Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 37 (1996), No. 2, 387-390.

Graded Rings of Complete Intersection Local Rings with Finite Residue Fields

Vijaylaxmi Trivedi

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, India. e-mail: vija@tifrvax.tifr.res.in

In this paper we prove that there are only finitely many graded rings arising as the associated graded rings of complete intersection local rings, with a given finite residue field k, which have a given dimension and multiplicity. More generally, if (R, m) is a Cohen-Macaulay (in fact a generalized Cohen-Macaulay) local ring with finite residue field, then given an integer e > 0 the set

S_{e}(R) = \{\overline{gr_{m}(R/I)} | I gen. by a part of a s.o.p. of R and e(R/I) = e\}

is finite, where e(R/I) denotes the multiplicity of the ring R/I, and \overline{gr_{m}(R/I)} denotes the isomorphism class of graded rings determined by gr_{m}(R/I). In particular, we see that only finitely many numerical functions can arise as Hilbert functions of such quotient rings R/I. This question, for arbitrary residue fields, was raised in [3].

We recall the following definition.

Definition. A Noetherian local ring (R, m) is generalized Cohen-Macaulay if ℓ(H_{m}^{i}(R)) < ∞ for 0 ≤ i < dim(R).

The following result is a trivial consequence of the result given in [4, Appendix, lemma 15].

Lemma 1. Let (R, m) be a generalized CM ring of dimension d. Then there exists C_{R} ∈ N such that for any system of parameters x_{1}, …, x_{d} of R, we have

ℓ_{R} (\frac{R}{< x_{1}, …, x_{d} >}) − e(< x_{1}, …, x_{d} >, R) ≤ C_{R}.  □

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1996 Heldermann Verlag, Berlin

Lemma 2. Let (R, m) be a generalized Cohen-Macaulay local ring of dimension ≥ r ≥ 1. Let I =< f_{1}, …, f_{r} > such that f_{1}, …, f_{r} do not form part of a system of parameters. Let

m(N) = inf{e(m, R/ < f_{1} + g_{1}, …, f_{r} + g_{r} >) | g_{1}, …, g_{r} ∈ m^{N} and f_{1} + g_{1}, …, f_{r} + g_{r} form part of a s.o.p.}.

Then

lim_{N→∞} m(N) = ∞.

Remark. We note that, as dim R ≥ 1, such g_{1}, …, g_{r} exist for each N ≥ 1.

Proof. Considering the faithfully flat extension R ⟶ R[X]_{m[X]}, where X is an indeterminate, one can assume that the residue field of R is infinite. For given N > 0, let g_{1}, …, g_{r} ∈ m^{N} such that f_{1}+g_{1}, …, f_{r} +g_{r} form a part of a system of parameters. Let I =< f_{1}+g_{1}, …, f_{r} + g_{r} >. Let q =< f_{r+1}, …, f_{d} > be an ideal of R which gives a reduction ideal of m in the quotient ring R/I. Then e(m, R/I) = e(q + I/I, R/I). By lemma 1, one can find a constant C_{R} depending only on R such that

ℓ_{R} (\frac{R}{I + q}) ≤ e(q + I, R) + C_{R} ≤ e (\frac{q + I}{I} , \frac{R}{I}) + C_{R},

where the second inequality follows from [2, theorem 14.9]. Now, since f_{1}, …, f_{r} do not form part of a s.o.p., the elements f_{1}, …, f_{r}, …, f_{d} cannot form a s.o.p.. Therefore dim R/(< f_{1}, …, f_{r} > +q) > 0, which implies that

ℓ_{R} (\frac{R}{< f_{1}, …, f_{r} > +q + m^{N}} ) ≥ N.

But since the canonical map

\frac{R}{I + q} ⟶ \frac{R}{< f_{1}, …, f_{r} > +q + m^{N}}

is surjective, we have ℓ_{R}(R/I + q) ≥ N. Hence

lim_{N→∞} m(N) = ∞.  □

Now we quote the following result [3, corollary 5] to prove the main result of this paper.

Theorem 1. Let (R, m) be a generalized Cohen-Macaulay local ring. Let A be a graded R/m-algebra and for r ∈ N let

S_{A}(R)_{r} = {(f_{1}, …, f_{r}) ∈ m^{⊕r} | f_{1}, …, f_{r} is a part of a s.o.p. and gr_{m}(R/ < f_{1}, …, f_{r} >) ≅ A}.

Then S_{A}(R)_{r} is an open subset of m^{⊕r} with respect to m-adic topology.

Theorem 2. Let (R, m) be a generalized Cohen-Macaulay local ring with finite residue field. Then given e > 0, the set

S_{e}(R) = \{\overline{gr_{m}(R/I)} | I gen. by a part of a s.o.p. of R and e(R/I) = e\}

is finite, where e(R/I) denotes the multiplicity of the ring R/I, and \overline{gr_{m}(R/I)} denotes the isomorphism class of graded rings determined by gr_{m}(R/I).

Proof. Without loss of generality one can assume that R is a complete local ring. Fix integers e > 0 and 0 ≤ r ≤ dim R.

Claim. R is a compact set with respect to m-adic topology.

