aa r X i v : . [ m a t h . C O ] F e b BINOMIAL REGULAR SEQUENCES AND FREE SUMS
WINFRIED BRUNSA
BSTRACT . Recently several authors have proved results on Ehrhart series of free sumsof rational polytopes. In this note we treat these results from an algebraic viewpoint.Instead of attacking combinatorial statements directly, we derive them from structuralresults on affine monoids and their algebras that allow conclusions for Hilbert and Ehrhartseries. We characterize when a binomial regular sequence generates a prime ideal or evennormality is preserved for the residue class ring.
1. I
NTRODUCTION
Recently several authors have proved results on Ehrhart series of free sums of rationalpolytopes; see Beck and Hos¸ten [1], Braun [5] and Beck, Jayawant, and McAllister [2].In this note we treat these results from an algebraic viewpoint. Instead of attacking com-binatorial statements directly, we derive them from structural results on affine monoidsand their algebras that allow conclusions for Hilbert and Ehrhart series. This procedurefollows the spirit of the monograph [6] to which the reader is referred for affine monoidsand their algebras.Our approach is best explained by the motivating example, namely free sums of ra-tional polytopes and their Ehrhart series. The
Ehrhart series of a rational polytope P isthe (formal) power series E P = (cid:229) ¥ k = E ( P , k ) t k where E ( P , k ) counts the lattice points inthe homothetic multiple kP ; see Beck and Robbins [3] for a gentle introduction to thefascinating area of Ehrhart series.One says that R = conv ( P ∪ Q ) is the free sum of the rational polytopes P and Q if0 ∈ P ∩ Q , the vector subspaces R P and R Q intersect only in 0, and ( Z m ∩ R R ) = ( Z m ∩ R P ) + ( Z m ∩ R Q ) . It has been proved in [2, Theorem 1.4] that the Ehrhart series of the three polytopes arerelated by the equation E R = ( − T ) E P E Q ( ∗ ) if and only if at least one of the polytopes P and Q is described by inequalities of type a x + · · · + a m x m ≤ b with a , . . . , a n ∈ Z and b ∈ { , } .We approach the validity of equation ( ∗ ) by considering the Ehrhart monoid E ( P ) = { ( x , k ) : x ∈ kP ∩ Z m } = R + ( P × { } ) ∩ Z m + . The Ehrhart series is the Hilbert series of E ( P ) or, equivalently, of the monoid algebra K [ E ( P )] over a field K , and therefore standard techniques for computing Hilbert seriescan be applied. Ehrhart monoids are normal: if nx ∈ E ( P ) for some x in the group Z E ( P ) and n ∈ Z + , n >
0, then x ∈ E ( P ) . The normality of a monoid M is equivalent to thenormality of K [ M ] . The free sum arises from the free join by a projection along the line through the rep-resentatives of the origins in P and Q , respectively, in the free join. The algebraic coun-terpart of the projection is the passage from the direct sum E ( P ) ⊕ E ( Q ) to a quotient M . By Corollary 2.5, M is automatically an affine monoid in this situation. However,the crucial question is whether M is naturally isomorphic to E ( R ) , and this is the case ifand only if M is normal. In terms of monoid algebras, the quotient is given by residueclasses modulo a binomial. Therefore the validity of ( ∗ ) can be seen as a special case ofthe preservation of normality modulo a binomial in a normal monoid algebra, for whichTheorem 3.3 provides a necessary and sufficient condition.In [2, Corollary 5.8] the intersection of R P and R Q in 0 has been generalized to theintersection of the affine hulls aff ( P ) and aff ( Q ) in a single rational point z ∈ P ∩ Q , andthe corresponding generalization of ( ∗ ) follows by entirely the same argument (Corollary3.7).Our discussion above shows that it is worthwhile to characterize when a binomial (ormore generally a regular sequence of binomials) in an affine monoid domain generates aprime ideal (Theorem 2.1 and Corollary 2.3), or when even normality is preserved modulosuch a binomial (Theorem 3.3 and Corollary 3.4). Also the main reduction step in Brunsand R¨omer [9] is of this type.It would be possible to mold the results of this note in the language of monoids andcongruences, but the ring-theoretic environment is much richer in notions and methods,and results like Hochster’s theorem on the Cohen-Macaulay property of normal affinemonoid domains could hardly be formulated in pure monoid theory.This work was initiated by discussions with Serkan Hos¸ten about [1] and then drivenby the desire to prove the results of [2] and [5] in an algebraic way. We are grateful toMatthias Beck for directing our attention to these papers, and we thank Benjamin Braun,Serkan Hos¸ten, Tyrrell McAllister and Matteo Varbaro for their careful reading of a pre-liminary version and valuable suggestions.2. I NTEGRALITY An affine monoid is a finitely generated submonoid of a group Z m . It is positive if x , − x ∈ M implies x =
