aa r X i v : . [ m a t h . A C ] J a n GR ¨OBNER-NICE PAIRS OF IDEALS
MIRCEA CIMPOEAS¸ AND DUMITRU I. STAMATE
Abstract.
We introduce the concept of a Gr¨obner nice pair of ideals in a poly-nomial ring and we present some applications.
Introduction
One feature of a Gr¨obner basis is that it extends a system of generators for anideal in a polynomial ring so that several invariants or algebraic properties are easierto read. It is a natural question to ask how to obtain a Gr¨obner basis for an idealobtained by performing basic algebraic operations. A first situation we discuss inthis note is when a Gr¨obner basis for the sum of the ideals I and J is obtainedby taking the union of two Gr¨obner bases for the respective ideals. In that casewe say that ( I, J ) is a Gr¨obner nice pair ( G -nice pair, for short). In Theorem 1.1we prove that for any given monomial order, ( I, J ) is a G -nice pair if and only ifin( I + J ) = in( I ) + in( J ), which is also equivalent to having in( I ∩ J ) = in( I ) ∩ in( J ).Given the ideals I and J in a polynomial ring, they could be a G -nice pair for some,for any, or for no monomial order. Situations of G -nice pairs of ideals have naturallyoccurred in the literature, especially related to ideals of minors, e.g. [1, 6, 7].One application of the new concept is in Corollary 1.7: assume J is any ideal inthe polynomial ring S and let f , . . . , f r in S be a regular sequence on S/J . Thenthe sequence in( f ) , . . . , in( f r ) is regular on S/ in( J ) if and only if in( J, f , . . . , f r ) =in( J ) + (in( f ) , . . . , in( f r )).We introduced the notion of a Gr¨obner nice pair in an attempt to unify somecases of distributivity in the lattice of ideals in a polynomial ring S . For instance,one consequence of Proposition 1.8 is that if ( J, E ) , ( J, E ′ ) and ( J, E ∩ E ′ ) are G -nicepairs, then ( J + E ) ∩ ( J + E ′ ) = J + ( E ∩ E ′ ) if and only if in(( J + E ) ∩ ( J + E ′ )) =in( J ) + in( E ∩ E ′ ). Moreover, in Proposition 1.11 we show that when ( E i ) i ∈ Λ is afamily of monomial ideals such that ( J, E i ) is G -nice for all i ∈ Λ, then ( J, T i ∈ Λ E i )is a G -nice pair and T i ∈ Λ ( J + E i ) = J + ( T i ∈ Λ E i ).One problem we raise in Section 2 is how to efficiently transform a pair of ideals( J, E ) into a G -nice pair ( J, F ) so that F ⊇ E . We show that in general there isno minimal such ideal F so that also J + F = J + E , see Example 2.4. However,if E is a monomial ideal then we may consider b E the smallest monomial ideal in S containing E and so that ( J, b E ) is G -nice. Furthermore, when J is a binomial ideal(i.e. it is generated by binomials) then J + E = J + b E , see Corollary 2.9. Mathematics Subject Classification.
Primary 13P10, 13F20; Secondary 13P99.
Key words and phrases.
Gr¨obner basis, polynomial ring, S-polynomial, regular sequence. uchberger’s criterion ([4, 3, 5]) asserts that a set of polynomials form a Gr¨obnerbasis for the ideal they generate if and only if for any two elements in the set their S -polynomial reduces to zero with respect to the given set. Having this fact in mind,a special class of Gr¨obner nice pairs is introduced in Section 3. Namely, if G J is aGr¨obner basis for J , then we say that the ideal E is S -nice with respect to G J if forall f ∈ G J and g ∈ E their S -polynomial S ( f, g ) ∈ E . To check that property it isenough to verify it for g in some Gr¨obner basis for E , see Proposition 3.4. A goodproperty is that if E i is S -nice with respect to G J for all i ∈ Λ, then so is T i ∈ Λ E i .This way, given G J , for any ideal (resp. monomial ideal E ) we can define the S -nice (monomial) closure: e E (resp. E ♯ ) is the smallest (monomial) ideal whichis S -nice w.r.t. G J and we show how to compute it in Proposition 3.10. While J + e E = J + E , it is not always the case that J + E ♯ = J + E . In Proposition 3.12we show that if G J consists of binomials and E is a monomial ideal, then b E = e E .We provide many examples for the notions that we introduce.1. Gr¨obner-nice pairs of ideals
Let K be any field. Throughout this paper, we usually denote by S the polynomialring K [ x , . . . , x n ], unless it is stated otherwise. For an ideal I in S and a monomialorder ≤ on S , the set of all Gr¨obner bases of I with respect to the given monomialorder will be denoted Gr¨ob ≤ ( I ), or simply Gr¨ob( I ) when there is no risk of confusion.As a piece of notation, with respect to a fixed monomial order, for f in S its leadingterm is denoted LT( f ), and its leading monomial in( f ).The following result is at the core of our work. Theorem 1.1.
