Effective Localization Using Double Ideal Quotient and Its Implementation
aa r X i v : . [ m a t h . A C ] F e b Effective Localization Using Double IdealQuotient and Its Implementation
Yuki Ishihara ∗ Kazuhiro Yokoyama † Abstract
In this paper, we propose a new method for localization of polynomialideal, which we call ”Local Primary Algorithm”. For an ideal I and aprime ideal P , our method computes a P -primary component of I afterchecking if P is associated with I by using double ideal quotient ( I : ( I : P )) and its variants which give us a lot of information about localizationof I . In commutative algebra, the operation of localization by a prime ideal is well-known as a basic tool. To realize it on computer algebra systems, we proposenew effective localization using double ideal quotient (DIQ) and its variants forideals, in a polynomial ring over a field. Here, by the words localization , wemean the saturation or the contraction of localized ideals.It is well-known that localization of ideal can be computed through its pri-mary decomposition. In more detail, for an ideal I of a polynomial ring K [ X ] = K [ x , . . . , x n ] over a field K and a multiplicatively closed set S in K [ X ], once aprimary decomposition Q of I is known, the localization (i.e. the contraction oflocalized ideal) of I by S can be computed by IK [ X ] S ∩ K [ X ] = T Q ∈Q ,Q ∩ S = ∅ Q (see Remark 3). Algorithms of primary decomposition have been much stud-ied, for example, by [3], [2], [8] and [5]. However, in practice, as such primarydecomposition tends to be very time-consuming, use of primary decompositionis not an efficient way and we need an efficient direct method without primarydecomposition. Toward a direct method of localization, for a given ideal I anda prime ideal P , first we provide several criteria for checking if a primary ideal Q can be a P -primary component of I , and then present a direct method named Local Primary Algorithm (LPA) which computes a P -primary component of I .Our method applies different procedures for two cases; isolated and embedded.Both cases use double ideal quotient and its variants as a tool for generatingand checking primary components. Of course, if we know all associated primesdisjoint from a multiplicatively closed set, we get its localization without com-puting other primary components. ∗ Graduate School of Science, Rikkyo University 3-34-1 Nishi-Ikebukuro, Toshima-ku,Tokyo, Japan 171-8501, [email protected] † Department of Mathematics, Rikkyo University 3-34-1 Nishi-Ikebukuro, Toshima-ku,Tokyo, Japan 171-8501, [email protected] I and J , we call an ideal ( I : ( I : J )) double ideal quotient inthe paper. Double ideal quotient appears in [10] to check associated primes orcompute equidimensional hull, and in [2], to compute equidimensional radical.We survey other properties of double ideal quotient and find that it and itsvariants have useful information about localization. For instance, for ideals I , J and a primary decomposition Q of I , a variant of DIQ ( I : ( I : J ) ∞ ) coincideswith T Q ∈Q ,J ⊂ IK [ X ] √ Q ∩ K [ X ] Q .To check the practicality of criteria on LPA , we made an implementationon the computer algebra system Risa/Asir [7] and demonstrate the performancein several examples. To evaluate effectiveness coming from its speciality, wecompare timings of it to ones of a general algorithm of primary decompositionin Risa/Asir.For practical implements we devise several efficient techniques for improvingour LPA. (For efficient computation of ideal quotient and saturation, see [10] and[4]). First, instead of computing the equidimensional hull hull( I + P m ), we usehull( I + P [ m ] G ) where P [ m ] G = ( f m , . . . , f mr ) for some generator G = { f , . . . , f r } of P . Second, we use a maximal independent set of P for computing hull( Q )where Q is a P -hull-primary ideal. Since a maximal independent set U of P isone of I + P m , we obtain hull( I + P m ) = ( I + P m ) K [ X ] K [ U ] × ∩ K [ X ]. Moreover,we also use U at the first step of LPA; use IK [ X ] K [ U ] × ∩ K [ X ] instead of I .By these efficient techniques, our experiment shows certain practicality of ourdirect localization method. Throughout this paper, we denote a polynomial ring K [ x , . . . , x n ] by K [ X ],where K is a computable field (e.g. the rational field Q or a finite field F p ) andwe denote the set of variables { x , . . . , x n } by X . We write ( f , . . . , f t ) K [ X ] forthe ideal generated by elements f , . . . , f t in K [ X ]. If the ring is obvious, wesimply use ( f , . . . , f t ). When we simply say I is an ideal, it means the I is anideal of K [ X ]. Moreover, we denote the radical of I by √ I . Here we give the definition of primary decomposition and that of localizationwhich seem slightly different from standard ones. We also give fundamentalnotions and properties related to localization.
Definition 1.
