NUMERICAL MONOIDS (I)

Seminar Series in Mathematics: Algebra 2003, 1 8 NUMERICAL MONOIDS (I) Introduction The numerical monoids are simple to define and naturally appear in various parts of mathematics, e.g. as the values monoids of the algebroid branches of the plane algebraic curves or as supporting structures in the explicit computation of the cyclotomic polynomials or in combinatorial problems arising in the additive arithmetics. Important structural results in the general theory of the numerical monoids were obtained by Sylvester, Frobenius, Kunz, Herzog, Fröberg, Selmer, Rödseth, and the impact in geometry was studied by Abhyankar, Moh, Garsia, Stöhr, Waldi, Angemüller... However, in spite of their simple definition, the study of the numerical monoids proved difficult. For instance, except for a few particular cases, no general closed arithmetic formulas are known (they simple seem not to exist). Instead such formulas, various computing algorithms were developed and subsequently simplified. The aim of this lecture is to introduce the basic notions and tools in the study of the numerical monoids. 1 Numerical monoids Proposition 1.1. Let M be a submonoid of the additive monoid (N, +). The following are equivalent: (i) There exists n N such that n + N M. (ii) The complement M := N M is finite. (iii) M is finitely generated by a system of relatively prime natural numbers. (iv) M contains two consecutive nonzero natural numbers. This lecture was held by Şerban Bărcănescu Institute of Mathematics, Bucharest, Romania e-mail: sbarcan@imar.ro 1

2 Numerical Monoids The obvious proof of the proposition is omitted. Definition 1.2. A monoid M verifying one of the equivalent conditions of Proposition 1.1 is called numerical. Proposition 1.3. Let M be a numerical monoid and put: Then: (i) ϕ M = f M + 1. f M = max M and ϕ M = min{n N n + N M}. (ii) For G M = {x M x < f M } we have: M = G M (ϕ M + N). (iii) M G M = and M G M = {0, 1, 2,..., f M }. Definition 1.4. With the above notations, f M is called the Frobenius number and ϕ M is called the Euler number of the numerical monoid M. Remarks 1.5. (i) In spite of the relation (i), Proposition 1.3, the numbers f M and ϕ M have different roles in the theory of the numerical monoids. (ii) The name of ϕ M is due to the fact that, in particular cases, it actually coincides with the value of the Euler totient function. (iii) The monoid (N, +) itself is numerical with f N = 1. Now, let M be a numerical monoid and K be a fixed field. The unique minimal generating system {g 1, g 2,..., g t } (t 2) of M (cf. (iii) Proposition 1.1) produces the following representations of the monoid algebra of M over K : (1) K[M] = K[X g 1, X g 2,..., X g t ] K[X] = k[n], showing that k[m] is a 1 dimensional Cohen-Macaulay domain. (2) Let ψ : K[Y 1, Y 2,..., Y t ] K[M] be the surjective K algebra homomorphism given by ψ(y j ) = X g i, j = 1, 2,..., t. ψ is graded for deg Y j = g j (all j s) so ker ψ is an homogeneous prime ideal called the toric ideal of M. It obviously contains among its generators the binomials Y g i d ij j g j d ij Yi, 1 i < j t,

Numerical Monoids 3 where d ij = gcd(g i, g j ). Therefore K[M] is a 1 dimensional affine K algebra, defining as such an algebraic curve Γ(M) A t K, called the monomial curve associated to M. Its name comes from the fact that, in an affine coordinate system (x 1,..., x t ), Γ(M) is parametrically given by monomials: x j = τ g j, τ K and j = 1, 2,..., t. The study of these curves (in particular, the fact that they are indeed toric varieties) is the geometric counterpart of the study of the numerical monoids and constitutes the subject of other lectures in this conference. 2 Zeta Functions A numerical monoid M has a subjacent combinatorial structure given by the partial order M defined by x, y M : x M y if and only if z M and x + z = y. Contrary to the natural order on the ambient N (which is an infinite chain, therefore gives N the structure of a generic distributive lattice), the order M on N is only a poset structure (not even a lattice, in general). The chain order obviously extends the order M, which therefore inherits the property of having finite descending subchains (i.e. is a good order on N). (M, M ) is not normally embedded in (N, ) (or else k[m] would be regular, which is the case for M = N only). To the monoidal order M we associate the reduced incidence algebra K[[M]], defined as the m adic completion of the localization k[m] m, m being the ideal generated by X g1, X g2,..., X gt. Obviously k[[m]] is a subalgebra of the power series algebra k[[x]]. The elements of k[[m]] are called incidence functions on M. The main incidence function on M is its zeta function defined by ζ M = α M X α. In this function many ennumerative properties of M are gathered: for instance ζ 1 M := µ M is the Möbius function on M (i.e. the reduced Euler characteristics of each chain complex (α), α M), ζ M 1 = λ M is the chain-ennumerating function on M, etc. The computation of ζ M is usually the first step in the study of the ennumerative structure of any monoid. In our context we have the following obvious expressions: Proposition 2.1. Let M be a numerical monoid with minimal generating system {g 1, g 2,..., g t }. Then ζ M is the power series development of any of the following rational functions:

