Math. Proc. Camb. Phil. Soc. 121 (1997) 437–441

SHIFTS IN RESOLUTIONS OF MULTIGRADED MODULES

Srikanth Iyengar

Abstract. Upper bounds are established on the shifts in a minimal resolution of a multigraded module. Similar bounds are given on the coefficients in the numerator of the Backelin-Lescot rational expression for multigraded Poincaré series.

Let K be a field and S = K[x_{1}, …, x_{n}] the polynomial ring with its natural n-grading. When I is an ideal generated by monomials in the variables x_{1}, …, x_{n}, the ring R = S/I is n-graded for the induced grading. We denote by deg[y] the degree in ℤ^{n} of a homogeneous element y in an n-graded R-module.

An n-graded finite R-module M is known to have a free resolution with i′th module ⨁_{j} Re_{i,j}, where the basis elements are n-homogeneous, and differentials ∂_{i} of degree 0, given by matrices whose entries are monomials in the variables. Furthermore, the resolution may be chosen such that Ker ∂_{i−1} is minimally generated by {∂_{i}(e_{i,j})}_{j}. Such a minimal resolution is unique up to n-graded isomorphism.

1. Shifts

For F a minimal resolution of M, the Betti numbers β_{i} = rank(F_{i}) and the shifts

deg[e_{i,j}] = a_{i,j} = (a_{i,j}(1), …, a_{i,j}(n)) ∈ ℤ^{n}

are invariants of M. We consider ℤ^{n} with the product order: c < d if c(k) ≤ d(k) for all 1 ≤ k ≤ n and c ≠ d. Every set {c_{1}, …, c_{m}} ⊂ ℤ^{n} has a least upper bound c = lub{c_{1}, …, c_{m}}, where c(k) = max{c_{1}(k), …, c_{m}(k)}. When a ∈ ℤ^{n}, we denote the monomial x_{1}^{a(1)} ⋯ x_{n}^{a(n)} by x^{a}. We write ⌊u⌋ for the greatest integer less than or equal to u and ⌈u⌉ for the least integer greater than or equal to u.

In the recent paper [2], Bruns and Herzog have obtained upper bounds on the shifts in F. We present a new, more direct approach to this problem, which leads to sharper bounds in the case of infinite resolutions.

Theorem 1. Let I be an ideal in the ring S = K[x_{1}, …, x_{n}], minimally generated by monomials x^{b_{1}}, …, x^{b_{m}} and set R = S/I. For an n-graded finite R-module M with shifts a_{i,j}, there are inequalities

a_{i,j} ≤ a + ⌊\frac{i}{2}⌋b for all i, j ≥ 1,

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 13A02, 13D02.

where b = lub{b_{1}, …, b_{m}} and a = lub{a_{1,1}, …, a_{1,β_{1}}}.

If I ≠ 0, then for each even i there is a k such that a_{i,j}(k) < a(k) + ⌊\frac{i}{2}⌋b(k).

The statement of Theorem 1 was suggested by the work of Bruns and Herzog [2]. As noted in [2, (3.2)], the case I = 0 of the theorem has the following consequence:

Corollary. Let M be a finite n-graded module, over the polynomial ring S, with shifts a_{i,j}. If cx^{a}, where c ∈ K, is a non-zero entry of some ∂_{i} in the multigraded resolution of M, then

x^{a} | lcm(x^{a_{1,1}}, …, x^{a_{1,β_{1}}} ).

In particular, the entries in the multigraded resolution of a monomial ideal divide the least common multiple of the generators of the ideal. □

The following example shows that, for each n, the bounds obtained in Theorem 1 are the best possible, in terms of a and b.

Example 1. Let R = K[x_{1}, …, x_{n}]/(x_{1} ⋯ x_{n}) so that b = (1, …, 1). The R-module M = R/(x_{n}^{2})R has a minimal resolution

F : ⋯ ⟶ R ⟶^{x_{1}···x_{n−1}} R ⟶^{x_{n}} R ⟶^{x_{1}···x_{n−1}} R ⟶^{x_{n}^{2}} R ⟶ 0.

