is well-known that Γ + ( I ) = Γ + ( I ) (see for instance [4, p. 58]).Let w ∈ Z n > and let f ∈ O n , f =
0. We define d w ( f ) = min {h w , k i : k ∈ supp ( f ) } , where h w , k i denotes the usual scalar product. If f = d w ( f ) = + ∞ . We say that a non-zero f ∈ O n is weighted homogeneous with respect to w when h w , k i = d w ( f ) , for all k ∈ supp ( f ) . Example 4.14.
Let us consider the submodule M of O generated by the columns of the followingmatrix: x y xy x + y xy x + y x yx y − xy xy − x − y x + y − x y . We observe that rank ( M ) = Λ M = { ( , ) , ( , ) , ( , ) } . Using Singular [14] we verified that M , M and M have finite colength and e ( M ) = e ( M ) = e ( M ) = I = M = M = ( x y , xy , x + y ) . We have e ( I ) = = e ( M ) . Since M ⊆ I , it fol-lows that I = M . Hence e ( M , M ) = e ( M , M ) = e ( M , M ) = e ( I ) =
11. This fact shows that δ ( M ) = e ( M ) + e ( M , M ) + e ( M ) = e ( I ) = = δ ( M ) = δ ( M ) . Therefore M is integrallydecomposable, by Proposition 4.5.By Theorem 4.9, the integral closure of M is expressed as M = (cid:8) h ∈ I ⊕ I ⊕ I : rank ( M + O h ) = (cid:9) . Let L = ( x + y , xy , x y , x , y ) . We observe that IL = L , therefore I is a reduction of L . Hence L ⊆ I . Let us see that equality holds.Let f = x + y . We observe that f is weighted homogeneous with respect to w = ( , ) .Let N denote the ideal of O generated by all monomials x k y k , where k , k ∈ Z > , such that d w ( x k y k ) = k + k >
11. Then L = ( f ) + N .Let g ∈ I . In particular Γ + ( g ) ⊆ Γ + ( I ) = Γ + ( I ) = Γ + ( x , y ) . Let g denote the part of lowestdegree with respect to w in the Taylor expansion of g , and let g = g − g . Then d w ( g ) >
10 and d w ( g ) >
11. In particular g ∈ N ⊆ L . Then g ∈ L if and only if g ∈ L .We may assume that supp ( g ) ⊆ { ( , ) , ( , ) } , as otherwise g ∈ L . If supp ( g ) is equal to { ( , ) } or to { ( , ) } , then the ideal ( f , g ) has finite colength and e ( f , g ) =
10, which is acontradiction, since ( f , g ) ⊆ I and e ( I ) =
11. Therefore g = α x + β y , for some α , β ∈ C \ { } .If α = β , we would have that ( f , g ) is an ideal of finite colength and e ( f , g ) =
10. Therefore α = β , which means that g ∈ ( f ) ⊆ L . Therefore I ⊆ L .By Theorem 4.9, we have that M = Z ( M ) ∩ ( I ⊕ I ⊕ I ) . The module Z ( M ) can be computed bymeans of Lemma 4.11. Thus we obtain that Z ( M ) is generated by the columns of the matrix −
11 1 . We have seen before that I = L . Let us remark that { x + y , xy , y } is a minimal system ofgenerators of L . Then, by intersecting the modules Z ( M ) and L ⊕ L ⊕ L , we finally obtain that M is generated by the columns of the following matrix: x + y xy y x + y xy y x + y xy y x + y xy y . NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 19
In the next subsection we will introduce an important class of modules that are integrally de-composable.4.2.
Newton non-degenerate modules.
Let us fix coordinates x , . . . , x n for C n . Let M be a sub-module of O pn and let us identify M with any matrix of generators of M . We recall that M i is theideal of O n generated by the elements of i -th row of M . We define the Newton polyhedron of M as Γ + ( M ) = Γ + (cid:0) p ∏ i = M i (cid:1) = Γ + ( M ) + · · · + Γ + ( M p ) = { k + · · · + k p : k i ∈ Γ + ( M i ) , for all i } . We denote by F c ( Γ + ( M )) the set of compact faces of Γ + ( M ) (see [2, p. 408] or [3, p. 397] fordetails).Let I be an ideal of O n . We denote by I the ideal by all monomials x k such that k ∈ Γ + ( I ) . Werefer to this ideal as the term ideal of I . If I is the zero ideal, then we set Γ + ( I ) = /0 and I = monomial if it admits a generating system formed by monomials.It is known that if I is a monomial ideal, then I = I (see [16, p. 141], [27, p. 11], or [44, p. 219]).The ideals I for which I is generated by monomials are characterized in [40] and are called Newtonnon-degenerate ideals (see also [3], [4], or [44, p. 242]).In [2], the first author introduced and studied the notion of Newton non-degenerate modules ofmaximal rank. Here we extend this concept to modules of submaximal rank.Let f ∈ O n and let f = ∑ k a k x k be the Taylor expansion of f around the origin. If ∆ is anycompact subset of R n > , then we denote by f ∆ the polynomial resulting as the sum of all terms a k x k such that k ∈ ∆ . If no such k exist, then we set f ∆ = Definition 4.15.