We fix a monomial order in the polynomial ring S . Let J and E beideals in S . The following conditions are equivalent:(a) in( J + E ) = in( J ) + in( E ) ;(b) for any G J ∈ Gr¨ob( J ) and G E ∈ Gr¨ob( E ) , we have G J ∪ G E ∈ Gr¨ob( J + E ) ;(c) there exist G J ∈ Gr¨ob( J ) and G E ∈ Gr¨ob( E ) , such that G J ∪ G E ∈ Gr¨ob( J + E ) ;(d) in( J ∩ E ) = in( J ) ∩ in( E ) ;(e) for any f ∈ J and g ∈ E , there exists h ∈ J ∩ E such that in( h ) =lcm(in f, in g ) ;(f ) for any = h ∈ J + E , there exist f ∈ J and g ∈ E with h = f − g and in( f ) = in( g ) .Proof. We set I = J + E .(a) ⇒ (b): Let G J = { f , . . . , f r } ∈ Gr¨ob( J ) and G E = { g , . . . , g p } ∈ Gr¨ob( E ).Clearly, G J ∪G E generates the ideal I . Since in( J ) = (in( f ) , . . . , in( f r )) and in( E ) =(in( g ) , . . . , in( g p )), by (a), it follows that in( I ) = (in( f ) , . . . , in( f r ) , in( g ) , . . . , in( g p ))and therefore, G J ∪ G E is a Gr¨obner basis of I .(b) ⇒ (c) is trivial.(c) ⇒ (a): Let G J = { f , . . . , f r } ∈ Gr¨ob( J ) and G E = { g , . . . , g p } ∈ Gr¨ob( E ),such that G J ∪ G E ∈ Gr¨ob( J + E ). It follows that in( J + E ) = in( J ) + in( E ), asrequired. a) ⇒ (d): Since always in( J ∩ E ) ⊆ in( J ) ∩ in( E ), it remains to prove the otherinclusion. Let m be any monomial in in( J ) ∩ in( E ). Hence there exist f ∈ J and g ∈ E such that m = in( f ) = in( g ). If f = g then m ∈ in( J ∩ E ) and we are done.Otherwise, since f − g ∈ J + E , by (a), in( f − g ) ∈ in( J ) + in( E ). If in( f − g ) ∈ in( J ), then there exists f ∈ J with LT( f − g ) = LT( f ), and we set g = 0. Ifin( f − g ) ∈ in( E ), then there exists g ∈ E with LT( f − g ) = LT( g ), and we set f = 0. In either case, we have in( f − f ) = in( g − g ) = m . If f − f = g − g we are done. Otherwise, since in(( f − f ) − ( g − g )) < in( f − g ), we repeat theabove procedure for f − f and g − g . By Dickson’s Lemma ([4, Theorem 1.9]) thisprocess eventually stops, hence m ∈ in( J ∩ E ).Condition (e) is a restatement of (d).(e) ⇒ (a): It is enough to prove that in( J + E ) ⊆ in( J ) + in( E ). Let p ∈ J + E and write p = f − g , with f ∈ J and g ∈ E . If LT( f ) = LT( g ) then in( p ) = in( f )or in( p ) = in( g ) and we are done. Assume LT( f ) = LT( g ). By (e), there exists h ∈ J ∩ E with LT( h ) = LT( f ) = LT( g ). We let f = f − h and g = g − h . We notethat p = f − g = f − g , in( f ) < in( f ) and in( g ) < in( g ). If LT( f ) = LT( g ) thenin( p ) ∈ in( J ) + in( E ), arguing as above. Otherwise, we repeat the same procedurefor f and g until it stops.(f) ⇒ (a) is obvious.(d) ⇒ (f): Let h ∈ I . We write h = f − g with f ∈ J and g ∈ E . If h ∈ J or h ∈ E then there is nothing to prove.If in( f ) = in( g ), we are done. Otherwise, assume in( f ) = in( g ). We distinguishtwo cases, depending whether in( f ) = in( h ) or in( f ) > in( h ).If in( h ) = in( f ) = in( g ), according to (d) we have that in( h ) ∈ in( J ) ∩ in( E ) =in( J ∩ E ). Therefore, there exists w ∈ J ∩ E with LT( w ) = LT( g ). We write h = ( f − w ) − ( g − w ). Note that f − w ∈ J , g − w ∈ E and in( g − w ) < in( g ) = in( h ).It follows that in( h ) = in( f − w ) > in( g − w ) so we are done. Note that g − w = 0,otherwise h would be in J , a contradiction.If in( h ) < in( f ) = in( g ) =: m , since in( h ) = in( f − g ) < m , it follows thatLT( f ) = LT( g ). Also, m ∈ in( J ) ∩ in( E ) = in( J ∩ E ). Let w ∈ J ∩ E within( w ) = m . We can assume LT( w ) = LT( f ) = LT( g ). We write h = f − g = f − g ,where f = f − w and g = g − w . Arguing as before, 0 = f ∈ J , 0 = g ∈ E ,in( f ) < m and in( g ) < m . If in( f ) = in( g ) we are done. If in( f ) = in( g ) = in( h )we are done by the discussion of the former case. Otherwise, we apply the sameprocedure for f and g , until it eventually stops. (cid:3) Remark 1.2.
The equivalence of the conditions (a), (d) and (e) in Theorem 1.1 wasproved in a different way by A. Conca in [1, Lemma 1.3] under the extra hypothesisthat the ideals J and E are homogeneous. Definition 1.3.
Let S be a polynomial ring with a fixed monomial order. If theideals J and E of S fulfill one of the equivalent conditions of Theorem 1.1, we saythat ( J, E ) is a
Gr¨obner nice (G-nice) pair of ideals.
Remark 1.4.