Let I be an ideal of K [ X ] . A set Q of primary ideals is called ageneral primary decomposition of I if I = T Q ∈Q Q . A general primary decompo-sition Q is called a primary decomposition of I if the decomposition I = T Q ∈Q Q is an irredundant decomposition. For a primary decomposition of I, each pri-mary ideal is called a primary component of I . The prime ideal associated witha primary component of I is called a prime divisor of I and among all primedivisors, minimal prime ideals are called isolated prime divisors of I and othersare called embedded prime divisors of I . A primary component of I is calledisolated if its prime divisor is isolated and embedded if its prime divisor is em-bedded. We denote by Ass( I ) and Ass iso ( I ) the set of all prime divisors of I and the set of all isolated prime divisors respectively. efinition 2. Let I be an ideal of K [ X ] and S a multiplicatively closed set in K [ X ] . We denote the set { f ∈ K [ X ] | f s ∈ I for some s ∈ S } by IK [ X ] S ∩ K [ X ] , and call it the localization of I with respect to S . For a multiplicativelyclosed set K [ X ] \ P , where P is a prime ideal, we denote simply by IK [ X ] P ∩ K [ X ] . We assume a multiplicatively closed set S always does not contain . Remark 3.
Given a primary decomposition Q of an ideal I , the localiza-tion of I by S is expressed as T Q ∈Q ,Q ∩ S = ∅ Q . Moreover, it is also equal to ( I : ( T P ∈ Ass( I ) ,P ∩ S = ∅ P ) ∞ ) . Thus if we know all primary components or allassociated primes, then we can compute localizations of I for any computable multiplicatively closed sets S . (We are thinking mainly about cases where S isfinitely generated or the complement of a prime ideal. In these cases, we can de-cide efficiently whether Q and S intersect or not). However, this method is not adirect method since it computes unnecessary primary components or associatedprimes. Lemma 4.
Let I be an ideal and P a prime divisor of I . If S is a multiplica-tively closed set with P ∩ S = ∅ and Q is a P -primary ideal, then the followingconditions are equivalent. ( A ) Q is a primary component of I ( B ) Q is a primary component of IK [ X ] S ∩ K [ X ] Proof.
First, ( A ) implies ( B ) from Proposition 4.9 in [1] . For primary de-compositions Q of I and Q ′ of IK [ X ] S ∩ K [ X ] with Q ∈ Q ′ , we obtain { Q ′ ∈ Q | Q ′ ∩ S = ∅} ∪ Q ′ is also a primary decomposition of I . Hence,( B ) implies ( A ). Definition 5 ([1], Chapter 4) . Let I be an ideal. A subset P of Ass( I ) is said tobe isolated if it satisfies the following condition: for a prime divisor P ′ ∈ Ass( I ) ,if P ′ ⊂ P for some P ∈ P , then P ′ ∈ P . Lemma 6 ([1], Theorem 4.10) . Let I be an ideal and P an isolated set containedin Ass( I ) . For a multiplicatively closed set S = K [ X ] \ S P ∈P P and a primarydecomposition Q of I , IK [ X ] S ∩ K [ X ] = T Q ∈Q , √ Q ∈P Q . Lemma 7.
Let Q be a primary decomposition of I and Q ∈ Q . For a multi-plicatively closed set S , the following conditions are equivalent. ( A ) IK [ X ] S ∩ K [ X ] ⊂ IK [ X ] √ Q ∩ K [ X ] . ( B ) Q ∩ S = ∅ .Proof. Show ( A ) implies ( B ). As IK [ X ] √ Q ∩ K [ X ] ⊂ Q , IK [ X ] S ∩ K [ X ] = T Q ′ ∈Q ,Q ′ ∩ S = ∅ Q ′ ⊂ Q . Since Q is irredundant, IK [ X ] S ∩ K [ X ] has √ Q -primarycomponent. Thus, Q ∩ S = ∅ . Now, we show ( B ) implies ( A ). Then, √ Q ∩ S = ∅ and Q ′ ∩ S = ∅ for any Q ′ ∈ Q s.t. Q ′ ⊂ √ Q . Thus, IK [ X ] √ Q ∩ K [ X ] = T Q ′ ⊂√ Q Q ′ implies IK [ X ] S ∩ K [ X ] ⊂ IK [ X ] √ Q ∩ K [ X ].Next we introduce the notion of pseudo-primary ideal. Definition 8.
Let Q be an ideal. We say Q is pseudo-primary if √ Q is a primeideal. In this case, we also say √ Q -pseudo-primary. Definition 9.
Let I be an ideal and P an isolated prime divisor of I . For P = { P ′ ∈ Ass( I ) | P is the unique isolated prime divisor contained in P ′ } nd S = K [ X ] \ S P ′ ∈P P ′ , we call Q = IK [ X ] S ∩ K [ X ] the P -pseudo-primarycomponent of I . This definition is consistent with one in [8]. We note thatthe P -pseudo-primary component is determined uniquely and has the P -isolatedprimary component of I as component. Remark 10.
Every P -pseudo-primary component of I is a P -pseudo-primaryideal. Let Q P is the P -pseudo-primary component of I . Then I = T P ∈ Ass iso ( I ) Q P ∩ I ′ for some I ′ s.t. Ass iso ( I ′ ) ∩ Ass iso ( I ) = ∅ . This decomposition is called apseudo-primary decomposition in [8], where it is computed by separators fromgiven Ass iso ( I ) . Meanwhile, we introduce another method to compute it by usingdouble ideal quotient in Lemma 32. Definition 11.