Let M be a non-zero submodule of O pn and let r = rank ( M ) . Let [ m i j ] be a p × m matrix of generators of M , where p m .(1) ([2, 3.6]) First assume r = p . We say that M is Newton non-degenerate when (cid:8) x ∈ C n : rank [( m i j ) ∆ i ( x )] < p (cid:9) ⊆ (cid:8) x ∈ C n : x · · · x n = (cid:9) , for any ∆ ∈ F c ( Γ + ( M )) , where we write ∆ as ∆ = ∆ + · · · + ∆ p with ∆ i being a compactface of Γ + ( M i ) , for all i = , . . . , p .(2) Now assume r < p . We say that M is Newton non-degenerate when M L is Newton non-degenerate, as a submodule (of rank r ) of O rn , for any L ∈ Λ M .In particular, if I is an ideal of O n and g , . . . , g s denotes a generating system of I , then I is New-ton non-degenerate if and only if { x ∈ C n : ( g ) ∆ ( x ) = · · · = ( g s ) ∆ ( x ) = } ⊆ { x ∈ C n : x · · · x n = } ,for any ∆ ∈ F c ( Γ + ( I )) .The following result follows from [2, 3.7, 3.8] and it characterizes the Newton non-degeneracyof submodules of O pn of maximal rank. Theorem 4.16. [2]
Let M ⊆ O pn be a submodule of rank p. Then the following conditions areequivalent: ( ) M is Newton non-degenerate. ( ) I p ( M ) is a Newton non-degenerate ideal and Γ + ( I p ( M )) = Γ + ( M ) . ( ) M = M ⊕ · · · ⊕ M p .If furthermore, λ ( O pn / M ) < ∞ , then the previous conditions are equivalent to the following: ( ) e (cid:0) I p ( M ) (cid:1) = n !V n (cid:0) Γ + ( M ) (cid:1) . ( ) M i is Newton non-degenerate, for all i = , . . . , p, and e ( M ) = δ ( M ) . As an immediate consequence of Theorem 4.16 the following result follows.
Corollary 4.17.
Let M be a submodule of O pn and let r = rank ( M ) . Then the following conditionsare equivalent: ( ) M is Newton non-degenerate. ( ) I r ( M L ) = ∏ i ∈ L M i , for all L ∈ Λ M . ( ) M { i ,..., i r } = M i ⊕ · · · ⊕ M i r , for all ( i , . . . , i r ) ∈ Λ M . ( ) M is integrally decomposable and M i is Newton non-degenerate, for all i = , . . . , p. Therefore, we see from the previous result that if M is Newton non-degenerate, then it is inte-grally decomposable. The converse does not hold in general, as Example 4.14 shows.From the results of the previous section we obtain the following combinatorial interpretation forthe analytic spread of Newton non-degenerate modules of maximal rank. Corollary 4.18.
Let M ⊆ O pn be a Newton non-degenerate module of rank p, then ℓ ( M ) = max (cid:8) dim ( ∆ ) : ∆ ∈ F c ( Γ + ( M )) (cid:9) + p . Proof.
We may assume R has infinite residue field and then ℓ ( M ) = ℓ ( M ) (see Remark 3.7). More-over M = M ⊕ · · · ⊕ M p , since M is Newton non-degenerate. Therefore ℓ ( M ) = ℓ ( M ⊕ · · · ⊕ M p ) = ℓ ( M · · · M p ) + p −
1, where the last equality is an application of Corollary 3.10. By [1, Theorem2.3] we have ℓ ( M · · · M p ) = max (cid:8) dim ( ∆ ) : ∆ ∈ F c (cid:0) Γ + ( M · · · M p ) (cid:1) (cid:9) + . Since Γ + ( M ) = Γ + ( M · · · M p ) the result follows. (cid:3) Example 4.19.
