PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I)

William Heinzer, L. J. Ratliff, Jr. and Kishor Shah

Abstract. Emmy Noether showed that every ideal in a Noetherian ring admits a decomposition into irreducible ideals. In this paper we explicitly calculate this decomposition in a fundamental case. Specifically, let R be a commutative ring with identity, let x_{1}, …, x_{d} (d > 1) be an R -sequence, let X = (x_{1}, …, x_{d})R, and let I be a monomial ideal (that is, a proper ideal generated by monomials x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}}) such that Rad(I) = Rad(X). Then the main result gives a canonical and unique decomposition of I as an irredundant finite intersection of ideals of the form (x_{1}^{n_{1}}, …, x_{d}^{n_{d}})R, where the exponents n_{1}, …, n_{d} are positive integers. Specifically, if z_{1}, …, z_{m} are the monomials in (I : X) − I, and if z_{j} = x_{1}^{a_{j,1}−1} ⋯ x_{d}^{a_{j,d}−1}, then I = ∩{(x_{1}^{a_{j,1}}, …, x_{d}^{a_{j,d}})R; j = 1, …, m}. We also calculate the decomposition of the ideals I^{[k]} generated by the k -th powers of the monomial generators of I. The methods we use are algebraic, but they were suggested by the geometry of lattices.

1. Introduction. Throughout this paper, R is a commutative ring with identity

1 ≠ 0, x_{1}, …, x_{d} (d > 1) is an R -sequence, X = (x_{1}, …, x_{d})R, and I is a monomial

ideal (that is, a proper ideal generated by monomials x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}}) such that Rad(I)

= Rad(X).

It is known (for example, see [HRS2, (3.15)]) that in a regular local ring R

of altitude two, irreducible ideals are parameter ideals. Therefore in altitude two

regular local rings, Emmy Noether’s fundamental decomposition theorem [N, Satz

IV] shows that each open ideal in R is a finite intersection of parameter ideals (but

of course the x’s may vary). One consequence of our main result, (4.1), is that a

similar statement holds for open monomial ideals in a Cohen-Macaulay local ring.

Monomial ideals are important in several areas of current research, and they

have been studied in their own right in several papers (for example, [EH] and [T]),

so many useful results are known about such ideals. In the present paper we are

1991 Mathematics Subject Classification. AMS (MOS) Subject Classification Numbers: Primary: 13A17, 13C99. Secondary: 13B99, 13H99.

The first author’s research on this paper was supported in part by the National Science Foundation, Grant DMS-9101176.

Typeset by AMS-TEX 2 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

interested in giving an explicit decomposition of I as an irredundant finite intersection of parameter ideals. We do this in Section 2 for the special case when I = X^{n} (n a positive integer), and it is shown that X^{n} is the irredundant intersection of the ( \begin{matrix}n+d−2 \\ d−1 \end{matrix} ) parameter ideals (x_{1}^{a_{1}}, …, x_{d}^{a_{d}})R, where a_{1}, … a_{d} are positive integers that sum to n + d − 1.

To extend this result to an arbitrary monomial ideal I (such that Rad(I) = Rad(X)), in Section 3 we introduce and study the J -corner-elements of a monomial ideal J. We show that they are the monomials in (J : X) − J, that there are only finitely many of them, and that if (x_{1}, …, x_{d−1})R ⊆ Rad(J), then their J -residue classes are a minimal basis, in any order, of (J : X)/J. Also, if Q is an open monomial ideal in a regular local ring (R, M) of altitude two, then v((Q : X)/Q) = v(Q) − 1 (where v(J) denotes the number of elements in a minimal basis of the ideal J), and if t is an integer such that v(Q) − 1 ≤ t ≤ 2v(Q) − 1, then Q can be chosen such that v(Q : X) = t. Finally, we give a geometric interpretation of I -corner-elements, an algebraic construction of them, and then close this section with several examples of such elements.

In Section 4 we show that if z_{1}, … z_{m} are the I -corner-elements, then I is the irredundant intersection of the m parameter ideals P (z_{j}) = (x_{1}^{a_{j,1}}, … x_{d}^{a_{j,d}})R, where z_{j} = x_{1}^{a_{j,1}−1} ⋯ x_{d}^{a_{j,d}−1}. Three interesting corollaries are: ∪{Ass(R/I^{n}); n ≥ 1} ⊆ Ass(R/X); and, if R is a Gorenstein local ring with maximal ideal M, if X is generated by a system of parameters, and if I is open, then v((I : M)/I) = v((I : X)/I), and I is irreducible if and only if there exists exactly one I -corner-element, and then I is generated by a system of parameters. Also, unique factorization holds in the sense that if I = ∩{P (z_{j}); j = 1, …, m} = ∩{P (w_{i}); i = 1, …, n}, then n = m and {z_{1}, … z_{m}} = {w_{1}, …, w_{n}}. Further, if k is a positive integer and I^{[k]} is the ideal generated by the k-th powers of the monomial generators of I, then I^{[k]} = ∩{(P (z_{j}))^{[k]}; j = 1, …, m} and the I^{[k]} -corner-elements are the m monomials z_{j}^{(k)} = x_{1}^{ka_{j,1}+k−1} ⋯ x_{d}^{ka_{j,d}+k−1}.

In Section 5 a related decomposition of I as an irredundant finite intersection of irreducible ideals is proved. Specifically, with the notation of the preceding paragraph, if R is local with maximal ideal M and if Q_{j} is maximal in S_{j} = {Q; Q is an ideal in R, P (z_{j}) ⊆ Q, and z_{j} ∉ Q} for j = 1, …, m, then each Q_{j} is irreducible, ∩{Q_{j}; j = 1, …, m} is an irredundant intersection, and (∩{Q_{j}; j =

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 3

1, …, m}) ∩ (I : X) = I + M(I : X). It then follows that if R is a regular local ring and X = M, then Q_{j} = P (z_{j}) for j = 1, … m.

Finally, in Section 6 we show that: if I is irreducible, then I is a parameter ideal; I is a parameter ideal if and only if I has exactly one corner-element; and, if R is a Gorenstein local ring of altitude d, then I is a parameter ideal if and only if I is irreducible. Also, the parameter ideals that are minimal with respect to containing I are the ideals P (z), where z is an I -corner-element.

The authors have been fascinated by the historic and fundamental decomposition theorems of Emmy Noether, and this fascination gave rise to the results in [HRS1, HRS2, HRS3, HRS4] and the present paper. We are pursuing further topics in this area (in particular, in [HMRS]), and we hope this theory turns out to be fascinating and useful to others.