So a = (0, …, 0, 2) and there are inequalities

a_{i,1} = \{ \begin{matrix}a + ⌊\frac{i}{2}⌋b & for odd i ≥ 1, \\ a + ⌊\frac{i}{2}⌋b − (0, …, 0, 1) & for even i ≥ 2. \end{matrix}

Thus the shifts attain the upper bound of Theorem 1 when i is odd, and are equal to them in all but the last coordinate when i is even.

Remark. Theorem 1 coincides with [2, (3.1)] when I = 0 and gives sharper bounds than [2, (3.4)] when I ≠ 0. Indeed, for M and R as in the example above, [2, (3.4)] yields a_{i,1} ≤ a + ⌊\frac{i}{2}⌋b + (1, …, 1).

Proof of Theorem 1. We have to show that for each k there is an r such that a_{i,j}(k) ≤ a_{1,r}(k) + ⌊\frac{i}{2}⌋b(k). The inequality is clear for i = 1. For i ≥ 2 it is enough to prove:

(1) If i is even, then for each k there exists an s such that a_{i,j}(k) ≤ a_{i−1,s}(k) + b(k).

(2) If i is odd, then for each k there exists an r such that a_{i,j}(k) ≤ a_{i−2,r}(k) + b(k). Indeed, if i is odd, then repeated application of (2) yields the desired inequality. If i

even, then using (1) find an s such that a_{i,j}(k) ≤ a_{i−1,s}(k) + b(k). When i = 2 we are done, else (i − 1) ≥ 3 is odd, so there is an r with a_{i−1,s}(k) ≤ a_{1,r}(k) + ⌊\frac{i−1}{2}⌋b(k). Thus,

a_{i,j}(k) ≤ a_{1,r}(k) + ⌊\frac{i−1}{2}⌋b(k) + b(k) = a_{1,r}(k) + ⌊\frac{i}{2}⌋b(k).

The proof of (1) and (2) uses the following observation which is verified directly:

(3) If F is a free R-module and u ∈ F, then x_{k}^{b(k)+1}u = 0 implies x_{k}^{b(k)} u = 0.

Fix a segment F_{i}⟶^{ν}F_{i−1}⟶^{μ}F_{i−2} of the minimal resolution F. Let ν(e_{i,j}) = ∑_{s} ν_{sj}e_{i−1,s} and μ(e_{i−1,j}) = ∑_{s} μ_{sj}e_{i−2,s}. Note that

(4)  a_{i,j} = deg[ν(e_{i,j})] = deg[ν_{sj}] + a_{i−1,s} for all s with ν_{sj} ≠ 0.

If i ≥ 2 is even and (1) fails, then choose k such that for all s, 1 ≤ s ≤ β_{i−1}, we have a_{i,j}(k) > a_{i−1,s}(k) + b(k). Together with (4) this implies that, for d = b(k), the monomial ν_{sj} is of the form x_{k}^{d+1} ν′_{sj} for all s. Set

w = ∑_{s} ν′_{sj}e_{i−1,s} ∈ F_{i−1} and y = x_{k}^{d} w ∈ F_{i−1}.

Note that x_{k}^{d+1}μ(w) = μ(ν(e_{i,j})) = 0. In view of (3), this shows that x_{k}^{d} μ(w) = 0. Therefore, μ(y) = 0, that is, y ∈ Kerμ. But now ν(e_{i,j}) = x_{k} y ∈ 𝔪 Kerμ, where 𝔪 = (x_{1}, …, x_{n}) is the homogeneous maximal ideal of R. As F is minimal, this is a contradiction.

This completes the proof of (1).

Assuming i ≥ 3 is odd and (2) fails, choose k such that

(5)  a_{i,j}(k) > a_{i−2,r}(k) + b(k) for all 1 ≤ r ≤ β_{i−2} .

