Binomial ideals and congruences on N n
BBINOMIAL IDEALS AND CONGRUENCES ON N n DEDICATED TO PROFESSOR ANTONIO CAMPILLOON THE OCCASION OF HIS 65TH BIRTHDAY.
LAURA FELICIA MATUSEVICH AND IGNACIO OJEDA
Abstract. A congruence on N n is an equivalence relation on N n that is compatible with theadditive structure. If k is a field, and I is a binomial ideal in k [ X , . . . , X n ] (that is, an idealgenerated by polynomials with at most two terms), then I induces a congruence on N n bydeclaring u and v to be equivalent if there is a linear combination with nonzero coefficients of X u and X v that belongs to I . While every congruence on N n arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless,the link between a binomial ideal and its corresponding congruence is strong, and one may thinkof congruences as the underlying combinatorial structures of binomial ideals. In the currentliterature, the theories of binomial ideals and congruences on N n are developed separately. Theaim of this survey paper is to provide a detailed parallel exposition, that provides algebraicintuition for the combinatorial analysis of congruences. For the elaboration of this survey paper,we followed mainly [10] with an eye on [5] and [13]. Preliminaries
In this section we introduce our main objects of study: binomial ideals and monoid congru-ences, and recall some basic results.Throughout this article, k [ X ] := k [ X , . . . , X n ] is the commutative polynomial ring in n vari-ables over a field k . In what follows we write X u for X u X u · · · X u n n , where u = ( u , u , . . . , u n ) ∈ N n , where here and henceforth, N denotes the set of nonnegative integers.1.1. Binomial ideals.
In this section we begin our study of binomial ideals. First of all, we recall that a binomial in k [ X ] is a polynomial with at most two terms, say λ X u + µ X v , where λ, µ ∈ k and u , v ∈ N n .We emphasize that, according to this definition, monomials are binomials. Definition 1. A binomial ideal of k [ X ] is an ideal of k [ X ] generated by binomials. Throughout this article, we assume that the base field k is algebraically closed. The reasonfor this is that some desirable results are not valid over an arbitrary field. These include thecharacterization of binomial prime ideals (Theorem 34), and the fact that associated primesof binomial ideals are binomial (see, e.g. Proposition 50). This failure can be seen even inone variable: the ideal (cid:104) X + 1 (cid:105) ⊂ R [ X ] is prime, but does not conform to the description inTheorem 34; the ideal (cid:104) X − (cid:105) ⊂ R [ X ] has the associated prime (cid:104) X + X + 1 (cid:105) , which is notbinomial. It is also worth noting that the characteristic of k plays a role when studying binomialideals, as can be seen by the different behaviors presented by (cid:104) X p − (cid:105) ⊂ k [ X ] depending onwhether the characteristic of k is p . Key words and phrases.
Binomial ideals. Graded Algebras. Congruences. Semigroup ideals. Toric ideals.Primary decomposition.The first author was partially supported by NSF grant DMS-1500832.The second author was partially supported by the project MTM2015-65764-C3-1, National Plan I+D+I, andby Junta de Extremadura (FEDER funds) - FQM-024. a r X i v : . [ m a t h . A C ] M a r LAURA FELICIA MATUSEVICH AND IGNACIO OJEDA
The following result is an invaluable tool when studying binomial ideals.
Proposition 2.
Let I ⊂ k [ X ] be an ideal. The following are equivalent:(1) I is a binomial ideal.(2) The reduced Gr¨obner basis of I with respect to any monomial order on k [ X ] consists ofbinomials.(3) A universal Gr¨obner basis of I consists of binomials.Proof. If I has a binomial generating set, the S -polynomials produced by a step in the Buch-berger algorithm are necessarily binomials. (cid:3) Since the Buchberger algorithm for computing Gr¨obner bases respects the binomial condition,Gr¨obner techniques are particularly effective when working with these objects. In particular, itcan be shown that some important ideal theoretic operations preserve binomiality. For instance,it is easy to show that eliminating variables from binomial ideals results in binomial ideals.
Corollary 3.
Let I be a binomial ideal of k [ X ] . The elimination ideal I ∩ k [ X i | i ∈ σ ] is abinomial ideal for every nonempty subset σ ⊂ { , . . . , n } .Proof. The intersection is generated by a subset of the reduced Gr¨obner basis of I with respectto a suitable lexicographic order. (cid:3) Example . Let ϕ : k [ X, Y, Z ] → k [ T ] be the k − algebra morphism such that X → T , Y → T and Z → T . It is known that ker( ϕ ) = (cid:104) X − T , Y − T , Z − T (cid:105)∩ k [ X, Y, Z ]. As a consequence of Corollary 3,ker( ϕ ) is a binomial ideal. In fact, ker( ϕ ) is the ideal generated by { Y − XZ, X Y − Z , X − Y Z } , as can be checked by executing the following code in Macaulay2 ([8]): R = QQ[X,Y,Z,T]I = ideal(X-T^3,Y-T^4,Z-T^5)eliminate(T,I)
Taking ideal quotients is a fundamental operation in commutative algebra. We can now showthat some ideal quotients of binomial ideals are binomial.
Corollary 5. If I is a binomial ideal of k [ X ] , and X u is a monomial, then ( I : X u ) is abinomial ideal.Proof. Recall that if { f , . . . , f (cid:96) } is a system of generators for I ∩ (cid:104) X u (cid:105) , then { f / X u , . . . ,f (cid:96) / X u } is a system of generators for ( I : X u ). Thus, the binomiality of ( I : X u ) follows if weshow that I ∩ (cid:104) X u (cid:105) is binomial.Introducing an auxiliary variable T , we have that I ∩ (cid:104) X u (cid:105) = ( T I + (1 − T ) (cid:104) X u (cid:105) ) ∩ k [ X ] . Since
T I + (1 − T ) (cid:104) X u (cid:105) is a binomial ideal, Corollary 3 implies that I ∩ (cid:104) X u (cid:105) is also binomial,as we wanted. (cid:3) We remark that the ideal quotient of a binomial ideal by a binomial is not necessarily bino-mial, and neither is the ideal quotient of a binomial ideal by a monomial ideal. When takingcolon with a single binomial, the above proof breaks because the product of two binomials isnot a binomial in general; indeed, (cid:0) (cid:104) X − (cid:105) : (cid:104) X − (cid:105) (cid:1) = (cid:104) X + X + 1 (cid:105) ⊂ k [ X ] . In the case of taking ideal quotient by a monomial ideal, say J = (cid:104) X u , . . . X u r (cid:105) , insteadof a single monomial, what makes the argument invalid is that the ideal ( I : J ) is equal to INOMIAL IDEALS AND CONGRUENCES ON N n ∩ ri =1 ( I : (cid:104) X u i (cid:105) ), and the intersection of binomial ideals is not necessarily binomial, as thefollowing shows: (cid:104) X − (cid:105) ∩ (cid:104) X − (cid:105) = (cid:104) X − X + 2 (cid:105) ⊂ k [ X ].1.2. Graded algebras.