2. Parametric Decompositions of Powers of an R -Sequence. The main result in this section, (2.4), shows that if X is an ideal generated by an R -sequence, then X^{n} is the irredundant intersection of ( \begin{matrix}n+d−2 \\ d−1 \end{matrix} ) parameter ideals. To prove this, we need a few preliminary results, so we begin with these.

(2.1) Definition. Let R be a ring, let x_{1}, …, x_{d} (d > 1) be an R -sequence, and let X = (x_{1}, …, x_{d})R. Then:

(2.1.1) (2.1.1) (2.1.1) A monomial (in x_{1}, … x_{d}) is a power product x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}}, where e_{1}, …, e_{d} are nonnegative integers (so a monomial is either a nonunit or the element 1), and a monomial ideal is a proper ideal generated by monomials.

(2.1.2) (2.1.2) (2.1.2) A parameter ideal (in x_{1}, …, x_{d}) is an ideal of the form (x_{1}^{a_{1}}, …, x_{d}^{a_{d}})R, where a_{1}, …, a_{d} are positive integers (so the parameter ideal (x_{1}^{a_{1}}, …, x_{d}^{a_{d}})R is a monomial ideal generated by the R -sequence x_{1}^{a_{1}}, …, x_{d}^{a_{d}}). If f = x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}} is a monomial, then we let P(f) denote the parameter ideal (x_{1}^{e_{1}+1}, …, x_{d}^{e_{d}+1})R. (Note that if f = 1, then P (f) = X.) And if a_{1}, … a_{d} are positive integers, then we let P(a_{1}, …, a_{d}) denote the parameter ideal (x_{1}^{a_{1}}, …, x_{d}^{a_{d}})R (so P (a_{1}, …, a_{d}) = P (f), where f = x_{1}^{a_{1}−1} ⋯ x_{d}^{a_{d}−1}).

(2.2) Remark. Let f and g be monomials. Then:

(2.2.1) (2.2.1) (2.2.1) If f_{1}, …, f_{n} are monomials then f ∈ (f_{1}, …, f_{n})R if and only if f ∈ f_{i}R for some i = 1, …, n.

4 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

(2.2.2) (2.2.2) (2.2.2) If f ∈ gR, then there exists a monomial k (possibly k = 1) such that f = gk.

(2.2.3) (2.2.3) (2.2.3) If h is a monomial such that fh = gh, then f = g.

(2.2.4) (2.2.4) (2.2.4) If fx_{j} = gx_{i} for some i ≠ j in {1, …, d}, then f ∈ x_{i}R and g ∈ x_{j}R.

Proof. It is shown in [T, Theorem 1] that if r ∈ R and rf ∈ (f_{1}, …, f_{n})R, then either f ∈ f_{i}R for some i = 1, …, n or r ∈ (x_{1}, …, x_{d})R. (2.2.1) readily follows from this.

(2.2.2)–(2.2.4) readily follow by the “independence” of power products in an R -sequence (that is, x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}} = x_{1}^{a_{1}} ⋯ x_{d}^{a_{d}} if and only if a_{i} = e_{i} for i = 1, …, d), □

(2.3) Lemma. Let f and g be monomials. Then g ∈ P (f) (see (2.1.2)) if and only if f ∉ gR.

Proof. Let f = x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}}. Then f ∉ x_{i}^{e_{i}+1}R for i = 1, …, d, since e_{i} < e_{i} + 1 for i = 1, …, d, so (2.2.1) shows that f ∉ P (f). Therefore if g ∈ P (f), then f ∉ gR.

For the converse assume that g ∉ P (f) and let g = x_{1}^{a_{1}} ⋯ x_{d}^{a_{d}}. Then a_{i} < e_{i} + 1 for i = 1, …, d, so e_{i} ≥ a_{i} for i = 1, …, d, hence f ∈ gR, □

(2.4), the main result in this section, extends [HRS2, (3.5)] (where it is shown that in a regular local ring (R, M = (x, y)R), M^{n} = ∩{(x^{n+1−i}, y^{i})R; i = 1, …, n}).

Concerning the ideals P (a_{1}, …, a_{d}) in (2.4), see (2.1.2).

(2.4) Theorem. Let X be an ideal that is generated by an R -sequence x_{1}, …, x_{d} (d > 1) and let n be a positive integer. Then X^{n} = ∩{P (a_{1}, …, a_{d}); a_{1} + ⋯ + a_{d} = n + d − 1} and this intersection is irredundant. Therefore X^{n} is the irredundant intersection of ( \begin{matrix}n+d−2 \\ d−1 \end{matrix} ) parameter ideals.

Proof. If n = 1, then this is clear, so it will be assumed that n > 1.

Let J = ∩{P (a_{1}, …, a_{d}); a_{1}+⋯+a_{d} = n+d−1}. Then since X^{n} is generated by the monomials x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}}, where e_{1}, …, e_{d} are nonnegative integers that sum to n, to show that X^{n} ⊆ J it suffices to show that each such monomial is in J. For this, fix f = x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}} and consider any of the ideals P (a_{1}, …, a_{d}). Then e_{i} ≥ a_{i} for some i = 1, …, d (since otherwise n = e_{1}+⋯+e_{d} < n+d = (e_{1}+1)+⋯+(e_{d}+1) ≤ a_{1}+⋯+a_{d} = n+d−1, and this is a contradiction), so f ∈ P (a_{1}, …, a_{d}). Therefore f ∈ J, so it follows that X^{n} ⊆ J.

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 5

For the opposite inclusion, since Rad(P (a_{1}, …, a_{d})) = Rad(X) for all positive integers a_{1}, …, a_{d} that sum to n + d − 1, [T, Lemma 6] shows that J is generated by monomials, so it suffices to show that if f is a monomial in X − X^{n}, then f is not in J. For this let f = x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}} ∈ X^{k} − X^{k+1}, where e_{1} + ⋯ + e_{d} = k and 1 ≤ k < n. Then since k < n there exists a nonnegative integer h such that (e_{1} + 1) + ⋯ + (e_{d−1} + 1) + (e_{d} + 1 + h) = n + d − 1, and since x_{1}, …, x_{d} is an R -sequence, it follows from (2.2.1) that f ∉ P ((e_{1} + 1), …, (e_{d−1} + 1), (e_{d} + 1 + h)). Therefore it follows that J ⊆ X^{n}, so J = X^{n}.