Let M be the submodule of O generated by the columns of the following matrix M = " x xy y y x x y xy x + x y . We observe that rank ( M ) = I ( M ) is a Newton non-degenerate ideal. Moreover I ( M ) =( xy , x y , x y , x y , x ) = M M . Therefore M = M ⊕ M , by Corollary 4.17 and ℓ ( M ) = ℓ ( M ) =
3, by Corollary 4.18.Analogously to Definition 4.6, for a submodule of O pn we introduce the following objects. Definition 4.20.
Let M ⊆ O np and let r = rank ( M ) . We define H M = ∑ ( i ,..., i r ) ∈ Λ M M i · · · M i r , NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 21 and C ( M ) = Z ( M ) ∩ (cid:0) M ⊕ · · · ⊕ M p (cid:1) , where we recall that Z ( M ) = { h ∈ R p : rank ( M ) = rank ( M + Rh ) } .We remark that H M is a monomial ideal and Γ + ( H M ) = Γ + ( J M ) . Therefore H M = J M , where J M is the ideal of O n generated by the monomials x k such that k ∈ Γ + ( J M ) . We also remark that C ( M ) ⊆ C ( M ) . The following result follows from Theorem 4.9 and Corollary 4.17. Corollary 4.21.
Let M be a submodule of O pn and let r = rank ( M ) . Consider the following condi-tions. ( ) M is Newton non-degenerate. ( ) I r ( M ) = J M . ( ) M = C ( M ) .Then (1) ⇒ (2) ⇒ (3) . Moreover, if r = p, then these implications become equivalences. Remark 4.22. (1) The implication (3) ⇒ (2) in Corollary 4.21 does not hold in general, asshown in Example 4.24. Analogous to Theorem 4.9, in a wide variety of examples ofmodules M ⊆ O with rank ( M ) =
2, we have checked that M is Newton non-degeneratewhenever I r ( M ) = J M . However we have not still found a proof or a counterexample of theimplication (2) ⇒ (1) of Corollary 4.21 in general.(2) We remark that the advantage of Corollary 4.21 over Theorem 4.9 is that it is usually easy toverify if a module is Newton non-degenerate via Theorem 4.16. Moreover, C ( M ) admitsa faster computation than C ( M ) as we can use convex-geometric methods to compute theintegral closure of monomial ideals.In the following example we use Corollary 4.21 to compute the integral closure of a family ofmodules. Example 4.23.
Let us consider the submodule M ⊆ O generated by the columns of the followingmatrix: x a xy y a y a x a xyx a + y a xy + x a y a + xy , where a ∈ Z > . We remark that rank ( M ) =
2. Let J = ( x a , xy , y a ) . The ideal J is integrally closedand M = M = M = J . An elementary computation shows that I ( M ) = ( xy a + − x a , x a + y − y a , x y ) and that I ( M ) is Newton non-degenerate. Moreover J M = ( x a , x y , y a ) and then I ( M ) = J M , since Γ + ( I ( M )) = Γ + ( J M ) . Therefore, by Corollary 4.21, we conclude that M = C ( M ) = C ( M ) . Given any element h = ( h , h , h ) ∈ O , we have that rank ( M ) = rank ( M + O h ) if and only if h = h + h (Lemma 4.12). Therefore M = (cid:8) h ∈ M ⊕ M ⊕ M : rank ( M ) = rank ( M + O h ) (cid:9) = (cid:8) [ h h h + h ] T ∈ O : h , h ∈ J (cid:9) . Therefore, a minimal generating system of M is given by the columns of the following matrix x a xy y a x a xy y a x a xy y a x a xy y a . Example 4.24.
Let M be the submodule of O generated by the columns of the following matrix " x x yx ( x + y ) y ( x + y ) . We observe that rank ( M ) =
1. The ideal I ( M ) is given by I ( M ) = ( x , x y , x ( x + y ) , y ( x + y )) = M + M = J M . We have Γ + ( I ( M )) = Γ + ( x , y ) . Let ∆ denote the unique compact face of dimension 1 of Γ + ( x , y ) . Hence ( x ) ∆ = ( x y ) ∆ = ( x ( x + y )) ∆ = x ( x + y ) and ( y ( x + y )) ∆ = y ( x + y ) . Sincethe line of equation y = − x is contained in the set of solutions of the system x ( x + y ) = y ( x + y ) = I ( M ) is Newton degenerate. Therefore I ( M ) = J M (otherwise I ( M ) would bea reduction of the monomial ideal J M and hence I ( M ) would be Newton non-degenerate, which isnot the case). Let us observe that I ( M ) = ( x ( x + y ) , y ( x + y )) + m . Therefore, by Corollary 2.12and applying Singular [14] (see Remark 4.13), we deduce that M = M .By computing explicitly a generating system of C ( M ) = Z ( M ) ∩ ( M ⊕ M ) , we also obtainthat C ( M ) = M and hence C ( M ) = M . Then (3) ; (2) in Corollary 4.21. Remark 4.25.