0. For a field K the monoid algebra K [ M ] is a finitely generated K -subalgebra of the Laurent polynomial ring K [ Z m ] . We write X x for the (Laurent) mono-mial with exponent vector x . Since the subgroup gp ( M ) of Z m generated by M is isomor-phic to Z d for d = rank M = rank gp ( M ) , the subalgebra K [ gp ( M )] ⊂ K [ Z m ] is a Laurentpolynomial ring in its own right . For an extensive treatment of affine monoids and theiralgebras we refer the reader to Bruns and Gubeladze [6], in particular to Chapter 4.A (multi)grading on a monoid M is a Z -linear map deg : gp ( M ) → Z d for some d > H ( M , g ) = { x ∈ M : deg x = g } is finite for all g , we can define theHilbert series H M ( T ) = (cid:229) g ∈ Z d H ( M , g ) T g where T stands for indeterminates T , . . . , T d and T g = T g . . . T g d d . See [6, Ch. 6] forthe basic theorems on Hilbert series. A priori, H M ( T ) lives in the Z [ T , . . . , T d ] -module Z [[ T , . . . , T d ]] of formal Laurent series. INOMIAL REGULAR SEQUENCES AND FREE SUMS 3
Every grading on M is the specialization of the fine grading in which deg is simply thegiven embedding gp ( M ) ֒ → Z m . We denote the Hilbert series of the fine grading by H M .Since M can be recovered from H M , it is justified to call it the generating function of M .We say that M is positively (multi)graded if deg ( M ) is a positive submonoid of Z d andthe elements of K ⊂ K [ M ] are the only ones of degree 0. This implies the finiteness of theHilbert function. By the classical theorem of Hilbert-Serre, H M ( T ) is the Laurent seriesexpansion of a rational function (with respect to the positive submonoid deg ( M ) ).A few more pieces of terminology and notation: we say that a nonzero x ∈ Z n is uni-modular if x generates a direct summand. The cone generated by A ⊂ R n is denoted bycone ( A ) , and aff ( A ) is the affine subspace spanned by A .For the basic theory of zerodivisors, R -sequences and depth in Noetherian rings werefer the reader to Bruns and Herzog [7]. Theorem 2.1.
Let K be a field, M an affine monoid, and x , y ∈ M noninvertible, x = y.Then the following statements (1) and (2) are equivalent: (1) X x − X y generates a prime ideal in K [ M ] . (2) (a) X x , X y is a K [ M ] -sequence; (b) gp ( M ) / Z ( x − y ) is torsionfree.Moreover, if j : M → M ′ is a surjective homomorphism onto an affine monoid M ′ with rank M ′ = rank M − and j ( x ) = j ( y ) , then (1) and (2) are equivalent to (3) K [ M ′ ] = K [ M ] / ( X x − X y ) under the induced homomorphism.Finally, if in this situation M ′ is positively multigraded and j ( z ) = for all nonzero z ∈ M,then (1), (2), and (3) are equivalent to (4) H M ′ = ( − T g ) H M with respect to the induced grading on M, g = deg j ( x ) .Proof. Let us start with the implication (2) = ⇒ (1). First we prove that no monomial is azerodivisor modulo X x − X y if (2)(a) holds. In fact, suppose that X z is such a zerodivisor.Then it is contained in an associated prime ideal P of X x − X y . But P is an associatedprime ideal of any nonzero element of R = K [ M ] it contains (since R is an integral do-main). Therefore P is an associated prime ideal of X z as well. Associated prime idealsof monomials are generated by monomials [6, 4.9], and so P contains both X x and X y together with X x − X y . This is a contradiction since X x , X y is a regular sequence in thelocalization R P .It follows that ( X x − X y ) is the contraction of its extension to the Laurent polynomialring K [ gp ( M )] (see [6, 4.C]). So it is enough that ( X x − X y ) K [ gp ( M )] is a prime ideal.This follows from (2)(b) since ( X x − X y ) K [ gp ( M )] = ( − X y − x ) K [ gp ( M )] and X y − x is anindeterminate in K [ gp ( M )] after a suitable choice of a basis of gp ( M ) .For the converse we first derive (2)(a). If ( X x − X y ) is a prime ideal, then no monomialcan be a zerodivisor modulo X x − X y . On the other hand, if X y were a zerodivisor modulo X x , then it would be contained in an associated prime ideal P of X x . But such P ismonomial and also an associated prime ideal of X x − X y . Thus it would be equal to ( X x − X y ) , which is not monomial.