Also, this intersection is irredundant, since if {a_{1}, …, a_{d}} and {b_{1}, …, b_{d}} are distinct sets of positive integers that sum to n + d − 1, then b_{i} > a_{i} for some i = 1, …, d, so x_{1}^{b_{1}−1} ⋯ x_{d}^{b_{d}−1} ∈ P (a_{1}, …, a_{d}), hence it follows that x_{1}^{b_{1}−1} ⋯ x_{d}^{b_{d}−1} ∈ ∩{P (a_{1}, …, a_{d}); a_{1}+⋯+a_{d} = n+d−1 and a_{i} ≠ b_{i} for some i}, and x_{1}^{b_{1}−1} ⋯ x_{d}^{b_{d}−1} ∉ P (b_{1}, …, b_{d}), by (2.3), so x_{1}^{b_{1}−1} ⋯ x_{d}^{b_{d}−1} ∉ ∩{P (a_{1}, …, a_{d}); a_{1}+⋯+a_{d} = n+d−1}.

For the final statement, each ideal P (a_{1}, …, a_{d}) is a parameter ideal, by (2.1.2). And the preceding paragraph shows that they are distinct for distinct d-tuples (a_{1}, …, a_{d}) of positive integers. To compute the number of these ideals, since we are only interested in the number of ideals, it may be assumed that X = (x_{1}, …, x_{d})R is the maximal ideal M in a regular local ring (R, M). Then by [HRS2, (2.3.2) and (2.4)] the number of ideals is d(X^{n}) = dim_{R/M}((X^{n} : X)/X^{n}) =

dim_{R/M}(X^{n−1}/X^{n}) = v(X^{n−1}) = ( \begin{matrix}n+d−2 \\ d−1 \end{matrix} ), □

(2.5) Corollary. If R is a Gorenstein local ring and altitude (R) = d, then X^{n} = ∩ {P (a_{1}, …, a_{d}); a_{1} + ⋯ + a_{d} = n + d − 1} is an irredundant intersection of ( \begin{matrix}n+d−2 \\ d−1 \end{matrix} ) irreducible ideals. Proof. If R is Gorenstein, then each open parameter ideal is irreducible, so the conclusion follows from (2.4), □

(2.6) Remark. It follows from (2.4) that the cardinality of {x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}}; e_{1}, …, e_{d} are positive integers that sum to n + d − 1\} is ( \begin{matrix}n+d−2 \\ d−1 \end{matrix} ).

3. J-Corner-Elements. We now want to extend (2.4) to an arbitrary monomial ideal I such that Rad(I) = Rad(X). (It should be noted that Rad(I) = Rad(X) is a necessary condition to extend (2.4), since the radical of each parameter ideal is the radical of X, and in (4.1) we show that this condition is also sufficient.) To

6 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

accomplish this extension, we have found it useful to use “corner-elements”. So in this section we introduce such elements and derive some of their basic properties, and then use some of these properties in the proof of (4.1) to give the desired extension of (2.4).

We think “corner-elements will be of interest and use in other problems, so in this section we prove several results concerning them. Specifically, we show in (3.2) and (3.7) that if J is a monomial ideal, then there exist only finitely many J -corner-elements, that they are the monomials in (J : X) − J, and that if (x_{1}, …, x_{d−1})R ⊆ Rad(J), then the J -residue classes of these corner-elements are a minimal basis, in any order, of (J : X)/J. We then apply these results to the case when J is an open monomial ideal in a regular local ring (R, M) of altitude two, give a geometric interpretation of I -corner-elements and an algebraic construction of them, and then close this section with several examples of such elements.

We begin with the definition.

(3.1) Definition. Let J be a monomial ideal. Then a J -corner-element is a monomial z such that z ∉ J and zx_{i} ∈ J for i = 1, …, d.

(The name “corner-element” is suggested by the geometric interpretation in (3.13), where a corner-element is an element z = x^{a}y^{b} with coordinates (a, b) such that (a, b + 1), (a + 1, b), and (a + 1, b + 1) are the coordinates of points in I and z ∉ I.)

Concerning (3.1), note that 1 is the unique X -corner-element (since each nonunit monomial is in X). Also, if J is a monomial ideal and 1 is a J -corner-element, then 1x_{i} ∈ J for i = 1, …, d, so J = X.

In (3.2) we characterize the J -corner-elements and show that there are only finitely many of them. (It follows from (3.2) that J uniquely determines its cornerelements. The converse of this is proved in (4.2) when Rad(J) = Rad(X).)

(3.2) Proposition. If J is a monomial ideal, then the J -corner-elements are the monomials in (J : X) − J. Also, if z, z′ are distinct J -corner-elements, then zR ⊈ z′R and z′R ⊈ zR, so there exist only finitely many J -corner-elements.

Proof. Let C be the set of J -corner-elements (so each element in C is a monomial). Then it is clear from (3.1) that C ⊆ (J : X) − J. And if z is a monomial in

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 7

(J : X) − J, then z ∉ J and zx_{i} ∈ J for i = 1, …, d, so z is a J -corner-element, hence z ∈ C. Therefore C is the set of monomials in (J : X) − J.

Now let z and z′ be distinct J -corner-elements and suppose that zR ⊆ z′R. Then (2.2.2) shows that z = z′f for some monomial f (and f ≠ 1, since z ≠ z′). But this implies that z = z′f ∈ J (since z′ is a J -corner-element), and this is a contradiction. Therefore zR ⊈ z′R and, similarly, z′R ⊈ zR.

Finally, the ideal generated by the J -corner-elements (viewed as elements in Z_{k}[x_{1}, …, x_{d}], where k is the characteristic of R and where Z_{k} is the ring generated by the identity of R) is finitely generated, so since there are no inclusion relations among the ideals they generate, (2.2.1) shows that there are only finitely many of them, □

(3.3) Corollary. Let J be a monomial ideal and let z_{1}, …, z_{m} be the J -corner-elements. Then for j = 1, …, m it holds that (z_{1}, …, z_{j−1}, z_{j+1}, …, z_{m})R ⊆ P (z_{j}) and z_{j} ∉ P (z_{j}). Therefore ∩{P (z_{j}); j = 1, …, m} is an irredundant intersection of parameter ideals.

Proof. (It follows from (3.2) that there are only finitely many J -corner-elements. Also, if m = 1 and z_{1} = 1, then P (z_{1}) = X, (0) (the ideal generated by the empty set) is contained in X, and 1 ∉ P (1) = X, so the conclusion holds in this case.)

Fix j ∈ {1, …, m}. Then it follows from (3.2) that if i ∈ {1, …, j − 1, j + 1, …, m}, then z_{j} ∉ z_{i}R, so (2.3) shows that z_{i} ∈ P (z_{j}) (hence (z_{1}, …, z_{j−1}, z_{j+1}, …, z_{m})R ⊆ P (z_{j})) and z_{j} ∉ P (z_{j}). This shows that ∩{P (z_{j}); j = 1, …, m} is an irredundant intersection, and (2.1.2) shows that the ideals P (z_{j}) are parameter ideals, □

In (3.4) we specify the X^{n} -corner-elements.

