Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 2, 579-581.

On a Theorem of F. S. Macaulay on Colon Ideals^{∗}

Dedicated to my teacher Prof. William Heinzer

J. K. Verma

Department of Mathematics, IIT Bombay Powai, Mumbai, India - 400076 e-mail: jkv@math.iitb.ernet.in

In §86 of his monograph [4] of 1916, F. S. Macaulay proved the following

Theorem 1.1. Let R be the polynomial ring k[X_{1}, X_{2}, …, X_{n}] over a field k. Let I be a height n ideal of R generated by homogeneous polynomials f_{1}, f_{2}, …, f_{n} of degree d_{1}, d_{2}, …, d_{n} respectively. Set M = (X_{1}, X_{2}, …, X_{n})R and δ = d_{1} + d_{2} + … + d_{n} − n + 1. Then for all integers t = 0, 1, …, δ

I : m^{t} = I + m^{δ−t}.

P. Griffiths [2] used Macaulay’s theorem in Hodge theory of smooth hypersurfaces in projective space. He also provided a proof of this theorem by using the local duality theorem in the complex case involving the Grothendieck residue symbol.

The objective of this note is to generalize Macaulay’s theorem to Gorenstein standard graded algebras over a field.

A crucial result used in our proof is a theorem of Macaulay about Hilbert series of Gorenstein graded algebras. For a modern proof see the corollary 4.4.6 of [1]. This has been generalized for Gorenstein graded rings over Artin rings in [3]. We shall use the zero dimensional case of this.

Theorem 1.2. Suppose that S = ⨁_{n=0}^{∞} S_{n} is a standard Gorenstein graded algebra over an Artin local ring S_{0}. Let δ = max{n|S_{n} ≠ 0}. Then λ(S_{i}) = λ(S_{δ−i}) for all i = 1, 2, …, δ.

∗Presented at the III National Conference on Commutative Algebra and Algebraic Geometry, Indian Institute of Science, Bangalore, October 16–21, 2000 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

We recall certain standard facts from [1] about local cohomology. Let R = ⊕_{n=0}^{∞}R_{n} be a standard graded algebra over a field k. Let M denote the irrelevant ideal ⊕_{n=1}^{∞}R_{n}. The a-invariant of R is defined by a(R) = max{n| [H_{M}^{d}(R)]_{n} ≠ 0}. If R is Cohen-Macaulay and f is a homogeneous degree d nonzerodivisor in R then a(R/fR) = a(R) + d. By Serre’s theorem on cohomology of projective space the a-invariant of a polynomial ring over a field in n variables is −n. This can also be seen by the Grothendieck-Serre formula.

Theorem 1.3. (Grothendieck-Serre) Let R = ⊕_{n=0}^{∞}R_{n} be a d-dimensional finitely generated graded algebra over an artinian local ring R_{0}. Let M denote the irrelevant ideal of R. Let λ(M) denote length of an R_{0} module M. Put H(n) = λ(R_{n}) and P (n) the Hilbert polynomial for the Hilbert function H(n). Then for all integers n

P (n) − H(n) = ∑_{i=0}^{d}(−1)^{i}λ (H_{M}^{i}(R)_{n})

If R is a polynomial ring over a field in d variables then P (n) = \binom{n+d−1}{d−1} and P (−d + i) = H(−d + i) for all i > 0, but P (−d) ≠ H(−d). Hence a(R) = −d. Thus If f = f_{1}, f_{2}, …, f_{d} is a regular sequence with deg(f_{i}) = r_{i} for all i = 1, 2, …, d then a(R/f) = r_{1} +r_{2} +…+r_{d} −d.

Theorem 1.4. Let R = ⨁_{n=0}^{∞} R_{n} be a standard Gorenstein graded algebra of dimension d over a field R_{0} = k. Let M denote the maximal homogeneous ideal of R. Let I =(f_{1}, f_{2}, …, f_{d}) be an M-primary ideal generated by forms f_{i} of degree r_{i} for i = 1, 2, …, d. Put δ = a(R) + r_{1} + r_{2} + ⋯ + r_{d} + 1. Then for t = 0, 1, …, δ

I : M^{t} = I + M^{δ−t}

Proof. We know that a(R/I)=a(R)+r_{1}+⋯+r_{d}. As R/I is Artin, a(R/I)=max{n| (R/I)_{n} ≠ 0}. Let N denote the maximal homogeneous ideal of R/I. Thus we need to prove that

0 : N^{t} = N^{δ−t}.

Apply induction on t. As S is a zero-dimensional Gorenstein ring, S = E_{S}(k) where E denotes the injective hull of k over S. Thus Hom(−, S) is an exact functor. Consider the exact sequence

0⟶N⟶S⟶S/N⟶0.

This gives the exact sequence

0⟶ Hom_{S}(S/N, S)⟶ Hom_{S}(S, S)⟶ Hom_{S}(N, S)⟶0.

By Matlis duality Hom_{S}(S, S) = S^{∗} where S^{∗} denotes the completion of S in the N-adic topology. As S is Artin, it is complete in this topology. As Hom_{S}(S/N, S) = (0 : N) and λ(Hom_{S}(P, S)) = λ(P) for any finite length S-module P, we obtain λ(0 : N) = λ(S/N) = 1. But N^{δ−1} ⊂ (0 : N) and N^{δ−1} ≠0. Hence λ(N^{δ−1})=λ(0 : N)=1. Therefore N^{δ−1} =(0 : N).

Assume that the result has been proved for t − 1. To prove it for t, consider the exact sequence,

0⟶N^{t−1}/N^{t}⟶S/N^{t}⟶S/N^{t−1}⟶0.

As Hom_{S}(−, S) is an exact functor, we get the exact sequence

0⟶ Hom(S/N^{t−1}, S)⟶ Hom(S/N^{t}, S)⟶ Hom(N^{t−1}/N^{t}, S)⟶0.

Therefore λ(0 : N^{t}) = λ(0 : N^{t−1}) + λ(N^{t−1}/N^{t}).

By induction, λ(N^{δ−t+1}) = λ(0 : N^{t−1}). Hence λ(0 : N^{t}) = λ(N^{δ−t+1}) + λ(N^{t−1}/N^{t}). Since the Hilbert function of the Gorenstein graded ring S is symmetric, λ(S_{i}) = λ(S_{δ−i−1}) for each i = 1, 2, …, δ − 1. Thus λ(N^{t−1}/N^{t}) = λ(N^{δ−t}/N^{δ−t+1}). Therefore λ(0 : N^{t}) = λ(N^{δ−t+1}) + λ(N^{δ−t}/N^{δ−t+1}). This proves (0 : N^{t}) = N^{δ−t}. □

References

[1] Bruns, W.; Herzog, J.: Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge 1993. Zbl 0788.13005

[2] Griffiths, P. A.: Complex analysis and algebraic geometry. Bull. Amer. Math. Soc. (N.S.) 1(4) (1979), 595–626. Zbl 0427.14005

[3] Johnston, B.; Verma, J. K.: On the length formula of Hoskin and Deligne and associated graded rings of two-dimensional regular local rings. Math. Proc. Cambridge Philos. Soc. 111 (1992),423–432. Zbl 0758.13011

[4] Macaulay, F. S.: The algebraic theory of modular systems. Cambridge University Press 1916, reissued with an introduction by P. C. Roberts 1994. Zbl 0802.13001

Received July 6, 2001