4 Numerical Monoids (i) ζ M = α G M X α + X ϕ M 1 1 X, (ii) ζ M = f M j=0 Xj β M Xβ + X ϕ M 1 1 X, (iii) ζ M = 1 1 X β M Xβ, (iv) ζ M = P (X)(1 X) g1... (1 X) g t, for some P Z[X]. The expression (iv) of ζ M comes from the embedded representation (2), 1, of the monoid algebra K[M]. The relevance of the explicit computation for ζ M will become apparent in the following section. 3 Symmetric numerical monoids We are now ready to introduce one of the main objects of study in the whole theory of the numerical monoids. Proposition 3.1. Let M be a numerical monoid. The following are equivalent: (i) G M = M = 1 2 ϕ M. (ii) The function δ : M G M given by δ(β) = f M β is a decreasing bijection (in the natural order on N). (iii) The monoid algebra K[M] is Gorenstein. Proof. The equivalence of (i) and (ii) is obvious in virtue of Proposition 1.3 (iii), with the remark that {0, 1, 2,..., f M } = f M + 1 = ϕ M. To prove (ii) (iii) we use Stanley s criterion for Cohen-Macaulay graded domains, which, in this context, becomes: K[M] is Gorenstein if and only if h Z and ζ M (X 1 ) = ( 1) 1 X h ζ M (X), where 1 = dim K[M] (the undeterminacy of the integer h is due to the undeterminacy of the canonical module ω K[M] up to a shift in grading). Now, using Proposition 2.1, we immediately see that: ζ M (X 1 ) = ( 1) 1 X fm ( X f M β 1 ) + X ϕm. 1 X β M

Numerical Monoids 5 Comparing this with proposition 2.1 (i), it follows that K[M] is Gorenstein if and only if: X α = X f M β, α G M which is precisely (ii) in the enounce. β M Definition 3.2. A numerical monoid M verifying one of the equivalent conditions in Proposition 3.1 is called symmetric. Remarks 3.3. (i) The result of Proposition 3.1 is due to E. Kunz. (ii) If M is symmetric it necessarily follows that ϕ M is even. Therefore ϕ M odd M is not symmetric. Now, let α M + = M {0}. Then X α is a non-zerodivisor in k[m], therefore K[M]/(X α ) is a graded artinian algebra. Proposition 3.4. (Fröberg) Let and let T (M) = {β M β + γ M, γ M + } h : T (M) K[M]/(X α ), h(β) = X β+α ( mod X α ). Then Im h is a K linear basis in Soc(K[M]/(X α )). Proof. See [V], ch. 10, Thm. 10.2.10. Corollary 3.5. Let τ(m) be the type of the Cohen-Macaulay algebra K[M] (i.e. the rank of the canonical module ω K[M] ). Then τ(m) = dim K Soc(K[M]/(X α )) = T (M). Corollary 3.6. K[M] is Gorenstein (i.e. T (M) = 1 T (M) = {f M }. M is symmetric) if and only if 4 Principal residues We define now the notion of residue modulo a numerical monoid M, which proves useful in inductive arguments. Let us first remark that the abelian group generated by M is simply Z and the congruence x y( mod M) if and only if y x M is not an equivalence relation but an order relation, namely the monoidal order M. The definition of the classes mod M as x + M, x N, reduces the