(3.4) Corollary. If n > 1 is a positive integer, then the X^{n} -corner-elements are the ( \begin{matrix}n+d−2 \\ d−1 \end{matrix} ) generators x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}} of X^{n−1} (so e_{1}, …, e_{d} are nonnegative integers such that e_{1} + ⋯ + e_{d} = n − 1).

Proof. By (3.2) the X^{n} -corner-elements are the monomials in X^{n−1} − X^{n} (since X^{n} : X = X^{n−1}), and since X is generated by an R -sequence of length d it follows that there are ( \begin{matrix}n+d−2 \\ d−1 \end{matrix} ) distinct such elements, □

8 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

It follows from (3.5) that if z is a J -corner-element, then the d elements zx_{1}, …, zx_{d} are members of distinct principal ideals generated by monomials in J.

(3.5) Proposition. Let f and g be monomials and let i ≠ j ∈ {1, …, d}. If fx_{i} ∈ gR and fx_{j} ∈ gR, then f ∈ gR.

Proof. If fx_{i} ∈ gR and fx_{j} ∈ gR, then (2.2.2) shows that there exist monomials h_{i}, h_{j} such that fx_{i} = gh_{i} and fx_{j} = gh_{j}. Then fx_{i}x_{j} = gh_{i}x_{j} = gh_{j}x_{i}, so h_{i}x_{j} = h_{j}x_{i} by (2.2.3). (2.2.4) then shows that h_{i} ∈ x_{i}R, so h_{i} = kx_{i} for some monomial k by (2.2.2). Therefore fx_{i}x_{j} = gh_{i}x_{j} = g(kx_{i})x_{j}, so f = gk ∈ gR by (2.2.3), □

(3.6) Corollary. If J is a monomial ideal that has a corner-element, and if f_{1}, …, f_{n} are monomials that generate J, then n ≥ d.

Proof. This follows immediately from (3.5), □

In (3.7) it is shown that if (x_{1}, …, x_{d−1})R ⊆ Rad(J), then the J -residue classes of the J -corner-elements are a minimal basis, in any order, of (J : X)/J. (In this regard, note that if J : X = J, then there are no J -corner-elements, and the empty set does generate the ideal (J : X)/J = J/J. On the other hand, if Rad(J) = Rad(X) then J : X ≠ J.)

(3.7) Theorem. Let J ≠ X be a monomial ideal such that (x_{1}, …, x_{d−1})R ⊆ Rad(J). Then the J -residue classes of the J -corner-elements are a minimal basis, in any order, of (J : X)/J.

Proof. (By “minimal basis”, we mean a basis such that no proper subset is a generating set of the ideal.) Since Rad((x_{1}, …, x_{d−1})R) ⊆ Rad(J), [T, Theorem 6] shows that J : X is a monomial ideal, so it follows from (3.2) that the J -residue classes of the J -corner-elements generate (J : X)/J.

Let C = {z_{1}, …, z_{m}} be the set of J -corner-elements (C is a finite set by (3.2)) and suppose that there exists a permutation π1, …, πm of 1, …, m such that z_{π1} ∈ (z_{π2}, …, z_{πm})R. Then z_{π1} ∈ z_{πk}R for some k = 2, …, m by (2.2.1), so (2.2.2) shows that z_{π1} = z_{πk}f for some monomial f (f ≠ 1, since z_{1}, …, z_{m} are distinct). However, this implies that z_{π1} = z_{πk}f ∈ J (since z_{πk} is a J

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 9

-corner-element), and this is a contradiction. Therefore z_{π1} ∉ (z_{π2}, …, z_{πm})R for all permutations π1, …, πm of 1, …, m. And no z_{j} in in J, so it follows from (2.2.1) that the J -residue classes of z_{1}, …, z_{m} are a minimal basis, in any order, of (J : X)/J, □

In (3.8) we consider the case when J = Q is open in a regular local ring R of altitude two. ((3.8) was noted in [HRS2, (3.3)] for the case M = (x, y)R, and therein a homological proof using [HS, (2.1)] was sketched for an arbitrary open ideal (in an altitude two regular local ring). (3.8) gives a non-homological proof for an arbitrary R -sequence of length two, but only for the case of an open monomial ideal.)

(3.8) Corollary. Let (R, M) be a regular local ring of altitude two, let x, y be an R -sequence, and let Q ≠ (x, y)R be an open monomial ideal in x and y, say v(Q) = n. Then v((Q : X)/Q) = n − 1.

Proof. Let Q = (f_{1}, …, f_{n})R and lexicographically order the f_{i} by saying that f_{i} < f_{j} (for f_{i} = x^{a}y^{b} and f_{j} = x^{c}y^{e}) if either a < c or a = c and b < e. Then it may be assumed that f_{1} < f_{2} < ⋯ < f_{n}. Therefore, since v(Q) = n and Q is open, it follows that there exist positive integers h, k, h_{2} < ⋯ < h_{n−1} (h_{n−1} ≤ h − 1), and k_{n−1} < ⋯ < k_{2} (k_{2} ≤ k − 1) such that f_{1} = y^{h}, f_{n} = x^{k}, and f_{i} = x^{k−k_{i}}y^{h−h_{i}} for i = 2, …, n−1. For j = 1, …, n−1 let z_{j} = x^{k−k_{j+1}−1}y^{h−h_{j}−1} (with h_{1} = 0 = k_{n}). Then z_{j} ∉ Q, z_{j}x = x^{k−k_{j+1}}y^{h−h_{j}−1} ∈ f_{j+1}R ⊆ Q, and z_{j}y = x^{k−k_{j+1}−1}y^{h−h_{j}} ∈ f_{j}R ⊆ Q, so z_{j} ∈ (Q : X) − Q for j = 1, …, n − 1. Thus each z_{j} is a Q -corner-element, and the geometric interpretation in (3.13) shows that every Q -corner-element is one of these z_{1}, …, z_{n−1}. Therefore the conclusion follows from (3.7), □

For the next corollary of (3.7) we need the following definition.

(3.9) Definition. If J is a monomial ideal, then c(J) denotes the number of J -corner-elements.

(3.10) Corollary. With the notation of (3.8), there exists a polynomial p(x) of degree two such that p(n) = c(Q^{n}) for large n.

Proof. It is well known that there exists a polynomial q(x) of degree two such that q(n) = v(Q^{n}) for large n. But Q^{n} is a monomial ideal, so the conclusion follows immediately from (3.7) and (3.8) with p(x) = q(x) − 1, □

In (3.11) it is shown that if v(Q) = n, where Q is as in (3.8), then n − 1 ≤ v(Q : X) ≤ 2n − 1 and for each integer t between n − 1 and 2n − 1 the ideal Q can be chosen so that v(Q : X) = t.