Let I be an ideal and Q a primary decomposition of I . We call hull( I ) = T Q ∈Q , dim( Q )=dim( I ) Q the equidimensional hull of I . Since every pri-mary component Q satisfying dim( Q ) = dim( I ) is isolated, hull( I ) is determinedindependently from choice of primary decompositions. For a given I , hull( I ) can be computed in several manners. For instance, itcan be computed by Ext functors [2] or a regular sequence contained in I [10]. Proposition 12 ([2], Theorem 1.1. [10], Proposition 3.41) . Let I be an idealand u ⊂ I be a c -length regular sequence, where c is the codimension of I . Then hull( I ) = (( u ) : (( u ) : I )) = ann K [ X ] (Ext cK [ X ] ( K [ X ] /I, K [ X ])) . Definition 13.
Let I be an ideal. We say that I is hull-primary if hull( I ) isa primary ideal. For a prime ideal P , we say a hull-primary ideal I is P -hull-primary if P = hull( √ I ) . Since a pseudo-primary ideal has the unique isolated component, we obtainthe following remark.
Remark 14.
A pseudo-primary ideal is hull-primary.
By the definition of the P -pseudo-primary component of I , it is easy to provethe following lemma. Lemma 15.
Let P be an isolated prime divisor of I and Q a P -pseudo-primarycomponent of I . Then, Q is a P -hull-primary and hull( Q ) is the isolated P -primary component of I . Using Lemma 15 and a variant of double ideal quotient , we generate theisolated P -primary component of I in Section 5. Lemma 16.
Let Q be a primary ideal. Let I and J be ideals. If IJ ⊂ Q and J
6⊂ √ Q , then I ⊂ Q . In particular, if I ∩ J ⊂ Q and J
6⊂ √ Q , then I ⊂ Q .Proof. Let f ∈ I and g ∈ J \ √ Q . Since Q is √ Q -primary, f g ∈ IJ ⊂ Q andthus f ∈ Q . Lemma 17.
Let I be a P -hull-primary and Q a P -primary ideal. If I ⊂ Q ,then hull( I ) ⊂ Q .Proof. Let Q be a primary decomposition of I and J = T Q ′ ∈Q ,Q ′ =hull( I ) Q ′ .Then I = hull( I ) ∩ J ⊂ Q and J P . Since Q is P -primary, we obtainhull( I ) ⊂ Q by Lemma 16. 4inally, we recall the famous Prime Avoidance Lemma. Lemma 18 ([1], Proposition 1.11) . (i) Let P , . . . , P n be prime ideals and let I be an ideal contained in S ni =1 P i . Then, I ⊂ P i for some i . (ii) Let I , . . . , I n be ideals and let P be a prime ideal containing T ni =1 I i . Then P ⊃ I i for some i . If P = T ni =1 I i , then P = I i for some i . We introduce fundamental properties of ideal quotient. The first two can beseen in several papers and books ([1], Lemma 4.4. [4], Lemma 4.1.3. [10], aremark before Proposition 3.56). The last two are direct consequences of thefirst two.
Lemma 19.
Let I and J be ideals, Q a primary ideal and Q a primary decom-position of I . Then, ( Q : J ) = Q , if J
6⊂ √
Q,K [ X ] , if J ⊂ Q, √ Q -primary ideal properly containing Q , if J Q, J ⊂ √ Q, ( Q : J ∞ ) = ( Q : √ J ∞ ) = ( Q , if J
6⊂ √
Q,K [ X ] , if J ⊂ √ Q, ( I : J ) = \ Q ∈Q ,J Q Q ∩ \ Q ∈Q ,J Q,J ⊂√ Q ( Q : J ) , ( I : J ∞ ) = ( I : √ J ∞ ) = \ Q ∈Q ,J Q Q. Double Ideal Quotient (DIQ) is an ideal of shape ( I : ( I : J )) where I and J are ideals. For an ideal I and its primary decomposition Q , we divide Q intothree parts: Q ( J ) = { Q ∈ Q | J
6⊂ √ Q } , Q ( J ) = { Q ∈ Q | J ⊂ Q } , Q ( J ) = { Q ∈ Q | J Q, J ⊂ √ Q } . Then, our DIQ is expressed precisely by components of them. The followingproposition can be proved directly from Lemma 19. We omit an easy but tediousproof.
Proposition 20.