Gradings play a big role when studying binomial ideals. The main result of this section isthat a ring is a quotient of a polynomial ring by a binomial ideal if and only if it has a specialkind of grading (Theorem 8).Recall that a k − algebra of finite type R is graded by a finitely generated commutative monoid S if R is a direct sum R = (cid:77) a ∈ S R a of k − vector spaces and the multiplication of R satisfies the rule R a R a (cid:48) = R a + a (cid:48) . Example . Observe that k [ X ] = (cid:76) u ∈ N n Span k { X u } . Remark . Let I be any ideal of k [ X ] and let π be the canonical projection of k [ X ] onto R := k [ X ] /I . Let S be the set of all one-dimensional subspaces Span k { π ( X u ) } of R ; if thekernel of π contains monomials, we adjoin to S the symbol ∞ associated to the monomials inker( π ). The set S is a commutative monoid with the operationSpan k { π ( X u ) } + Span k { π ( X v ) } = Span k { π ( X u X v ) } = Span k { π ( X u + v ) } and identity element Span k { } = Span k { π ( X ) } . Note that if X v ∈ ker( π ), then for any othermonomial X u , X u X v ∈ ker( π ). In other words,Span k { π ( X u ) } + ∞ = ∞ . We point out that the set { Span k { π ( X ) } , . . . , Span k { π ( X n ) }} generates S as a monoid. Thereis a natural k − vector space surjection (cid:77) Span k { π ( X u ) }∈ S X u (cid:54)∈ ker π Span k { π ( X u ) } → R. (1)We observe that if (1) is an isomorphism of k − vector spaces, then R is finely graded by S ,meaning that R is S − graded and every graded piece has dimension at most 1.The following result provides the first link between binomial ideals and monoids. Theorem 8. A k − algebra R of finite type admits a presentation of the form k [ X ] /I , where I is a binomial ideal, if and only if R can be finely graded by a finitely generated commutativemonoid.Proof. First assume that R admits a grading of the given type by a finitely generated com-mutative monoid S . Let f , . . . , f n be k − algebra generators of R . Without loss of generality,we may assume that f , . . . , f n are homogeneous. Denote a i the degree of f i , for i = 1 , . . . , n .Since R is (finely) graded by the monoid generated by { a , . . . , a n } , we may assume that S isgenerated by { a , . . . , a n } .Give k [ X ] an S − grading by setting the degree of X i to be a i , and consider the surjection k [ X ] → R given by X i (cid:55)→ f i , which is a graded ring homomorphism. The kernel of this mapis a homogeneous ideal of k [ X ], and is therefore generated by homogeneous elements. On theother hand, by the fine grading condition, for any two monomials X u , X v ∈ k [ X ] with the same S − degree, neither of which maps to zero in R , there is a scalar λ ∈ k ∗ such that the binomial X u − λ X v ∈ k [ X ] maps to zero in R . Thus, the kernel of the above surjection is generated bybinomials.Conversely, by Remark 7, it suffices to show that the map (1) is injective. We need to provethat if Σ is a nonempty subset of { Span k { π ( X u ) } ∈ S | X u (cid:54)∈ ker π } , then the image of Σ in R LAURA FELICIA MATUSEVICH AND IGNACIO OJEDA is linearly independent. This follows if we show that if f = (cid:80) ri =1 λ i X u i ∈ I with λ , . . . , λ r ∈ k ∗ and X u i / ∈ I for all 1 ≤ i ≤ r , then there exist 1 ≤ j ≤ r and λ ∈ k ∗ such that X u − λ X u j ∈ I (in other words, π ( X u ) = π ( X u j )). To see this, note that since I is a binomial ideal, it hasa k − vector space basis consisting of binomials, and therefore we can write f = (cid:80) (cid:96)i =1 µ i B i ,where µ , . . . , µ (cid:96) ∈ k ∗ and each B i is a binomial in I with two terms, neither of which is in I (the latter by the assumption on f ). The monomial X u must appear in at least one of thebinomials B , . . . , B (cid:96) , say B i . Of course, the second monomial appearing in B i has the sameimage under π as X u . If this second monomial in B i is one of the X u , . . . , X u r , we are done.Otherwise, the second term of B i must appear in another of the binomials B i , say B i . Notethat both monomials in B i have the same image under π as X u . If the second monomial of B i is one of the X u , . . . , X u r , again, we are done. Otherwise, continue in the same manner.Since we only have finitely many binomials to consider, this process must stop, and produce amonomial X u j such that π ( X u ) = π ( X u j ). (cid:3) Congruences on monoids and binomial ideals
We now start our study of monoid congruences, and their relationship to binomial ideals. Weshow how binomial ideals induce congruences, and how any congruence can arise this way. Wealso address the question of when two different binomial ideals give rise to the same congruence.
Definition 9.
Let S be a commutative monoid. A congruence ∼ on S is an equivalencerelation on S which is additively closed: a ∼ b ⇒ a + c ∼ b + c for a , b and c ∈ S . The following result, which follows directly from the definition, gives a first indication thatcongruences on commutative monoids are analogous to ideals in commutative rings.
Proposition 10. If ∼ is a congruence on a commutative monoid S , then S/ ∼ is a commutativemonoid. (cid:3) Let φ : S → S (cid:48) be a monoid morphism.The kernel of φ is defined asker φ := (cid:8) ( a , b ) ∈ S × S | φ ( a ) = φ ( b ) (cid:9) . Note that if φ is a monoid morphism, the relation on S determined by ker φ ⊂ S × S is actuallya congruence. Moreover, every congruence on S arises in this way: if ∼ is a congruence on S , then ∼ can be recovered as the congruence induced by the kernel of the natural surjection S → S/ ∼ .We write cong( S ) ⊂ P ( S × S ) for the set of congruences on S ordered by inclusion. (Here P indicates the power set.) We say that S is Noetherian if every nonempty subset of cong( S )has a maximal element (equivalently, cong( S ) satisfies the ascending chain condition). Thefollowing is an important result in monoid theory. Theorem 11.
A commutative monoid S is Noetherian if and only if S is finitely generated. The fact that a Noetherian monoid is finitely generated is the hard part of the proof. It isdue to Budach [4], and is the main result in Chapter 5 in Gilmer’s book [7], where it appearsas Theorem 5.10. Brookfield has given a short and self contained proof in [2]. We will justprovide a proof of the converse, namely, that finitely generated monoids are Noetherian (see [7,Theorem 7.4]), after Theorem 13.Set S be a commutative monoid finitely generated by A = { a , . . . , a n } . The monoid mor-phism π : N n −→ S ; e i (cid:55)−→ a i , i = 1 , . . . , n, (2)where e i denotes the element in N n whose i − th coordinate is 1 with all other coordinates 0, issurjective and gives a presentation S = N n / ∼ INOMIAL IDEALS AND CONGRUENCES ON N n by simply taking ∼ = ker π . Unless stated otherwise, we write [ u ] for the class of u ∈ N n modulo ∼ . Remark . In what follows, all monoids considered are commutative and finitely generated.Given a monoid S , the semigroup algebra k [ S ] := (cid:76) a ∈ S Span k { χ a } is the direct sum withmultiplication χ a χ b = χ a + b . (This terminology is in wide use, even though the algebra k [ S ]would be more precisely named a “monoid algebra”.) Theorem 13.
Let A = { a , . . . , a n } be a generating set of a monoid S , and consider thepresentation map π : N n → S induced by A . We define a map of semigroup algebras ˆ π : k [ N n ] = k [ X ] → k [ S ] ; X u (cid:55)→ χ π ( u ) . (3) Let I A := (cid:104) X u − X v | π ( u ) = π ( v ) (cid:105) ⊆ k [ X ] . (4) Then ker ˆ π = I A , so that k [ S ] ∼ = k [ X ] /I A . Moreover, I A is spanned as a k − vector space by { X u − X v | π ( u ) = π ( v ) } .Proof. By construction, I A ⊆ ker ˆ π . To prove the other inclusion, give k [ X ] an S − grading bysetting deg( X i ) = π ( e i ) = a i . Then the map ˆ π is graded (considering k [ S ] with its natural S − grading), and therefore its kernel is a homogeneous ideal of k [ X ]. Note that X u and X v have the same S − degree if and only if π ( u ) = π ( v ).We observe that ker ˆ π contains no monomials, so any polynomial in ker ˆ π has at least twoterms. Let f be a homogeneous element of ker ˆ π . Then there are λ, µ ∈ k ∗ and u , v ∈ N n suchthat f = λ X u + µ X v + g , with g a homogeneous polynomial with two fewer terms than f .Since f is homogeneous, we have that π ( u ) = π ( v ), and therefore X u − X v ∈ I A ⊂ ker ˆ π . Then f − λ ( X u − X v ) is a homogeneous element of ker ˆ π , and has fewer terms than f . Continuingin this manner, we conclude that f ∈ I A . Since ker ˆ π is a homogeneous ideal, we see that I A ⊇ ker ˆ π , and therefore I A = ker ˆ π .For the final statement, we note that any binomial ideal in k [ X ] is spanned as a k − vectorspace by the set of all of its binomials. Since I A contains no monomials and is S − graded,any binomial in I A is of the form X u − λ X v , where λ ∈ k ∗ and π ( u ) = π ( v ). But then X u − X v ∈ I A , and again using that I A contains no monomials, we see that λ = 1. Thisimplies that { X u − X v | π ( u ) = π ( v ) } is the set of all binomials of I A , which implies that it isa k − spanning set for this ideal. (cid:3) We are now ready to prove that finitely generated monoids are Noetherian.
Proof of Theorem 11, reverse implication.
Let S be a finitely generated monoid, and considera presentation S = N n / ∼ , where ∼ is a congruence on N n . In this proof, for u ∈ N n , we denoteby [ u ] the equivalence class of u with respect to ∼ .Let ≈ be a congruence on S , and let (cid:39) be the congruence on N n given by setting theequivalence class of u ∈ N n with respect to (cid:39) to be the set (cid:83) { v ∈ N n | [ u ] ≈ [ v ] } [ v ]. Then thecongruence (cid:39) is such that S/ ≈ = N n / (cid:39) .Now let ≈ and ≈ be two congruences on S and consider the natural surjections π i : N n → N n / (cid:39) i for i = 1 ,
2. Then if ≈ ⊆ ≈ (as subsets of S × S ), we have that I A ⊆ I A , where theseideals are defined as in (4) by considering the generating sets A i = { π i ( e j ) | j = 1 , . . . , n } , i =1 , , respectively. We conclude that Noetherianity of the monoid S follows from the fact that k [ X ] is a Noetherian ring. (cid:3) In order to continue to explore the correspondence between congruences and binomial ideals,we introduce some terminology.
LAURA FELICIA MATUSEVICH AND IGNACIO OJEDA
Definition 14.