(3.11) Proposition. With the notation of (3.8) assume that v(Q) = n. Then n − 1 ≤ v(Q : X) ≤ 2n − 1, and for each intermediate integer t there exists an ideal Q such that v(Q) = n and v(Q : X) = t.

Proof. (3.8) shows that (Q : X)/Q is generated by v(Q) − 1 = n − 1 elements, so it follows that Q : X can be generated by the preimages of these n − 1 elements together with the n generators of Q. Therefore n − 1 ≤ v(Q : X) ≤ 2n − 1.

Now let t be a given positive integer such that n − 1 ≤ t ≤ 2n − 1 and let s be the integer such that t = (n − 1) + s, so 0 ≤ s ≤ n. For i = 1, …, s let f_{i} = x^{2(i−1)}y^{n+s−2i}, for i = s + 1, …, n let f_{i} = x^{s+i−1}y^{n−i}, and let Q = (f_{1}, …, f_{n})R. Then the Q -corner-elements are the elements z_{j} = x^{2j−1}y^{n+s−2j−1} (for j = 1, …, s) and the elements z_{j} = x^{s+j−1}y^{n−1−j} (for j = s + 1, …, n − 1). If s = 0, then f_{1} ∈ z_{1}R, f_{i} ∈ z_{i−1}R ∩ z_{i}R for i = 2, …, n − 1, and f_{n} ∈ z_{n−1}R, and if s > 0, then f_{i} ∈ z_{i−1}R for i = s + 1, …, n and f_{i} ∉ (z_{1}, …, z_{n−1})R for i = 1, …, s, so by (2.2.1) (and (3.8)) it readily follows that f_{1}, …, f_{s}, z_{1}, … z_{n−1} is a minimal basis of Q : X so v(Q : X) = s + n − 1 = t, □

(3.12) Corollary. Let (R, M = (x, y)R) be a regular local ring of altitude two, let Q ≠ M be an open monomial ideal in x and y, and let n = v(Q). Then v((Q : M)/Q) = n − 1, n − 1 ≤ v(Q : M) ≤ 2n − 1, and for each intermediate integer t there exists an ideal Q such that v(Q) = n and v(Q : M) = t.

Proof. This follows immediately from (3.8) and (3.11), since M = X, □

In (3.13) we give a geometric interpretation of the I -corner-elements for a monomial ideal I in an R -sequence x, y of length two such that Rad(I) = Rad((x, y)R).

(3.13) Geometric Interpretation. Assume that d = 2, let x = x_{1} and y = x_{2}, let f_{1}, …, f_{n} be a minimal basis of I (where the f_{l} are monomials in x and y, say f_{l} = x^{i_{l}}y^{j_{l}}), and assume that Rad(I) = Rad((x, y)R). Lexicographically order

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 11

the f_{l} (as in the proof of (3.8)) and assume that f_{1} < ⋯ < f_{n}. Plot the n points (i_{l}, j_{l}) (corresponding to the f_{l}) in the first quadrant of the xy-plane. Then for each of these n points draw the horizontal line segment connecting (i_{l}, j_{l}), (i_{l} + 1, j_{l}), (i_{l} + 2, j_{l}), …, and draw the vertical line segment connecting (i_{l}, j_{l}), (i_{l}, j_{l} + 1), (i_{l}, j_{l} + 2), . . .. (Then it is clear that there is a one-to-one correspondence from the set D = {(a, b); a ≥ i_{l} and b ≥ j_{l} for some l = 1, …, n} to a subset M of the set of monomials in Q, and it follows from (2.2.1) that, in fact, every monomial in Q is in M.) Since (i_{l}, j_{l}) < (i_{l+1}, j_{l+1}), (i_{l+1}, j_{l}) are the coordinates of the intersection of the rightward extending horizontal line segment thru (i_{l}, j_{l}) with the ascending vertical line segment thru (i_{l+1}, j_{l+1}). Then z_{l} = x^{i_{l+1}−1}y^{j_{l}−1} ∉ Q, z_{l}y has coordinates on the rightward extending horizontal line segment thru (i_{l}, j_{l}) (so z_{l}y ∈ Q), and z_{l}x has coordinates on the ascending vertical line segment thru (i_{l+1}, j_{l+1}) (so z_{l}x ∈ Q), hence z_{l} is a Q -corner-element. And since a Q -corner-element must correspond to some (a, b) with 0 ≤ a < i_{n} and 0 ≤ b < j_{1}, it is readily checked that all Q -corner-elements are obtained in this way, so there are exactly n − 1 of them, where n = v(Q).

(3.14) Algebraic Construction. Let x_{1}, …, x_{d} be an R -sequence and let I be a monomial ideal such that Rad(I) = Rad((x_{1}, …, x_{d})R). Then the following is an algebraic construction of the I -corner-elements. (For ease of description it will be said that deg(f) = n if f = x_{1}^{e_{1}} ⋯ x_{d}^{e_{d}} and e_{1} + ⋯ + e_{d} = n.) Let S be the set of monomials (in x_{1}, …, x_{d}) that are not in I (so S is a finite set, since for i = 1, …, d there exists a positive integer n_{i} such that x_{i}^{n_{i}} ∈ I). Let w = max{n; n = deg(f) for some f ∈ S}. For j = 1, …, w let D_{j} = {f ∈ S; deg(f) = j}, let C_{w} = D_{w}, and for j = 1, …, w − 1 let C_{j} = {f ∈ D_{j}; fx_{i} ∉ D_{j+1} for i = 1 …, d} (possibly some of the sets C_{j} are empty for j < w). Then C_{l} ∪ ⋯ ∪ C_{w} is the set of I -corner-elements (and this union is disjoint).

Proof. Let f ∈ C_{j} for some j = 1, …, w. Then f ∉ I (since f ∈ C_{j} ⊆ S) and for i = 1, …, d it holds that deg(fx_{i}) = j +1. If j = w, then fx_{i} ∉ S (for no element in S has degree greater than w = j), and if j < w, then fx_{i} ∉ D_{j+1} = {g ∈ S; deg(g) = j + 1} (by the definition of C_{j}). Therefore in either case (j = w or j < w)fx_{i} ∉ S for i = 1, …, d, so fx_{i} ∈ I, hence f is an I -corner-element. Therefore C_{1} ∪⋯∪C_{w} ⊆ C = {f; f is an I -corner-element}.

