Homology, Homotopy and Applications, vol.2, No.2, 2000, pp.17–27 ISSN 1512–0139

TRUNCATIONS OF THE RING OF NUMBER-THEORETIC FUNCTIONS

JAN SNELLMAN

(communicated by Winfried Bruns)

Abstract We study the ring Γ of all functions ℕ^{+} → K, endowed with the usual convolution product. Γ, which we call the ring of number-theoretic functions, is an inverse limit of the “truncations”

Γ_{n} = { f ∈ Γ ∀m > n : f(m) = 0 } .

Each Γ_{n} is a zero-dimensional, finitely generated K-algebra, which may be expressed as the quotient of a finitely generated polynomial ring with a stable (after reversing the order of the variables) monomial ideal. Using the description of the free minimal resolution of stable ideals given by Eliahou-Kervaire, and some additional arguments by Aramova-Herzog and Peeva, we give the Poincaré-Betti series for Γ_{n}.

1. Introduction

Cashwell and Everett [2] studied “the ring of number-theoretic functions”

Γ = { f ℕ^{+} → K }  (1)

where ℕ^{+} is the set of positive natural numbers (we denote by ℕ the set of all natural numbers) and K is a field containing the rational numbers. Γ is endowed with component-wise addition and multiplication with scalars, and with the convolution (or Cauchy) product

fg(n) = ∑_{\begin{matrix}(a,b)∈(ℕ^{+})×(ℕ^{+}) \\ ab=n \end{matrix}} f(a)g(b)  (2)

With these operations, Γ becomes a commutative K-algebra. It is immediate that it is a local domain; less obvious is the fact that it is a unique factorisation domain. Cashwell and Everett proved this in [2] using the isomorphism

\begin{matrix}Φ : Γ & → K[[X]] \\ f & ↦ ∑ f(n)x_{1}^{α_{1}}x_{2}^{α_{2}} ⋯ \end{matrix}  (3)

where X = {x_{1}, x_{2}, x_{3}, … }, K[[X]] is the “large” power series ring of all functions from the free abelian monoid M = [X] (the free abelian monoid generated by X) to K, and where the summation extends over all n = p_{1}^{α_{1}}p_{2}^{α_{2}} ⋯ ∈ ℕ^{+}. Here, and henceforth, we denote by p_{i} the i’th prime number, with p_{1} = 2, and by P the set of all prime numbers. That (3) is an isomorphism is immediate from the following isomorphism of commutative monoids, implied by the fundamental theorem of arithmetics:

(ℕ^{+}, ·) ≃ ∐_{p∈P} (ℕ, +)  (4)

The author was supported by grants from Svenska Institutet and Kungliga Vetenskapsakademin, and by grant n. 231801F from Centre International des Etudiants et Stagiaires. Received 1 November 1999, revised 19 January 2000; published on 31 January 2000. 1991 Mathematics Subject Classification: 13Dxx,10.00 Key words and phrases: Ring of number theoretic functions, Poincaré-Betti series, stable monomial ideals \begin{matrix}c \\ ◯ \end{matrix} 2000, Jan Snellman. Permission to copy for private use granted.

The following number-theoretic functions are of particular interest (whenever possible, we use the same notation as in [2]):

1. The multiplicative unit ϵ given by ϵ(1) = 1, ϵ(n) = 0 for n > 1,

2. λ : ℕ^{+} → ℕ given by λ(1) = 0, λ(q_{1} ⋯ q_{l}) = l if q_{1}, …, q_{l} are any (not necessarily distinct) prime numbers.

3. λ̃ : ℕ^{+} → ℕ given λ̃(1) = 0, λ̃(p_{1}^{a_{1}} … p_{r}^{a_{r}}) = ∑ a_{r}p_{r}.

4. The Möbius function μ(1) = 1, μ(n) = (−1)^{v} if n is the product of v distinct prime factors, and 0 otherwise,

5. For any i ∈ ℕ^{+}, χ_{i}(p_{i}) = 1, and χ_{i}(m) = 0 for m ≠ p_{i}. Note that under the isomorphism (3), Φ(χ_{i}) = x_{i}.

The topic of this article is the study of the “truncations” Γ_{n}, where for each n ∈ ℕ^{+},

Γ_{n} = { f ∈ Γ m > n ⟹ f(m) = 0 }  (5)

With the modified multiplication given by

fg(n) = ∑_{\begin{matrix}(a,b)∈\{1,...,n\}×\{1,...,n\} \\ ab=n \end{matrix}} f(a)g(b)  (6)

Γ_{n} becomes a K-algebra, isomorphic to Γ/J_{n}, where J_{n} is the ideal

J_{n} = { f ∈ Γ ∀m ⩽ n : f(m) = 0 } .

If we define

π_{n} : Γ → Γ_{n}  (7)
π_{n}(f)(m) = \begin{cases}f(m) & m ⩽ n \\ 0 & m > n \end{cases}  (8)

then π_{n} is a K-algebra epimorphism, and J_{n} is the kernel of π_{n}. We note furthermore that J_{n} is generated by monomials in the elements χ_{i}.