The chosen monomial order is essential. For example, let J = ( x + y ), E = ( x ) and I = J + E = ( x , y ) as ideals in S = K [ x, y ]. If x > y , then n( J ) = in( E ) = ( x ), but in( I ) = ( x , y ). Thus the pair ( J, E ) is not G -nice. Onthe other hand, if y > x , then in( I ) = in( J ) + in( E ) = ( x , y ), hence the pair ( J, E )is G -nice.On the other hand, some pairs of ideals are never G -nice, regardless of the mono-mial order which is used. Indeed, let J = ( x + y ) and E = ( xy ) in S = K [ x, y ].It is easy to see that S ( x + y , xy ) equals either y (if x > y ), or x (if x < y ),and in either case in( S ( x + y , xy )) / ∈ (in( J ) , in( E )).Here are some examples of classes of Gr¨obner nice pairs of ideals. Example 1.5. (1) If J ⊆ E ⊆ S , then the pair ( J, E ) is G -nice.(2) If J and E are monomial ideals in S , then the pair ( J, E ) is G -nice.(3) If J and E are ideals in S whose generators involve disjoint sets of variables,then the pair ( J, E ) is G -nice.(4) (Conca [1]) Let X = ( x ij ) be an n × m matrix of indeterminates and let Z be a set of consecutive rows (or columns) of X . For t an integer with1 ≤ t ≤ min { n, m } we let J = I t ( X ) be the ideal in S = K [ X ] generated bythe t -minors of X . Also, let E = I t − ( Z ) ⊂ S . Then ( J, E ) is a G -nice pairof ideals, according to [1, Proposition 3.2].More generally, [1, Proposition 3.3] states that when Y is a ladder , J = I t ( X ) and E = I t − ( Y ∩ Z ), then the pair ( J, E ) is G -nice. We refer to [1]for the unexplained terminology and further details on this topic.(5) Ideals generated by various minors in a generic matrix are a source of G -nicepairs of ideals, see [6, Lemma 4.2], [7, Lemma 2.10].The following results characterize situations when one of the ideals of the G -pairis generated by a regular sequence. Proposition 1.6.
Let J be an ideal in the polynomial ring S and let f ∈ S whichis regular on S/J . The following conditions are equivalent:(a) in(
J, f ) = in( J ) + (in( f )) ;(b) in( f ) is regular on S/ in( J ) .Proof. We note that since f is regular on S/J we get thatin( J ∩ ( f )) = in( f J ) = in( f ) in( J ) . By Theorem 1.1(d), property (a) is equivalent to in( J ∩ ( f )) = in( J ) ∩ (in( f )). Thatin turn is equivalent to in( f ) in( J ) = in( J ) ∩ (in( f )), which is a restatement of thecondition (b), since S is a domain. (cid:3) Corollary 1.7.
Let J be any ideal in the polynomial ring S and let f , . . . , f r in S be a regular sequence on S/J . Then the sequence in( f ) , . . . , in( f r ) is regular on S/ in( J ) if and only if in( J, f , . . . , f r ) = in( J ) + (in( f ) , . . . , in( f r )) . In particular, if f , . . . , f r is a regular sequence on S , then in( f ) , . . . , in( f r ) isregular on S if and only if { f , . . . , f r } is a Gr¨obner basis for ( f , . . . , f r ) .Proof. This follows from Proposition 1.6 by induction on r . (cid:3) he G -nice condition is also connected to the distributivity property in the latticeof ideals of S , as the following result shows. Proposition 1.8.
Let
J, E and E ′ be ideals in the polynomial ring S such that ( J, E ) and ( J, E ′ ) are G -nice pairs. The following conditions are equivalent:(a) ( J + E ) ∩ ( J + E ′ ) = J + ( E ∩ E ′ ) and ( J, E ∩ E ′ ) is a G -nice pair of ideals;(b) in(( J + E ) ∩ ( J + E ′ )) = in( J ) + in( E ∩ E ′ ) .Proof. (a) ⇒ (b) is straightforward.(b) ⇒ (a): We denote I = J + E and I ′ = J + E ′ . We have thatin( J ) + in( E ∩ E ′ ) ⊆ in( J + ( E ∩ E ′ )) ⊆ in( I ∩ I ′ )and thus, by (b), these inclusions are in fact equalities. In particular, in( J ) + in( E ∩ E ′ ) = in( J + ( E ∩ E ′ )), hence the pair ( J, E ∩ E ′ ) is G -nice.On the other hand, since J + ( E ∩ E ′ ) ⊆ I ∩ I ′ and in( J + ( E ∩ E ′ )) = in( I ∩ I ′ ),it follows that I ∩ I ′ = J + ( E ∩ E ′ ). (cid:3) The two parts of condition (a) in Proposition 1.8 are independent, as the followingexample shows.
Example 1.9. (1) Let J = ( x + y + z ), E = ( xy, y + yz ) and E ′ = ( xy, y + yz + x + y + z ) be ideals in S = K [ x, y, z ]. Then I = J + E = J + E ′ =( J, xy ).On S we consider the reverse lexicographic monomial order (or revlex, forshort) with x > y > z . We have in( I ) = ( x , xy, y ), in( E ) = in( E ′ ) =( xy, y ) and one can check with Singular ([2]) that in( E ∩ E ′ ) = ( xy, y ).Therefore, ( J, E ) and (
J, E ′ ) are G -nice pairs of ideals, ( J + E ) ∩ ( J + E ′ ) = J + ( E ∩ E ′ ) = I , but the pair ( J, E ∩ E ′ ) is not G -nice.(2) In S = K [ x, y, z, t ] let J = ( x + y + z , xy − t ), E = ( − y zt + xz + t , x yz − zt + xyt , x zt + y z − x yt + yz , yzt − xy t − x z − x zt , − zt − y z + xyt − y z − y z + x t − z ) and E ′ = ( xz − yzt , y zt − z t , y z + z + x yzt , y z t + y z t + x zt , − z t − yz t − x y zt ).We set I = J + E and I ′ = J + E ′ .We claim that the inclusion J + ( E ∩ E ′ ) ⊂ I ∩ I ′ is strict. Indeed,considering the reverse lexicographic order on S with x > y > z > t , one cancheck with Singular ([2]) that y z t ∈ in( I ∩ I ′ ), but y z t / ∈ in( J +( E ∩ E ′ )).However, one can verify that the pair ( J, E ∩ E ′ ) is G -nice.The following result is a dual form of Proposition 1.8. Proposition 1.10.
