Graded algebras with cyclotomic Hilbert series
aa r X i v : . [ m a t h . A C ] M a y Graded algebras with cyclotomic Hilbert series
Alessio Borz`ı, Alessio D’Al`ı
Abstract
Let R be a positively graded algebra over a field. We say that R is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has allof its roots on the unit circle. Such rings arise naturally in commuta-tive algebra, numerical semigroup theory and Ehrhart theory. If R isstandard graded, we prove that, under the additional hypothesis that R is Koszul or has an irreducible h -polynomial, Hilbert-cyclotomic al-gebras coincide with complete intersections. In the Koszul case, thisis a consequence of some classical results about the vanishing of devi-ations of a graded algebra. The
Hilbert series of a positively graded k -algebra R is a prominent objectin commutative algebra. It encodes the information on how many forms ofdegree d are contained in R for each possible d , and has been the object ofintense study since the late nineteenth century. Some simple inductive rea-soning shows that the Hilbert series can be expressed as a rational function.Many properties of the graded algebra are reflected into the numerator ofsuch expression. In the last few years, several authors have been investigatingthe behaviour of the roots of this polynomial [7, 16, 3], often focusing on thecombinatorially interesting case when such roots are all real. For the com-binatorial consequences of real-rootedness, we direct the interested reader tothe survey [5]. Mathematics Subject Classification . Primary: 13D40; Secondary: 13A02, 16S37,20M14, 13H10. In this paper, when not specified differently, we express the Hilbert series of a gradedalgebra R as a rational function reduced to lowest terms ; in particular, it makes sense tospeak of the numerator. Main Definition (Definition 2.1) . Let R be a positively graded k -algebra.We say that R is Hilbert-cyclotomic (or simply cyclotomic ) if the numeratorof its reduced Hilbert series is Kronecker, i.e. has all of its roots on the unitcircle.Cyclotomic graded algebras have been considered in lattice polytope the-ory, where they are related to Ehrhart-positivity [17, 6], and numerical semi-group theory [10, 20, 15, 4, 22]. Moreover, every graded complete intersectionis cyclotomic (see the discussion before Proposition 2.4 for a more precisestatement).Generally speaking, the cyclotomic condition cannot be enough to char-acterize complete intersections: for instance, Gr¨obner deformation preservesthe Hilbert series but not necessarily the complete intersection property.Even under the stronger hypothesis that the given algebra is a graded Cohen–Macaulay domain, there exist cyclotomic algebras which fail to be completeintersections, as shown by Stanley [23, Example 3.9] (see also Example 2.5herein). However, if we restrict our focus to numerical semigroup rings , it isyet unknown whether cyclotomic algebras and complete intersections coin-cide, as we now explain.A numerical semigroup S is an additive submonoid of N with finite com-plement N \ S . For an introduction to numerical semigroups, see [21]. The semigroup polynomial of S is defined as P S ( x ) = 1+( x − P g ∈ N \ S x g . It is aneasy exercise to check that the semigroup polynomial P S ( x ) is the numeratorof the reduced Hilbert series of the semigroup ring k [ S ]. Ciolan, Garc´ıa-S´anchez and Moree [10] call the numerical semigroup S cyclotomic if P S ( x )has all of its roots in the unit circle, i.e. the ring k [ S ] is Hilbert-cyclotomic.The original motivation for this notion comes from the following folkloreresult in number theory (see for instance [20, Theorem 1]): if p and q are dis-tinct primes and h p, q i is the numerical semigroup generated by p and q , then P h p,q i ( x ) = Φ pq ( x ), where Φ n ( x ) is the n -th cyclotomic polynomial. More gen-erally, if a and b are two coprime integers, then P h a,b i ( x ) = Q n | ab, n ∤ a, n ∤ b Φ n ( x ) . As we have seen above, if k [ S ] is a complete intersection, then S is cyclo-tomic. It was verified in [10] that every cyclotomic numerical semigroup withFrobenius number up to 70 is a complete intersection (where the Frobeniusnumber of S is max( N \ S )). This motivates the following conjecture:2 onjecture 1.1 (Ciolan, Garc´ıa-S´anchez, Moree [10]) . A numerical semi-group S is cyclotomic if and only if k [ S ] is a complete intersection. Now, let us go back to the more general setting where R is a positivelygraded k -algebra. In the spirit of Conjecture 1.1 and of a question by Stan-ley [23, p. 64], it is of interest to find additional hypotheses under which thecyclotomic condition for R becomes equivalent to being a complete intersec-tion. The main result of this paper shows that this is the case for Koszulalgebras , a class of quadratic standard graded algebras enjoying many desir-able homological properties (for an overview, we refer the interested readerto, e.g., [11] and [12]).