To describe the main idea of this paper, we need a few additional definitions. First, for any n ∈ ℕ^{+} we denote by r(n) ∈ ℕ the largest integer such that p_{r(n)} ⩽ n. In other words, r(n) is the number of prime numbers ⩽ n (this number is often denoted π(n)). Secondly, for a monomial m = x_{1}^{α_{1}} ⋯ x_{w}^{α_{w}}, we define the support Supp(m) as the set of positive integers i such that α_{i} > 0. We define max(m) and min(m) as the maximal and minimal elements in the support of m.

Definition 1.1. A monomial ideal I ⊂ K[x_{1}, …, x_{r}] is said to be strongly stable if whenever m is a monomial such that x_{j}m ∈ I, then x_{i}m ∈ I for all i ⩽ j. If this condition holds at least for all i ⩽ j = max(m) then I is said to be stable.

We can now state our main theorem:

Theorem 1.2. Let n ∈ ℕ^{+} and r = r(n). Then the following holds:

(I) Γ_{n} ≃ \frac{K[x_{1},...,x_{r}]}{I_{n}}, where I_{n} is a strongly stable monomial ideal, with respect to the reverse

order of the variables. (II) Γ_{n} is artinian, with dim_{K}(Γ_{n}) = n. Furthermore, if it is given the natural grading

with |χ_{i}| = 1, then its Hilbert series is ∑_{i} d_{i}t^{i} where d_{i} is the number of w ⩽ n with λ(w) = i.

(III) There is a 1-1 bijection between the minimal monomial generators of I_{n} of minimal support v, and the solutions in non-negative integers to the equation

log n − log p_{v} < ∑_{i=v}^{r} b_{i} log p_{i} ⩽ log n  (9)

(IV) If we denote by C_{n,v} the number of such solutions, then the Poincaré-Betti series of the free minimal resolution of K as a cyclic module over Γ_{n} is the following rational function:

P (Tor_{∗}^{Γ_{n}}(K, K), t) = \frac{(1 + t)^{r}}{1 − t^{2} (∑_{i=1}^{r}(1 + t)^{(i−1)}C_{n,r−i+1})}  (10)

We will show this result, and also give the graded Poincaré-Betti series. For this, we define the number C_{n,v,d} which counts the number of minimal generators of I_{n} of minimal support v and total degree d. We determine some elementary properties of the numbers C_{n,v,d} and C_{n,v}.

2. The ring of number-theoretic functions and its truncations

2.1. Norms, degrees, and multiplicativity

For a monomial M ∋ m = x_{1}^{a_{1}} … x_{n}^{a_{n}} we define the weight of m as w(m) = p_{1}^{a_{1}} … p_{n}^{a_{n}} (we put w(1) = 1). Hence w gives a bijection between M and ℕ^{+}. Furthermore, we can define a term order on M by m > m′ iff w(m) > w(m′). If we define the initial monomial in(f) of f ∈ K[[X]] as the monomial in Supp(f) minimal with respect to >, then in(f) is easily seen to correspond to the norm N(α) of a number-theoretic function α, defined as the smallest n such that α(n) ≠ 0. Here, we must use w and Φ to identify M and ℕ^{+} and K[[X]] and Γ. As observed in [2], the norm is multiplicative: N(αβ) = N(α)N(β).

Cashwell and Everett also define the degree D(α) to mean the smallest d such that there exists an n with λ(n) = d and α(n) ≠ 0. This corresponds the smallest total degree of a monomial in Supp(f). Furthermore, the norm M(α), defined as the smallest integer n with λ(n) = D(α), α(n) ≠ 0, corresponds to the initial monomial of f under the term order obtained by refining the total degree partial order with the term order >.

A multiplicative function is an element α ∈ Γ such that α(1) = 1 and α(ab) = α(a)α(b) whenever a and b are relatively prime. Cashwell and Everett observes that a multiplicative function is necessarily a unit in Γ. One can further observe that if α is multiplicative, then f = Φ(α) can be written

f(x_{1}, x_{2}, x_{3}, …) = f_{1}(x_{1})f_{2}(x_{2})f_{3}(x_{3}) ⋯

where each f_{i}(x_{i}) ∈ K[[x_{i}]] is invertible. In particular, the constant function Γ ∋ ν_{0} with ν_{0}(n) = 1 for all n, corresponds to

∑_{m∈M} m = \frac{1}{1 − x_{1}} \frac{1}{1 − x_{2}} \frac{1}{1 − x_{3}} ⋯

Since the Möbius function is defined to be the inverse of this function, we get that it corresponds to

(1 − x_{1})(1 − x_{2})(1 − x_{3}) ⋯ = 1 − (∑_{i=1}^{∞} x_{i}) + (∑_{i<j} x_{i}x_{j}) − ( ∑_{i<j<k} x_{i}x_{j}x_{k}) + ⋯

2.2. Truncations of the ring of number-theoretic functions Let n, n′ ∈ ℕ^{+}, n′ > n. Then there is a K-algebra epimorphism

φ_{n}^{n′} : Γ_{n′} → Γ_{n}
φ_{n}^{n′}(f)(m) = \begin{cases}f(m) & m ⩽ n \\ 0 & m > n \end{cases}

Hence, the Γ_{n}’s form an inverse system.