A binomial ideal is said to be unital if it is generated by binomials of the form X u − λ X v with λ equal to either or . A binomial ideal is said to be pure if does not containany monomial. Corollary 15.
A relation ∼ on N n is a congruence if and only if there exists a pure unitalideal I ⊂ k [ X ] such that u ∼ v ⇐⇒ X u − X v ∈ I .Proof. If ∼ is a congruence on N n , then N n / ∼ is a (finitely generated) monoid. Consider thenatural surjection π : N n → N n / ∼ , and let A = { π ( e ) , . . . , π ( e n ) } . Use this informationto construct I A as in (4). By Theorem 13 and its proof, the ideal I A satisfies the requiredconditions.For the converse, let I a pure unital ideal of k [ X ] such that u ∼ v ⇐⇒ X u − X v ∈ I .Clearly, ∼ is reflexive and symmetric. For transitivity, it suffices to observe that X u − X w =( X u − X v ) + ( X v − X w ) ∈ I , for every u , v and w such that u ∼ v and v ∼ w . Finally, as I is an ideal, it follows that X w ( X u − X v ) = X u + w − X v + w ∈ I , for every X u − X v ∈ I and X w ∈ k [ X ]. We conclude that ∼ is a congruence. (cid:3) We review some examples of pure unital binomial ideals and their associated congruences.We remark in particular that different binomial ideals may give rise to the same congruence.
Example . (i) The ideal I = (cid:104) X − Y (cid:105) ⊂ k [ X, Y ] defines a congruence ∼ on N with N / ∼ = N .(1) The ideal I = (cid:104) X − Y, Y − (cid:105) ⊂ k [ X, Y ] defines a congruence ∼ on N such that N / ∼ = Z / Z .(2) The ideal I = (cid:104) X − Y (cid:105) ⊂ k [ X, Y ] defines a congruence ∼ on N such that N / ∼ isisomorphic to the submonoid S of Z ⊕ Z / Z generated by (1 ,
0) and (1 , S = { , a, b } where the sum is defined as follows:+ 0 a b a ba a b bb b b b The ideal I = (cid:104) X − Y, Y − Y (cid:105) ⊂ k [ X, Y ] determines a congruence ∼ on N such that S ∼ = N / ∼ .An arbitrary binomial ideal J of k [ X ] induces a congruence ∼ J on N n defined as u ∼ J v ⇐⇒ there exists λ ∈ k ∗ such that X u − λ X v ∈ J. (5)Note that this ideal defines the same congruence as the pure unital binomial ideal I = (cid:104) X u − X v | there exists λ ∈ k ∗ such that X u − λ X v ∈ J (cid:105) . Example . (1) Let J = (cid:104) X − Y, Y (cid:105) ⊂ k [ X, Y ]. The congruence ∼ J induced by J on N is exactly thesame that one in Example 16 3.(2) The congruence ∼ (cid:104) X,Y (cid:105) on N is the same as the induced by I = (cid:104) X − Y, X − X (cid:105) on N . Note that (cid:104) X, Y (cid:105) is a monomial ideal, while I contains no monomials.If a binomial ideal I contains monomials, then the exponents of all monomials in I form asingle equivalence class in the congruence ∼ I . This equivalence class satisfies an absorptionproperty, as in the definition below. Definition 18.
A non-identity element ∞ in a monoid S is nil if a + ∞ = ∞ , for all a ∈ S . INOMIAL IDEALS AND CONGRUENCES ON N n For example, the “formal” element ∞ introduced in Remark 7 is nil, since it corresponds tothe monomial class. Note that a monoid S can have at most one nil element: if ∞ , ∞ (cid:48) ∈ S areboth nil, then ∞ + ∞ (cid:48) = ∞ (cid:48) because ∞ (cid:48) is nil, and ∞ (cid:48) + ∞ = ∞ because ∞ is nil. Since S iscommutative, ∞ = ∞ (cid:48) .As we have noted above, if I is a binomial ideal that contains monomials, then the class ofmonomial exponents is a nil element for the congruence ∼ I . The converse of this assertion isfalse: if J is a binomial ideal containing monomials, then the ideal I produced by Corollary 15for the congruence ∼ J has no monomials and has a nil element (since J contains monomials,and therefore ∼ J does). On the other hand, if ∼ is a congruence on N n with a nil element ∞ ,then there exists a binomial ideal J in k [ X ] that contains monomials, and such that ∼ = ∼ J . Tosee this, let I be the ideal produced by Corollary 15 for ∼ , and consider J = I + (cid:104) X e | [ e ] = ∞(cid:105) ,noting that adding this particular monomial ideal does not change the underlying congruence.We make this more precise in Proposition 19. Proposition 19.
Let I ⊂ k [ X ] be a binomial ideal. If J is a binomial ideal of k [ X ] such that I ⊂ J and ∼ J = ∼ I , then N n / ∼ I has a nil ∞ and J = I + (cid:104) X e | [ e ] = ∞(cid:105) .Proof. As I ⊂ J , there is a binomial X u − λ X v ∈ J \ I . Since ∼ I = ∼ J , necessarily X u , X v ∈ J ; inparticular N n / ∼ J = N n / ∼ I has a nil ∞ . We claim that the ideal J is equal to I + (cid:104) X e | [ e ] = ∞(cid:105) .To see that J contains I + (cid:104) X e | [ e ] = ∞(cid:105) , we note that I ⊂ J . Also, we know that J containsa monomial X u , and so [ u ] = ∞ . If e ∈ N n is such that [ e ] = ∞ = [ u ], then X u − µ X e ∈ J forsome µ ∈ k ∗ , and since X u ∈ J , we see that X e ∈ J . For the reverse inclusion, it is enough tosee that any binomial in J belongs to I + (cid:104) X e | [ e ] = ∞(cid:105) . But as before, if X u − λ X v ∈ J \ I ,then X u , X v ∈ J , and therefore [ u ] = [ v ] = ∞ , because a monoid can have at most one nilelement. (cid:3) A monoid ideal E of N n is a proper subset such that E + N n ⊆ E ; Figure 1 shows a typicalexample. (2,5) (3,4) (4,2)n m Figure 1.
The integer points in shaded area form a monoid ideal of N .Let E ⊆ N n be a monoid ideal of N n . The Rees congruence on N n modulo E is thecorrespondence ∼ on N n defined by u ∼ v ⇐⇒ u = v or both u and v ∈ E . Notice that theRees congruence on N n modulo E is the same as the induced by ∼ M E with M E = (cid:104) X e | e ∈ E (cid:105) .Monoid ideals and nil elements are related as follows. Lemma 20.
Let S be a monoid. Then S has a nil element if and only if for any presentation N n / ∼ of S there exists a monoid ideal E of N n such that ∼ contains the Rees congruence on N n modulo E . In this case, [ e ] = ∞ , for any e ∈ E . LAURA FELICIA MATUSEVICH AND IGNACIO OJEDA
Proof.
Let N n / ∼ be a presentation of S given by a monoid surjection π : N n → S .For the direct implication, assume that ∞ ∈ S is a nil. Then E := π − ( ∞ ) is a monoid idealof N n . Indeed, given u ∈ N n and e ∈ E we have that π ( e + u ) = π ( e ) + π ( u ) = ∞ + π ( u ) = ∞ , so that e + u ∈ E . Note that E (cid:54) = N n since nil elements are nonzero. Moreover, by construction,if e , e (cid:48) ∈ E, then e ∼ e (cid:48) , which means that ∼ contains the Rees congruence on N n modulo E .Conversely, let E be a monoid ideal of N n such that ∼ contains the Rees congruence on N n modulo E . We claim that the class [ e ] (for any e ∈ E ) is a nil in S = N n / ∼ . To see this,let e ∈ E , u ∈ N n . Then π ( e ) + π ( u ) = π ( e + u ). Since E is a monoid ideal, e + u ∈ E .This implies that e ∼ e + u (or equivalently, π ( e ) = π ( e + u )) because ∼ contains the Reescongruence modulo E . To complete the proof of our claim, we need to show that [ e ] ( e ∈ E )is not the zero class. This follows from the fact that E (cid:54) = N n . (cid:3) Our next goal is to prove Theorem 22, which is a more precise version of Proposition 19. Withthat result in hand, we will be able to introduce the binomial ideal associated to a congruencein Definition 25.