Let
J, E and E ′ be ideals in the polynomial ring S such that thepairs ( J, E ) and ( J, E ′ ) are G -nice. The following conditions are equivalent:(a) J ∩ E + J ∩ E ′ = J ∩ ( E + E ′ ) and the pair ( J, E + E ′ ) is G -nice.(b) in( J ∩ E + J ∩ E ′ ) = in( J ) ∩ in( E + E ′ ) .Proof. (a) ⇒ (b): We have that in( J ∩ E + J ∩ E ′ ) = in( J ∩ ( E + E ′ )) = in( J ) ∩ in( E + E ′ ), since the pair ( J, E + E ′ ) is G -nice. b) ⇒ (a): We have thatin( J ) ∩ in( E + E ′ ) = in( J ∩ E + J ∩ E ′ ) ⊆ in( J ∩ ( E + E ′ )) ⊆ in( J ) ∩ in( E + E ′ ) , hence the inequalities in this chain become equalities. It follows that in( J ∩ ( E + E ′ )) = in( J ) ∩ in( E + E ′ ) and thus, by Theorem 1.1(d), the pair ( J, E + E ′ ) is G -nice.Also, in( J ∩ E + J ∩ E ′ ) = in( J ∩ ( E + E ′ )), therefore J ∩ E + J ∩ E ′ = J ∩ ( E + E ′ ). (cid:3) Given E a monomial ideal, we denote by G ( E ) its unique minimal set of monomialgenerators. Clearly, G ( E ) ∈ Gr¨ob( E ) for any monomial order. Proposition 1.11.
Let J be any ideal in the polynomial ring S and let ( E i ) i ∈ Λ bea family of monomial ideals in S such that the pair ( J, E i ) is G -nice for all i ∈ Λ .We set I i = J + E i for all i ∈ Λ , I = T i ∈ Λ I i and E = T i ∈ Λ E i . Then ( J, E ) is a G -nice pair and I = J + E .Also, if G J ∈ Gr¨ob( J ) then G J ∪ G ( E ) is a Gr¨obner basis of I .Proof. Since the E i ’s are monomial ideals then E is a monomial ideal, too. We havein( E i ) = E i for all i ∈ Λ and in( E ) = E . Obviously, in( J ) + E ⊆ in( J + E ). On theother hand, in( J + E ) ⊆ T i ∈ Λ in( J + E i ) = T i ∈ Λ (in( J ) + E i ) = in( J ) + T i ∈ Λ E i =in( J ) + E . Thus, the pair ( J, E ) is G -nice.Since J + E ⊆ I and in( J + E ) = in( I ), it follows that I = J + E . The lastassertion follows immediately. (cid:3) The following proposition shows that the G -nice property behaves well withrespect to taking sums of ideals. For any positive integer m we denote [ m ] = { , . . . , m } . Proposition 1.12.
Let E , . . . , E m be ideals in S such that ( E i , E j ) is a G -nicepair for all ≤ i, j ≤ m . Let X ⊂ [ m ] . We denote E X = P i ∈ X E i and E X c = P j ∈ [ m ] \ X E j . Then ( E X , E X c ) is a G -nice pair of ideals.Proof. For all i we pick a Gr¨obner basis G i ∈ Gr¨ob( E i ). We claim that G Y = S i ∈ Y G i is a Gr¨obner basis of E Y , for any Y ⊆ [ m ].If | Y | = 1, there is nothing to prove. Assume | Y | ≥ f, g ∈ G Y . If f, g ∈ G i , for some i ∈ Y , then S ( f, g ) → G i S ( f, g ) → G Y
0. If f ∈ G i and g ∈ G j , with i = j in Y , then S ( f, g ) → G i ∪ G j
0, since G i ∪ G j is aGr¨obner basis of E i + E j . It follows that S ( f, g ) → G Y
0. Thus, G Y is a Gr¨obnerbasis of E Y , which proves our claim.In follows that G [ m ] = G X ∪ G X c is a Gr¨obner basis for E [ m ] = E X + E X c , andtherefore the pair ( E X , E X c ) is G -nice. (cid:3) Corollary 1.13. If J is any ideal and ( E i ) i ∈ Λ is a family of monomial ideals, suchthat the pair ( J, E i ) is G -nice for all i ∈ Λ , then the pair ( J, P i ∈ Λ E i ) is G -nice.Proof. Note that P i ∈ Λ E i can be written as the sum of finitely many terms in thesum. On the other hand, any two monomial ideals form a G -nice pair, so theconclusion follows by Proposition 1.12. (cid:3) . Creating Gr¨obner-nice pairs
Let J be an ideal in S . Given any ideal E ⊂ S such that ( J, E ) is not a G -nicepair, we are interested in finding ideals F in S , “close” to E , such that(1) J + F = J + E and ( J, F ) is a G -nice pair . Lemma 2.1. If ( J, F ) is a G -nice pair with E ⊇ F so that J + E = J + F , then ( J, E ) is a G -nice pair.Proof. We have in( J + E ) = in( J + F ) = in( J ) + in( F ) ⊆ in( J ) + in( E ), hencein( J + E ) = in( J ) + in( E ) and ( J, E ) is a G -nice pair. (cid:3) Based on Lemma 2.1, for (1) we should look at ideals F ⊇ E . In general, ( J, J + E )is a G -nice pair, so in the worst case we may take F = J + E . But sometimes, it isalso the best choice, as the following example shows. Example 2.2.