Lemma 2.1. \underleftarrow{lim} Γ_{n} ≃ Γ.

Proof. Given any f ∈ Γ, the sequence (π_{1}(f), π_{2}(f), π_{3}(f), …) is coherent. Conversely, given any coherent sequence (g_{1}, g_{2}, g_{3}, …), we can define g : ℕ → K by g(m) = g_{i}(m) where i ⩾ m.

As a side remark, we note that

Lemma 2.2. The decreasing filtration

J_{1} ⊋ J_{2} ⊋ J_{3} ⊋ ⋯  (11)

is separated, that is, ∩_{n}J_{n} = (0).

Definition 2.3. We define

I_{n} = K[[X]] { m ∈ M w(m) > n } ,  (12)

that is, as the monomial ideal in K[[X]] generated by all monomials of weight strictly higher than n. We put A_{n} = \frac{K[[X]]}{I_{n}}.

Proposition 2.4. A K-basis of A_{n} is given by all monomials of weight ⩽ n. Hence A_{n} is an artinian algebra, with dim_{K}(A_{n}) = n. Putting r = r(n), we have that

A_{n} = \frac{K[[X]]}{I_{n}} ≃ \frac{K[x_{1}, …, x_{r}]}{I_{n} ∩ K[x_{1}, …, x_{r}]}  (13)

Proof. As a vector space, K[[X]] ≃ U ⊕ I_{n}, where U consists of all functions supported on monomials of weight ⩽ n. It follows that A_{n} ≃ U as K vector spaces. Of course, there are exactly n monomials of weight ⩽ n. Finally, if s > r then w(x_{s}) = p_{s} > n, hence x_{s} ∈ I_{n}.

We will abuse notations and identify I_{n} and its contraction I_{n} ∩ K[x_{1}, …, x_{r}].

Lemma 2.5. Γ_{n} ≃ A_{n}.

Proof. Since A_{n} has a K-basis is given by all monomials of weight ⩽ n, the two K-algebras are isomorphic as K-vector spaces. The multiplication in A_{n} is induced from the multiplication in K[[X]], with the extra condition that monomials of weight > n are truncated. This is the same multiplication as in Γ_{n}.

Proposition 2.6. I_{n} is a strongly stable ideal, with respect to the reverse order of the variables.

Proof. We must show that if m ∈ I_{n}, and x_{i} |m, then mx_{j}/x_{i} ∈ I for i ⩽ j ⩽ r. We have that w(mx_{j}/x_{i}) = w(m)p_{j}/p_{i} > w(m) > n.

Part I of the main theorem is now proved.

We give K[x_{1}, …, x_{r}] an ℕ^{2}-grading by giving the variable x_{i} bi-degree (1, p_{i}). Since each I_{n} is bihomogeneous, this grading is inherited by A_{n}.

Theorem 2.7. The bi-graded Hilbert series of A_{n} is given by

A_{n}(t, u) = ∑_{i,j} c_{ij}t^{i}u^{j},

where c_{ij} is the number of p_{1}^{a_{1}} … p_{r}^{a_{r}} ⩽ n with ∑ a_{r} = i and ∑ a_{r}p_{r} = j. Furthermore,

A_{n}(t, 1) = ∑_{i} d_{i}t^{i}
A_{n}(1, u) = ∑_{j} e_{j}u^{j}

where d_{i} is the number of w ⩽ n with λ(w) = i, and e_{i} is the number of w ⩽ n with λ̃(w) = i. In particular, the t^{1}-coefficient of A_{n}(t, 1) is the number of prime numbers ⩽ n.

Proof. The monomial x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} has bi-degree (∑_{i=1}^{n} a_{i}, ∑ a_{i}p_{i}).

This establishes part II of the main theorem.

3. Minimal generators for I_{n}

Let n ∈ ℕ^{+}, and let r = r(n). We have that

x_{1}^{a_{1}} … x_{r}^{a_{r}} = m ∈ I_{n} ⟺ w(m) > n ⟺ ∏_{i=1}^{r} p_{i}^{a_{i}} > n.  (14)

We denote by G(I_{n}) the set of minimal monomial generators of I_{n}. For m = x_{1}^{a_{1}} … x_{r}^{a_{r}} to be an element of G(I_{n}) it is necessary and sufficient that m ∈ I_{n} and that for 1 ⩽ v ⩽ r, x_{v} |m ⟹ m/x_{v} ∉ I_{n}. In other words,

1 ⩽ j ⩽ n, a_{j} > 0 ⟹ n < ∏_{i=1}^{r} p_{i}^{a_{i}} ⩽ p_{j}n.  (15)