Definition 21. An augmentation ideal for a given binomial ideal I ⊂ k [ X ] is a maximalideal of the form I aug := (cid:104) X i − λ i | λ i ∈ k ∗ , i = 1 , . . . , n (cid:105) such that I ∩ I aug is a binomial ideal. We point out that, given a binomial ideal I , an augmentation ideal for I may or may notexist (see [10, Example 9.13] for a binomial ideal without an augmentation ideal). The followingresult is the N n -version of [10, Theorem 9.12]. Theorem 22. If I (cid:96) ⊃ . . . ⊃ I is a chain of distinct binomial ideals of k [ X ] inducing the samecongruence on N n , then (cid:96) ≤ . Moreover, if (cid:96) = 1 then I is pure and I is not: I = I ∩ I aug for an augmentation ideal for I .Proof. By Proposition 19, all we need to show is that if (cid:96) = 1, then I = I ∩ I aug , where I aug is an augmentation ideal for I . Denote by ∼ the congruence induced by I (and I ).Assume (cid:96) = 1, so that I does not have monomials, and I does. In particular, we may selecta monomial X e ∈ I , and its equivalence class [ e ] with respect to ∼ is a nil element, that wedenote ∞ . For each 1 ≤ i ≤ n , consider the monomial X i = X e i . Since [ e i ] + ∞ = ∞ , thereexists λ i ∈ k ∗ such that X i X e − λ i X e ∈ I (because I and I induce the same congruence).We now define I aug = (cid:104) X i − λ i | i = 1 , . . . , n (cid:105) , and claim that I ∩ I aug = I , which in particularshows that I aug is an augmentation ideal for I .By construction, ( I : X e ) ⊇ I aug . Note that ( I : X e ) (cid:54) = (cid:104) (cid:105) , as I contains no monomials.Thus, since I aug is maximal, ( I : X e ) = I aug , and we conclude that I aug contains I . This,and I ⊃ I , imply that I ∩ I aug ⊇ I . Moreover, I aug (cid:54)⊇ I because I has monomials, while I aug does not. Consequently I (cid:41) I ∩ I aug ⊇ I . Now the equality I ∩ I aug = I will followfrom Proposition 19 if we show that I ∩ I aug is binomial (since the fact that I and I inducethe same congruence ∼ implies that the congruence induced by I ∩ I aug is also ∼ ). To seethat I ∩ I aug is binomial, we use the argument from [5, Corollary 1.5]. Introduce an auxiliaryvariable t , and consider the binomial ideal J = I + tI aug + (1 − t ) (cid:104) X u | [ u ] = [ e ] (cid:105) ⊂ k [ X , t ].Since, by Proposition 19, I = I + (cid:104) X u | [ u ] = ∞ = [ e ] (cid:105) , we have that J ∩ k [ X ] = I ∩ I aug .Now apply Corollary 3. (cid:3) INOMIAL IDEALS AND CONGRUENCES ON N n Example . If I = (cid:104) X − Y, Y (cid:105) ⊂ k [ X, Y ], then I = I ∩ (cid:104) X − , Y − (cid:105) = (cid:104) X − Y, Y − Y (cid:105) .This can be verified as follows. R = QQ[X,Y];I1 = ideal(X-Y, Y^2);Iaug = ideal(X-1,Y-1);I0 = intersect(I1,Iaug);mingens I0;
Note that these ideals already appeared in Examples 17 (i) and 16 3.
Remark . The previous results highlight one way in which two different binomial ideals in k [ X ] induce the same congruence on N n , namely if one contains the other, the congruence hasa nil element, and the larger ideal contains monomials corresponding to the nil class, while thesmaller ideal has no monomials.There is another way to produce binomial ideals inducing the same congruence. Let I be abinomial ideal in k [ X ], and let µ , . . . , µ n ∈ k ∗ . Consider the ring isomorphism k [ X ] → k [ X ]given by X i (cid:55)→ µ i X i for i = 1 , . . . , n . (This kind of isomorphism is known as rescaling thevariables .) Then the image of I is a binomial ideal, which induces the same congruence as I .Indeed, the effect on I of rescaling the variables is to change the coefficients of the binomialsin I by a nonzero multiple, which does not alter the exponents of those monomials.In Theorem 22, the ideal I can be made unital by rescaling the variables, by using that k isalgebraically closed if necessary. The ideal obtained this way equals the ideal introduced in (4).We are now ready to introduce the binomial ideal associated to a congruence in N n . Definition 25.
Given a congruence ∼ on N n , denote by I ∼ the unital binomial ideal of k [ X ] which is maximal among all proper binomial ideals inducing ∼ . We say that I ∼ is the binomialideal associated to ∼ . To close this section, we introduce one final notion.
Definition 26.
Let ∼ and ∼ be congruences on N n . The intersection ∼ of ∼ and ∼ ,denoted ∼ = ∼ ∩ ∼ , is the congruence on N n defined by u ∼ v if and only if u ∼ v and u ∼ v . From the point of view of equivalence relations, the equivalence classes of ∼ ∩ ∼ form apartition of N n which is the common refinement of the partitions induced by ∼ and ∼ . Thefollowing result motivates the use of the intersection notation and terminology: the intersectionof congruences corresponds to the ideal generated by the binomials in the intersection of theirassociated binomial ideals. Proposition 27.
Let ∼ , ∼ and ∼ be congruences on N n whose associated ideals in k [ X ](Definition 25) are I ∼ , I ∼ and I ∼ , respectively. Then ∼ = ∼ ∩ ∼ if and only if I ∼ ⊆ I ∼ ∩ I ∼ ,and the equality holds if and only if I ∼ ∩ I ∼ is a binomial ideal.Proof. The statement u ∼ v if and only if u ∼ v and u ∼ v is exactly the same as X u − X v ∈ I ∼ if and only if X u − X v ∈ I ∼ and X u − X v ∈ I ∼ . The direct implication of the last statementfollows from Theorem 22 and its converse is trivially true because I ∼ is a binomial ideal. (cid:3) The following example illustrates the last statement above.
Example . Let ∼ and ∼ be the congruences on N such that u ∼ v if u − v ∈ Z (2 , − u ∼ v if u − v ∈ Z (3 , − Q [ X, Y ] associated to ∼ and ∼ are I ∼ = (cid:104) X − Y (cid:105) and I ∼ = (cid:104) X − Y (cid:105) , respectively. Clearly, the binomial idealassociated to ∼ = ∼ ∩ ∼ is I ∼ = (cid:104) X − Y (cid:105) . Whereas, I ∼ ∩ I ∼ = (cid:104) X + X Y − XY − Y (cid:105) : R = QQ[X,Y];I1 = ideal(X^2-Y^2);I2 = ideal(X^3-Y^3);intersect(I1,I2); Toric, lattice and mesoprime ideals
This section is devoted to the (finitely generated abelian) monoids contained in a group.Let ( G, +) be a finitely generated abelian group and let A = { a , . . . , a n } be a given subsetof G , we consider the subsemigroup S of G generated by A , that is to say, S = N a + . . . + N a n . Since 0 ∈ N , the semigroup S is actually a monoid. We may define a surjective monoid map asfollows deg A : N n −→ S ; u = ( u , . . . , u n ) (cid:55)−→ deg A ( u ) = n (cid:88) i =1 u i a i . (6)In the literature, this map is called the factorization map of S and accordingly, the fiberdeg − A ( a ) is called the set of factorizations of a ∈ S .Clearly deg A ( − ) determines a congruence on N n ; in fact, it is the congruence on N n whosepresentation map is precisely deg A ( − ) (cf. (2)). Therefore, if (cid:91) deg A is the map defined in (3),namely, (cid:91) deg A : k [ N n ] = k [ X ] → k [ S ] ; X u (cid:55)→ χ deg A ( u ) , by Theorem 13, we have that I A = ker( (cid:91) deg A ) is spanned as a k − vector space by the set ofbinomials { X u − X v | u , v ∈ N n with deg A ( u ) = deg A ( v ) } . (7)Observe that k [ X ] is S − graded via deg( X i ) = a i , i = 1 , . . . , n . This grading is known asthe A− grading on k [ X ]. The semigroup algebra k [ S ] = ⊕ a ∈ S Span k { χ a } also has a natural S − grading. Under these gradings, the map of semigroup algebras (cid:91) deg A is a graded map. Hence,the ideal I A = ker( (cid:91) deg A ) is S − homogeneous. Proposition 29.