Let I and J be ideals. Then, ( I : ( I : J )) = \ Q ∈Q ( J ) Q : \ Q ′ ∈Q ( J ) Q ′ ∩ \ Q ′ ∈Q ( J ) ( Q ′ : J ) ∩ \ Q ∈Q ( J ) Q : \ Q ′ ∈Q ( J ) Q ′ ∩ \ Q ′ ∈Q ( J ) ( Q ′ : J ) , p ( I : ( I : J )) = \ P ∈ Ass( I ) ,J ⊂ P P. Corollary 21 ([10], Corollary 3.4) . Let I be an ideal and P a prime ideal.Then, P belongs to Ass( I ) if and only if P ⊃ ( I : ( I : P )) .Proof. We note P ⊃ ( I : ( I : P )) if and only if P ⊃ p ( I : ( I : P )). ByProposition 20, p ( I : ( I : P )) = T P ′ ∈ Ass( I ) ,P ⊂ P ′ P ′ . If P ∈ Ass( I ), then p ( I : ( I : P )) = T P ′ ∈ Ass( I ) ,P ⊂ P ′ P ′ ⊂ P . On the other hand, if P ⊃ p ( I : ( I : P )),then there is P ′ ∈ Ass( I ) s.t. P ′ ⊂ P and P ′ ⊃ P . Thus P = P ′ ∈ Ass( I ).Replacing ideal quotient with saturation in DIQ, we have the following. Proposition 22.
Let Q be a primary decomposition of I . Then, ( I : ( I : J ) ∞ ) = \ Q ∈Q ,J ⊂ IK [ X ] √ Q ∩ K [ X ] Q, (1)( I : ( I : J ∞ ) ∞ ) = \ Q ∈Q ,J ⊂ √ IK [ X ] √ Q ∩ K [ X ] Q, (2)( I : ( I : J ∞ )) = \ Q ∈Q ( J ) ( Q : \ Q ′ ∈Q ( J ) Q ′ ) ∩ \ Q ∈Q ( J ) ( Q : \ Q ′ ∈Q ( J ) Q ′ ) . (3) We call them the first saturated quotient , the second saturated quotient , and the third saturated quotient respectively.Proof. Here, we give an outline of the proof. The formula (1) can be proved bycombining the equation( I : ( I : J ) ∞ ) = ( I : p ( I : J ) ∞ ) = T Q ∈Q , T Q ′∈Q J ) √ Q ′ ∩ T Q ′∈Q J ) √ Q ′ Q Q by Lemma 19 and the following equivalence(1-a) J ⊂ IK [ X ] √ Q ∩ K [ X ].(1-b) T Q ′ ∈Q ( J ) √ Q ′ ∩ T Q ′ ∈Q ( J ) √ Q ′
6⊂ √ Q .for each Q ∈ Q . The second formula (2) can be proved by combining theequation ( I : ( I : J ∞ ) ∞ ) = ( I : ( I : J m ) ∞ ) = T Q ∈Q ,J m ⊂ IK [ X ] √ Q ∩ K [ X ] Q for asufficiently large m from the first formula (1), and the following equivalence(2-a) J m ⊂ IK [ X ] √ Q ∩ K [ X ] for a sufficiently large m .(2-b) J ⊂ q IK [ X ] √ Q ∩ K [ X ].for each Q ∈ Q . The third formula (3) can be proved directly from Lemma 19.Now, we explain some details. We show (1-a) implies (1-b). If T Q ′ ∈Q ( J ) √ Q ′ ∩ T Q ′ ∈Q ( J ) √ Q ′ ⊂ √ Q ,then by Lemma 18, √ Q ′ ⊂ √ Q for some Q ′ ∈ Q ( J ) ∪ Q ( J ). Since Q ′ ⊂√ Q ′ ⊂ √ Q , we obtain IK [ X ] √ Q ∩ K [ X ] = T Q ′′ ∈Q ,Q ′′ ⊂√ Q Q ′′ ⊂ Q ′ . However,since Q ′ ∈ Q ( J ) ∪Q ( J ), we obtain J Q ′ and this contradicts J ⊂ IK [ X ] √ Q ∩ K [ X ] ⊂ Q ′ . 6how (1-b) implies (1-a). Let Q ′ ∈ Q contained √ Q . Since T Q ′′ ∈Q ( J ) √ Q ′′ ∩ T Q ′′ ∈Q ( J ) √ Q ′′
6⊂ √ Q , we obtain Q ′
6∈ Q ( J ) ∪ Q ( J ) and Q ′ ∈ Q ( J ). Hence, J ⊂ Q ′ and J ⊂ T Q ′ ⊂√ Q Q ′ = IK [ X ] √ Q ∩ K [ X ].Trivially, (2-a) implies (2-b) since J ⊂ √ J m ⊂ q IK [ X ] √ Q ∩ K [ X ]. Show(2-b) implies (2-a). For Q ∈ Q ( J ) ∪ Q ( J ), let m Q = min { m | J m ⊂ Q } and m = max { m Q | Q ∈ Q ( J ) ∪ Q ( J ) } . Then, ( I : J ∞ ) = ( I : J m ). Since IK [ X ] √ Q ∩ K [ X ] = T Q ′ ∈Q ,Q ′ ⊂√ Q Q ′ , we obtain Q ′ ∈ Q ( J ) ∪ Q ( J ) for any Q ′ ∈ Q contained in √ Q . Thus, we obtain J m ⊂ IK [ X ] √ Q ∩ K [ X ].Using the first saturated quotient, we devise criteria for primary componentin Section 4. The second saturated quotient can be used to isolated primedivisor check and generate an isolated primary component in Section 5. Thethird saturated quotient gives another prime divisor criterion (Criterion 5 inSection 4) other than Corollary 19 by the following proposition. Proposition 23.