Theorem A (Theorem 3.9) . If R is a Koszul algebra, then R is Hilbert-cyclotomic if and only if it is a complete intersection. Moreover, we prove that complete intersections and cyclotomic algebrascoincide also under the assumption that R is standard graded and its h -polynomial is irreducible over Q . This is in line with a result in the forth-coming article [4]: see Question 4.1 and the discussion preceding it. Recallthat the Kronecker polynomials which are irreducible over Q are preciselythe cyclotomic polynomials Φ m ( x ). Theorem B (Theorem 4.3) . Let R be a standard graded algebra. Then h ( R, x ) = Φ m ( x ) if and only if m is prime and R is a hypersurface of degree m . In this paper, a graded algebra will always be a commutative finitely gener-ated N -graded algebra R = L i ∈ N R i with R = k , where k is a field. Wewill write the reduced Hilbert series of R as H ( R, x ) = N R ( x ) /D R ( x ). If R is generated by its degree 1 part, we will say that R is standard graded . Inthis case, as is customary, we will call N R the h -polynomial of R and denoteit by h ( R, x ). Definition 2.1.
Let R be a graded algebra. We say that R is Hilbert-cyclotomic (or simply cyclotomic ) if the numerator of its reduced Hilbertseries is Kronecker, i.e. has all of its roots on the unit circle.3 emark 2.2.
Besides the connection to numerical semigroups highlighted inthe introduction, the cyclotomic condition has also been studied in Ehrharttheory. Braun and Liu [17, 6] call a polytope h ∗ -unit-circle-rooted if itsEhrhart ring is Hilbert-cyclotomic. Remarkably, such polytopes are Ehrhart-positive [6, Corollary 1.4].A degree s polynomial f ( x ) = P si =0 a i x i with integer coefficients is saidto be palindromic if a i = a s − i for every i ∈ { , . . . , s } , or equivalently if f ( x ) = x s f (1 /x ). For every n >
1, one has that the cyclotomic polynomialΦ n ( x ) := Q ( j,n )=1 (cid:0) x − e πij/n (cid:1) is palindromic. Since a Kronecker polynomialis a product of cyclotomic polynomials and the palindromic property is pre-served under taking products, it follows that every Kronecker polynomial f with f (1) = 0 is palindromic.Now let R be a graded algebra of Krull dimension d . Since the order ofthe pole of H ( R, x ) at x = 1 equals d (see for instance [1, Chapter 11]), itfollows that N R (1) = 0. Thus, from the above observations we infer that Remark 2.3.
If the graded algebra R is cyclotomic, then the numerator N R of its reduced Hilbert series is palindromic.The rest of this section is devoted to a quick exploration of how the cy-clotomic condition relates to complete intersections and Gorenstein algebras,as summarized by the following diagram:completeintersection GorensteincyclotomicFigure 1: The single arrow holds for Cohen-Macaulay do-mains (Proposition 2.7), whereas the dashed arrow is the sub-ject of Conjectures 1.1 and 2.6 and holds for standard gradedalgebras that are Koszul (Theorem 3.9) or have irreducible h -polynomial (Theorem 4.3).Let R be a (graded) complete intersection, i.e. a quotient of a positivelygraded polynomial ring S = k [ x , . . . , x n ] by an ideal generated by a homo-geneous regular sequence f , . . . , f e . Setting d i := deg x i and m j := deg f j ,4ne shows that the Hilbert series of R can be written in the (non-reduced)form H ( R, x ) = (1 − x m ) . . . (1 − x m e )(1 − x d ) . . . (1 − x d n ) , see for instance [23, Corollary 3.3]. Hence, it follows that Proposition 2.4.
Every graded complete intersection is cyclotomic.
The converse of Proposition 2.4 does not hold even under the hypoth-esis that the given algebra is a Cohen–Macaulay standard graded domain,as already observed by Stanley [23, Example 3.9]. Examples of cyclotomicnon-complete intersection Cohen–Macaulay standard graded domains can befound in any dimension d ≥
2: it is enough to adjoin variables to the followingexample provided by Aldo Conca.