Use the notation introduced above, and assume that a , . . . , a n are nonzero.The following are equivalent:(1) The fibers of map deg A ( − ) are finite.(2) deg − A ( ) = { (0 , . . . , } .(3) S ∩ ( − S ) = { } , that is to say, a ∈ S and − a ∈ S ⇒ a = .(4) The relation a (cid:48) (cid:22) a ⇐⇒ a (cid:48) − a ∈ S is a partial order on S .Proof. Before we proceed with the proof, we note that if one of the a i is zero, then this resultis false. For example, let G = Z , A = { a = 0 , a = 1 } . Then S = N , for which 3 and 4 hold,but deg A ( − ) does not satisfy either 1 or 2.1 ⇒ u ∈ deg − A ( ), then for every (cid:96) ∈ N , (cid:96) u ∈ deg − A ( ). If u (cid:54) = (0 , . . . , − A ( )is infinite.1 ⇐ N n has finitely many minimalelements with respect to the partial order given by coordinatewise ≤ . Suppose that deg − A ( a )is infinite. Then by Dickson’s Lemma there exists u ∈ deg − A ( a ) which is not minimal, andtherefore there is also v ∈ deg − A ( a ) such that v ≤ u coordinatewise. We conclude that u − v ∈ N n is a nonzero element of deg − A ( ). INOMIAL IDEALS AND CONGRUENCES ON N n ⇒ u , v ∈ N n be such that deg A ( u ) = a and deg A ( v ) = − a . Then deg A ( u + v ) = ,so that u + v = (0 , . . . , u = v = (0 , . . . , a = .2 ⇐ u ∈ deg − A ( ). If u , u ∈ N n are such that u = u + u , then by 3 deg A ( u ) =deg A ( u ) = . Repeatedly applying this argument, we conclude that if u (cid:54) = 0, so in particularit has a nonzero coordinate, then there exists 1 ≤ i ≤ n such that a i = deg A ( e i ) = , acontradiction.3 ⇔ (cid:22) is always reflexive and transitive. The fact that (cid:22) is antisymmetric isequivalent to 3. (cid:3) Remark . If the conditions of Proposition 29 hold, the monoid S generated by A is saidto be positive . When S is positive, m = (cid:104) X , . . . , X n (cid:105) is the only S − homogeneous maximalideal in k [ X ]. Recall that a graded ideal m in a graded ring R is a graded maximal ideal or ∗ maximal ideal if the only graded ideal properly containing m is R itself. Graded rings with aunique graded maximal ideal are known as graded local rings or ∗ local rings . Many resultsvalid for local rings are also valid for graded local rings, starting with Nakayama’s Lemma.In particular, the minimal free resolution of any finitely generated A− graded k [ X ] − module iswell-defined (see [3, Section 1.5] and [1]).All the monoids in this section are contained in a group. The next result characterize thecondition for a monoid to be contained in a group. To state it we need to introduce the followingconcepts. Definition 31.
Let ∼ be a congruence on N n . We will say that a ∈ N n / ∼ is cancellable if b + a = c + a ⇒ b = c , for all b , c ∈ N n / ∼ . A monoid is said to be cancellative ifall its elements are cancellable. A congruence ∼ on N n is cancellative if the monoid N n / ∼ iscancellative. In the part 3 of Example 16, an example of non-cancellative monoid is exhibited.
Proposition 32.
A (finitely generated commutative) monoid is contained in a group if andonly if it is cancellative.Proof.
The direct implication is clear. Conversely, if S = N n / ∼ is a cancellative finitelygenerated commutative monoid, then ∼ can be extended on Z n as follows: u ∼ v if u + e ∼ v + e for some (any) e ∈ N n such that u + e and v + e ∈ N n . Since G = Z n / ∼ has a natural groupstructure and S ⊆ G , we are done. (cid:3) The above result shows that our definition of cancellative congruence is equivalent to theusual one (see [7, p. 44]).3.1.
Toric ideals and toric congruences.
Suppose now that G is torsion-free and let G ( A ) denote the subgroup of G generated by A .Since G is torsion-free, then G ∼ = Z m , for some m . Thus, the semigroup S is isomorphic to asubsemigroup of Z m . In this case, S is said to be an affine semigroup and the ideal I A iscalled the toric ideal associated to A .Without loss of generality, we may assume that a i ∈ Z d , for every i = 1 , . . . , n , with d =rank( G ( A )) ≤ m . Moreover, one can prove that, if A generates a positive monoid (see Remark30), there exists a monoid isomorphism under which a i is mapped to an element of N n , i =1 , . . . , n (see, e.g. [3, Proposition 6.1.5]), which justifies the use of the term “positive”. Lemma 33. If A = { a , . . . , a n } ⊂ Z d , then I A is prime. Proof.
By hypothesis, we have that k [ S ] is isomorphic to the subring k [ t a , . . . , t a n ] of theLaurent polynomial ring k [ Z d ] = k [ t ± , . . . , t ± d ]; in particular, k [ S ] ∼ = k [ X ] /I A is a domain.Therefore I A is prime. (cid:3) Theorem 34.
Let I be a binomial ideal of k [ X ] . The ideal I is prime if and only if there exists A = { a , . . . , a r } ⊂ Z d such that I = I A k [ X ] + (cid:104) X r +1 , . . . , X n (cid:105) , up to permutation and rescaling of variables.Proof. Suppose that I is prime. If I contains monomials, there exists a set of variables,say X r +1 , . . . , X n (by permuting variables if necessary), such that I is equal to I (cid:48) k [ X ] + (cid:104) X r +1 , . . . , X n (cid:105) where I (cid:48) is a pure prime binomial ideal of k [ X , . . . , X r ]. Therefore, with-out loss of generality, we may suppose I = I (cid:48) and r = n . Now, by Theorem 8, k [ X ] /I ∼ = k [ S ] = (cid:76) a ∈ S Span k { χ a } , for some commutative monoid S generated by A = { a , . . . , a n } . Recall thatthe above isomorphism maps X i to λ i χ a i for some λ i ∈ k ∗ , i = 1 , . . . , n . So, by rescalingvariables if necessary, we may assume λ i = 1 for every i . Now, if S is not contained in a group,by Proposition 32, there exist a , a (cid:48) and b ∈ S such that a + b = a (cid:48) + b and a (cid:54) = a (cid:48) . Thus, X v ( X u − X u (cid:48) ) ∈ I , but X u − X u (cid:48) (cid:54)∈ I , where u ∈ deg − A ( a ) , u (cid:48) ∈ deg − A ( a (cid:48) ) and v ∈ deg − A ( b ).So, since I is prime, we have that X v ∈ I which is a contradiction. On other hand, if G ( S ) hastorsion, there exist two different elements a and a (cid:48) ∈ S such that n a = n a (cid:48) for some n ∈ N .Therefore X n u − X n u (cid:48) ∈ I , where u ∈ deg − A ( a ) and u (cid:48) ∈ deg − A ( a (cid:48) ). Since k is algebraicallyclosed and I is prime, X u − ζ n X u (cid:48) ∈ I , where ζ n is a n − th root of unity; in particular, a = a (cid:48) which is a contradiction. Putting all this together, we conclude that S is an affine semigroup.The opposite implication is a direct consequence of Lemma 33. (cid:3) Definition 35.
A congruence ∼ on N n is said to be toric if the ideal I ∼ is prime. The following result proves that our definition agrees with the one given in [10].
Corollary 36.
A congruence ∼ on N n is toric if and only if the non-nil elements of N n / ∼ form an affine semigroup.Proof. The direct implication follows from Theorem 34. Conversely, we assume that the non-nilelements of N n / ∼ form an affine semigroup S. In this case, we have that [ u ] + [ v ] = ∞ implies[ u ] = ∞ or [ v ] = ∞ , for every u and v ∈ N n , because S is contained in a group and groups haveno nil element. Therefore, since N n / ∼ is generated by the classes [ e i ] modulo ∼ , i = 1 , . . . , n ,we obtain that S is generated by A = { [ e i ] (cid:54) = ∞ | i = 1 , . . . , n } Now, applying Theorem 34again, we conclude that I ∼ is a prime ideal. (cid:3) Lattice ideals and cancellative congruences.
Consider now a subgroup L of Z n and define the following congruence ∼ on N n : u ∼ v ⇐⇒ u − v ∈ L . Clearly, N n / ∼ is contained in the group Z n / L and the associated ideal I ∼ is equal to I L := { X u − X v | u − v ∈ L} . The subgroups of Z n are also called lattices. This justifies the term “lattice” in the followingdefinition. Definition 37.
Let L be a subgroup of Z n and ρ : L → k ∗ be a group homomorphism. Thelattice ideal corresponding to L and ρ is I L ( ρ ) := (cid:104) X u − ρ ( u − v ) X v | u − v ∈ L(cid:105) . INOMIAL IDEALS AND CONGRUENCES ON N n An ideal I of k [ X ] is called a lattice ideal if there is subgroup L ⊂ Z n and a group homomor-phism ρ : L → k ∗ such that I = I L ( ρ ) . Observe that the ideal I L above is a lattice ideal for the group homomorphism ρ : L → k ∗ such that ρ ( u ) = 1 , for every u ∈ L . Moreover, given a subgroup L of Z n , we have that thecongruence on N n defined by a lattice ideal I L ( ρ ) is the same as the congruence on N n definedby I L , for every group homomorphism ρ : L → k ∗ .Let us characterize the cancellative congruences on N n in terms of their associated binomialideals. In order to do this, we first recall the following result from [5]. Proposition 38. [5, Corollary 2.5] If I is a pure binomial ideal of k [ X ] , then there is a uniquegroup morphism ρ : L ⊆ Z n → k ∗ such that I : ( (cid:81) ni =1 X i ) ∞ = I L ( ρ ) . Observe that from Proposition 38, it follows that no monomial is a zero divisor modulo alattice ideal.
Corollary 39.