Let I and J be ideals. Then p ( I : ( I : J ∞ )) = T P ∈ Ass( I ) ,J ⊂ P P. Proof.
Let Q be a primary decomposition of I . By Proposition 22 (3), p ( I : ( I : J ∞ )) = \ Q ∈Q ( J ) s ( Q : \ Q ′ ∈Q ( J ) Q ′ ) ∩ \ Q ∈Q ( J ) s ( Q : \ Q ′ ∈Q ( J ) Q ′ ) . Since Q is minimal, we obtain Q T Q ′ ∈Q ( J ) Q ′ for any Q ∈ Q ( J ) and Q T Q ′ ∈Q ( J ) Q ′ for any Q ∈ Q ( J ). Thus, by Lemma 19, p ( I : ( I : J ∞ )) = \ Q ∈Q ( J ) s ( Q : \ Q ′ ∈Q ( J ) Q ′ ) ∩ \ Q ∈Q ( J ) s ( Q : \ Q ′ ∈Q ( J ) Q ′ )= \ Q ∈Q ( J ) p Q ∩ \ Q ∈Q ( J ) p Q = \ P ∈ Ass( I ) ,J ⊂ P P. In this section, we present several criteria for primary component which checkif a P -primary ideal Q is a primary component of I or not without computingprimary decomposition of I based on the first saturated quotient. We firstpropose a general criterion applicable to any primary ideals. Later, we proposesome specialized criteria aiming for isolated primary components and maximalones. Finally, we add criteria for prime divisors. We use the first saturated quotient to check if a given primary ideal is a com-ponent or not. We introduce a key notion saturated quotient invariant . Definition 24.
Let I and J be ideals. We say that J is saturated quotientinvariant of I if ( I : ( I : J ) ∞ ) = J. I . Lemma 25.
Let I be an ideal and J a proper ideal of K [ X ] . Then, the followingconditions are equivalent. ( A ) J = IK [ X ] S ∩ K [ X ] for some multiplicatively closed set S . ( B ) J is saturated quotient invariant of I .Proof. Let Q be a primary decomposition. Show ( A ) implies ( B ). From Propo-sition 22 (1),( I : ( I : IK [ X ] S ∩ A ) ∞ ) = \ Q ∈Q ,IK [ X ] S ∩ K [ X ] ⊂ IK [ X ] √ Q ∩ K [ X ] Q. (1)By Lemma 7, IK [ X ] S ∩ K [ X ] ⊂ IK [ X ] √ Q ∩ K [ X ] if and only if Q ∩ S = ∅ .Thus, \ Q ∈Q ,IK [ X ] S ∩ K [ X ] ⊂ IK [ X ] √ Q ∩ K [ X ] Q = \ Q ∈Q ,Q ∩ S = ∅ Q, (2)Combining (1), (2) and IK [ X ] S ∩ K [ X ] = T Q ∈Q ,Q ∩ S = ∅ Q by Remark 3, weobtain ( I : ( I : IK [ X ] S ∩ A ) ∞ ) = IK [ X ] S ∩ K [ X ].Next, show ( B ) implies ( A ). From Proposition 22 (1),( I : ( I : J ) ∞ ) = \ J ⊂ IK [ X ] √ Q ∩ K [ X ] Q = J. (3)Let P = {√ Q | Q ∈ Q , J ⊂ IK [ X ] √ Q ∩ K [ X ] } . We may assume P 6 = ∅ ,otherwise P = ∅ and J = K [ X ]. Then P is isolated since if P ′ ∈ Ass( I )and P ′ ⊂ P for some P ∈ P , then J ⊂ IK [ X ] P ∩ K [ X ] ⊂ IK [ X ] P ′ ∩ K [ X ]and P ′ ∈ P . Let S = K [ X ] \ S P ∈P P . By Lemma 6, IK [ X ] S ∩ K [ X ] = T Q ∈Q , √ Q ∈P Q = T J ⊂ IK [ X ] √ Q ∩ K [ X ] Q . By (3), we obtain IK [ X ] S ∩ K [ X ] = J .Based on Lemma 25, we have the following criterion for primary component. Theorem 26 (Criterion 1) . Let I be an ideal and P a prime divisor of I .For a P -primary ideal Q , if Q ( I : P ∞ ) , then the following conditions areequivalent. ( A ) Q is a P -primary component for some primary decomposition of I . ( B ) ( I : P ∞ ) ∩ Q is saturated quotient invariant of I .Proof. Show ( A ) implies ( B ). Let Q be a primary decomposition. Let P = { P ′ ∈ Ass( I ) | P P ′ or P ′ = P } and S = K [ X ] \ S P ′ ∈P P ′ . Then S is a multiplicatively closed set and ( I : P ∞ ) ∩ Q ⊂ IK [ X ] S ∩ K [ X ] since( I : P ∞ ) ∩ Q = T Q ′ ∈Q ,P Q ′ Q ′ ∩ Q . For each Q ′ ∈ Q with Q ′ ∩ S = ∅ , there is P ′ ∈ P such that √ Q ′ ⊂ P ′ , i.e. √ Q ′ ∈ P . Thus, ( I : P ∞ ) ∩ Q ⊃ IK [ X ] S ∩ K [ X ]and ( I : P ∞ ) ∩ Q = IK [ X ] S ∩ K [ X ] . By Lemma 25, IK [ X ] S ∩ K [ X ] is saturatedquotient invariant of I .Show ( B ) implies ( A ). By Lemma 25, there is a multiplicatively closed set S such that ( I : P ∞ ) ∩ Q = IK [ X ] S ∩ K [ X ]. Let Q be a primary decompositionof I . We know IK [ X ] S ∩ K [ X ] = T Q ′ ∈Q ,Q ′ ∩ S = ∅ Q ′ . By the assumption, Q ( I : P ∞ ) and thus ( I : P ∞ ) ∩ Q has a P -primary component. Then neither T Q ′ ∈Q ,Q ′ ∩ S = ∅ Q ′ nor ( I : P ∞ ) has a P -primary component. Hence,8 = ( I : P ∞ ) ∩ Q ∩ T Q ′ ∈Q ,Q ′ ∩ S = ∅ Q ′ = T Q ′ ∈Q ,P Q ′ Q ′ ∩ Q ∩ T Q ′ ∈Q ,Q ′ ∩ S = ∅ Q ′ is a primary decomposition and Q is its P -primary component. Next, we propose criteria for primary components having special propertieswhich can be applied for particular prime divisors. These criteria may be com-puted more easily than the general one. If Q is a primary ideal whose radical is an isolated divisor P of an ideal I , thenwe don’t need to compute ( I : P ∞ ) since the P -primary component of I is thelocalization of I by P . Theorem 27 (Criterion 2) . Let I be an ideal and P an isolated prime divisorof I . For a P -primary ideal Q , the following conditions are equivalent. ( A ) Q is the isolated P -primary component of I . ( B ) ( I : ( I : Q ) ∞ ) = Q .Proof. Show ( A ) implies ( B ). Let S = K [ X ] \ P . By Lemma 25, Q = IK [ X ] S ∩ K [ X ] is saturated quotient invariant of I and thus ( I : ( I : Q ) ∞ ) = Q . Next,we show ( B ) implies ( A ). By Lemma 25, there is a multiplicatively closed set S s.t. IK [ X ] S ∩ K [ X ] = Q . Since Q is primary, IK [ X ] S ∩ K [ X ] is the isolated P -primary component. Each isolated prime divisor is minimal in Ass( I ). On the contrary, we consider”maximal prime divisor” and propose the following criterion for it. Definition 28.
Let P be a prime divisor of I . We say P is maximal if thereis no prime divisor P ′ of I containing P properly. Theorem 29 (Criterion 3) . Let I be an ideal and P a maximal prime divisorof I . For P -primary ideal Q , the following conditions are equivalent. ( A ) Q is a P -primary component of I . ( B ) ( I : P ∞ ) ∩ Q = I .Proof. Show ( A ) implies ( B ). Let Q be a primary decomposition of I with Q ∈ Q . Since P is maximal in Ass( I ), ( I : P ∞ ) = T Q ′ ∈Q , √ Q ′ P Q ′ = T Q ′ ∈Q ,Q ′ = Q Q ′ . Thus, ( I : P ∞ ) ∩ Q = T Q ′ ∈Q ,Q ′ = Q Q ′ ∩ Q = I . Next, weshow ( B ) implies ( A ). Let Q ′ be a primary decomposition of ( I : P ∞ ). Since Q ′ does not have P -primary component, Q ′ ∪ { Q } is a primary decompositionof I . The general case can be reduced to maximal case via localization by maximalindependent set (See [4] the definition of maximal independent and its compu-tation). Letting S = K [ U ] × = K [ U ] \ { } , we obtain the following as a specialcase of Lemma 4. 9 heorem 30 (Criterion 4) . Let I be an ideal and P a prime divisor of I . If U is a maximal independent set of P in X and Q is a P -primary ideal , then thefollowing conditions are equivalent. ( A ) Q is a primary component of I . ( B ) Q is a primary component of IK [ X ] K [ U ] × ∩ K [ X ] . Here, we add a criterion for prime divisor based on the third saturated quotient.
Theorem 31 (Criterion 5) . Let I be an ideal and P a prime ideal. Then, thefollowing conditions are equivalent. ( A ) P ∈ Ass( I ) . ( B ) P ⊃ ( I : ( I : P )) . ( C ) P ⊃ ( I : ( I : P ∞ )) .Proof. By Corollary 21, ( A ) is equivalent to ( B ). By Proposition 23, p ( I : ( I : P )) = p ( I : ( I : P ∞ )) = T P ′ ∈ Ass( I ) ,P ⊂ P ′ P ′ . Thus, equivalence be-tween ( A ) and ( C ) is proved by the similar way of Corollary 21.Next, we devise criteria for isolated prime divisor based on the second satu-rated quotient. Lemma 32.