Definition 3.1. For n, v, d positive integers, we define:

C_{n} = #G(I_{n})  (16)
C_{n,v} = # { m ∈ G(I_{n}) min(m) = v }  (17)
C_{n,v,d} = # { m ∈ G(I_{n}) min(m) = v, |m| = d }  (18)

Theorem 3.2. C_{n,v} is the number of solutions (b_{1}, …, b_{r}) ∈ ℕ^{r} to the equation

log n − log p_{v} < ∑_{i=v}^{r} b_{i} log p_{i} ⩽ log n.  (19)

Equivalently, C_{n,v} is the number of integers x such that n/p_{v} < x ⩽ n and such that no prime factors of x are smaller than p_{v}.

Similarly, C_{n,v,d} is the number of solutions (b_{1}, …, b_{r}) ∈ ℕ^{r} to the system of equations

log n − log p_{v} < ∑_{i=v}^{r} b_{i} log p_{i} ⩽ log n

∑_{i=1}^{r} b_{i} = d − 1.

(20)

or equivalently, C_{n,v,d} is the number of integers x such that n/p_{v} < x ⩽ n and such that no prime factors of x are smaller than p_{v}, and with the additional constraint that λ(x) = d.

Proof. We have that a_{v} > 0, a_{w} = 0 for w < v. Hence equation (15) implies that

n < ∏_{j=v}^{r} p_{i}^{a_{i}} ⩽ p_{v}n.

Putting b_{v} = a_{v} − 1, b_{j} = a_{j} for j > v we can write this as

n < p_{v} ∏_{j=v}^{r} p_{i}^{b_{i}} ⩽ p_{v}n ⟺ n/p_{v} < ∏_{j=v}^{r} p_{i}^{b_{i}} ⩽ n

from which (19) follows by taking logarithms. This implies (20) as well.

We have now proved part III of the main theorem.

Homology, Homotopy and Applications, vol. 2, No. 2, 2000 22

Figure 1: The numbers C_{n} and C_{n,i}.

n  Σ  i = 1  i = 2  3  4  5  6  7  8  9  10
2  1  1
3  3  2  1
4  3  2  1
5  6  3  2  1
6  6  3  2  1
7  10  4  3  2  1
8  10  4  3  2  1
9  11  5  3  2  1
10  11  5  3  2  1
11  16  6  4  3  2  1
12  16  6  4  3  2  1
13  22  7  5  4  3  2  1
14  22  7  5  4  3  2  1
15  23  8  5  4  3  2  1
16  23  8  5  4  3  2  1
17  30  9  6  5  4  3  2  1
18  30  9  6  5  4  3  2  1
19  38  10  7  6  5  4  3  2  1
20  38  10  7  6  5  4  3  2  1
21  39  11  7  6  5  4  3  2  1
22  39  11  7  6  5  4  3  2  1
23  48  12  8  7  6  5  4  3  2  1
24  48  12  8  7  6  5  4  3  2  1
25  50  13  9  7  6  5  4  3  2  1
26  50  13  9  7  6  5  4  3  2  1
27  51  14  9  7  6  5  4  3  2  1
28  51  14  9  7  6  5  4  3  2  1
29  61  15  10  8  7  6  5  4  3  2  1
30  61  15  10  8  7  6  5  4  3  2  1

Figure 2: The numbers C_{n,i,g}.

n  i = 1  i = 2  3  4  5  6  7  8  9
2  1
3  2  1
4  u + 1  1
5  u + 2  2  1
6  2 u + 1  2  1
7  2 u + 2  3  2  1
8  u2 + u + 2  3  2  1
9  u2 + 2 u + 2  u + 2  2  1
10  u2 + 3 u + 1  u + 2  2  1
11  u2 + 3 u + 2  u + 3  3  2  1
12  2 u2 + 2 u + 2  u + 3  3  2  1
13  2 u2 + 2 u + 3  u + 4  4  3  2  1
14  2 u2 + 3 u + 2  u + 4  4  3  2  1
15  2 u2 + 4 u + 2  2 u + 3  4  3  2  1
16  u3 + u2 + 4 u + 2  2 u + 3  4  3  2  1
17  u3 + u2 + 4 u + 3  2 u + 4  5  4  3  2  1
18  u3 + 2 u2 + 3 u + 3  2 u + 4  5  4  3  2  1
19  u3 + 2 u2 + 3 u + 4  2 u + 5  6  5  4  3  2  1
20  u3 + 3 u2 + 2 u + 4  2 u + 5  6  5  4  3  2  1
21  u3 + 3 u2 + 3 u + 4  3 u + 4  6  5  4  3  2  1
22  u3 + 3 u2 + 4 u + 3  3 u + 4  6  5  4  3  2  1
23  u3 + 3 u2 + 4 u + 4  3 u + 5  7  6  5  4  3  2  1
24  2 u3 + 2 u2 + 4 u + 4  3 u + 5  7  6  5  4  3  2  1
25  2 u3 + 2 u2 + 5 u + 4  4 u + 5  u + 6  6  5  4  3  2  1
26  2 u3 + 2 u2 + 6 u + 3  4 u + 5  u + 6  6  5  4  3  2  1
27  2 u3 + 3 u2 + 6 u + 3  u2 + 3 u + 5  u + 6  6  5  4  3  2  1
28  2 u3 + 4 u2 + 5 u + 3  u2 + 3 u + 5  u + 6  6  5  4  3  2  1
29  2 u3 + 4 u2 + 5 u + 4  u2 + 3 u + 6  u + 7  7  6  5  4  3  2  1
30  2 u3 + 5 u2 + 4 u + 4  u2 + 3 u + 6  u + 7  7  6  5  4  3  2  1