12 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

And if g ∈ C, then g ∉ I, so g ∈ S, so g ∈ D_{j}, where j = deg(g). Also, deg(gx_{i}) = j + 1 and gx_{i} ∈ I for i = 1, …, d, so gx_{i} ∉ D_{j+1}. Therefore g ∈ C_{j}, so C ⊆ C_{1} ∪ ⋯ ∪ C_{w}, □

(3.15) Remark. With the notation of (3.14) let f be a monomial that is not in I. Then there exists a monomial g (possibly g = 1) such that fg is an I -corner-element.

Proof. It may be assumed that f is not an I -corner-element, so fx_{i} ∉ I for some i = 1, …, d. Let T = {g; g is a monomial in x_{1}, …, x_{d} and fg ∉ I}. Then T is a finite set (since T is contained in the finite set S of (3.14)), so let g ∈ T such that the sum of its exponents is greater than or equal to the sum of the exponents of the other monomials in T. Then fgx_{i} ∈ I for i = 1, …, d, by the maximality of the sum of the exponents of g, so fg is an I -corner-element, □

Before giving some examples of I -corner-elements, we first prove one more result concerning them. (Some additional properties are given in (4.11)-(4.12).)

(3.16) Proposition. Let I ⊂ J be monomial ideals such that Rad(I) = Rad(X). Then some I -corner-element is in J.

Proof. There exists a monomial f ∈ J − I, by hypothesis. Then (3.15) shows that there exists a monomial g (possibly g = 1) such that fg is an I -corner-element, and it is clear that fg ∈ J, □

This section will be closed with several examples of Q -corner-elements for an open monomial ideal Q in a regular local ring.

(3.17) Example. Let (R, M = (x, y)R) be a regular local ring of altitude two, let x_{1} = x and x_{2} = y, and let Q = (y^{9}, xy^{7}, x^{3}y^{4}, x^{5}y^{2}, x^{11})R, so f_{1} = y^{9}, f_{2} = xy^{7}, f_{3} = x^{3}y^{4}, f_{4} = x^{5}y^{2}, f_{5} = x^{11}. Then the Q -corner-elements are z_{1} = y^{8}, z_{2} = x^{2}y^{6}, z_{3} = x^{4}y^{3}, and z_{4} = x^{10}y. (This can be checked by using either (3.13) or (3.14).) Therefore (Q : M)/Q = (y^{8}, x^{2}y^{6}, x^{4}y^{3}, x^{10}y)R/Q by (3.7).

(3.18) Example. Let (R, M = (x, y, z)R) be a regular local ring of altitude three, let x_{1} = x, x_{2} = y, and x_{3} = z, and let Q = (z^{4}, y^{2}z^{3}, y^{3}, xyz, xy^{2}, x^{2})R. Then the Q -corner-elements are yz^{3}, y^{2}z^{2}, xz^{3}, and xy. (This can be checked by writing

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 13

down the sets S, D_{j}, and C_{j} (for j = 1, …, 4) of (3.14). Thus S = {z, z^{2}, z^{3}, y, yz, yz^{2}, yz^{3}, y^{2}, y^{2}z, y^{2}z^{2}, x, xz, xz^{2}, xz^{3}, xy} (in lexicographic order), D_{1} = {z, y, x}, D_{2} = {z^{2}, yz, y^{2}, xz, xy}, D_{3} = {z^{3}, yz^{2}, y^{2}z, xz^{2}}, and D_{4} = {yz^{3}, y^{2}z^{2}, xz^{3}}. Then C_{4} = D_{4}, C_{3} = ∅ (since at least one of fx, fy, fz is in D_{4} for each f ∈ D_{3}), C_{2} = {xy} (since at least one of fx, fy, fz is in D_{3} for f ∈ {z^{2}, yz, y^{2}, xz} and none of xyx, xy^{2}, xyz is in D_{3}) and C_{1} = ∅ (since at least one of fx, fy, fz is in D_{2} for each f ∈ D_{1}).) (3.19) Example. Let (R, M = (w, x, y, z)R) be a regular local ring of altitude four, let x_{1} = w, x_{2} = x, x_{3} = y, and x_{4} = z, and let Q = (z^{5}, yz^{4}, y^{2}z^{2}, y^{3}, xz^{2}, xyz, x^{3}z, x^{3}y^{2}, x^{4}, w)R. Then the Q -corner-elements are z^{4}, yz^{3}, y^{2}z, x^{2}z, x^{2}y^{2}, and x^{3}y. (This can be checked by using (3.14).) (3.20) Example. Let (R, M = (w, x, y, z)R) be a regular local ring of altitude four, let x_{1} = w, x_{2} = x, x_{3} = y, and x_{4} = z, and let Q = (x^{d}, y^{c}, x^{b}, wxyz, w^{a})R, where a > 1, b > 1, c > 1, d > 1 are integers. Then the Q -corner-elements are x^{b−1}y^{c−1}z^{d−1}, w^{a−1}y^{c−1}z^{d−1}, w^{a−1}x^{b−1}z^{d−1}, and w^{a−1}x^{b−1}y^{c−1}. (This can be checked by using (3.14).) (3.21) Example. Let (R, M = (x_{1}, …, x_{d})R) be a regular local ring of altitude d and let Q = (x_{1}^{a_{1}}, …, x_{d}^{a_{d}})R, where the a_{i} are positive integers. Then Q is irreducible, so by (4.3) there is only one Q -corner-element, namely z = x_{1}^{a_{1}−1} ⋯ x_{d}^{a_{d}−1}.

4. Parametric Decompositions of Monomial Ideals. (2.4) (together with (3.4)) shows that X^{n} is the irredundant finite intersection of the parameter ideals P (z), where z is an X^{n} -corner-element. The main result in this section, (4.1), generalizes this to an arbitrary monomial ideal I such that Rad(I) = Rad(X). And in (4.10) we show that such a decomposition is unique. (4.1) Theorem. Let I be a monomial ideal such that Rad(I) = Rad(X) and let z_{1}, …, z_{m} be the I -corner-elements. Then I = ∩{P (z_{j}); j = 1, …, m} is a decomposition of I as an irredundant intersection of parameter ideals. Proof. Let J = ∩{P (z_{j}); j = 1, …, m}. Then (3.3) shows that J is the irredundant intersection of the m parameter ideals P (z_{j}).

Now let f be a monomial in I and suppose that f ∉ P (z_{j}) for some j = 1, …, m. Then z_{j} ∈ fR ⊆ I, by (2.3), and this contradicts the fact that z_{j} ∉ I (since z_{j} is

14 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

an I -corner-element). Therefore I ⊆ J.