Let I be an ideal and P an isolated prime divisor of I . If Q is the P -pseudo-primary component of I , then ( I : ( I : P ∞ ) ∞ ) = Q .Proof. Let Q be a primary decomposition of I . By Proposition 22 (2),( I : ( I : P ∞ ) ∞ ) = T Q ∈Q ,P ⊂ √ IK [ X ] √ Q ∩ K [ X ] Q. Thus it is enough to show that the following statements are equivalent for each Q ∈ Q .(1-a) P ⊂ q IK [ X ] √ Q ∩ K [ X ].(1-b) P is the unique isolated prime divisor which is contained in √ Q .Show (1-a) implies (1-b). As q IK [ X ] √ Q ∩ K [ X ] ⊂ √ Q , we know P ⊂ √ Q .Then, suppose there is another isolated prime divisor P ′ contained in √ Q . Weobtain q IK [ X ] √ Q ∩ K [ X ] = \ Q ′ ∈Q ,Q ′ ⊂√ Q p Q ′ ⊂ P ′ . However, this implies P ⊂ P ′ and contradicts that P ′ is isolated. It is easy toprove that (1-b) implies (1-a). Theorem 33 (Criterion 6) . Let I be an ideal and P a prime ideal containing I . Then, the following conditions are equivalent. ( A ) P is an isolated prime divisor of I . ( B ) ( I : ( I : P ∞ ) ∞ ) = K [ X ] .Proof. Show ( A ) implies ( B ). By Lemma 32, ( I : ( I : P ∞ ) ∞ ) = Q = K [ X ].Show ( B ) implies ( A ). By Proposition 22 (2),( I : ( I : P ∞ ) ∞ ) = T Q ∈Q ,P ⊂ √ IK [ X ] √ Q ∩ K [ X ] Q = K [ X ]10or a primary decomposition Q of I . Then, there is an isolated prime divisor P ′ containing P . Since √ I ⊂ P ⊂ P ′ and P ′ is isolated, this implies P = P ′ isisolated.Since each prime divisor of I contains I , Theorem 33 directly induces thefollowing. Corollary 34 (Criterion 7) . Let I be an ideal and P a prime divisor of I .Then, (i) P is isolated if ( I : ( I : P ∞ ) ∞ ) = K [ X ] , (ii) P is embedded if ( I : ( I : P ∞ ) ∞ ) = K [ X ] . In this section, we devise Local Primary Algorithm (LPA) which computes P -primary component of I . Our method applies different procedures for two cases;isolated and embedded. Algorithm 1 shows the outline of LPA. Its terminationcomes from Proposition 35. We remark that, for given prime divisors disjointfrom a multiplicatively closed set S , we can compute all primary componentsdisjoint from S by LPA. Then their intersection gives the localization by S . First, we introduce several ways to generate primary component through equidi-mensional hull computation.
Proposition 35 ([2], Section 4. [6], Remark 10) . Let I be an ideal and P a prime divisor of I . For any positive integer m, I + P m is P-hull-primary,and for a sufficiently large integer m , hull( I + P m ) is a P -primary componentappearing in a primary decomposition of I . We can use Criteria for Primary Component to check m is large enough ornot. If P is an isolated prime divisor, then the component is computed directlyby using the second saturated quotient. By Lemma 15 and Lemma 32, we obtainthe following theorem. Theorem 36.
Let I be an ideal and P an isolated prime divisor of I . Then hull(( I : ( I : P ∞ ) ∞ )) is the isolated P -primary component of I . lgorithm 1 General Frame of Local Primary Algorithm
Input: I : an ideal, P : a prime ideal Output: ( a P -primary component of I if P is a prime divisor of I ” P is not a prime divisor” otherwise if P is a prime divisor of I ( Criterion 5 ) then if P is isolated ( Criteria 6,7 ) then Q ← the P -pseudo-primary component of I ( Lemma 32 ) Q ← hull( Q ) ( Theorem 36 ) return Q is the isolated P primary component else m ← while Q is not primary component of I ( Criteria 1,3,4 ) do Q ← a P -hull-primary ideal related to m ( Proposition 35, Lemma38 ) Q ← hull( Q ) m ← m + 1 end while return Q is an embedded P -primary component end if else return ” P is not a prime divisor” end if We introduce practical technique for implement LPA.
Let G = { f , . . . , f r } be a generator of P . Usually we take { f e f e · · · f e r r | e + · · · + e r = m } as a generator of P m for a positive integer m . However, thisgenerator has ( r + m − r − m ! elements and it becomes difficult to compute hull( I + P m )when m becomes large. To avoid the explosion of the number of the generator,we can use P [ m ] G = ( f m , . . . , f mr ) instead. Lemma 37.