If a_{i−1,s}(k) ≥ a_{i,j}(k) for some s, then from (5) we get a_{i−1,s}(k) > a_{i−2,r}(k) + b(k), for all r. Since (i − 1) ≥ 2 is even, this contradicts (1). Thus, for each s we have, a_{i,j}(k) > a_{i−1,s}(k). Hence, using (4), we write each monomial ν_{sj} in the form x_{k} ν′_{sj}.

Two applications of (4) yield a_{i,j} = deg[ν_{sj}] + deg[μ_{rs}] + a_{i−2,r} whenever ν_{sj} ≠ 0 and μ_{rs} ≠ 0. With (5) this shows that, for d = b(k), the monomial ν_{sj}μ_{rs} is of the form x_{k}^{d+1} λ_{sr} for all r and s. Consider the elements

w = ∑_{r,s} λ_{sr}e_{i−2,r} ∈ F_{i−2} and y = ∑_{s} ν′_{sj}e_{i−1,s} ∈ F_{i−1}.

A straightforward computation shows that x_{k}^{d} w = μ(y). As in the proof of (1), we see that x_{k}^{d+1}w = μ(ν(e_{i,j})) = 0. By (3), this implies μ(y) = x_{k}^{d} w = 0. Thus y ∈ Kerμ and hence ν(e_{i,j}) = x_{k} y ∈ 𝔪 Kerμ. Again, we get a contradiction to the minimality of F.

Thus (2) holds. To complete the proof of Theorem 1 we need to prove that when I ≠ 0 and i is even, a_{i,j} < a + ⌊\frac{i}{2}⌋b.

Assume to the contrary. Choose j such that a_{i,j} = a + ⌊\frac{i}{2}⌋b. As F is minimal, there is a ν_{s,j} ≠ 0. From (4) we get a_{i,j} = deg[ν_{s,j}] + a_{i−1,s}. Hence

a + ⌊\frac{i}{2}⌋b = deg[ν_{s,j}] + a_{i−1,s} ≤ deg[ν_{s,j}] + a + ⌊\frac{i−1}{2}⌋b = deg[ν_{s,j}] + a + ⌊\frac{i}{2}⌋b − b

Therefore deg[ν_{s,j}] ≥ b and so ν_{s,j} = 0. This contradicts the choice of ν_{s,j}.

Thus Theorem 1 is proved. □

2. Poincaré Series

For each i ≥ 0, the K-vector space Tor_{i}^{R}(M, K) is multigraded with homogeneous basis elements of degrees a_{i,j} for j = 1, …, β_{i}. The multigraded Poincaré series of M in the variables t and s_{1}, …, s_{n} is defined by P_{M}^{R}(s, t) = ∑_{i⩾0}r_{i}(s)t^{i}, where

r_{i}(s) = ∑_{d∈ℤ^{n}} dim_{K} Tor_{i}^{R}(M, K)_{d}s^{d}

is a Laurent polynomial in ℤ[s_{1}^{±1}, …, s_{n}^{±1}]. Note that by setting s_{1} = ⋯ = s_{n} = u one obtains the usual Poincaré series P_{M}^{R}(u, t) of the graded R-module associated to the n-graded R-module M. We denote by deg_{k}(f) the degree of f ∈ ℤ[s_{1}^{±1}, …, s_{n}^{±1}] in the variable s_{k}.

Theorem of Backelin and Lescot [1,5]. If M is finitely generated, then P_{M}^{R}(s, t) is a rational function of the form

P_{M}^{R}(s, t) = \frac{∑_{i=0}^{p} f_{i}^{M}(s)t^{i}}{1 + ∑_{j=1}^{q} g_{j}^{R}(s)t^{j}} ,

with f_{i}^{M} ∈ ℤ[s_{1}^{±1}, …, s_{n}^{±1}] and g_{j}^{R} ∈ ℤ[s_{1}, …, s_{n}]. The denominator is independent of M and satisfies q ≤ ∑_{k} b(k) and deg_{k}(g_{j}^{R}) ≤ b(k) for k = 1, …, n where b = lub{b_{1}, …, b_{m}}.