Example 3.3. The first few I_{n}’s are as follows: I_{2} =(x_{1}^{2}), I_{3} =(x_{1}^{2}, x_{2}^{2}, x_{1}x_{2}), I_{4} =(x_{1}^{3}, x_{2}^{2}, x_{1}x_{2}), I_{5} =(x_{1}^{3}, x_{2}^{2}, x_{1}x_{2}, x_{3}^{2}, x_{1}x_{3}, x_{2}x_{3}).

We tabulate C_{n,i} and C_{n,i,d}, the latter in form of the polynomial u^{−2} ∑_{j} C_{n,i,j}u^{j} in the tables 1 and 2.

Theorem 3.4. (1) C_{n,v} = 0 for v > r(n)

(2) ∀n ∈ ℕ : ∀v ⩽ r(n) : C_{n,1+r(n)−v} ⩾ v,

(3) ∀n ∈ ℕ : C_{n} ⩾ \binom{r(n)+1}{2},

(4) ∀v ∈ ℕ : ∃N : ∀n ⩾ N : C_{n,1+r(n)−v} = v.

(5) If n is even, then C_{n,v} = C_{n−1,v} for all v,

(6) C_{n,1} = ⌈n/2⌉.

Proof. (1) Obvious.

(2) and (3) It suffices to show that for any subset S ⊂ {1, …, r} of cardinality 1 or 2, there is an m ∈ G(I_{n}) with Supp(m) = S. If S = {i} then there is an unique positive integer a such that p_{i}^{b−1} ⩽ n < p_{i}^{b}, and m = x_{i}^{b} is the desired generator. If S = {i, j} with i < j then we claim that there is a positive integer a such that x_{i}^{a}x_{j} ∈ G(I_{n}). Namely, choose b such that p_{i}^{b−1} ⩽ n < p_{i}^{b}, then since p_{i} < p_{j} one has n < p_{i}^{b−1}p_{j}. Hence x_{i}^{b−1}x_{j} ∈ I_{n}, so it is a multiple of some minimal generator. By the definition of b, this minimal generator must be of the form x_{i}^{a}x_{j} for some a, which establishes the claim.

(6) We must show that the number of solutions in ℕ^{r} to

\frac{n}{2} < ∏_{i=1}^{r} p_{i}^{b_{i}} ⩽ n

is precisely ⌈\frac{n}{2}⌉. Obviously, any integer ∈ (\frac{n}{2}, n] fits the bill; there are ⌈\frac{n}{2}⌉ of those.

(5) The case v = 1 follows from (6). Hence, it suffices to show that if v > 1, x ∈ (\frac{n}{p_{v}} , n] ∩ ℕ, and if x has no prime factor < p_{v}, then x ∈ (\frac{n−1}{p_{v}}, n − 1] ∩ ℕ. The only way this can fail to happen is if x = n, but then x is even, and has the prime factor 2 = p_{1} < p_{v}, a contradiction.

(4) For large enough n, the only integers x ⩽ n with all prime factors ⩾ 1 + r(n) − v are p_{1+r(n)−v}, …, p_{r(n)}. There is v of these, and they are all > \frac{n}{p_{v}}.

Theorem 3.5. 1. C_{n,v,d} = 0 for v > r(n), and for d < 2, 2. ∀v ∈ ℕ : ∃N : ∀n ⩾ N : C_{n,1+r(n)−v,2} = v, C_{n,1+r(n)−v,d} = 0 for d ≠ 2,

3. \binom{r(n)}{2} = # \{ m ∈ ℕ^{+} m ⩽ n, λ(m) = 2 \}.

Proof. The first and the last assertions are obvious. The second one follows from the proof of (4) in the previous lemma.