(2)(b) follows from (1) since the primeness of the extension of ( X x − X y ) to K [ gp ( M )] evidently implies that gp ( M ) / Z ( x − y ) is torsionfree [6, 4.32]; also see Remark 2.2(c). WINFRIED BRUNS
For the equivalence of (3) to (1) and (2) we note that the natural surjection from K [ M ] to K [ M ′ ] factors through K [ M ] / ( X x − X y ) . Since rank M ′ = rank M −
1, one has dim K [ M ′ ] = dim K [ M ] − j is a height 1 prime ideal. Sothe natural isomorphism K [ M ′ ] = K [ M ] / Ker j turns into K [ M ′ ] = K [ M ] / ( X x − X y ) if andonly if Ker j = ( X x − X y ) .For statement (4) to make sense, we need that j ( z ) = z ∈ M . This assumptionimplies that we indeed obtain a multigrading on M by setting deg z = deg j ( z ) . Theequivalence of (4) follows by the same argument: one has H K [ M ] / ( X x − X y ) = ( − T g ) H K [ M ] since X x − X y is homogeneous of degree g , and H K [ M ′ ] = ( − T g ) H K [ M ] if and only if thetwo algebras are isomorphic. (cid:3) Remark 2.2. (a) Condition (3) has been formulated in view of the applications below. If(1) holds, then K [ M ] / ( X x − X y ) is automatically an affine monoid domain K [ M ′ ] whoseunderlying monoid is the image of M in gp ( M ) / Z ( x − y ) [6, 4.32].(b) It is not hard to see that monomials X x , . . . , X x n form a K [ M ] -sequence if and onlyif X x i , X x j is a K [ M ] -sequence for all i = j . Nevertheless condition (2)(a) is not easy tocheck in general. If K [ M ] is Cohen-Macaulay or satisfies at least Serre’s condition ( S ) ,for example if M is normal, then (2)(a) is equivalent to the fact that there is no facet F of cone ( M ) with x , y / ∈ F , or, in other words, every facet contains at least one of x or y . Indeed, in a ring satisfying ( S ) the associated prime ideals of non-zerodivisorshave height 1, and the height 1 monomial prime ideals are exactly those spanned by themonomials X z , z / ∈ F , for some facet F of cone ( M ) [6, 4.D].(c) One should note that K [ Z d ] / I is not only a domain, but even a regular domain if I is generated by binomials X x − X y , . . . , X x n − X y n such that x − y , . . . , x n − y n gen-erate a rank n direct summand. By induction it is enough to prove the claim for n = x = x , y = y . With respect to a of basis of Z d containing y − x as the first element, K [ Z d ] = K [ Y ± , . . . , Y ± d ] with Y = X y − x , and K [ Z d ] / ( − Y ) arises from the regulardomain K [ Y , . . . , Y d ] / ( − Y ) by the inversion of the monomials in Y , . . . , Y d .For a finite subset A ⊂ Z m let the (automatically positive) monoid M ( A ) over A be thesubmonoid of Z m + generated by the vectors ( x , ) ∈ Z m + , x ∈ A . This type of monoidwill play a special role later on, but is useful already now for the construction of examples. x yuw zv u x = y z vw x = yu = vw z F IGURE
1. Successive identification of lattice pointsThe geometry behind Theorem 2.1 is illustrated by Figure 1. We start from the monoid M ( A ) where A is the set of vertices of the direct product of the unit 2-simplex and theunit 1-simplex. The monoid M ′ arising from the identification of x and y is then defined INOMIAL REGULAR SEQUENCES AND FREE SUMS 5 by the 5 lattice points of the quadrangle in the middle, and if we further identify u and v ,we end with the line segment on the right with its 4 lattice points. The polytopes in themiddle and on the right are obtained from their left neighbors by projection along the linethrough the identified points, indicated by x = y and u = v .We generalize the theorem to sequences of more than two elements, leaving the gener-alization of (3) and (4) to the reader. Corollary 2.3.