Example 2.5.
Let R = k [ s , s t , s t , s t , t ] ⊆ k [ s, t ]. Then R is a2-dimensional standard graded domain which is Cohen–Macaulay in everycharacteristic (one checks via some characteristic-free Gr¨obner basis compu-tation that the system of parameters { s , t } is a regular sequence for R ).Since the h -polynomial of R is (1+ x ) , one has that R is cyclotomic; however,it is not a complete intersection.It turns out that investigating the case of one-dimensional domains (notnecessarily standard graded) is essentially equivalent to solving Conjecture1.1. In fact, from [24, Proposition 3.1] every graded domain of Krull dimen-sion one over an algebraically closed field k is isomorphic to the semigroupalgebra k [Γ] of some additive submonoid Γ of N . Further, every submonoidΓ of N is isomorphic to the numerical semigroup S = Γ / gcd(Γ) [21, Propo-sition 2.2]. Hence, we can reformulate Conjecture 1.1 in purely algebraicterms: Conjecture 2.6 (Conjecture 1.1, algebraic version) . Every cyclotomic gradeddomain of Krull dimension one over an algebraically closed field is a completeintersection.
We close this section by discussing the relation between the cyclotomiccondition and the Gorenstein property. A famous theorem by Stanley [23,Theorem 4.4] states that a Cohen–Macaulay graded domain R is Gorensteinif and only if N R ( x ) is palindromic. Recalling Remark 2.3, we hence obtainthe following result: 5 roposition 2.7. Let R be a Cohen–Macaulay graded domain. If R is cy-clotomic, then R is Gorenstein. Proposition 2.7 generalizes both [20, Theorem 5] for numerical semigroupsand [17, Corollary 2.2.9] for lattice polytopes. The converse of Proposi-tion 2.7, however, does not hold even in these more specific settings. Onecan consider for instance the semigroup rings of the numerical semigroups S k = h k, k + 1 , . . . , k − i with k ≥ The goal of this section is to prove that, for Koszul algebras, the cyclotomicproperty characterizes complete intersections. We begin by some definitions.
Definition 3.1.
Let R be a standard graded k -algebra. We say that R is Koszul if the minimal graded free resolution of k as an R -module is linear,i.e. Tor Ri ( k , k ) j = 0 whenever i = j . Definition 3.2.
Let R be a graded k -algebra. The Poincar´e series of k asan R -module is P ( R, x ) = P + ∞ i =0 β Ri ( k ) x i , where β Ri ( k ) := dim k Tor Ri ( k , k ).The following remark can be found for instance in [19, Remark 1]. Remark 3.3.
Given a formal series P ( x ) = 1 + P + ∞ i =1 a i x i with a i ∈ Z , thereexist unique integers e i ∈ Z such that P ( x ) = + ∞ Y i =1 (1 − x i ) e i . (1)If P ( x ) is just a polynomial, then it is Kronecker if and only if e i = 0 for i ≫
0. For a proof of this fact, see [10, Lemma 12].The factorization in (1) was used by Ciolan, Garc´ıa-S´anchez and Moree[10] to define the cyclotomic exponent sequence of a numerical semigroup.This notion can be generalized as follows.
Definition 3.4.
Let R be a graded algebra and let N R ( x ) be the numeratorof its reduced Hilbert series. Since N R (0) = 1, we can factor N R as in (1).The integers e i will be called the cyclotomic exponent sequence of R and willbe denoted by e i ( R ). 6 consequence of Remark 3.3 is the following equivalence. Corollary 3.5.
Let R be a graded algebra. The following conditions areequivalent:1. R is cyclotomic;2. e i ( R ) = 0 for i ≫ . Definition 3.6.
Let R be a graded k -algebra and let P ( R, x ) be the Poincar´eseries of the residue field k as an R -module. We write P ( R, x ) = + ∞ Y i =1 (1 + x i − ) ε i − + ∞ Y i =1 (1 − x i ) ε i and call the integers ( ε i ) i ∈ N so obtained the sequence of deviations of R .An interesting feature of deviations is their ability to tell whether or not R is a complete intersection. The strongest version of this result is Halperin’srigidity theorem, see for instance [2, Theorem 7.3.4]. For our aims, however,a weaker statement originally due to Gulliksen [14] will suffice (see also [2,Theorem 7.3.3]): Theorem 3.7.