4. Poincaré series

In [3], a minimal free multi-graded resolution of a I over S is given, where S = K[x_{1}, …, x_{r}] is a polynomial ring, and I ⊂ (x_{1}, …, x_{r})^{2} is a stable ideal. As a consequence, the following formula for the Poincaré-Betti series is derived:

P (Tor_{∗}^{S}(I, K), t) = ∑_{a∈G(I)}(1 + t)^{max(a)−1}  (21)

where G(I) is the minimal generating set of I. Since the resolution is multi-graded, (21) can be modified to yield a formula for the graded Poincaré-Betti series (we here consider S as ℕ-graded, with each variable given weight 1):

P (Tor_{∗,∗}^{S}(I, K), t, u) = ∑_{a∈G(I)} u^{|a|}(1 + t)^{max(a)−1}  (22)

We will use the following variant of this result:

Theorem 4.1 (Eliahou-Kervaire). Let I ⊂ (x_{1}, …, x_{r})^{2} ⊂ K[x_{1}, …, x_{r}] = S be a stable monomial ideal. Put

b_{i,d} = # { m ∈ G(I) max(m) = i, |m| = d }  (23)
b_{i} = # { m ∈ G(I) max(m) = i }  (24)

Then

P (Tor_{∗}^{S}(I, K), t) = ∑_{i=1}^{r} b_{i}(1 + t)^{(i−1)}  (25)

P (Tor_{∗,∗}^{S}(I, K), t, u) = ∑_{i=1}^{r} ((1 + tu)^{(i−1)} ∑_{j} b_{i,j}u^{j}) .  (26)

For the Betti-numbers we have that

β_{q} = dim_{K} (Tor_{q}^{S}(I, K)) = ∑_{i=1}^{r} b_{i}\binom{i − 1}{q}.  (27)

From Proposition 2.6 we have that the ideals I_{n} are stable after reversing the order of the variables. Hence, replacing max by min, and hence b_{i} with C_{n,1+r−i}, we get:

Corollary 4.2. Let n ∈ ℕ^{+}, r = r(n), S = K[x_{1}, …, x_{r}]. Then

P (Tor_{∗}^{S}(I_{n}, K), t) = ∑_{i=1}^{r} C_{n,1+r−i}(1 + t)^{(i−1)}  (28)

P (Tor_{∗,∗}^{S}(I_{n}, K), t, u) = ∑_{i=1}^{r}(1 + tu)^{(i−1)} ∑_{j} C_{n,1+r−i,j}u^{j}.  (29)

For the Betti-numbers we have that

β_{q} = ∑_{i=1}^{r} C_{n,1+r−i}\binom{i − 1}{q}.  (30)

In [6, 1] it is shown that if S = K[x_{1}, …, x_{r}] and I is a stable monomial ideal in S, then S/I is a Golod ring. Hence, from a result of Golod [4] (see also [5]), it follows that

P (Tor_{∗}^{S/I}(K, K), t) = \frac{(1 + t)^{r}}{1 − t^{2}P (Tor_{∗}^{S}(I, K), t)}  (31)

Regarding S as an ℕ-graded ring, one can show that in fact

P (Tor_{∗}^{S/I}(K, K), t, u) = \frac{(1 + ut)^{r}}{1 − t^{2}P (Tor_{∗}^{S}(I, K), t, u)}  (32)

The following theorem is an immediate consequence:

Theorem 4.3 (Herzog-Aramova, Peeva). Let S = K[x_{1}, …, x_{r}], and suppose that I is a stable monomial ideal in S. Put

\begin{aligned}b_{i,d} & = # \{ x ∈ G(I) max(x) = i, |x| = d \} \\ b_{i} & = # \{ x ∈ G(I) max(x) = i \} \end{aligned}

Then, for R = S/I, we have that

P (Tor_{∗}^{R}(K, K), t) = \frac{(1 + t)^{r}}{1 − t^{2} ∑_{i=1}^{r}(1 + t)^{(i−1)} ∑_{j} b_{i}}  (33)
P (Tor_{∗}^{R}(K, K), t, u) = \frac{(1 + t)^{r}}{1 − t^{2} ∑_{i=1}^{r}(1 + tu)^{(i−1)} ∑_{j} b_{i,j}u^{j}}  (34)

Specialising to the case of A_{n}, we obtain:

Corollary 4.4. Let n ∈ ℕ^{+}, and let r = r(n). Regard A_{n} as a naturally graded K-algebra, with each x_{i} given weight 1, and regard K as a cyclic A-module. Then

P (Tor_{∗}^{A_{n}}(K, K), t) = \frac{(1 + t)^{r}}{1 − t^{2} ∑_{i=1}^{r}(1 + t)^{(i−1)}C_{n,r−i+1}}  (35)
P (Tor_{∗}^{A_{n}}(K, K), t, u) = \frac{(1 + ut)^{r}}{1 − t^{2} (∑_{i=1}^{r} ((1 + tu)^{(i−1)} ∑_{j} C_{n,r−i+1,j}u^{j}))}  (36)

Part IV of the main theorem is now proved.