With K and M as in Theorem 2.1, let x , . . . , x n , n ≥ , be noninvertibleelements of M. Then the following statements (1) and (2) are equivalent: (1) X x − X x , . . . , X x n − − X x n is a K [ M ] -sequence and generates a prime ideal P. (2) (a) X x , . . . , X x n is a K [ M ] -sequence; (b) x − x , . . . , x n − − x n generate a rank n − direct summand of gp ( M ) .Proof. For the proof of the implication (1) = ⇒ (2)(a) let Q be the prime ideal of K [ M ] generated by all noninvertible monomials. Since the associated prime ideals of monomialideals are themselves monomial, X x , . . . , X x n is a K [ M ] -sequence if and only if it is a K [ M ] Q -sequence, and the latter property follows from depth K [ M ] Q ′ ≥ n for all primeideals Q ′ ⊃ ( X x , . . . , X x n ) [7, 1.6.19].A prime ideal Q ′ ⊃ ( X x , . . . , X x n ) contains the regular sequence X x − X x , . . . , X x n − − X x n of length n − P . Moreover Q contains X x n and / ∈ P .This implies depth K [ M ] Q ′ ≥ n , and (2)(a) has been verified. (2)(b) follows since theextension of P to K [ gp ( M )] is a prime ideal [6, 4.32].For (2) = ⇒ (1) we use induction for which the starting case n = P ′ = ( X x − X x , . . . , X x n − − X x n − ) ; by induction K [ M ] / P ′ is an affine monoiddomain K [ M ′ ] (see Remark 2.2(a)). The only critical condition is whether X x n − , X x n is a K [ M ′ ] -sequence since (2)(b) of the theorem is evidently satisfied. Let Q ′ be a prime idealin K [ M ′ ] containing X x n − , X x n , and let Q be its preimage in K [ M ] . Then Q contains thetotal sequence X x , . . . , X x n , and we conclude depth K [ M ] Q ≥ n . But modulo the regularsequence X x − X x , . . . , X x n − − X x n − of length n − n −
2, andtherefore depth K [ M ′ ] Q ′ ≥
2. This makes it impossible that X n is a zerodivisor modulo X x n − in K [ M ′ ] . (cid:3) Remark 2.4. (a) For the proof of the implication (1) = ⇒ (2)(a) of Corollary 2.3 we haveonly used that X x − X x , . . . , X x n − − X x n , X x n is a K [ M ] -sequence. The converse does alsohold.Since ( X x − X x , . . . , X x n − − X x n , X x n ) = ( X x , . . . , X x n ) , the same argument that hasbeen used for (1) = ⇒ (2)(a) shows that X x − X x , . . . , X x n − − X x n , X x n is a K [ M ] Q ′ -sequence. The only problem is to lift regularity of the sequence to K [ M ] . We can nolonger use the fine grading, but it is sufficient that there is a multigrading for which(i) Q ′ is the ideal generated by the noninvertible homogeneous elements, and (ii) X x − X x , . . . , X x n − − X x n , X x n are homogeneous. Then we are dealing with homogeneous el-ements in the ∗ maximal ideal Q ′ of the ∗ local ring K [ M ] . See [7, 1.5.15(c)] that coversthe case of positive Z -gradings; however, it is solely relevant that the grading group istorsionfree (Bourbaki [4, Ch. 4, §
3, no. 1]).
WINFRIED BRUNS
It remains to find a suitable grading. To this end we let U be the saturation of Z ( x − x ) + · · · + Z ( x n − − x n ) in gp ( M ) . Then G = gp ( M ) / U is torsionfree, and the naturalhomomorphism gp ( M ) → G is the right choice.(b) We have assumed in Theorem 2.1 and in Corollary 2.3 that x , . . . , x n are nonin-vertible. If one allows that one of the x i is a unit in M , then (2)(a) makes no senseanymore since the definition of K [ M ] -sequence comprises the condition ( x , . . . , x n ) = K [ M ] . But dropping this requirement and keeping only that x i is not a zerodivisor modulo ( x , . . . , x i − ) for i = , . . . , n is not the way out.The ideal P generated by the X x i − − X x i is independent of the order of the x i , andespecially its primeness does not depend on the order. However, the second property in(1), namely that the generators form a K [ M ] -sequence, may be order sensitive if one of the x i is a unit and we have left the shelter of the “roof” Q ′ above. For a concrete example set M = M ( A ) where A is the set of the vertices 3-dimensional unit cube, and x , y , z are chosenas indicated in Figure 2. Then X x − X y , X y − X z , X z −
1, corresponding to x , y , z , ∈ M , is xy z F IGURE
2. The unit cubenot a K [ M ] -sequence, although the permutation X z − , X y − X z , X x − X y , correspondingto 0 , z , y , x ∈ M , is a K [ M ] -sequence, and both sequences generate the same prime ideal P .In fact, K [ M ] / P is isomorphic to the polynomial ring in one variable over K .This not really a surprise: in a non-local situation the fact that an ideal P is generatedby a regular sequence of length 3 does not imply that every length 3 sequence generating P is regular.The order that just made the generators of P a K [ M ] -sequence does always work: (2)(b)alone is equivalent to (1), provided x is a unit. Under this assumption all argumentsremain essentially unchanged, except that the set of monomial ideals containing x , . . . , x n is automatically empty.(c) Binomial regular sequences in polynomial rings K [ Z m + ] have been investigated inFischer, Morris and Shapiro [10] and Fischer and Shapiro [11].We now turn to a situation in which the conditions of Theorem 2.1 are automaticallysatisfied. Corollary 2.5.
Let L,M and N be affine monoids, j : M ⊕ N → L a surjective homomor-phism with rank L = rank M + rank N − , and suppose that j ( x ) = j ( y ) for x ∈ M, y ∈ N,x = or y = . Furthermore assume that nonzero j ( z ) = for all z ∈ M ⊕ N and let L bepositively multigraded such that deg j ( x ) is a unimodular element of the grading group.Then K [ L ] ∼ = K [ M ⊕ N ] / ( X x − X y ) and H L = ( − T g ) H M ⊕ N with respect to the gradingon M ⊕ N induced by the grading on L.
INOMIAL REGULAR SEQUENCES AND FREE SUMS 7
Proof.