Let J = ( x − y ) and E = ( x ). We consider on S = K [ x, y ] therevlex order with x > y . Let I = J + E . Then in( I ) = I = ( x , y ). Let F ⊇ E bean ideal with J + F = I and in( I ) = in( J ) + in( F ). We claim that F = I .Indeed, since y ∈ in( F ), there exists f ∈ F with LT( f ) = y . It follows that f = y + ax + by + c , where a, b, c ∈ K . Since y , f ∈ I , we get that ax + by + c ∈ I ,and therefore a = b = c = 0. Thus y ∈ F = I . Remark 2.3.
For J and E ideals in S , assume the ideal F satisfies (1) and F ⊇ E .In order to find an ideal E ′ ⊆ S such that E ⊆ E ′ ⊆ F and ( J, E ′ ) is a G -nice pair,where E ′ is as small as possible, a natural approach is the following. Set I = J + E .We write in( I ) = in( J ) + in( E ) + ( m , . . . , m s ) , where m , . . . , m s are the monomials in G (in( I )) \ (in( J ) + in( E )). Since in( I ) =in( J + F ) = in( J )+in( F ), it follows that m , . . . , m s ∈ in( F ). We choose g , . . . , g s ∈ F such that in( g i ) = m i for 1 ≤ i ≤ s . Let E = E + ( g , . . . , g s ). Then J + E = J + E = I andin( I ) = in( J ) + in( E ) + ( m , . . . , m s ) ⊆ in( J ) + in( E ) ⊆ in( J + E ) = in( I ) . Thus (
J, E ) is a G -nice pair of ideals.The following example shows that given the ideal I there does not always exist aminimal ideal E ′ (eventually containing E ) such that ( J, E ′ ) is a G -nice pair with I = J + E ′ . Example 2.4. (1) We consider the lexicographic order on S = K [ x, y ] inducedby x > y . Let J = ( y ) ⊂ I = ( y , x − y , xy ). Then in( I ) = ( y , x , xy ).We define the sequence ( α k ) k ≥ by α = 5 and α k +1 = 3 α k − k ≥
1. Let g k = xy − y α k and E k = ( x − y , xy − y α k ). One can easily check that J + E k = I ,and in( E k ) = ( x , xy, y α k − ). Therefore, ( J, E k ) is a G -nice pair of ideals for all k .Since xg k − y ( x − y ) = y − xy α k = y − xyg k − y α k − , it follows that y − y α k − ∈ E k . Then g k +1 = g k + y α k − y α k − = g k + y α k − ( y − y α k − ). Hence E k ( E k +1 for ll k . We also note that ∩ k ≥ E k = ( x − y ), ( J, ( x − y )) is a G -nice pair of ideals,and J + ( x − y ) ( I .(2) In S = K [ x, y, z ] ordered lexicographically (with x > y > x ) we let J =( x − y , z ), E = ( xy ), F = ( xy, y , z ). Denote I = J + E = J + F = ( x − y , z , xy ).We note that in( I ) = ( x , xy, y , z ) = in( J ) + in( F ). Therefore, ( J, E ) is not a G -nice pair, while ( J, F ) is one. We set E k = ( xy, y + z k +1 + z k + · · · + z , z k +2 ). Then E ( E k +1 ( E k ( F , while ( J, E k ) is a G -nice pair for all k ≥ Definition 2.5.
Consider the ideal J ⊂ S and G J ∈ Gr¨ob( J ) . A normal form withrespect to G J is a map N F ( −|G J ) : S → S , which satisfies the following conditions:(i) N F (0 |G J ) = 0 ;(ii) if N F ( g |G J ) = 0 then in( N F ( g |G J )) / ∈ in( J ) ;(iii) g − N F ( g |G J ) = P f ∈G J c f f , where c f ∈ S and in( g ) ≥ in( c f f ) , for all f ∈ G J with c f = 0 .Moreover, N F is called a reduced normal form , if for any f ∈ S no monomial of N F ( f |G J ) is contained in in( G J ) . Given the ideals
J, E in S , G J ∈ Gr¨ob( J ) and N F ( −|G J ) a normal form withrespect to G J , we denote N F ( E |G J ) = ( N F ( g |G J ) : g ∈ E ). Proposition 2.6.