It should be noted that the assumption that M is n-graded is essential. Indeed, Jacobsson [3, Corollary 2.3] constructs a graded module M over an n-graded ring R such that P_{M}^{R}(u, t) is not a rational function.

Using Theorem 1, we establish upper bounds on deg_{k}(f_{i}) and simplify a crucial step in the proof the Backelin and Lescot’s result.

Theorem 2. If M has no free direct summand, then there are inequalities:

deg_{k}(f_{i}^{M}) ≤ a(k) + ⌈\frac{i}{2}⌉b(k) for 1 ≤ k ≤ n 0 ≤ i ≤ p.

Proof. We keep the notation from the proof of Theorem 1, and write f_{i} and g_{i} for f_{i}^{M} and g_{i}^{R} respectively. Thus, a_{1,j} = deg[μ_{sj}] + a_{0,s} for all s with μ_{sj} ≠ 0. As M has no free summands, for each s there is a j with ν_{sj} ≠ 0 and hence, a_{0,s} ≤ a_{1,j}. Coupled with Theorem 1, this yields deg_{k}(r_{i}) ≤ a(k) + ⌊\frac{i}{2}⌋b(k) for all i ≥ 0. By Backelin’s result deg_{k}(g_{i}) ≤ b(k) for all i ≥ 1. In view of these inequalities we get

deg_{k}(∑_{j=0}^{i−1} r_{j}g_{i−j}) ≤ max_{j<i}\{deg_{k}(r_{j}g_{i−j})\} ≤ max_{j<i}\{a(k)+⌊\frac{j}{2}⌋b(k)+b(k)\} ≤ a(k)+⌊\frac{i−1}{2}⌋b(k).

The expression for P_{M}^{R} yields f_{i} = r_{i} + ∑_{j=0}^{i−1} r_{j}g_{i−j}. Therefore

deg_{k}(f_{i}) ≤ max\{a(k) + ⌊\frac{i}{2}⌋b(k), a(k) + ⌊\frac{i−1}{2}⌋b(k)\} ≤ a(k) + ⌈\frac{i}{2}⌉b(k).

This proves Theorem 2. □

The following example, adapted from [4, p. 44], shows that the degree of t in the numerator term of P_{M}^{R}(s, t) admits no upper bound depending only on a and b.

Example 2. If R = K[x_{1}, x_{2}]/(x_{1}^{2}, x_{2}^{2}), then b = (2, 2) and P_{K}^{R} = 1/(1 − ts_{1})(1 − ts_{2}), cf.[7, Theorem 4]. Let F be a minimal resolution of K. As R is self-injective, the dual complex F^{∗} = Hom_{R}(F, R) has H_{0}(F^{∗}) = K and H_{−i}(F^{∗}) = 0 for i ≠ 0.

Splicing F and F^{∗} we get the exact complex G : ⋯ → F_{1} → F_{0} → F_{0}^{∗} → F_{1}^{∗} → … The complex G_{≥(p−2)} obtained by truncating G at F_{(p−2)}^{∗} is a minimal free resolution of M_{p} = H_{0}(G_{≥(p−2)}). With u_{i} = ∑_{j+k=i} s_{1}^{−j}s_{2}^{−k}, we get for each p ≥ 3, an expression

P_{M_{p}}^{R}(s, t) = ∑_{i=0}^{p−2} u_{i}t^{p−i−2} + \frac{t^{p−1}}{(1 − ts_{1})(1 − ts_{2})},

so that M_{p} has a = (0, 0) while the numerator of the Poincaré series of M_{p} over R has degree p in the variable t. □

A key step in Lescot’s argument in [5] is based on a spectral sequence argument. We use Theorem 1 to give a proof that does not involve spectral sequences.