We must verify the conditions (1)(a) and (b) of the theorem. For (a) the verificationis a trivial exercise. For (b) let G be the grading group of L . The grading on L inducesgradings on M and N via the embedding of M and N , respectively, into M ⊕ N . Considerthe homomorphism M ⊕ N → G ⊕ G , ( u , v ) ( deg u , deg v ) . Under this homomorphism x − y = ( x , − y ) goes to the unimodular element ( deg x , − deg y ) of G ⊕ G . Therefore x − y is unimodular in gp ( M ⊕ N ) . (cid:3) As a special case of Corollary 2.5 we can consider the free sum of point configurations.Following [2] let A , B ⊂ R m . We say that A ∪ B is the free sum of A and B if 0 ∈ A ∩ B and the vector subspaces R A and R B of R m intersect only in 0. The relationship between M ( A ∪ B ) and M ( A ) ⊕ M ( B ) is given by part (1) of the next corollary in terms of monoidalgebras. Corollary 2.6.
Let A and B be finite subsets of Z m such that A ∪ B is the free sum of A andB. Set x = ( , ) ⊕ , y = ⊕ ( , ) . Then (1) K [ M ( A ∪ B )] ∼ = K [ M ( A ) ⊕ M ( B )] / ( X x − X y ) ; (2) H M ( A ∪ B ) = ( − T m + ) H M ( A ) H M ( B ) .Proof. We set M = M ( A ) , N = M ( B ) and L = M ( A ∪ B ) . Then the natural embeddings M ⊂ L and N ⊂ L induce a surjective homomorphism M ⊕ N → L , ( x , k ) ⊕ ( y , l ) ( x + y , k + l ) . Both x and y go to ( , ) ∈ L ⊂ Z m + , and therefore to a unimodular element ingp ( L ) . It only remains to apply Corollary 2.5. (cid:3) Remark 2.7.
We have formulated Corollary 2.6 for the fine grading. Since every othergrading is a specialization of the fine grading, the formula in (2) holds for every coarsergrading as well. In particular it holds for the standard grading on M ( A ) , M ( B ) and M ( A ∪ B ) in which deg ( x , k ) = k ∈ Z .The formula in (2) was stated (for the standard grading) in [1, Lemma 10] without thefactor 1 − T m + . Therefore some of the results in [1] need an analogous correction, but thisonly concerns the denominators of the Hilbert series appearing there, and the statementsabout the numerator polynomials remain untouched.The construction of free sums has been generalized in [2] as follows. We considersubsets A and B of R m such that aff ( A ) and aff ( B ) meet in a single point p ; see Figure3 A B F IGURE
3. Intersection in a rational pointfor a very simple example. In this example H M ( A ∪ B ) = + T ( − T ) = ( − T ) H M ( A ) H M ( B ) in the standard grading. The “correction” 1 − T reflects that ( p , ) ∈ M ( A ) ∩ M ( B ) . WINFRIED BRUNS
Let A , B ⊂ Z m be finite and suppose that aff ( A ) and aff ( B ) meet exactly in p ; thennecessarily p ∈ Q m . We can form M ( A ) , M ( B ) and M ( A ∪ B ) . If p ∈ Z m ∩ A ∩ B , thenwe are in the situation of Corollary 2.6 after an affine-integral coordinate transformation.But the comparison of the monoids is already possible under a weaker assumption, assuggested by the example above and [2, Corollary 5.8], to which we will come back inCorollary 3.7. Corollary 2.8.
Let A , B ⊂ Z m be finite and suppose that aff ( A ) and aff ( B ) intersect in asingle point p . Furthermore suppose ( k p , k ) ∈ M ( A ) ∩ M ( B ) for the smallest k > suchthat kp ∈ Z m . Set x = ( k p , k ) ⊕ and y = ⊕ ( k p , k ) . Then (1) K [ M ( A ∪ B )] ∼ = K [ M ( A ) ⊕ M ( B )] / ( X x − X y ) ; (2) H M ( A ∪ B ) = ( − T ( kp , k ) ) H M ( A ) H M ( B ) .Proof. Since rank M ( A ∪ B ) = rank M ( A ) + rank M ( B ) − x − y needs a differentargument: it holds since ( k p , k ) has coprime entries. (Note that we have not defined k bythe condition that k p ∈ M ( A ) ∩ M ( B ) .) (cid:3)
3. N
ORMALITY
Let P ⊂ R m be a rational polytope. Then the (ordinary) Ehrhart function is given by E ( P , k ) = ( kP ∩ Z d ) and the corresponding generating function E P (cid:229) ¥ k = E ( P , k ) T k is the Ehrhart series . Inorder to interpret the Ehrhart series as a Hilbert series one forms the monoid E ( P ) = { ( x , k ) : x ∈ kP ∩ Z m } ⊂ Z m + . By Gordan’s lemma E ( P ) is an affine monoid, and the Ehrhart series of P is just thestandard Hilbert series of E ( P ) . We define the multigraded or fine Ehrhart series (orlattice point generating function) of P by E P = H E ( P ) . It is tempting to interpret the results in Section 2 as statements about Ehrhart series.Such an interpretation is indeed possible and will be given below, but it requires furtherhypotheses. Let us consider the situation of Corollary 2.6 and rational polytopes P and Q in R m , such that 0 ∈ P ∩ Q , the vector subspaces R P and R Q intersect only in 0, and ( Z m ∩ R R ) = ( Z m ∩ R P ) + ( Z m ∩ R Q ) . Then we say that conv ( P ∪ Q ) is the (convex) free sum of P and Q (see Henk, Richter-Gebert and Ziegler [12] for further information). The free sum of polytopes can be con-structed from the free join by projecting along the line through the representatives of theorigins in the free join. Figure 4 illustrates this construction.We would like to conclude that E R = ( − T m + ) E P E Q . This conclusion is equivalentto the fact that E ( R ) arises from E ( P ) ⊕ E ( Q ) via the construction in Corollary 2.5. Ingeneral this is not the case, even if the evidently necessary conditions are satisfied. INOMIAL REGULAR SEQUENCES AND FREE SUMS 9 x y x = y F IGURE
4. From the free join of two line segments to their free sum
Example 3.1.