With notation as above, we have:(i) J + E = J + N F ( E |G J ) ;(ii) ( J, N F ( E |G J )) is a G -nice pair of ideals;(iii) if N F is a reduced normal form and E = N F ( E |G J ) , then any h ∈ J + E can be written as h = f + g , such that f ∈ J , g ∈ E and no monomial of g is contained in in( J ) ;(iv) if ( E i ) i ∈ Λ is a family of ideals with E i = N F ( E i |G J ) for all i ∈ Λ , then N F \ i ∈ Λ E i |G J ! ⊆ \ i ∈ Λ E i . Proof. ( i ) : If h ∈ J + E , then h = N F ( h |G J )+ P f ∈G J c f f , and therefore N F ( h |G J ) ∈ J + E .( ii ) : Since, by ( i ), J + E = J + N F ( E |G J ), it is enough to consider the case E = N F ( E |G J ) and to prove that in( J + E ) ⊆ in( J ) + in( E ). Let h ∈ J + E .Then h = j + g where j ∈ J and g ∈ E . We also write g = j + g , where welet g = N F ( g |G J ) ∈ N F ( E |G J ) = E . Thus h = ( j + j ) + g . If g = 0, thenin( h ) ∈ in J . Otherwise, by the definition of the normal form, in( g ) / ∈ in( J ) whichimplies in( h ) ∈ in( J ) + in( E ).( iii ) : For any h ∈ J + E , the decomposition h = ( h − N F ( h |G J )) + N F ( h |G J )satisfies the required condition.( iv ) is straightforward. iven the ideals J, E ⊆ S we introduce the sets E J,E = { F ⊆ S : E ⊆ F, J + E = J + F, ( J, F ) is a G -nice pair of ideals } , E mJ,E = { F ∈ E J,E : F is a monomial ideal } The previous discussion shows that the set E J,E may not have a minimal element.However, when E is a monomial ideal and E mJ,E = ∅ , then the latter set has aminimum. Definition 2.7.
Let J be an ideal in S , and E a monomial ideal in S . The G -nicemonomial closure of E with respect to J is the (monomial) ideal b E = \ E ⊆ F, F monomial ideal , ( J,F ) is G -nice F. The ideal b E naturally depends on the ideal J , although this is not reflected in thenotation. We prefer not to complicate the notation since it will be clear from thecontext what J is. Proposition 2.8.
Let J be any ideal in S , and E a monomial ideal in S . Then ( J, b E ) is a G -nice pair. Moreover, if E mJ,E = ∅ , then b E is the smallest element in E mJ,E with respect to inclusion.Proof. The first part follows from Proposition 1.11. For the second assertion, notethat if F ∈ E mJ,E then J + E ⊆ J + b E ⊆ J + F , hence J + E = J + b E , b E ∈ E mJ,E andit is its smallest element. (cid:3) Corollary 2.9.
With notation as in Proposition 2.8, if J is a binomial ideal, then J + E = J + b E .Proof. Note that if b is any binomial in S and m is any monomial in S , then their S -polynomial S ( b, m ) is a monomial. Also, since J is a binomial ideal, it has aGr¨obner basis G J consisting of binomials. Therefore, we can define the monomialideal F which extends E by adding the monomials S ( b, m ) where m ∈ G ( E ) and b ∈ G J . Then F ∈ E mJ,E , and we apply Proposition 2.8. (cid:3) The ideal b E can be computed as follows. Remark 2.10.
Let J ⊂ S be an ideal, and let E ⊂ S be a monomial ideal. Let F ⊂ S be any monomial ideal such that ( J, F ) is a G -nice pair and E ⊆ F . Let G be the minimal monomial generators of in( J + E ) which are not in in( J ) nor inin( E ). Clearly, G ⊂ F . We let E = E + ( G ). If ( J, E ) is G -nice, then b E = E .Else, we argue as above and we get a chain of monomial ideals E ⊆ E ⊆ · · · ⊆ F .By notherianity, this chain stabilizes at some point E i = E i +1 = · · · and we get b E = E i . Proposition 2.11.
Let J be a binomial ideal and let ( E i ) i ∈ Λ be a family of monomialideals. Assume F i ⊇ E i are monomial ideals such that ( J, F i ) is a G -nice pair and J + E i = J + F i , for all i ∈ Λ . Then \ i ∈ Λ ( J + E i ) = J + \ i ∈ Λ b E i = J + \ i ∈ Λ F i . roof. Using the Corollary 2.9 we have that J + E i = J + b E i = F + F i for all i ∈ Λ,hence \ i ∈ Λ ( J + E i ) = \ i ∈ Λ ( J + b E i ) = \ i ∈ Λ ( J + F i ) . On the other hand, by Proposition 2.8, ( J, b E i ) is a G -nice pair for all i . Now usingProposition 1.11 we get that T i ∈ Λ ( J + b E i ) = J + T i ∈ Λ b E i and T i ∈ Λ ( J + F i ) = J + T i ∈ Λ F i . (cid:3) Example 2.12.
We consider the revlex order with x > y > z on S = K [ x, y, z ].Let J = ( x + y + z ) and E = ( xy ) be ideals in S . Note that G = { x + y + z , xy, y + yz } is a Gr¨obner basis of I = J + E . Therefore, in( I ) = ( x , xy, y )strictly includes in( J ) + in( E ) = ( x , xy ), and the pair ( J, E ) is not G -nice.Let F ⊂ S be any monomial ideal such that the pair ( J, F ) is G -nice and E ⊆ F .Since in( J + E ) ⊆ in( J + F ) = ( x ) + F , it follows that ( xy, y ) ⊂ F . Let E =( xy, y ). We have ( x , xy, y , yz ) = in( J + E ′ ) ⊆ ( x ) + F . Thus yz ∈ F . Clearly, E = ( xy, y , yz ) ⊆ F . Since ( J, E ) is a G-nice pair, we conclude that E = b E .3. A special class of Gr¨obner-nice pairs of ideals
To verify if a set is a Gr¨obner basis implies computing the S -polyonomial of anytwo elements in the set and testing if it reduces to zero with respect to the givenset, see [4, 3]. Inspired by this, we propose the following. Definition 3.1.