A congruence ∼ on N n is cancellative if and only if I ∼ is a lattice ideal.Proof. By Proposition 32, ∼ is cancellative if and only if N n / ∼ is contained in a group G . Thus,the natural projection π : N n → N n / ∼ can be extended to a group homorphism ¯ π : Z n → G whose restriction to N n is π . Since the kernel, L , of ¯ π is a subgroup of Z n that defines the samecongruence as ∼ , we conclude that both ideals I ∼ and I L are equal. For the converse, we firstnote that the congruence on N n defined by a lattice ideal I L ( ρ ) is the same as the congruenceon N n defined by I L , for every group homomorphism ρ : L ⊂ Z n → k ∗ (see the comment afterequation (5)). Now, it suffices to note that if I ∼ = I L for some subgroup L of Z n , then N n / ∼ is contained in Z n / L . (cid:3) Observe that a lattice ideal I L is not prime in general. Indeed, I = (cid:104) X − Y (cid:105) is a latticeideal corresponding to the subgroup of Z generated by (2 , −
2) which is clearly not prime. Letus give a necessary and sufficient condition for a lattice ideal to be prime.
Definition 40.
Let L be subgroup of Z n and set Sat( L ) := ( Q ⊗ Z L ) ∩ Z n = { u ∈ Z n | d u ∈ L for some d ∈ Z } . Clearly,
Sat( L ) is subgroup of Z n and it is called the saturation of L . We say that L is saturated if L = Sat( L ) . Proposition 41.
A lattice ideal I L ( ρ ) is prime if and only if L is saturated.Proof. By using the same argument as in the proof of Corollary 39, we obtain that Z n / ∼ = Z n / L , where ∼ the congruence defined by I L ( ρ ) on Z n . Now, since Z n / L is the groupgenerated by N n / ∼ , and Z n / L is torsion-free if and only if L is saturated, we obtain thedesired equivalence. (cid:3) Notice that the congruence defined by L is contained in the congruence defined by Sat( L ). Infact, Sat( L ) defines the smallest toric congruence on N n containing the congruence defined by L on N n . Therefore, we may say each cancellative congruence has exactly one toric congruenceassociated .The primary decomposition of a lattice ideal I L ( ρ ) can be completely described in terms of L and ρ . Let us reproduce this result. For this purpose, we need additional notation. Definition 42. If p is a prime number, we define Sat p ( L ) and Sat (cid:48) p ( L ) to be the largest sub-lattices of Sat( L ) containing L such that Sat p ( L ) / L has order a power of p and Sat (cid:48) p ( L ) / L has order relatively prime to p. If p = 0 , we adopt the convention that Sat p ( L ) = L and Sat (cid:48) p ( L ) = Sat( L ) . Theorem 43. [5, Corollaries 2.2 and 2.5]
Let char( k ) = p ≥ and consider a group morphsim ρ : L ⊆ Z n → k ∗ . If the order of Sat (cid:48) p ( L ) / L is g , there are g distinct group morphisms ρ , . . . , ρ g extending ρ to Sat (cid:48) p ( L ) and for each j ∈ { , . . . , g } a unique group morphism ρ (cid:48) j extending ρ to Sat( L ) . Moreover, there is a unique group morphism ρ (cid:48) extending ρ to Sat p ( L ) .The radical, associated primes and minimal primary decomposition of I L ( ρ ) ⊂ k [ X ] are: (cid:112) I L ( ρ ) = I Sat p ( L ) ( ρ (cid:48) ) , Ass( k [ X ] /I L ( ρ )) = { I Sat( L ) ( ρ (cid:48) j ) | j = 1 , . . . , g } and I L ( ρ ) = g (cid:92) j =1 I Sat (cid:48) p ( L ) ( ρ j ) where I Sat (cid:48) p ( L ) ( ρ j ) is I Sat( L ) ( ρ (cid:48) j ) -primary. In particular, if p = 0 , then I L ( ρ ) is a radical ideal. Theassociated primes I Sat( L ) ( ρ (cid:48) j ) of I L ( ρ ) are all minimal and have the same codimension rank( L ) . Mesoprime ideals and prime congruences.
Given δ ⊆ { , . . . , n } , set N δ := { ( u , . . . , u n ) ∈ N n | u i = 0 , for all i (cid:54)∈ δ } and define Z δ asthe subgroup of Z n generated by N δ . Morover, if δ = ∅ , by convention, then Z δ = { } ⊂ Z n . Definition 44.
Given δ ⊆ { , . . . , n } and a group homomorphism ρ : L ⊆ Z δ → k ∗ , a δ − mesoprime ideal is an ideal of the form I L ( ρ ) + p δ c with p δ c := (cid:104) X j | j (cid:54)∈ δ (cid:105) . By convention, p ∅ c = (cid:104) X , . . . , X n (cid:105) and p ∅ = (cid:104) (cid:105) .Example . (1) The ideal (cid:104) X − , X (cid:105) ⊂ k [ X , X ] is mesoprime for δ = { } (2) By Theorem 34, every binomial prime ideal is mesoprime, for a suitable δ .(3) Lattice ideals are mesoprime for δ = { , . . . , n } .Due to Theorem 43, a mesoprime ideal can be understood as a condensed expression thatincludes all the information necessary to produce the primary decomposition of the ideal simplyby using arithmetic arguments.Observe that the congruence on N n defined by I L + p δ c is the same as the congruence definedby I L ( ρ ) + p δ c , for every δ ⊆ { , . . . , n } and every group homomorphism ρ : L ⊆ Z δ → k ∗ . Lemma 46.
Let δ ⊆ { , . . . , n } . If I is a δ − mesoprime ideal, then I : X i = I, for all i ∈ δ .Equivalently, I : ( (cid:81) i ∈ δ X i ) ∞ = I .Proof. If I is a δ − mesoprime ideal, there exists ρ : L ⊆ Z δ → k ∗ such that I = I L ( ρ ) + p δ c . Let X i f ∈ I, i ∈ δ . We want to show that f ∈ I . So, without loss of generality, we may assumethat no term of f lies in p δ c . In this case, X i f ∈ I L ( ρ ). Now, by Proposition 38, we concludethat f ∈ I L ( ρ ), and hence f ∈ I . (cid:3) Definition 47.
A congruence ∼ on N n is said to be prime if the ideal I ∼ is mesoprime forsome δ ⊆ { , . . . , n } . Let us prove that this notion of prime congruence is the same as the usual one (see [7, p. 44]).
Proposition 48.
A congruence ∼ on N n is prime if and only if every element of N n / ∼ iseither nil or cancellable. INOMIAL IDEALS AND CONGRUENCES ON N n Proof. If ∼ is prime congruence on N n , then there exist δ ⊆ { , . . . , n } and a subgroup L ⊆ Z δ such that I ∼ = I L + p δ c . Let [ u ] be non-nil and let [ v ] and [ w ] ∈ N n / ∼ be such that [ v ] + [ u ] =[ v + u ] = [ w + u ] = [ w ] + [ u ]. In particular, X u ( X v − X w ) = X v + u − X w + v ∈ I ∼ . Since [ u ] isnon-nil, X u does not belong to I ∼ . Therefore, u ∈ { X i } i ∈ δ and, by Lemma 46, X v − X w ∈ I ∼ ,that is, [ v ] = [ w ]. So [ u ] is cancellable.Conversely, suppose that every element of N n / ∼ is either nil or cancellable, set δ = { i ∈{ , . . . , n } : [ e i ] is cancellable } . Clearly, j (cid:54)∈ δ if and only if X j ∈ I ∼ . So, there exist a binomialideal J in k [ { X i } i ∈ δ ] such that I ∼ = J k [ X ] + p δ c (if δ = ∅ , take J = (cid:104) (cid:105) ). Moreover, since[ e i ] is cancellable for every i ∈ δ, if X e i f = X i f ∈ J , for some i ∈ δ , then f ∈ J . Thus,by Proposition 38, J is lattice ideal of k [ { X i } i ∈ δ ] and, consequently, J k [ X ] is a lattice ideal.Therefore, I ∼ = J k [ X ] + p δ c is a δ − mesoprime ideal and we are done. (cid:3) Cellular binomial ideals
In this section we study the so-called cellular binomial ideals defined by D. Eisenbud andB. Sturmfels in [5]. Cellular binomial ideals play a central role in the theory of primary de-composition of binomials ideals (see [5] and also [6, 13, 14]). As in the previous section, wewill determine the congruences on N n corresponding to those ideals. We will also outline analgorithm to compute a decomposition of a binomial ideal into cellular binomial ideals whichwill produce (primary) decompositions of the corresponding congruences.Let us start by defining the notion of cellular ideal. Definition 49.