Let P ⊂ R be the lattice polytope spanned by the points − ( e + e + e ) , e i , e i + e j , i , j = , , i = j ( e i denotes the i -th unit vector). For Q we choose the interval [ − , ] ⊂ R . Consider P and Q as lattice polytopes in R = R ⊕ R . Then R = conv ( P ∪ Q ) is indeed the free sum of P and Q .Set A = P ∩ Z and B = Q ∩ Z . That E ( Q ) = M ( B ) holds for trivial reasons, and usingNormaliz [8] one checks that E ( P ) = M ( A ) . One even has R ∩ Z = A ∪ B . Nevertheless E ( R ) = M ( A ∪ B ) . This can be checked by Normaliz directly or by inspection of theEhrhart series: E P ⊕ Q = + T + T + T + T ( − T ) = + T + T + T + T ( − T ) = ( − T ) E P E Q . As we will see in Corollary 3.6, this inequality is not a surprise.In the following we will have to adjoin inverse elements to the affine monoid M ; see [6,p. 62]. Relative to [6] we use the shortcut M [ − G ] = M [ − ( G ∩ M )] for faces G of cone ( M ) .Extensions of type M [ − G ] appear naturally when localizations of monoid domains are tobe considered since the subsets G ∩ M of M are exactly those complementary to primeideals. The following characterization of regular localizations K [ M ] P is only implicitlygiven in [6]. Lemma 3.2.
Let M be an affine monoid of rank d, K a field, and P a prime ideal in K [ M ] .Let Q be the (automatically prime) ideal generated by all monomials X x ∈ P and let F bethe face of cone ( M ) spanned by all y ∈ M, X y / ∈ Q. Then the following are equivalent: (1) K [ M ] Q is a regular local ring; (2) K [ M ] P is a regular local ring; (3) M [ − F ] is isomorphic to Z d − n ⊕ Z n + for some n, ≤ n ≤ d.Proof. The implications (3) = ⇒ (2) = ⇒ (1) hold since regularity is preserved underlocalizations.For (1) = ⇒ (3) it is enough that R = K [ M [ − F ]] is a regular ring; see [6, 4.45]. Thisfollows from general principles that hold for ∗ local rings; see the discussion in [6, p. 208].Nevertheless a direct argument may be welcome. The crucial observation is that every(prime) ideal of R generated by monomials is contained in Q ′ = QR .First we show that R is normal. To this end let R be the normalization of R . It is itselfan affine monoid domain and a finitely generated R -module. The localization ( R / R ) Q ′ vanishes since R Q ′ is regular and thus normal. But then R / R vanishes since its supportwould have to contain a monomial prime ideal if it were empty. By [6, 4.45] factoriality of R is sufficient for (3), and it holds if all monomial height 1prime ideals P ′ are principal (Chouinard’s theorem [6, 4.56]). But this follows by thesame argument that shows normality: a monomial generating the extension of P ′ to thefactorial ring R Q ′ must generate P ′ itself. (cid:3) The key to results about Ehrhart series is the preservation of normality in the situationof Theorem 2.1. As we will see, normality depends on the height of monoid elementsover facets: every x ∈ M has a well-defined (lattice) height over a facet F of cone ( M ) , wedenote it by ht F ( x ) . It is the number of hyperplanes between F and x parallel to F thatpass through lattice points and do not contain F ; so ht F ( x ) = x ∈ F . Theorem 3.3.