Let
J, E be ideals in S and G J ∈ Gr¨ob( J ) . We say that E is S -nicewith respect to G J if for any f ∈ G J and g ∈ E we have S ( f, g ) ∈ E . Example 3.2. If J ⊆ E , then E is S -nice with respect to G J for any G J ∈ Gr¨ob( J ). Proposition 3.3.
Assume the ideal E is S -nice with respect to G J ∈ Gr¨ob( J ) . Then ( J, E ) is a G -nice pair of ideals.Proof. Let G E be any Gr¨obner basis for E . We claim that G E ∪ G J is a Gr¨obner basisfor E + J . Indeed, we only need to consider S -polynomials S ( f, g ) where f ∈ G J and g ∈ G E . Since S ( f, g ) ∈ E we infer that the former reduces to 0 w.r.t. G J ∪ G E .Applying Theorem 1.1(c) finishes the proof. (cid:3) Proposition 3.4.
Let
J, E be ideals in S and G J ∈ Gr¨ob( J ) . The following state-ments are equivalent:(a) the ideal E is S -nice with respect to G J ;(b) for any G E ∈ Gr¨ob( E ) , for any f ∈ G J and g ∈ G E one has that S ( f, g ) ∈ E ;(c) there exists G E ∈ Gr¨ob( E ) such that for any f ∈ G J and g ∈ G E one has that S ( f, g ) ∈ E .Proof. The implications (a) ⇒ (b) ⇒ (c) are clear. We suppose (c) holds. Withoutloss of generality, we may also assume that the polynomials in G J and G E are monic.Let f ∈ G J and g ∈ G E . We can write g = P pi =1 u i g i where g i ∈ G E and in( g ) ≥ n( u i g i ) for i = 1 , . . . , p and such that LT( g ) = LT( u g ). We set h = g − u g . Then S ( f, g ) = lcm(in( f ) , in( g ))in( f ) f − lcm(in( f ) , in( g ))in( g ) g = lcm(in( f ) , in( g ))in( f ) ( f − LT( f )) − lcm(in( f ) , in( g ))in( g ) ( g − LT( u g ))= lcm(in( f ) , in( g ))in( f ) ( f − LT( f )) − lcm(in( f ) , in( g ))in( g ) ( u g − LT( u g )) − lcm(in( f ) , in( g ))in( g ) h. Note that u g − LT( u g ) = LT ( u )( g − LT( g )) + ( u − LT ( u )) g . Thus, S ( f, g ) = lcm(in( f ) , in( g ))in( f ) ( f − LT( f )) − lcm(in( f ) , in( g ))in( g ) ( g − LT( g )) − lcm(in( f ) , in( g ))in( g ) (( u − LT ( u )) g + h )= lcm(in( f ) , in( g ))lcm(in( f ) , in( g )) S ( f, g ) − lcm(in( f ) , in( g ))in( g ) (( u − LT ( u )) g + h ) . Since S ( f, g ) , g , h ∈ E we obtain that S ( f, g ) ∈ E , too. This proves statement(a). (cid:3) The S -nice property is stable when taking intersections. Proposition 3.5.
Let J be an ideal in S and G J ∈ Gr¨ob( J ) . Assume that in thefamily of ideals ( E i ) i ∈ Λ each is S -nice with respect to G J . Then(a) the ideal T i ∈ Λ E i is S -nice with respect to G J ;(b) if ( E i , E j ) is a G -nice pair for all i, j ∈ Λ , then P i ∈ Λ E i is S -nice with respectto G J .Proof. (a): Let f ∈ G J and g ∈ T i ∈ Λ E i Since E i is S -nice w.r.t. G J we get that S ( f, g ) ∈ E i for all i ∈ Λ. This proves (a).(b): For all i ∈ Λ we pick G i ∈ Gr¨ob( E i ). Arguing as in the proof of Proposi-tion 1.12 we get that G = S i ∈ Λ G i is a Gr¨obner basis for P i ∈ Λ E i . Then for any f ∈ G J and any g ∈ G we have that S ( f, g ) ∈ P i ∈ Λ E i . Conclusion follows byProposition 3.4. (cid:3) An immediate consequence of the previous result is the following related form ofProposition 1.8.
Corollary 3.6.
Let
J, E and E ′ be ideals in S and G J ∈ Gr¨ob( J ) . Assume that E and E ′ are S -nice with respect to G J . Then the following conditions are equivalent:(a) ( J + E ) ∩ ( J + E ′ ) = J + ( E ∩ E ′ ) ;(b) in(( J + E ) ∩ ( J + E ′ )) = in( J ) + in( E ∩ E ′ ) . In view of Proposition 3.5, for any ideal we can define its S -nice (monomial)closure. efinition 3.7. Let J be any ideal in S and let G J ∈ Gr¨ob( J ) . For any ideal E ⊂ S we set e E = \ E ⊆ F, F is S -nice w.r.t. G J F and we call it the S -nice closure of E with respect to G J .Moreover, if E is a monomial ideal, we set E ♯ = \ E ⊆ F, F monomial ideal is S -nice w.r.t. G J F and we call it the S -nice monomial closure of E with respect to G J . As with Definition 2.7, we prefer not to complicate notation and include G J in it,as it will be clear from the context the Gr¨obner basis which is used. Remark 3.8.