A proper ideal I of k [ X ] is cellular if, for some δ ⊆ { , . . . , n } , we have that(1) I : ( (cid:81) i ∈ δ X i ) ∞ = I ; equivalently I : X i = I, for every i ∈ δ ,(2) there exists d i ∈ N such that X d i i ∈ I, for every i (cid:54)∈ δ .In this case, we say that I is cellular with respect to δ or, simply, δ − cellular. By convention,the ∅ − cellular ideals are the binomial ideals whose radical is (cid:104) X , . . . , X n (cid:105) . Observe that an ideal I of k [ X ] is cellular if, and only if, every variable of k [ X ] is eithera nonzerodivisor or nilpotent modulo I. In particular, prime, lattice, mesoprime and primaryideals are cellular.The following proposition establishes the relationship between cellular binomial and meso-prime ideals.
Proposition 50.
Let δ ⊆ { , . . . , n } . If I is a δ − cellular binomial ideal in k [ X ] , there exists agroup morphism ρ : L ⊆ Z δ → k ∗ such that(1) ( I ∩ k [ { X i } i ∈ δ ]) k [ X ] = I L ( ρ ) .(2) I + p δ c = I L ( ρ ) + p δ c .(3) √ I + p δ c = (cid:112) I L ( ρ ) + p δ c .(4) √ I = (cid:112) I L ( ρ ) + p δ c .In particular, the radical of a cellular binomial ideal is a mesoprime ideal, and the minimalassociated primes of I are binomial.Proof. If δ = ∅ , then I ∩ k [ { X i } i ∈ δ ]) = 0 and it suffices to take ρ : { } → k ∗ ; (cid:55)→
1. So,assume without loss of generality that δ (cid:54) = ∅ .In order to prove part (a), we first note that J := I ∩ k [ { X i } i ∈ δ ] is binomial by Corollary 3,and that J : ( (cid:81) i ∈ δ X i ) ∞ = J by the definition of cellular ideal. Thus, J k [ X ] : ( (cid:81) ni =1 X i ) ∞ = J k [ X ] and, by Proposition 38, there is a unique group morphism ρ : L ⊆ Z δ → k ∗ such that J k [ X ] = I L ( ρ ). Part (b) is an immediate consequence of (a).By part (b) and according to the properties of the radical, we have that (cid:112) I + p δ c = (cid:112) I L ( ρ ) + p δ c = (cid:113)(cid:112) I L ( ρ ) + p δ c ⊇ (cid:112) I L ( ρ ) + p δ c ⊇ (cid:112) I L ( ρ ) + p δ c . On other hand, given f ∈ (cid:112) I L ( ρ ) + p δ c , we can write f = h + (cid:80) i (cid:54)∈ δ g i X i where h e ∈ I L ( ρ ) forsome e >
0. Now, since I L ( ρ ) ⊆ I L ( ρ ) + p δ c = I + p δ c , we have that f e = (cid:0) h + (cid:80) i (cid:54)∈ δ g i X i (cid:1) e ∈ I + p δ c , that is to say, f ∈ √ I + p δ c . Thus, we obtain that √ I + p δ c = (cid:112) I L ( ρ ) + p δ c , as claimedin (c).For part (d), we observe that (cid:112) I L ( ρ ) + p δ c = (cid:113) I L ( ρ ) + (cid:104) X d i i | i (cid:54)∈ δ (cid:105) , and that I L ( ρ ) + (cid:104) X d i i | i (cid:54)∈ δ (cid:105) = ( I ∩ k [ { X i } i ∈ δ ]) k [ X ] + (cid:104) X d i i | i (cid:54)∈ δ (cid:105) ⊆ I ⊆ I + p δ c , for every d i ≥ , i (cid:54)∈ δ . Therefore, taking radicals, by part (c) we conclude that (cid:112) I L ( ρ ) + p δ c = √ I = √ I + p δ c .Now, the last statements are direct consequences of the definition of mesoprimary ideal andTheorem 43. (cid:3) In the following definition we introduce the concept of primary congruence on N n . We provethat our notion of primary congruence is equivalent to the one given in [7, p. 44]. Definition 51.
A congruence ∼ on N n is said to be primary if the ideal I ∼ is cellular. Definition 52.
Let ∼ be a congruence on N n . An element a ∈ N n / ∼ is said to be nilpotent if d a is nil, for some d ∈ N . Proposition 53.
A congruence ∼ on N n is primary if and only if every element of N n / ∼ isnilpotent or cancellable.Proof. If ∼ is a primary congruence on N n , the binomial associated ideal I ∼ is δ − cellular forsome δ ⊆ { , . . . , n } . Let [ u ] be a non-nilpotent element of N n / ∼ . Given [ v ] and [ w ] ∈ N n / ∼ such that [ v ] + [ u ] = [ v + u ] = [ w + u ] = [ w ] + [ u ], we have that X u ( X v − X w ) ∈ I ∼ . Since[ u ] is not nilpotent, ( X u ) d (cid:54)∈ I ∼ , for every d ∈ N . Therefore, no variable X i with i (cid:54)∈ δ divides X u and, by the definition of cellular ideal, we conclude that I ∼ : X u = I ∼ ; in particular, X v − X w ∈ I ∼ , that is, [ v ] = [ w ], and hence [ u ] is cancellable.Conversely, suppose that every element of N n / ∼ is nilpotent or cancellable. Set δ = { i ∈{ , . . . , n } : [ e i ] is cancellable } . Clearly, j ∈ δ if and only if X j is a nonzerodivisor modulo I ∼ and j (cid:54)∈ δ if and only if X d j j ∈ I ∼ , for some d j ≥
1. Therefore, I ∼ is a δ − cellular ideal (see theparagraph just after Definition 49). (cid:3) As a consequence, if ∼ is a primary congruence on N n , then, by Proposition 50, J := √ I ∼ is a mesoprime ideal. Therefore, associated to ∼ there is one and only one prime congruence, ∼ J , obtained by removing nilpotent elements.4.1. Cellular Decomposition of Binomial Ideals.Definition 54.
A cellular decomposition of an ideal I ⊆ k [ X ] is an expression of I as anintersection of cellular ideals with respect to different δ ⊆ { , . . . , n } , say I = (cid:92) δ ∈ ∆ C δ , (8) INOMIAL IDEALS AND CONGRUENCES ON N n for some subset ∆ of the power set of { , . . . , n } . Moreover, the cellular decomposition (8) issaid to be minimal if C (cid:48) δ (cid:54)⊇ (cid:84) δ ∈ ∆ \{ δ (cid:48) } C δ for every δ (cid:48) ∈ ∆ ; in this case, the cellular component C δ is said to be a δ − cellular component of I. Example . Every minimal primary decomposition of a monomial ideal I ⊆ k [ X ] into mono-mial ideals is a minimal cellular decomposition of I. Consequently, there is non-uniquenessfor cellular decomposition in general: consider for instance the following cellular (primary)decomposition (cid:104) X , XY (cid:105) = (cid:104) X (cid:105) ∩ (cid:104) X , XY, Y n (cid:105) , where n can take any positive integral value.Cellular decompositions of an ideal I of k [ X ] always exist. A simple algorithm for cellulardecomposition of binomial ideals can be found in [13, Algorithm 2], this algorithm forms part ofthe binomials package developed by T. Kahle and it is briefly described below. The interestedreader may consult [9] and [13] for further details.The following result is the key for producing cellular decompositions of binomial ideals intobinomial ideals. Lemma 56.
Let I be a proper binomial ideal in k [ X ] . If I is not cellular then there exists i ∈ { , . . . , n } and a positive integer d such that I = ( I : X di ) ∩ ( I + (cid:104) X di (cid:105) ) , with I : X di and I + (cid:104) X di (cid:105) binomial ideals strictly containing I. Proof. If I is not cellular, there exists at least one variable X i which is zerodivisor and notnilpotent modulo I . Then, by the Noetherian property of k [ X ], there is a positive integer d suchthat I : (cid:104) X di (cid:105) = I : (cid:104) X ei (cid:105) for every e ≥ d . We claim that I decomposes as ( I : X di ) ∩ ( I + (cid:104) X di (cid:105) ).Indeed, let f ∈ ( I : X di ) ∩ ( I + (cid:104) X di (cid:105) ) and let f = g + hX di for some g ∈ I . Then X di f = X di g + hX di and, thus hX di = X i f − X i g ∈ I . That is, h ∈ I : (cid:104) X di (cid:105) = I : (cid:104) X di (cid:105) . Hence, hX di ∈ I and,consequently, f ∈ I .It remains to see that both I : X di and I + (cid:104) X di (cid:105) are binomial ideals which strictly contain I . On the one hand, the ideal I + (cid:104) X di (cid:105) is binomial and I is strictly contained in it, as X i isnot nilpotent modulo I. On the other hand, I : X di is binomial by Corollary 5, and I is strictlycontained in I : X di because X i is a zerodivisor modulo I. (cid:3) Now, by Lemma 56, if I is not a cellular ideal then we can find two new proper ideals strictlycontaining I. If these ideals are cellular then we are done. Otherwise, we can repeat the sameargument with these new ideals, getting strictly increasing chains of binomial ideals. Since k [ X ]is a Noetherian ring, each one of these chains has to be stationary. So, in the end, we obtaina (redundant) cellular decomposition of I. Observe that this process does not depend on thebase field.