Let M be a normal affine monoid of rank d, and Suppose that x , y ∈ Msatisfy conditions (2)(a) and (b) of Theorem 2.1. Then the following are equivalent: (1) K [ M ] / ( X x − X y ) is normal. (2) If G is a subfacet of cone ( M ) such that x , y / ∈ G, then M [ − G ] ∼ = Z d − ⊕ Z + , and xor y has height over one of the exactly two facets F ′ , F ′′ containing G.Proof. As for Theorem 2.1, we start with the implication (2) = ⇒ (1). By Hochster’s the-orem K [ M ] is Cohen-Macaulay [6, 6.10], and thus Theorem 2.1 implies that K [ M ] / ( X x − X y ) is Cohen-Macaulay. Moreover, K [ M ] / ( X x − X y ) ∼ = K [ M ′ ] where M ′ is the image of M in gp ( M ) / Z ( x − y ) .It is enough to show that K [ M ′ ] satisfies Serre’s condition ( R ) since ( S ) follows fromCohen-Macaulayness (see [6, 4.F] for Serre’s conditions and normality). Sice K [ M ′ ] isa monoid domain, it is enough to check that the localizations with respect to monomialprime ideals of height 1 are regular [6, Exerc. 4.16]. Let P be such a prime ideal in K [ M ′ ] .The preimage Q in K [ M ] has height 2 and contains X x − X y . There are two cases todistinguish: (i) X x , X y / ∈ Q and (ii) X x , X y ∈ Q . In fact, Q contains either both monomialsor none.Somewhat surprisingly, case (i) does not imply any other condition on x and y thanthose occurring already in Theorem 2.1, which are satisfied by hypothesis. Let Q ′ be theideal generated by all monomials in Q . We have 0 = Q ′ = Q since Q ′ contains monomials,but X x , X y / ∈ Q . Therefore all monomials outside the facet F of cone ( M ) corresponding to Q ′ are inverted in the passage to K [ M ] Q . Since M is normal, M [ − F ] ∼ = Z d − ⊕ Z + , and x and y belong to Z d − because they are not in Q ′ . Since x − y is a basis element in gp ( M ) ,it is a basis element of the subgroup Z d − , and K [ Z d − ⊕ Z + ] / ( X x − X y ) is a regular´ring(see Remark 2.2(c)). Its localization K [ M ′ ] P is therefore also regular.Now we turn to case (ii). We write the subfacet G of cone ( M ) corresponding to Q asthe intersection of facets F ′ and F ′′ . Let Q ′ and Q ′′ be the corresponding height 1 primeideals. Since X x and X y cannot occur together in Q ′ or Q ′′ , one of them, say X x , liesin Q ′ and X y lies in Q ′′ . Since M [ − ( F ′ ∩ F ′′ )] ∼ = Z d − ⊕ Z + , the localization K [ M ] Q isa regular local ring. Choosing bases in the summands, we write K [ M [ − ( F ′ ∩ F ′′ )]] = K [ Z ± , . . . , Z ± d − , U , V ] . In this notation X x − X y = m U ht F ′ ( x ) − n V ht F ′′ ( y ) , m , n monomials in K [ Z ± , . . . , Z ± d − ] . The full localization K [ M ] Q is reached if we invert all elements in K [ Z ± , . . . , Z ± d − , U , V ] outside the prime ideal generated by U and V . The residue class ring modulo X x − X y INOMIAL REGULAR SEQUENCES AND FREE SUMS 11 is regular if (and only if) X x − X y ∈ Q Q \ ( Q Q ) , and this is equivalent to ht F ( x ) ≤ G ( y ) ≤ = ⇒ (2)one has to reverse the arguments just used inthe case (ii). First, the regularity of K [ M ′ ] P = K [ M ] Q / ( X x − X y ) implies the regularityof K [ M ] Q since the Krull dimension goes up by 1 and the number of generators of themaximal ideal by at most 1. Now Lemma 3.2 gives the structure of M [ − G ] . Moreover, assaid already, K [ M ′ ] P = K [ M ] Q / ( X x − X y ) is regular only if X x − X y ∈ Q Q \ ( Q Q ) . (cid:3) We draw consequences similar to those of Theorem 2.1.
Corollary 3.4.
Under the hypotheses of Corollary 2.3 the following are equivalent: (1) K [ M ] / ( X x − X x , . . . , X x n − − X x n ) is a normal domain; (2) for each face F such that rank M − dim F = n and x , . . . , x n / ∈ F, one has thefollowing: (a) M [ − F ] ∼ = Z d − n ⊕ Z n + ; (b) at least n − of the n nonzero numbers ht F i ( x j ) are equal to for the facetsF , . . . , F n containing F and j = , . . . , n.In particular, it is sufficient for (1) that all n nonzero heights ht F i ( x j ) are equal to in thesituation of (2).Proof. The equivalence of (1) and (2) follows by arguments entirely analogous to thoseproving the theorem, except that the critical localizations are now of type M [ − F ] = Z d − n ⊕ Z n + , and the regularity of the residue class ring modulo ( X x − X x , . . . , X x n − − X x n ) is the crucial condition.For the last statement we observe that the normal monoid M [ − G ] splits into a directsum of its unit group and a positive (normal) affine monoid of rank n [6, 2.26]. Thepositive component must be isomorphic to Z n + . In fact, the standard map [6, p. 59] sendsit surjectively and therefore isomorphically onto Z n + . (cid:3) Corollary 3.5.