Let R be a graded algebra. The following conditions areequivalent:1. R is a complete intersection;2. ε i ( R ) = 0 for i ≫ . Corollary 3.5 and Theorem 3.7 exhibit a formal similarity. Such a simi-larity becomes substantial when R is a Koszul algebra, as the following resultshows. Proposition 3.8.
Let R be a Koszul algebra of Krull dimension d . Then e i ( R ) = ( − ε ( R ) + d i = 1( − i ε i ( R ) i > . In particular, e i ( R ) = 0 for i ≫ ⇐⇒ ε i ( R ) = 0 for i ≫ . roof. Note that N R ( x ) = h ( R, x ) = (1 − x ) d H ( R, x ) because R is standardgraded. Further, since R is Koszul, from [12, Theorem 1] we have that H ( R, x ) P ( R, − x ) = 1. Now write + ∞ Y i =1 (1 − x i ) e i ( R ) = h ( R, x ) = (1 − x ) d H ( R, x ) = (1 − x ) d P ( R, − x ) == (1 − x ) d + ∞ Y j =1 (1 − x j ) ε j ( R )+ ∞ Y j =1 (1 − x j − ) ε j − ( R ) = (1 − x ) d + ∞ Y i =1 (1 − x i ) ( − i ε i ( R ) and the claim follows. Theorem 3.9. If R is a Koszul algebra, then R is cyclotomic if and only ifit is a complete intersection.Proof. The result follows directly from Proposition 3.8, Corollary 3.5 andTheorem 3.7.
Remark 3.10.
One may also show that, if R is Koszul and cyclotomic, thenthe Betti numbers of the residue field k as an R -module “do not grow toofast”, i.e. it holds thatcurv R ( k ) := lim sup n → + ∞ n p β Rn ( k ) ≤ . By [2, Corollary 8.2.2], this implies that R is a complete intersection; however,the proof of [2, Corollary 8.2.2] still relies on deviations. h -polynomial Let R be a cyclotomic graded algebra and assume that N R is irreducibleover Q . This means that N R ( x ) = Φ m ( x ) for some m ∈ N . Under thiscondition, in the case when R = k [ S ] for some numerical semigroup S (andhence N R equals the semigroup polynomial P S ), it is proved in [4] that then S = h p, q i for some primes p = q , and consequently m = pq . Since eachnumerical semigroup of the form h p, q i is a complete intersection, this impliesin particular that Conjecture 1.1 holds true when P S is irreducible. Thisprompts the following questions: 8 uestion 4.1. Let R be a graded algebra and assume that the numerator N R of its reduced Hilbert series is irreducible. Is it true that R is cyclotomicif and only if it is a complete intersection? Question 4.2.
Which cyclotomic graded algebras R have a Hilbert serieswhose numerator is irreducible, i.e. N R ( x ) = Φ m ( x ) for some m ?The aim of this section is to answer both of the above questions in thecase when R is standard graded. Theorem 4.3.
Let R be a standard graded algebra. Then h ( R, x ) = Φ m ( x ) if and only if m is prime and R is a hypersurface of degree m . As a consequence, Question 4.1 has a positive answer when R is standardgraded. To prove Theorem 4.3, we will need some auxiliary results. First,we recall some basic properties of cyclotomic polynomials. Lemma 4.4. (a)
Let m > . Then Φ m (1) = ( p if m = p k for some prime p otherwise. (b) For any prime p and any k ≥ , one has that Φ p k ( x ) = Φ p ( x p k − ) . The following lemma is a generalization of [23, Theorem 3.6].
Lemma 4.5.