Example 4.5. We consider the case n = 5, then r = r(n) = 3, so S = K[x_{1}, x_{2}, x_{3}] and I = I_{5} = (x_{1}^{3}, x_{1}x_{2}, x_{1}x_{3}, x_{2}^{2}, x_{2}x_{3}, x_{3}^{2}). We get that C_{5,1} = 3, C_{5,2} = 2, C_{5,3} = 1. According to our formulas^{1} we have

P_{I}^{S}(t) = 1 + 2(1 + t) + 3(1 + t)^{2} = 6 + 8t + 3t^{2}
P_{K}^{S/I} = \frac{(1 + t)^{r}}{1 − t^{2}P_{I}^{S}(t)} = \frac{1}{1 − 3t}

When we consider the grading by total degree, we have that C_{5,1,2} = 2, C_{5,1,3} = 1, C_{5,2,2} = 2, C_{5,3,2} = 1. Hence, our formulas yield

P_{I}^{S}(t, u) = u^{2} + 2u^{2}(1 + t) + (2u^{2} + u^{3})(1 + t)^{2}
= 5u^{2} + u^{3} + (6u^{2} + 2u^{3})t + (2 u^{2} + u^{3}) t^{2}
P_{K}^{S/I}(t, u) = −\frac{1 + tu}{u^{3}t^{2} + 2 t^{2}u^{2} + 2 tu − 1}

We list the first few Poincaré-Betti series P (Tor_{∗}^{A_{n}}(K, K), t, u) in table 3.

Conjecture 4.6. P (Tor_{∗}^{A_{n}}(K, K), t) = −\frac{(1+t)^{ℓ_{1}(n)}}{q_{n}(t)} , q_{n}(t) = ∑_{i=0}^{ℓ_{2}(n)} h_{i}(n)t^{i}, with

1. q_{n}(−1) ≠ 0,

2. ℓ_{1}(n) is the number of odd primes p such that p^{2} ⩽ n,

3. ℓ_{2}(n) = ℓ_{1}(n) + 1,

4. h_{0}(n) = −1,

5. h_{1}(n) = r(n) − ℓ_{1}(n),

6. h_{ℓ_{2}(n)}(n) = C_{n,1} = ⌈n/2⌉.

5. Acknowledgements

I am indebted to Johan Andersson for suggesting the idea of studying the homological properties of the truncations Γ_{n}. I thank the referee for suggesting a simplified proof of parts of Theorem 3.4.

1Here, we have used the abbreviation P_{I}^{S}(t) = P (Tor_{∗}^{S}(I, K), t), we will also write P_{K}^{S/I}(t) = P (Tor_{∗}^{S/I}(K, K), t) et cetera.