6 Numerical Monoids congruence x + M = y + M to the equality x = y. Therefore we cannot define a quotient monoid N/M. What we can do is only to define a local independence mod M on the natural numbers, in the sense of their incompatibility in the monoidal order M and a local congruence mod M in the sense of the compatibility in M. The order M being good, each (maximal) chain has an initial vertex. The chains passing through some α M all have the initial verex in 0(zero), so only the chains passing through elements β M may contribute to the local congruence (independence) mod M. We have the following result: Proposition 4.1. Let g 1 be the least generator of M and put σ = g 1 1. Then: (i) The set {1, 2,..., σ} is an antichain of the monoidal order M. (ii) For any β M each maximal M vertex in {1, 2,..., σ}. chain through β has the initial Proof. (i) is immediate by the definition of σ. For (ii) it suffices to show that, given β M, there is a j β {1, 2,..., σ} such that j β M β. Suppose not: then β σ, β σ + 1,..., β are all in M and, obviously, β > g 1. If β σ < g 1 then β g 1 < σ and, since β = (β g 1 ) + g 1, we would have β g 1 M β, in contradiction with our hypothesis. Therefore g 1 β σ, so g 1 < β σ g 1 + 1 β σ 2g 1 β. If β σ < 2g 1, since β = (β 2g 1 ) + 2g 1, we would again have a contradiction. Therefore 2g 1 < β σ 3g 1 β. Continuing inductively we would obtain ng 1 β for all n 1. Definition 4.2. With the notations in Proposition 4.1, the numbers {1, 2,..., σ} are called principal residues and their set is called the initial segment of the numerical monoid M. Fix now a numerical monoid M to which we adjoin a new generator w M, obtaining a new monoid H = Nw + M. Clearly M H, so H M and f H f M. In order to compare the Frobenius numbers f M and f H, we have first to compare M and H (since f M = max M and f H = max H). We have the following result: Proposition 4.3. With the above notations let (i) (M : w) is a numerical monoid. (ii) (M : w) = M : w. (M : w) = {m N mw M}.

Numerical Monoids 7 (iii) Let {1, 2,..., σ} be the initial segment of (M : w) (so σ + 1 = min{m N mw M}). Then H = M \ σ [(jw + M) M]. j=1 Proof. (i) is clear since there is m N such that mw > ϕ M therefore (m + N)w M. (ii) simply reads mw M if and only if m (M : w). To prove (iii) remark that H = M \ (H M) and that H = j 0 (jw + M). So we have to prove that: [(jw + M) M] = j 0 σ [jw + M) M]. This comes to show that each multiple mw wich belongs to M, either equals one of w, 2w,..., σw or is M than one of the principal mutiples. But: mw M m M : w m M : w and, according to Proposition 4.1 (ii), there exists j m {1, 2,..., σ} such that: j m (M,w) m i.e. α M : w such that j m + α = m. Then mw = j m w + αw and, since αw M, we get mw M j m w. 5 Arithmetic translation Finally, let us translate into arithmetic language the definition of the Frobenius number and the symmetry property of a numerical monoid. We omit the proofs of the following statements because they are mere reformulations of the previous ones. Proposition 5.1. Let M be a numerical monoid minimally generated by g 1, g 2,..., g t and let L M (X) = L M (X 1,..., X t ) be the associated linear form: Then j=1 L M (X) = X 1 g 1 + X 2 g 2 +... + X t g t Z[X 1,..., X t ]. f M = max{ν N the equation L M (X) = ν has no solution in N t } = = max{ ν N the equation L M (X) = ν+g 1 +...+g t has no solution in (N ) t }.

8 Numerical Monoids This result can be rephrased as follows: Proposition 5.2. Let ν N be a natural number such that: (i) the equation L M (X) = ν has no solution in N t, (ii) for each τ N, the equation L M (X) = ν + τ has solution in N t. Then ν = f M. By translating with g 1 + g 2 +... + g t the last characterization, we get: Proposition 5.3. Let ν N be a natural number such that: (i) the equation L M (X) = ν has no solution in (N ) t, (ii) for each τ N the equation L M (X) = ν + τ has solution in (N ) t. Then ν (g 1 + g 2 +... + g t ) = f M. Proposition 5.4. The following are equivalent: (i) The monoid M is symmetric, (ii) The equation L M (X) = f M β has solution in N t if and only if β M. Remarks 5.5. (i) In Proposition 5.4 (ii) the only if part may be dropped, since L M (X) = f M α obviously has no N t solutions for α M. (ii) In Proposition 5.2 (ii) the number N(τ) of the N t solutions for L M (X) = f M + τ is obviously finite (each component x j of a solution is (f M + τ)/g j, j = 1, 2,..., t). Therefore the ennumerative properties of M are also reflected in the Dirichlet-type series References D M (z) = N(τ) τ z, z C. τ N [H] J. Herzog, Generators and relations of Abelian Semigroups and Rings, Manuscripta Math., 3 (1970) [K1] E. Kunz, Ebene algebraische Kurven, Regensburger Trichter, 23(1991) [K2] E. Kunz, The values semigroups of a one-dimensional Gorenstein ring, Proc. AMS (1970) [V] R. H. Villareal, Monomial Algebras, Pure & Appl. Math. 238(2001).