Let R be a Noetherian normal domain. We note that the only general approach tocompute the integral closure of an arbitrary submodule M ⊆ R p is to compute the normalization R ( M ) of the Rees algebra R ( M ) . Indeed, by [39] we have [ R ( M )] = M and this algebra can becomputed via algorithms such as the one in [15], which is implemented in Macaulay2 under thecommand integralClosure .We note that Theorem 4.9 and Corollary 4.21 can be used to compute the effectively the integralclosure of integrally decomposable modules. Other algorithms that compute integral closures ofmodules under special conditions can be found in the literature (see for instance [46, 9.23]).The following two examples are motivated by Example 5.8 of [34]. Example 4.26.
Let us consider the submodule M ⊆ O generated by the columns of the followingmatrix x y x y x x + y y . The rank of M is 2 and I ( M ) = ( x , x y , y ) . Thus I ( M ) = m . By Corollary 2.12, we have M = Z ( M ) ∩ A ( M ) , where(4.2) A ( M ) = (cid:8) h = [ h h h ] T ∈ O : I ( M , h ) ⊆ m (cid:9) . NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 23
In general, the submodule Z ( M ) can be computed by using Singular [14], as explained in Remark4.13. In this case it is immediate to see that Z ( M ) = . In (4.2) the minors of size 2 of the matrix ( M , h ) are x h , yh − xh , y h , x ( h − h ) , yh − ( x + y ) h , xh − ( x + y ) h and y ( h − h ) . Then A ( M ) is equal to the intersection of the followingsubmodules of O : N = (cid:8) h = [ h h h ] T ∈ O : x h ∈ m (cid:9) N = (cid:8) h = [ h h h ] T ∈ O : yh − xh ∈ m (cid:9) N = (cid:8) h = [ h h h ] T ∈ O : y h ∈ m (cid:9) N = (cid:8) h = [ h h h ] T ∈ O : x ( h − h ) ∈ m (cid:9) N = (cid:8) h = [ h h h ] T ∈ O : yh − ( x + y ) h ∈ m (cid:9) N = (cid:8) h = [ h h h ] T ∈ O : xh − ( x + y ) h ∈ m (cid:9) N = (cid:8) h = [ h h h ] T ∈ O : y ( h − h ) ∈ m (cid:9) . Each of the above submodules can be computed with Singular. For instance, to obtain a generatingsystem of N we can use the following procedure. Let S denote the quotient ring O / m and let usconsider the submodule of S given by syz S ( − x − y , , y ) = { ( g , g , g ) ∈ S : ( − x − y ) g + yg = } . Once we have obtained a matrix of generators of syz S ( − x − y , , y ) with Singular, if B is anysubmodule of O whose image in S generates syz S ( − x − y , , y ) , then N = B + ( m ⊕ m ⊕ m ) .Therefore it follows that N is generated by the columns of the matrix y y xy − y x − xy + y y
00 0 y y − y y x + y . By computing a minimal generating system of Z ( M ) ∩ N ∩ · · · ∩ N , it follows that x xy y y
00 0 0 x y x xy y x + y y . We remark that M = M , M = M and M = ( x + y ) + m . Therefore, a computation with Singularshows that the module C ( M ) , which is defined as Z ( M ) ∩ ( M ⊕ M ⊕ M ) , is equal to M .However we have the strict inclusion I ( M ) ⊆ J M in this case, since J M = m . Hence we have(3) ; (2) in Theorem 4.9.The inequality I ( M ) = J M implies that M is not integrally decomposable, by Theorem 4.9. Ac-tually, none of the submodules M { , } , M { , } and M { , } are integrally decomposable, by Proposi-tion 4.5, since they δ ( M , ) = δ ( M , ) = δ ( M , ) = e ( M , ) = e ( M , ) = e ( M , ) = Example 4.27.