Proposition[5, Lemma 3]. For each k such that 1 ≤ k ≤ n, there is an equality of the form

\frac{P_{M}^{R}(s, t)}{P_{K}^{R}(s, t)} = h_{0} + h_{1}s_{k} + ⋯ + \frac{h_{r}s_{k}^{r}}{1 + ts_{k}}

with h_{i} ∈ ℤ[[s_{1}, …, ŝ_{k}, …, s_{n}, t]] for 1 ≤ i ≤ r.

Proof. After reordering, we can find an integer m such that deg_{k}(x^{b_{j}}) = 0 for 1 ≤ p ≤ m and deg_{k}(x^{b_{p}}) > 0 for p > m. Set

R_{k} = K[x_{1}, …, x̂_{k}, …, x_{n}]/(x^{b_{1}}, …, x^{b_{m}}) and A = R_{k}[x_{k}].

Then A is an n-graded ring and R ≅ A/x_{k}J as n-graded rings.

If {c_{i,j}} are the shifts of M considered as an A-module, Theorem 1 yields c_{i,j}(k) ≤ c(k) + ⌊\frac{i}{2}⌋d(k), where d = lub\{b_{1}, …, b_{m}\} and c = lub\{c_{1,1}, …, c_{1,β_{1}}\}. Clearly d(k) = 0 and c_{i,j}(k) ≤ c(k) for all i, j ≥ 1. Thus, with r′ = c(k), we get

(6)  P_{M}^{A}(s, t) = g_{0} + g_{1}s_{k} + ⋯ + g_{r′}s_{k}^{r′} with g_{i} ∈ ℤ[[t, s_{1}, ⋯, ŝ_{k}, ⋯, s_{n}]]

The remainder of the proof is as in [5].

As A = R_{k}[x_{k}] and R_{k} is an (n − 1)-graded ring

(7)  P_{K}^{A}(s, t) = (1 + s_{k}t)P_{K}^{R_{k}} with P_{K}^{R_{k}} ∈ ℤ[[s_{1}, …, ŝ_{k}, …, s_{n}, t]]

As Ker(Tor_{∗}^{A}(M, K) ⟶ Tor_{∗}^{R}(M, K)) is an n-graded subspace of Tor_{∗}^{A}(M, K), its ntuple Hilbert series B(s, t) is in ℤ[[s_{1}, …, ŝ_{k}, …, s_{n}, t]][s_{k}]. This series appears in the following expression

(8)  P_{M}^{R}(s, t)/P_{K}^{R}(s, t) = [P_{M}^{A}(s, t) − (1 + t)B(s, t)]/P_{K}^{A}(s, t)

which is the n-tuple version of [6, Chapter 4, (4.1)], and is due to the fact that A ⟶ R is a Golod homomorphism [6 , Theorem 2.3].

The proposition follows by combining relations (6), (7) and (8).

Acknowledgement. I am indebted to my thesis advisor Luchezar Avramov for the many helpful discussions concerning this paper.

References

1. J. Backelin, Les anneaux locaux à relations monomiales ont des séries de Poincaré-Betti rationelles, C. R. Acad. Sci. Paris, Sér. I Math. 295 (1982), 607–610.

2. W. Bruns and J. Herzog, On multigraded resolutions, Math. Proc. Cambridge Phil. Soc. 118 (1995), 245–257.

3. C. Jacobsson, Finitely presented graded Lie algebras and homomorphism of local rings, J. Pure Appl. Algebra 38 (1985), 243–253.

4. D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64.

5. J. Lescot, Séries de Poincaré des modules multi-gradués sur les anneaux monomiaux, Algebraic Topology. Rational Homotopy (Y. Félix, ed.), Lecture Notes Math. 1318, Springer-Verlag, 1986, pp. 155–161.

6. G. Levin, Lectures on Golod homorphisms, Reports 15, Matematiska Instit., Stockholms Univ., Stockholm, Sweden, 1976.

7. J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14-27.

Deparment of Mathematics, Purdue University, West Lafayette, Indiana 47907.

  E-mail address: iyengar@math.purdue.edu