Example . Consider the binomial ideal I = (cid:104) X Y − Z , X Y − Z , X − Y Z (cid:105) of Q [ X, Y, Z ] . By using [13, Algorithm 2] we obtain the following cellular decomposition, I = I ∩ I ∩ I , where I = (cid:104) Y − Z, X − Z (cid:105) I = (cid:104) Z , XZ, X − Y Z (cid:105) I = (cid:104) X − Y Z, XY Z − Z , XZ − Z , Z , Y (cid:105) . loadPackage "Binomials";R = QQ[X,Y,Z];I = ideal(X^4*Y^2-Z^6,X^3*Y^2-Z^5,X^2-Y*Z);binomialCellularDecomposition I As a final conclusion we may notice the following:
Corollary 58.
Let ∼ be a congruence on N n . A primary decomposition of ∼ can be obtainedby computing a cellular decomposition of I ∼ .Proof. It is a direct consequence of Proposition 27 by the definition of primary congruence. (cid:3) Mesoprimary ideals
The main objective of this section is to analyze the mesoprimary ideals and their corre-sponding congruence. Mesoprimary ideals were introduced by Thomas Kahle and Ezra Millerin [10] as an intermediate construction between cellular and primary binomial ideals. Kahleand Miller proved combinatorially that every cellular binomial can be decomposed into finitelymany mesoprimary ideals over an arbitrary field. However, not every decomposition of a bi-nomial ideal as an intersection of mesoprimary ideals is a mesoprimary decomposition in thesense of Kahle and Miller. These mesoprimary decompositions feature refined combinatorialrequirements, and currently there is no algorithm available to compute them. On the otherhand, decompositions of binomial ideals into mesoprimary ideals can be produced algorithmi-cally.. Despite of this, mesoprimary decompositions have been successfully used to solve openproblems (see [11] and [12]).The following preparatory result will be helpful in understanding what mesoprimary idealsare.
Proposition 59.
Let I be a δ − cellular binomial ideal in k [ X ] . If X u ∈ k [ { X i } i (cid:54)∈ δ ] \ I , then I : X u is a δ − cellular binomial ideal.Proof. First of all, we note that I : X u (cid:54) = (cid:104) (cid:105) because X u (cid:54)∈ I . Moreover, we have that I : X u is binomial by Corollary 5. Now, since I : ( (cid:81) i ∈ δ X i ) ∞ = I , then( I : X u ) : ( (cid:89) i ∈ δ X i ) ∞ = ( I : ( (cid:89) i ∈ δ X i ) ∞ ) : X u = I : X u . And, clearly, for every i (cid:54)∈ δ, X d i i ∈ I : X u for some d i ≥ I ⊆ I : X u . Putting all thistogether, we conclude that I : X u is a δ − cellular binomial ideal. (cid:3) If I is a δ − cellular binomial ideal, then the ideal ( I : X u )+ p δ c is δ − mesoprime by Propositions59 and 50(b). Moreover, there exists d i ≥ X d i i ∈ I for each i (cid:54)∈ δ . Thus there arefinitely many mesoprime ideals of the form ( I : X u ) + p δ c . These are the so-called mesoprimesassociated to I : Definition 60.
Let I be a δ − cellular binomial ideal in k [ X ] . We will say that I L ( ρ ) + p δ c is amesoprime ideal associated to I if there exist a monomial X u ∈ k [ { X i } i (cid:54)∈ δ ] such that (cid:0) ( I : X u ) ∩ k [ { X i } i ∈ δ ] (cid:1) k [ X ] = I L ( ρ ) . Now we may introduce the notion of mesoprimary ideal.
Definition 61.
A binomial ideal is said to be mesoprimary if it is cellular and it has onlyone associated mesoprime ideal. A congruence ∼ on N n is mesoprimary if I ∼ is a mesoprimaryideal of k [ X ]The following lemma clarifies the notion of mesoprimary ideal. Lemma 62. A δ − cellular binomial ideal I in k [ X ] is mesoprimary if and only if ( I : X u ) ∩ k [ { X i } i (cid:54)∈ δ ] = I ∩ k [ { X i } i (cid:54)∈ δ ] , for all X u ∈ k [ { X i } i (cid:54)∈ δ ] \ I .Proof. It suffices to note that I has two different associated mesoprimes if and only if thereexists X u ∈ k [ { X i } i (cid:54)∈ δ ] such that ( I : X u ) ∩ k [ { X i } i (cid:54)∈ δ ] (cid:54) = I ∩ k [ { X i } i (cid:54)∈ δ ] because, in this case,by Proposition 50, ( I : X u ) + p δ c and I + p δ c are two different associated mesoprimes to I . (cid:3) INOMIAL IDEALS AND CONGRUENCES ON N n Definition 63.
Let ∼ be a congruence on N n . An element a ∈ N n / ∼ is said to be partlycancellable if a + b = a + c (cid:54) = ∞ ⇒ b = c , for all cancellable b , c ∈ N n Proposition 64.
A congruence ∼ on N n is mesoprimary if and only if it is primary and everyelement in N n / ∼ is partly cancellable.Proof. If ∼ is a mesoprimary congruence on N n , then I = I ∼ is δ − cellular for some δ ⊆{ , . . . , n } . Thus, ∼ is primary. Moreover, ( I : X u ) ∩ k [ { X i } i ∈ δ ] = I ∩ k [ { X i } i ∈ δ ], for all X u ∈ k [ { X i } i (cid:54)∈ δ ] \ I (equivalently, for all u ∈ N n such that [ u ] is nilpotent and it is not anil). Therefore, if [ u ] ∈ N n / ∼ is nilpotent and [ v ] , [ w ] are cancellable elements such that[ u ] + [ v ] = [ u ] + [ w ] (cid:54) = ∞ , then X v − X w ∈ ( I : X u ) ∩ k [ { X i } i ∈ δ ] = I ∩ k [ { X i } i ∈ δ ] , that is to say [ v ] = [ w ] . So, [ u ] is partly cancellative.Conversely, suppose that ∼ is primary congruence on N n such that every element in N n / ∼ is partly cancellable. Since ∼ is primary, we have that I ∼ is δ − cellular, by setting δ = { i ∈{ , . . . , n } : [ e i ] is cancellable } . Now, if X u ∈ k [ { X i } i (cid:54)∈ δ ] \ I , we have that [ u ] is partly cancellable.Thus, for every X v − X w ∈ k [ { X i } i ∈ δ ], we have that X u ( X v − X w ) ∈ I ⇒ X v − X w ∈ I .Therefore, ( I : X u ) ∩ k [ { X i } i ∈ δ ] ⊆ I ∩ k [ { X i } i ∈ δ ]. Now, since the opposite inclusion is alwaysfulfilled, by Lemma 62, we are done. (cid:3) There are other intermediate constructions between cellular and primary ideals, such as theunmixed decomposition (see [5, 13] and, more recently, [6]). The following example shows thatunmixed cellular binomial ideals are not mesoprimary. Recall that an unmixed cellular binomialideal is a cellular binomial ideal with no embedded associated primes (see [13, Proposition 2.4]).
Example . Consider the unmixed cellular binomial I ⊂ k [ X, Y ] generated by { X − , Y ( X − , Y } . The ideal I is not mesoprimary, because( I : Y ) ∩ k [ X ] = (cid:104) X − (cid:105) (cid:54) = (cid:104) X − (cid:105) = I ∩ k [ X ] . loadPackage "Binomials";R = QQ[X,Y]I = ideal(X^2-1,Y*(X-1),Y^2)cellularBinomialAssociatedPrimes Ieliminate(I:Y,Y)eliminate(I,Y) We end this section by exhibiting the statement of Kahle and Miller which describes theprimary decomposition of a mesoprimary ideal, in order to give an idea of how useful would beto have an algorithm for the mesoprimary decomposition of a cellular binomial ideal.
Proposition 66 ([10, Corollary 15.2 and Proposition 15.4]) . Let I be a ( δ -cellular) mesopri-mary ideal, and denote by I L ( ρ ) the lattice ideal I ∩ k [ { X i } i ∈ δ ] . The associated primes of I areexactly the (minimal) primes of its associated mesoprime I + p δ c . Moreover, if I L ( ρ ) = ∩ gj =1 I j is the primary decomposition of I L ( ρ ) from Theorem 38, then I = g (cid:92) j =1 ( I + I j ) is the primary decomposition of I . Notice that the hypothesis k algebraically closed is only needed when Theorem 38 is applied. Acknowledgement
We thank the anonymous referees for their detailed suggestions and com-ments, which have greatly improved this article. The present paper is based on a course of lectures delivered by the second author at the EACA’s Third International School on Com-puter Algebra and Applications . He thanks theorganizers for giving him that opportunity.
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E-mail address : [email protected] Departamento de Matem´aticasUniversidad de ExtremaduraE-06071 Badajoz (SPAIN)
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