By Proposition 3.5, e E (resp. E ♯ ) is indeed the smallest ideal (resp.the smallest monomial ideal) in S which is S -nice with respect to G J . Clearly, E ⊆ e E ⊆ E ♯ , and also b E ⊆ E ♯ . By Example 3.2, the ideal J + E is S -nice w.r.t. G J , hence J + E ⊆ J + e E ⊆ J + E . The latter implies that J + E = J + e E. Example 3.9. In S = K [ x, y ] we consider J = ( x + y ) with G J = { x + y } and E = ( x ). If x > y then S ( x + y , x ) = y , and so e E = ( x , y ) = J + E . On theother hand, if y > x then S ( x + y , x ) = x ∈ E and thus e E = E = ( x ).In general, the following proposition is useful for computing e E and E ♯ . Proposition 3.10.
Let
J, E be ideals in S and G J ∈ Gr¨ob( J ) . Then(a) e E = P i ≥ E i , where the ideal E i is defined inductively as E = E and E i +1 = E i + ( S ( f, g ) : f ∈ G J , g ∈ E i ) for all i > .(b) If E is a monomial ideal, then E ♯ = P i ≥ F i , where F = E and F i + i is theideal generated by the monomial terms of the polynomials in e F i , for all i > .Proof. (a): If f ∈ G J and g ∈ G i , then S ( f, g ) ∈ E i +1 . Therefore, P i ≥ E i is S -nicewith respect to G J . Conversely, since E ⊂ e E and e E is S -nice w.r.t. G J , it followsthat S ( f, g ) ∈ e E for any f ∈ G J and g ∈ E . Therefore, E ⊆ e E . Inductively, we getthat E i ⊆ e E for any i ≥
0. This completes the proof.(b): If m ∈ F i is a monomial and f ∈ G J , then S ( f, m ) ∈ e F i ⊆ F i +1 . Therefore, P i ≥ F i is S -nice with respect to G J . Conversely, if m ∈ E is a monomial and f ∈ G J , then S ( f, m ) ∈ E ♯ . Since E ♯ is a monomial ideal,, any monomial which isin the support of S ( f, m ) is in E ♯ . Therefore, F ⊆ E ♯ . Inductively, we get F i ⊆ E ♯ for all i ≥
0. Thus P i ≥ F i = E ♯ . (cid:3) Example 3.11. In S = K [ x, y, z ] we consider the revlex order with x > y > z and the ideals J = ( x + y + z ), E = ( xy ), E = ( xy, y + yz ) and E =( xy, y + yz + x + y + z ). From Example 1.9 we have that J + E = J + E = J + E ,and that ( J, E ), ( J, E ) are G -nice pairs of ideals. Let G J = { x + y + z } ∈ Gr¨ob( J ).We claim that E is S -nice with respect to G J , but E is not. e note that G = { xy, y + yz } ∈ Gr¨ob( E ) and G = { xy, y + yz + x + y + z , x + xz } ∈ Gr¨ob( E ). Also, S ( x + y + z , xy ) = y + yz ∈ E and S ( x + y + z , y + yz ) = y ( y + z ) − x ( y + yz ) = y ( y + yz ) − xy ( xy + xz ) ∈ E .This proves the claim. Moreover, since S ( x + y + z , xy ) = y + yz we infer that,when computed with respect to G J , e E = E .Let U be a monomial ideal in S which is S -nice with respect to G J , and E ⊆ U .Since S ( x + y + z , xy ) = y + yz ∈ U one has that y , yz ∈ U . Let L =( xy, y , yz ). As S ( x + y + z , y ) = y + y z ∈ L and S ( x + y + z , yz ) = y z + yz ∈ L , it follows that L is S -nice with respect to G J and moreover, E ♯ = L .We also remark that J + E ( J + E ♯ . Proposition 3.12.
Let J be an ideal in S , G J ∈ Gr¨ob( J ) and E a monomial idealin S . Then(a) if there exists a monomial ideal F in S which is S -nice with respect to G J and J + E = J + F , then J + E ♯ = J + E ;(b) if J is a binomial ideal and G J ∈ Gr¨ob( J ) consists of binomials, then e E = E ♯ .Proof. Part (a) is clear. For (b) we let E = E and we note that S ( b, m ) is amonomial, for any binomial b and monomial m . Therefore, using the notationfrom Proposition 3.10 we obtain an ascending chain of monomial ideals E i +1 = E i + ( S ( f, g ) : f ∈ G J , g ∈ E i ). This shows that e E = P i ≥ E i is a monomial idealand we are done. (cid:3) Acknowledgement . We thank Aldo Conca for pointing our attention to his paper([1]). We gratefully acknowledge the use of the computer algebra system Singular([2]) for our experiments.The authors were partly supported by a grant of the Romanian Ministry of Edu-cation, CNCS–UEFISCDI under the project PN-II-ID-PCE-2011-3–1023.The paper is in final form and no similar paper has been or is being submittedelsewhere.
Note added in the proof . After this paper was finished, we learned from MatteoVarbaro that the equivalence of conditions (a) and (d) in Theorem 1.1 has also beenproved in the Ph.D. thesis of Michela Di Marca,
Connectedness properties of dualgraphs of standard graded algebras , University of Genova, December 2017.
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Prime splittings of determinantal ideals , Comm. Algebra (2018),2278–2296. Mircea Cimpoeas¸, Simion Stoilow Institute of Mathematics of the RomanianAcademy, Research unit 5, P.O.Box 1-764, Bucharest 014700, Romania
Email address : [email protected] Dumitru I. Stamate, Faculty of Mathematics and Computer Science, Universityof Bucharest, Str. Academiei 14, Bucharest, Romania, andSimion Stoilow Institute of Mathematics of the Romanian Academy, Researchgroup of the project ID-PCE-2011-1023, P.O.Box 1-764, Bucharest 014700, Romania
Email address : [email protected]@fmi.unibuc.ro