Finally, [T, Lemma 6] shows that J is a monomial ideal (since Rad(P (z_{j})) = Rad(X) for j = 1, …, m), so it suffices to show that each monomial that is not in I is not in J. For this, let f be a monomial that is not in I. Then (3.15) shows that there exists a monomial g (possibly g = 1) such that fg is an I -corner-element, so fg = z_{j} for some j = 1, …, m (since (3.2) shows that the I -corner-elements are finite in number and uniquely determined by I). Then f ∉ P (z_{j}), by (2.3), so it follows that I ⊇ J, hence I = J by the preceding paragraph, □

In (4.2) it is shown that the corner-elements of a monomial ideal I determine I when Rad(I) = Rad(X). (4.2) Corollary. If I and J are monomial ideals such that Rad(I) = Rad(X) = Rad(J) and if (I : X) − I = (J : X) − J, then I = J. Proof. If (I : X) − I = (J : X) − J, then I and J have the same corner-elements, by (3.2), so this follows immediately from (4.1), □

(4.3) Corollary. If Q is an open monomial ideal in a Gorenstein local ring R of altitude d > 1, then Q is irreducible if and only if there exists exactly one Q -corner-element, and then Q is generated by a system of parameters. Proof. Let m be the number of Q -corner-elements. Then Q is the irredundant intersection of m (open) parameter ideals, by (4.1). Since R is Gorenstein, an open parameter ideal is irreducible, so Q is the irredundant intersection of m (open) irreducible ideals. Since each such decomposition of Q has the same number of factors, m = 1 if and only if Q is irreducible.

For the final statement, if Q is irreducible, then Q = P (z) is generated by a system of parameters, where z is the Q -corner-element, □

The next corollary is closely related to (2.5) and (4.3). (4.4) Corollary. Let I and z_{1}, …, z_{m} be as in (4.1) and assume that R is a Gorenstein local ring of altitude d. Then I = ∩{P (z_{j}); j = 1, …, m} is an irredundant intersection of m irreducible ideals. Proof. If R is Gorenstein, then each open parameter ideal is irreducible, so the conclusion follows from (4.1), □

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 15

In [T, Theorem 8] it is shown (among other things) that if Rad(I) = Rad(X), then ∪{P; P ∈ Ass(R/I)} = ∪{Q; Q ∈ Ass(R/X)}. (4.1) yields a simple proof of the following closely related result.

(4.5) Corollary. If I is as in (4.1), then ∪{Ass(R/I^{n}); n ≥ 1} ⊆ Ass(R/X).

Proof. It is well known that if Y and Z are ideals that are generated by R -sequences such that Rad(Y) = Rad(Z), then Ass(R/Y) = Ass(R/Z). It therefore follows that if z_{1}, …, z_{m} are the I -corner-elements, then Ass(R/X) = Ass(R/P (z_{j})) for j = 1, …, m (since each P (z_{j}) is generated by powers of x_{1}, …, x_{d}). Therefore, since I = ∩{P (z_{j}); j = 1, …, m}, it follows that Ass(R/I) = Ass(R/(∩{P (z_{j}); j = 1, …, m})) ⊆ ∪{Ass(R/P (z_{j})); j = 1, …, m} = Ass(R/X). Finally, I^{n} is generated by monomials for all n ≥ 1, so it follows from what was just shown that Ass(R/I^{n}) ⊆ Ass(R/X), □

For the proof of the next corollary we need the following definition.

(4.6) Definition. If P is a prime divisor of an ideal I in a Noetherian ring, then D_{P}(I) denotes the number of P -primary ideals in a decomposition of I as an irredundant intersection of irreducible ideals.

Concerning (4.6), a classical result of E. Noether [N, Satz VII] says that D_{P} (I) is well defined (that is, D_{P} (I) is independent of the particular irredundant irreducible decomposition of I).

(4.7) Corollary. Let (R, M) be a Gorenstein local ring, let X be an ideal generated by a system of parameters x_{1}, …, x_{d}, and let Q be an open monomial ideal. Then v((Q : M)/Q) = v((Q : X)/Q).

Proof. Let m be the number of Q -corner-elements. Then (3.7) shows that v((Q : X)/Q) = m, and (4.1) shows that Q is the irredundant intersection of m parameter ideals. Since R is Gorenstein, each of these parameter ideals is irreducible, so Q is the irredundant intersection of m irreducible ideals, so D_{M}(Q) = m (see (4.6)). However, [HRS2, (2.4)] shows that D_{M}(Q) = v((Q : M)/Q). Therefore v((Q : X)/Q) = m = v((Q : M)/Q), □

(4.1) shows that the I -corner-elements determine a decomposition of I as an irredundant intersection of parameter ideals. (4.8) shows that the converse also

16 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

holds.

(4.8) Proposition. For j = 1, …, m let a_{j} = (a_{j,1}, …, a_{j,d}) be a d-tuple of positive integers and let I = ∩{P (a_{j}); j = 1, …, m} be a decomposition of I as an irredundant intersection of parameter ideals. Then the I -corner-elements are the m elements x_{1}^{a_{j,1}−1} ⋯ x_{d}^{a_{j,d}−1}.

Proof. (Note: Rad(I) = Rad(X), since Rad(P (a_{j})) = Rad(X) for j = 1, …, m.)

It will first be shown that each of the m elements z_{j} = x_{1}^{a_{j,1}−1} ⋯ x_{d}^{a_{j,d}−1} is an I -corner-element.

For this, note first that P (z_{j}) = P (a_{j}) for j = 1, …, m. Therefore z_{i} ∉ z_{j}R for all i ≠ j ∈ {1, …, m} (for if z_{i} ∈ z_{j}R, then P (a_{i}) = P (z_{i}) ⊆ P (z_{j}) = P (a_{j}), and this is a contradiction). Therefore (2.3) shows that z_{j} ∉ P (z_{j}) = P (a_{j}) (so z_{j} ∉ I) and that z_{j} ∈ P (z_{k}) = P (a_{k}) for k ∈ {1, …, j − 1, j + 1, …, m}. Also, z_{j}x_{i} ∈ x_{i}^{(a_{j.1}−1)+1}R ⊆ P (a_{j}) for i = 1, …, d, so z_{j}x_{i} ∈ ∩{P (a_{h}); h = 1, …, m} = I. Therefore z_{j} is an I -corner-element, so it follows that z_{1}, …, z_{m} are among the I -corner-elements.