Proof of the claim. For each i ∈ N, consider the canonical continuous surjective map ϕ_{i} : R/m^{i+1} ⟶ R/m^{i}, where each R/m^{i} is a discrete topological space with respect to its m-adic topology. Now if ∏_{i} R/m^{i} is considered with the canonical product topology then the following map is continuous:

\begin{matrix}ϕ : ∏_{n} R/m^{n} & ⟶ & ∏_{n} R/m^{n} \\ (x_{1}, x_{2}, …, x_{n}, …) & → & (x_{1} − ϕ_{1}(x_{2}), x_{2} − ϕ_{2}(x_{3}), …, x_{n} − ϕ_{n}(x_{n+1}) …). \end{matrix}

Since | R/m | < ∞, the set ∏_{n} R/m^{n} is compact and therefore its closed subset R = ϕ^{−1}(0) is compact. Hence the claim.

Now given a point \overline{x} = (x_{1}, …, x_{r}) ∈ R^{⊕r}, consider an open set U\overline{x} as follows:

Case (a): If (x_{1}, …, x_{r}) ∉ m^{⊕r} then for any g_{1}, …, g_{r} ∈ m the point (x_{1} + g_{1}, …, x_{r} + g_{r}) ∉ m^{⊕r}. In this case we choose the open set

U\overline{x} = \{(x_{1} + g_{1}, …, x_{r} + g_{r}) | g_{1} …, g_{n} ∈ m\} ⊆ R^{⊕r}.

Case (b): If (x_{1}, …, x_{r}) ∈ m^{⊕r} and x_{1}, …, x_{r} is a part of a s.o.p., then by theorem 1 above, there exists N_{1} ∈ N such that for any g_{i} ∈ m^{N_{1}}, we have an isomorphism of associated graded rings,

gr_{m} (\frac{R}{< x_{1} + g_{1}, …, x_{r} + g_{r} >}) ≅ gr_{m} (\frac{R}{< x_{1}, …, x_{r} >}) .

Let U\overline{x} = \{(x_{1} + g_{1}, …, x_{r} + g_{r}) | g_{1} …, g_{n} ∈ m^{N_{1}}\} ⊆ m^{⊕r}.

Case (c): If (x_{1}, …, x_{r}) ∈ m^{⊕r} and x_{1}, …, x_{r} is not a part of a s.o.p., then by lemma 2 above, there exists N_{2} ∈ N such that, whenever x_{1} + g_{1}, …, x_{r} + g_{r} form a part of a s.o.p., we have e(R/ < x_{1}+g_{1}, …, x_{r}+g_{r} >) > e, for all g_{i} ∈ m^{N_{2}}. In this case let U\overline{x} = \{(x_{1}+g_{1}, …, x_{r}+g_{r}) | g_{1} …, g_{n} ∈ m^{N_{2}}\}. So any (x_{1} + g_{1}, …, x_{r} + g_{r}) coming from U\overline{x}, either do not form a part of a s.o.p., or if they do, then we have e(R/ < x_{1} + g_{1}, …, x_{r} + g_{r} >) > e.

Now cover R^{⊕r} by such open sets \{U\overline{x}\}_{\overline{x}∈R^{⊕r}}. Since R^{⊕r} is compact, it has a finite subcover say, U_{1}, …, U_{k}. Now for any \overline{x} = (x_{1}, …, x_{r}) such that x_{1}, …, x_{r} form a part of a s.o.p. and e(R/ < x_{1}, …, x_{r} >) = e, the point \overline{x} belongs to some U_{i}, where obviously U_{i} is an open set of the type described in Case (b). This implies that the set

S_{r,e}(R) = \{\overline{gr_{m}(R/ < f_{1}, …, f_{r} >)} | f_{1}, …, f_{r} is a part of a s.o.p. and e(R/ < f_{1}, …, f_{r} >) = e}

is finite for all 0 ≤ r ≤ d. Hence the set S_{e}(R) is finite. □

Corollary 1. Given integers d, e and a finite field k, let C_{d,e} be the set of isomorphism classes of all complete intersection local rings of dimension d, of multiplicity e and with residue field k. Then the set

S_{d,e} = \{\overline{gr_{m}(A)} | (A, m) ∈ C_{d,e}\}

is finite.

Proof. It is enough to prove that the set

Ŝ_{d,e} = \{\overline{gr_{m}(A)} | (A, m) ∈ Ĉ_{d,e}\}

is finite, where Ĉ_{d,e} is the set of isomorphism classes of complete local rings in C_{d,e}. By [1], the embedding dimension of a Cohen-Macaulay local ring is bounded by a function, which depends only on multiplicity and dimension of the ring, more precisely

μ(m) ≤ e + (d − 1)

for all A ∈ Ĉ_{d,e}. By Cohen structure theorem, one can find a power series ring R over W (k) (the unique complete unramified discrete valuation ring with residue field k) such that for every A ∈ Ĉ_{d,e} we have an R-sequence f_{1}, …, f_{r} such that A ≅ R/ < f_{1}, …, f_{r} >. Hence the corollary follows by theorem 2, above. □

References

[1] Abhyankar, S. S.: Local rings of high embedding dimension. Amer. J. Math. 89 (1967), 1073–1077.

[2] Matsumura, H.: Commutative ring theory. Cambridge University Press (1992).

[3] Srinivas, V.; Trivedi, V.: The invariance of Hilbert functions of quotients under small perturbations. to appear in J. Algebra.

[4] Stückrad, J.; Vogel, W.: Buchsbaum Rings and Applications. VEB Deutscher Verlag d. Wissenschaften, Berlin 1986.

Received April 3, 1995; revised version September 5, 1995