Let R be a standard graded k -algebra such that H ( R, x ) = 1 + x + x + . . . + x s − (1 − x ) d (2) for some s > , d ≥ . Then R = k [ x , . . . , x d +1 ] / ( f ) for some homogeneouspolynomial f of degree s .Proof. Write R as the quotient of a standard graded polynomial ring S = k [ x , . . . , x n ] by a homogeneous ideal I ⊆ ( x , . . . , x n ) . Here n is the em-bedding dimension of R , which is strictly greater than d since R is not apolynomial ring itself. It follows from Hilbert’s syzygy theorem that theHilbert series of R can be written in a non-reduced way as H ( R, x ) = K ( R, x )(1 − x ) n , (3)9here K ( R, x ) = P i,j ( − i β Si,j ( R ) x j and β Si,j ( R ) := dim k Tor Si ( R, k ) j is the( i, j )-th Betti number of R as an S -module. Comparing Equations (2) and(3), we have that K ( R, x ) = (1 + x + x + . . . + x s − )(1 − x ) n − d = (1 − x s )(1 − x ) n − d − . Now, if n − d − >
0, it follows that the coefficient of x in K ( R, x ) is nonzero,which is impossible since I does not contain any linear form. Hence, n = d +1and K ( R, x ) = 1 − x s . It is left as an exercise to the reader to check that I must then be minimally generated by a single homogeneous polynomial ofdegree s . Lemma 4.6.
Let R be a standard graded k -algebra of Krull dimension d . Ifthe h -polynomial of R is palindromic of even degree s , then h ( R, > .Proof. The integer h ( R,
1) is the multiplicity of R [8, Definition 4.1.5, Corol-lary 4.1.9], and as such it is positive. Suppose by contradiction that h ( R, R as k [ x , . . . , x n ] /I for some homogeneous ideal I . Sincethe Hilbert series stays the same when passing to the initial ideal, we canassume without loss of generality that I is a monomial ideal. Moreover,since polarization preserves the h -polynomial, we can further assume that I is a squarefree monomial ideal; hence, I is the Stanley–Reisner ideal of somesimplicial complex ∆.Let D − h ( R, x ) = P si =0 h i x i . Byconstruction, the h -vector of ∆ is ( h , h , . . . , h D ), where h i = 0 if i > s .Knowing the h -vector of ∆ gives us access to its f -vector ( f − , f , . . . , f D − ),where f i is the number of i -dimensional faces of ∆. As shown for instance in[18, Corollary 1.15], the transformation is given by D X i =0 f i − ( x − D − i = D X i =0 h i x D − i . (4)In particular, f D − = P Di =0 h i = P si =0 h i = h ( R,
1) = 1. Since ∆ containsa ( D − D ( D − f D − ≥ D . Substituting x − y inside Equation (4), we find10hat f D − is the coefficient of y in P Di =0 h i ( y + 1) D − i . Hence, f D − = Dh + ( D − h + . . . + ( D − s ) h s = ( D − s ) s X i =0 h i + sh + ( s − h + . . . + h s − = ( D − s ) + s s X i =0 h i = D − s < D, where the third equality comes from the fact that h ( R, x ) is palindromic ofeven degree.
Proof of Theorem 4.3.
The “if” part is clear. Let us prove the “only if”.Suppose that R = k [ x , . . . , x n ] /I for some homogeneous ideal I containedin ( x , . . . , x n ) . By hypothesis we have that H ( R, x ) = Φ m ( x )(1 − x ) d (5)for some m > d = dim R ≥
0. Since Φ m ( x ) is palindromic of evendegree, applying Lemma 4.6 yields that Φ m (1) = 1. It follows from part(a) of Lemma 4.4 that m = p k for some prime p and k ≥
1. Now assumethat k > q = p k − . By part (b) of Lemma 4.4, one has thatΦ m ( x ) = Φ p ( x q ) = 1 + x q + x q + . . . + x ( p − q . Expanding Equation (5) at x = 0, we get that the standard graded algebra R contains d forms of degree1 and (cid:0) d + q − q (cid:1) + 1 forms of degree q , which is impossible. Hence, m = p . Nowapply Lemma 4.5. Acknowledgements
The second-named author was supported by the EPSRC grant EP/R02300X/1.The authors are grateful to Aldo Conca and Diane Maclagan for some usefuladvice on the structure of the present paper; Aldo Conca has also providedExample 2.5. The authors wish to thank also Benjamin Braun and WinfriedBruns for some fruitful discussions and insights. Many computations andexamples were carried out via the computer algebra systems Macaulay2 [13]and Normaliz [9]. 11 eferences [1] M. F. Atiyah and I. G. Macdonald.
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Alessio Borz`ı
Mathematics Institute, University of Warwick, Coventry CV47AL, United Kingdom.Alessio D’Al`ı