Let Q be a primary decomposition of I and Q ∈ Q . If √ Q -hull-primary ideal Q ′ satisfies I ⊂ Q ′ ⊂ Q , then ( Q \ { Q } ) ∪ { hull( Q ′ ) } is anotherprimary decomposition of I .Proof. By Lemma 17, we obtain I ⊂ Q ′ ⊂ hull( Q ′ ) ⊂ Q . Since I ∩ hull( Q ′ ) = I and Q ∩ hull( Q ′ ) = hull( Q ′ ), we obtain I = I ∩ hull( Q ′ ) = \ Q ′′ ∈Q ,Q ′′ = Q Q ′′ ∩ Q ∩ hull( Q ′ ) = \ Q ′′ ∈Q ,Q ′′ = Q Q ′′ ∩ hull( Q ′ ) . Thus, (
Q \ { Q } ) ∪ { hull( Q ′ ) } is an irredundant primary decomposition of I . Lemma 38.
For any positive integer m , I + P [ m ] G is P -hull-primary, and fora sufficiently large m , hull( I + P [ m ] G ) is a P -primary component appearing in aprimary decomposition of I . roof. As √ I + P = q I + P [ m ] G = P , I + P [ m ] G is P -hull-primary. By Theorem35, hull( I + P m ) is a P -primary component of I for a sufficiently large m . Since I ⊂ I + P [ m ] G ⊂ I + P m ⊂ hull( I + P m ), hull( I + P [ m ] G ) is a P -primary componentby Lemma 37. Next, we devise another computation of hull( I + P m ) based on maximal indepen-dent set (MIS) which is much efficient than computations based on Proposition12. Similarly, by this technique we can replace I with IK [ X ] K [ U ] × ∩ K [ X ] atthe first step of LPA. Lemma 39.
Let I be a P -hull-primary ideal. For a maximal independent set U of P , hull( I ) = IK [ X ] K [ U ] × ∩ K [ X ] .Proof. Let Q be a primary decomposition of I . Then, hull( I ) is the uniqueprimary component disjoint from K [ U ] × . Thus, IK [ X ] K [ U ] × ∩ K [ X ] = T Q ∈Q ,Q ∩ K [ U ] × = ∅ Q = hull( I ) . We made a preliminary implementation on a computer algebra system Risa/Asir[7] and apply it to several examples as naive experiments. Here we show sometypical examples. Timings are measured on a PC with Xeon E5-2650 CPU.First, we see an ideal whose embedded primary components are hard tocompute. Let I ( n ) = ( x ) ∩ ( x , y ) ∩ ( x , y , ( z + 1) n + 1). If n is considerablelarge, it is difficult to compute a full primary decomposition of I ( n ) though theisolated devisor ( x ) can be detected pretty easily. We apply Local Primary Al-gorithm (LPA) for this example to compute the isolated primary component for P = ( x ). We also see another example which is more valuable for mathematics.An ideal A k,m,n is defined in [9] and its primary decomposition has importantmeanings in Computer Algebra for Statistics. We consider an isolated primedivisor P = ( x , x , x , x ) of A , , in Q [ x ij | ≤ i ≤ , ≤ j ≤ I (100) I (200) I (300) I (400) I (500) A , , / P noro pd.syci dec 0.36 15.6 88.3 289 96.0 > P = ( x x − x x , x x − x x , x x − x x , x x − x x , x x − x x , x , x , x , x , x , x , x ) A , , in Q [ x ij | ≤ i ≤ , ≤ j ≤
4] and P = ( x x − x x , x x − x x , x x − x x , x x − x x , x , x , x , x , x , x , x , x , x ) of A , , in Q [ x ij | ≤ i ≤ , ≤ j ≤ I + P m ) is much high. On the other hand,we can see LPA-(Pm+MIS) has good effectiveness by its speciality comparinga full primary decomposition function noro pd.syci dec.Algorithm A , , / P A , , / P noro pd.syci dec 3.11 34.8LPA > In commutative algebra, the operation of ”localization by a prime ideal” iswell-known as a basic tool. However, its computation through primary decom-position is much difficult. Thus, we devise a new effective localization
LocalPrimary Algorithm (LPA) using Double Ideal Quotient(DIQ) and its variantswithout computing unnecessary primary components for localization. For ourconstruction of LPA, we devise several criteria for primary component based onDIQ and its variants. We take preliminary benchmarks for some examples toexamine certain effectiveness of LPA coming from its speciality. To make ourLPA very practical we shall continue to improve it through obtaining timingdata for a lot of larger examples.In future work, we are finding a way to compute ”sample points” of primedivisors. For localization it does not need all divisors; it is enough to find f P ∈ P ∩ S for each prime divisor P with P ∩ S = ∅ and we obtain IK [ X ] S ∩ K [ X ] = ( I : ( Q P ∩ S = ∅ f P ) ∞ ). Another work is to apply our primary componentcriteria to probabilistic or inexact methods for primary decomposition, such asnumerical ones. Probabilistic or inexact ways have low computational costs,however, they have low accuracy for outputs. Hence, our criterion using doubleideal quotient may help to guarantee their outputs. Finally, localization ingeneral setting, that is localization by a prime ideal not necessary associated isinteresting work. Acknowledgement:
The authors would like to thank the referees for theirhelpful comments to improve the presentation of this paper. The authors arealso grateful to Masayuki Noro for technical assistance with the computer ex-periments and coding on Risa/Asir. 14 eferenceseferences