Let L,M and N be normal affine monoids, j : M ⊕ N → L a surjectivehomomorphism with rank L = rank M + rank N − , and suppose that j ( x ) = j ( y ) forx ∈ M, y ∈ N, x = or y = . Then the following are equivalent: (1) K [ L ] = K [ M ⊕ N ] / ( X x − X y ) and K [ L ] is normal, (2) ht F ( x ) ≤ for all facets F of cone ( M ) or ht G ( y ) ≤ for all facets G of cone ( N ) .Proof. If (2) is satisfied, then x − y is unimodular in gp ( M ⊕ N ) , and we need no longerthink about the isomorphism K [ L ] = K [ M ⊕ N ] / ( X x − X y ) .In checking the equivalence of (1) and (2) in regard to normality, one notes that thecritical subfacets of cone ( M ⊕ N ) are exactly the intersections F ′ ∩ F ′′ where F ′ is theextension of a facet of cone ( M ) not containing x and F ′′ extends a facet of cone ( N ) notcontaining y , and all such pairs ( F ′ , F ′′ ) must be considered. (cid:3) We want to state consequences for Ehrhart series similar to Corollaries 2.6 and 2.8. Inthe situation of the free sum (and similarly in that analogous to Corollary 2.8) one alwayshas a homomorphism j : E ( P ) ⊕ E ( Q ) → E ( R ) where R = conv ( P ∪ Q ) . Set L = Im j . ByCorollary 2.5 we have H L = ( − T m + ) E P E Q . But L and E ( R ) generate the same conein R m + (since R = conv ( P ∩ Q ) ) and the same subgroup of Z m + (since ( Z m ∩ R R ) = ( Z m ∩ R P ) + ( Z m ∩ R Q ) ), and E ( R ) is normal. Therefore E ( R ) is the normalization of L , and the following statements are equivalent: (i) L is normal, (ii) L = E ( R ) , and (iii) H L = E R .After these preparations we obtain [2, Theorem 1.3]. It generalizes [5, Corollary 1]properly (see [2, Remark 3.5]). Corollary 3.6.
Let R ⊂ R m be a rational polytope that is the free sum of the rationalpolytopes P and Q, both containing . Then the following are equivalent: (1) At least in one of P or Q the origin has height ≤ over all facets; (2) E R = ( − T m + ) E P E Q . In the same way, as Corollary 2.8 generalizes Corollary 2.6, we can generalize Corol-lary 3.6 and thus generalize [2, Corollary 5.8], but we must also generalize the condition ( Z m ∩ R R ) = ( Z m ∩ R P ) ⊕ ( Z m ∩ R Q ) . To this end we say that a subset A of Z m is the Z -affine hull of B ⊂ Z m if A = (cid:8) a x + · · · + a n x n : n ≥ , x , . . . , x n ∈ B , a , . . . , a n ∈ Z , a + · · · + a n = (cid:9) . Note that the Z -affine hull is the subgroup generated by B if 0 ∈ B . Corollary 3.7.
Let P , Q ⊂ R m be rational polytopes such that aff ( P ) and aff ( Q ) meet in asingle point p ∈ P ∩ Q. Set R = conv ( P ∪ Q ) and suppose that aff ( R ) ∩ Z m is the Z -affinehull of ( aff ( P ) ∪ aff ( Q )) ∩ Z m . Furthermore let k be the smallest positive integer such thatk p ∈ Z m . Then the following are equivalent: (1) At least in one of E ( P ) or E ( Q ) the point ( k p , k ) has height ≤ over all facets; (2) E R = ( − T ( kp , k ) ) E P E Q . Finally we derive [9, Theorem 3] without using arguments on triangulations.
Corollary 3.8.
Let M be an affine monoid such that K [ M ] is Gorenstein and let X w , w ∈ M,generate the canonical module of K [ M ] . Furthermore let x , . . . , x n ∈ M noninvertibleelements such that w = x + · · · + x n . Then K [ M ] / ( X x − X x , . . . , X x n − − X x n ) is again aGorenstein normal affine monoid domain and has dimension rank M − ( n − ) .Proof. The point w is distinguished by the fact that it has height 1 over each facet. There-fore “height vectors” defined x , . . . , x n are 0-1-vectors with disjoint supports, and Corol-lary 3.4 applies. It yields that the residue class ring is a normal affine monoid domain,and the Gorenstein property is preserved modulo regular sequences. (cid:3) R EFERENCES [1] M. Beck and S. Hos¸ten.
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