Let us consider the submodule M ⊆ O generated by the columns of the followingmatrix: " x a y b x c y d , where a , b , c , d ∈ Z > . Let I = I ( M ) = ( x a + c , x a y d , y b + d ) . Since the ideals M and M are gener-ated by monomials we have, from Theorem 4.16, that M is decomposable ⇐⇒ M = M ⊕ M ⇐⇒ M is Newton non-degenerate ⇐⇒ I is Newton non-degenerate and Γ + ( I ) = Γ + ( M M ) . (4.3)Therefore M is not decomposable if and only if Γ + ( I ) is strictly contained in Γ + ( M M ) . We seethat Γ + ( M M ) = Γ + ( x a + c , x a y d , y b + d , x c y b ) . Let us observe that Γ + ( I ) = Γ + ( x a + c , y b + d ) if andonly if ad > bc .Let us suppose first that ad > bc . Then Γ + ( I ) is strictly contained in Γ + ( M M ) if and only if ( c , b ) lies below the line determined by the two vertices of Γ + ( I ) , which is to say that ad > bc .If ad < bc , then the Newton boundary of Γ + ( I ) is equal to the union of two segments and ( c , b ) belongs to the interior of Γ + ( I ) . Hence Γ + ( I ) = Γ + ( M M ) and this implies that M isdecomposable by (4.3).Thus we have shown that M is not decomposable if and only if ad > bc . In this case, we have Γ + ( I ) = Γ + ( x a + c , y b + d ) . Let w = ( b + d , a + c ) . By Corollary 2.12 we obtain that M = (cid:8) h = [ h h ] T ∈ O : d w ( x a h ) > ( a + c )( b + d ) , (4.4) d w ( y b h − x c h ) > ( a + c )( b + d ) and d w ( y d h ) > ( a + c )( b + d ) (cid:9) . Once positive integer values are assigned to a , b , c , d , it is possible to obtain a generating system of M with Singular [14] by following an analogous procedure as in Example 4.26.In the following example we show an example of a non-decomposable integrally closed sub-module of O of rank 2 whose ideal of maximal minors is not simple (that is, it is factorized as theproduct of two proper integrally closed ideals). Example 4.28.
Let M be the submodule of O generated by the columns of the following matrix: " x xy y y x + y y . We observe that e ( M ) = e ( M ) = e ( M , M ) =
2. Therefore δ ( M ) =
14. However e ( M ) =
22. Then M is not decomposable by Proposition 4.5. In particular, since M is a submoduleof O , M does not split as the direct sum of two proper ideals of O . Let I = I ( M ) = ( − xy + xy + y , − x y + y , − xy + x + x y ) . By Corollary 2.12 it follows that M = n h ∈ O : I ( M + O h ) ⊆ I ( M ) o . NALYTIC SPREAD AND INTEGRAL CLOSURE OF INTEGRALLY DECOMPOSABLE MODULES 25
An easy computation shows that I is Newton non-degenerate and Γ + ( I ) = Γ + ( x , xy , y ) . Theideal generated by all monomials x k y k such that ( k , k ) ∈ Γ + ( I ) is J = ( x , x y , x y , xy , y ) .Hence I = J and this implies that M = n h = [ h h ] T ∈ O : x h − y h , y h − y h , xyh − ( x + y ) h ∈ J o . Hence M = N ∩ N ∩ N , where N = n h = [ h h ] T ∈ O : x h − y h ∈ J o N = n h = [ h h ] T ∈ O : y h − y h ∈ J o N = (cid:8) h = [ h h ] T ∈ O : xyh − ( x + y ) h ∈ J (cid:9) . As in Example 4.26, using Singular [14] we obtain that N = " y x xy y x N = " y x xy y N = " y x y − x y + x y − xy x − x y + x y − x y + xy x y − xy xy y x + y . Using Singular again, we have N ⊆ N ∩ N . Therefore M = N . As we have discussed before, M is not decomposable and obviously it is integrally closed. However we have that I ( M ) = I ( M ) = ( x , x y , x y , xy , y ) = ( x , y )( x , x y , x y , y ) . That is, the ideal I ( M ) is not simple. We also refer to [23] for another type and wide class ofexamples of integrally closed and non-decomposable submodules N ⊆ O of rank 2 for which theideal I ( N ) is not simple (these examples are motivated by a question raised by Kodiyalam in [34,p. 3572] about the converse of Theorem 5.7 of [34]).A CKNOWLEDGEMENTS
This work started during the stay of the authors at the Mathematisches Forschungsinstitut Ober-wolfach in June 2018. The authors wish to thank this institution for hospitality and financialsupport. The first author also acknowledges the Department of Mathematical Sciences of NewMexico State University (Las Cruces, NM, USA) also for hospitality and financial support. Wealso acknowledge Prof. F. Hayasaka for informing us about the existence of his preprint [23]. Theauthors would like to thank the referees for their helpful suggestions and comments.R
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