\begin{matrix}n & Graded & Non − graded \\ 2 & − (tu − 1)^{−1} & − (t − 1)^{−1} \\ 3 & − (2 tu − 1)^{−1} & − (2 t − 1)^{−1} \\ 4 & −\frac{1+tu}{(u^{3}+u^{2})t^{2}+tu−1} & − (2 t − 1)^{−1} \\ 5 & −\frac{1+tu}{(u^{3}+2 u^{2})t^{2}+2 tu−1} & − (3 t − 1)^{−1} \\ 6 & −\frac{1+tu}{(2 u^{3}+u^{2})t^{2}+2 tu−1} & − (3 t − 1)^{−1} \\ 7 & −\frac{1+tu}{(2 u^{3}+2 u^{2})t^{2}+3 tu−1} & − (4 t − 1)^{−1} \\ 8 & −\frac{1+tu}{(u^{4}+u^{3}+2 u^{2})t^{2}+3 tu−1} & − (4 t − 1)^{−1} \\ 9 & −\frac{1+2 tu+t^{2}u^{2}}{(u^{5}+2 u^{4}+2 u^{3})t^{3}+(u^{4}+3 u^{3}+4 u^{2})t^{2}+2 tu−1} & −\frac{1+t}{5 t^{2}+3 t−1} \\ 10 & −\frac{1+2 tu+t^{2}u^{2}}{(u^{5}+3 u^{4}+u^{3})t^{3}+(u^{4}+4 u^{3}+3 u^{2})t^{2}+2 tu−1} & −\frac{1+t}{5 t^{2}+3 t−1} \\ 11 & −\frac{1+2 tu+t^{2}u^{2}}{(u^{5}+3 u^{4}+2 u^{3})t^{3}+(u^{4}+4 u^{3}+5 u^{2})t^{2}+3 tu−1} & −\frac{1+t}{6 t^{2}+4 t−1} \\ 12 & −\frac{1+2 tu+t^{2}u^{2}}{(2 u^{5}+2 u^{4}+2 u^{3})t^{3}+(2 u^{4}+3 u^{3}+5 u^{2})t^{2}+3 tu−1} & −\frac{1+t}{6 t^{2}+4 t−1} \\ 13 & −\frac{1+2 tu+t^{2}u^{2}}{(2 u^{5}+2 u^{4}+3 u^{3})t^{3}+(2 u^{4}+3 u^{3}+7 u^{2})t^{2}+4 tu−1} & −\frac{1+t}{7 t^{2}+5 t−1} \\ 14 & −\frac{1+2 tu+t^{2}u^{2}}{(2 u^{5}+3 u^{4}+2 u^{3})t^{3}+(2 u^{4}+4 u^{3}+6 u^{2})t^{2}+4 tu−1} & −\frac{1+t}{7 t^{2}+5 t−1} \\ 15 & −\frac{1+2 tu+t^{2}u^{2}}{(2 u^{5}+4 u^{4}+2 u^{3})t^{3}+(2 u^{4}+6 u^{3}+5 u^{2})t^{2}+4 tu−1} & −\frac{1+t}{8 t^{2}+5 t−1} \\ 16 & −\frac{1+2 tu+t^{2}u^{2}}{(u^{6}+u^{5}+4 u^{4}+2 u^{3})t^{3}+(u^{5}+u^{4}+6 u^{3}+5 u^{2})t^{2}+4 tu−1} & −\frac{1+t}{8 t^{2}+5 t−1} \\ 17 & −\frac{1+2 tu+t^{2}u^{2}}{(u^{6}+u^{5}+4 u^{4}+3 u^{3})t^{3}+(u^{5}+u^{4}+6 u^{3}+7 u^{2})t^{2}+5 tu−1} & −\frac{1+t}{9 t^{2}+6 t−1} \\ 18 & −\frac{1+2 tu+t^{2}u^{2}}{(u^{6}+2 u^{5}+3 u^{4}+3 u^{3})t^{3}+(u^{5}+2 u^{4}+5 u^{3}+7 u^{2})t^{2}+5 tu−1} & −\frac{1+t}{9 t^{2}+6 t−1} \\ 19 & −\frac{1+2 tu+t^{2}u^{2}}{(u^{6}+2 u^{5}+3 u^{4}+4 u^{3})t^{3}+(u^{5}+2 u^{4}+5 u^{3}+9 u^{2})t^{2}+6 tu−1} & −\frac{1+t}{10 t^{2}+7 t−1} \\ 20 & −\frac{1+2 tu+t^{2}u^{2}}{(u^{6}+3 u^{5}+2 u^{4}+4 u^{3})t^{3}+(u^{5}+3 u^{4}+4 u^{3}+9 u^{2})t^{2}+6 tu−1} & −\frac{1+t}{10 t^{2}+7 t−1} \\ 21 & −\frac{(1+tu)^{2}}{t^{3}u^{6}+3 u^{5}t^{3}+t^{2}u^{5}+3 t^{3}u^{4}+3 u^{4}t^{2}+4 t^{3}u^{3}+6 t^{2}u^{3}+8 t^{2}u^{2}+6 tu−1} & −\frac{1+t}{11 t^{2}+7 t−1} \\ 22 & −\frac{(1+tu)^{2}}{t^{3}u^{6}+3 u^{5}t^{3}+t^{2}u^{5}+4 t^{3}u^{4}+3 u^{4}t^{2}+3 t^{3}u^{3}+7 t^{2}u^{3}+7 t^{2}u^{2}+6 tu−1} & −\frac{1+t}{11 t^{2}+7 t−1} \\ 23 & −\frac{(1+tu)^{2}}{t^{3}u^{6}+3 u^{5}t^{3}+t^{2}u^{5}+4 t^{3}u^{4}+3 u^{4}t^{2}+4 t^{3}u^{3}+7 t^{2}u^{3}+9 t^{2}u^{2}+7 tu−1} & −\frac{1+t}{12 t^{2}+8 t−1} \\ 24 & −\frac{(1+tu)^{2}}{2 t^{3}u^{6}+2 u^{5}t^{3}+2 t^{2}u^{5}+4 t^{3}u^{4}+2 u^{4}t^{2}+4 t^{3}u^{3}+7 t^{2}u^{3}+9 t^{2}u^{2}+7 tu−1} & −\frac{1+t}{12 t^{2}+8 t−1} \\ 25 & −\frac{(1+tu)^{3}}{q(t,u)} & −\frac{(1+t)^{2}}{13 t^{3}+22 t^{2}+7 t−1} \end{matrix}

Figure 3: Graded and non-graded Poincaré-Betti series of the minimal free resolution of K over A_{n}.

References

[1] Annetta Aramova and Jürgen Herzog. Koszul Cycles and Eliahou-Kervaire Type Resolutions. Journal of Algebra, 181(2):347–370, 1996.

[2] E. D. Cashwell and C. J. Everett. The ring of number-theorethic functions. Pacific Journal of Mathematics, 9:975–985, 1959.

[3] S. Eliahou and M. Kervaire. Minimal resolutions of some monomial ideals. J. Algebra, 129:1–25, 1990.

[4] E. S. Golod. On the homology of some local rings. Soviet Math. Dokl., 3:745–749, 1962.

[5] T. Gulliksen and G. Levin. Homology of Local Rings, volume 20 of Queen’s Papers in Pure and Applied Mathematics. 1969.

[6] Irena Peeva. 0-Borel Fixed Ideals. Journal of Algebra, 184(3):945–984, 1996.

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Jan Snellman jans@matematik.su.se

School of Informatics University of Wales Dean Street, Bangor Gwynedd LL57 1UT Wales UK