Now let w be an I -corner-element. Then w ∉ I, so w ∉ P (z_{j}) = P (a_{j}) for some j = 1, …, m. Therefore z_{j} ∈ wR, by (2.3), so z_{j} = wg for some monomial g by (2.2.2). If g ≠ 1, then wg ∈ I, since w is an I -corner-element. But this implies that z_{j} ∈ I, and this contradicts the fact that z_{j} is an I -corner-element. Therefore g = 1, so w = z_{j}, so z_{1}, …, z_{m} are all the I -corner-elements, □

(4.9) Corollary. Let z_{1}, …, z_{m} be monomials such that z_{i} ∉ z_{j}R for i ≠ j ∈ {1, …, m}, let J = (z_{1}, …, z_{m})R, and let I = ∩{P (z_{j}); j = 1, …, m}. Then z_{1}, …, z_{m} are the I -corner-elements and I : X = I + J.

Proof. If ∩{P (z_{j}); j = 1, …, m} is an irredundant intersection, then it follows from (4.8) that the I -corner-elements are the elements z_{1}, …, z_{m}, so I : X = I + J by (3.2). Therefore it remains to show that this intersection is irredundant.

For this, suppose, on the contrary, that it is redundant. Then by resubscripting the z_{j}, if necessary, it may be assumed that I = ∩{P (z_{j}); j = 1, …, k} for some k < m (so z_{1}, …, z_{k} are the I -corner-elements, by (4.8)). Then z_{m} ∉ I, since z_{m} ∉ P (z_{m}) ⊇ I, so (3.15) shows that there exists a monomial g such that gz_{m} is an I -corner-element. Therefore gz_{m} = z_{i} for some i = 1, …, k, and this contradicts the hypothesis that z_{i} ∉ z_{j}R for i ≠ j ∈ {1, …, m}. Therefore the intersection is irredundant, □

In (4.10) we show that a decomposition as in (4.1) of a monomial ideal is unique.

(4.10) Theorem (Unique Factorization). Let z_{1}, …, z_{m} and w_{1}, …, w_{n} be monomials such that ∩{P (z_{j}); j = 1, …, m} = ∩{P (w_{i}); i = 1, …, n} are irredundant intersections of parameter ideals. Then n = m and {z_{1}, …, z_{m}} = {w_{1}, …, w_{n}}.

Proof. Let I = ∩{P (z_{j}); j = 1, …, m}. Then it follows from (4.8) that z_{1}, …, z_{m} are the I -corner-elements, so they are the monomials in (I : X) − I by (3.2). However, I = ∩{P (w_{i}); i = 1, …, n}, by hypothesis, so similar statements hold for w_{1}, …, w_{n} in place of z_{1}, …, z_{m}, hence it follows that n = m and that {w_{1}, …, w_{n}} = {z_{1}, …, z_{m}}, □

In (4.11) we note two additional results concerning I -corner-elements.

(4.11) Proposition. Assume that x_{1}, …, x_{d} is a permutable R -sequence, let I be a monomial ideal such that Rad(I) = Rad(X), let z_{1}, …, z_{m} be the I -corner -elements, and let f be a monomial. Then:

(4.11.1) (4.11.1) (4.11.1) fI = (∩{P (fz_{j}); j = 1, …, m}) ∩ fR and fz_{1}, …, fz_{m} are fI -corner-elements.

(4.11.2) (4.11.2) (4.11.2) I : fR = ∩{P (w_{j}); j = 1, …, k}, where z_{j} = w_{j}f for j = 1, …, k and z_{j} ∉ fR for j = k + 1, …, m (for some k ∈ {0, 1, …, m}) and w_{1}, …, w_{k} are the I : fR -corner-elements.

Proof. For (4.11.1), since each permutation of x_{1}, …, x_{d} is an R -sequence, each monomial is regular. Therefore since z_{j} ∈ (I : X) − I, by (3.2), it is readily checked that fz_{j} ∈ (fI : X) − fI for j = 1, …, m, so each fz_{j} is an fI -corner-element by (3.2). Also, P (fz_{j}) : fR = P (z_{j}), for if f = x_{1}^{b_{1}} ⋯ x_{d}^{b_{d}} and z_{j} = x_{1}^{a_{1}} ⋯ x_{d}^{a_{d}}, then P (fz_{j}) : fR = (x_{1}^{a_{1}+b_{1}+1}, …, x_{d}^{a_{d}+b_{d}+1})R : x_{1}^{b_{1}} ⋯ x_{d}^{b_{d}}R = (x_{1}^{a_{1}+1}, x_{2}^{a_{2}+b_{2}+1}, …, x_{d}^{a_{d}+b_{d}+1})R : x_{2}^{b_{2}} ⋯ x_{d}^{b_{d}}R = ⋯ = (x_{1}^{a_{1}+1}, …, x_{d}^{a_{d}+1})R = P (z_{j}). Therefore it follows that (∩{P (fz_{j}); j = 1, …, m})∩fR = f[(∩{P (fz_{j}); j = 1, …, m}) : fR] = f[∩{P (fz_{j}) : fR; j = 1, …, m}] = f[∩{P (z_{j}); j = 1, …, m}] = fI, the last equality by (4.1).

18 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

For (4.11.2), z_{j} ∈ fR if and only if f ∉ P (z_{j}), by (2.3). Therefore if z_{j} = fw_{j} for j = 1, …, k and if z_{j} ∉ fR (so f ∈ P (z_{j})) for j = k + 1, …, m, then it follows from (4.1) that I : fR = (∩{P (z_{j}); j = 1, …, m}) : fR = ∩{P (z_{j}) : fR; j = 1, …, m} = ∩{P (w_{j}f) : fR; j = 1, …, k} = ∩{P (w_{j}); j = 1, …, k}. Finally, it follows from (4.8) that if ∩{P (w_{j}); j = 1, …, k} is an irredundant intersection, then w_{1}, …, w_{k} are the I : fR -corner-elements, so it remains to show that this intersection is irredundant.

For this, suppose that the intersection is redundant, so (by resubscripting, if necessary) there exists h < k such that I : fR = ∩{P (w_{j}); j = 1, …, h} is an irredundant intersection, so w_{1}, …, w_{h} are the I : fR -corner-elements by (4.8). Therefore either (a) w_{k} ∈ I : fR; or, (b) w_{k} ∉ I : fR. If (b) holds, then gw_{k} is an I : fR -corner-element for some monomial g by (3.15), so gw_{k} = w_{j} for some j = 1, …, h (so g ≠ 1). Therefore gz_{k} = fgw_{k} = fw_{j} = z_{j}, so z_{j} = gz_{k} ∈ I (by the definition of I -corner-element, since g ≠ 1 is a monomial), and this contradicts the fact that z_{j} in an I -corner-element. Therefore (b) does not hold, so (a) holds, hence z_{k} = fw_{k} ∈ I, and this contradicts the fact that z_{k} is an I -corner-element. Therefore neither (a) nor (b) holds, so it follows that ∩{P (w_{j}); j = 1, …, k} is an irredundant intersection, hence w_{1}, …, w_{k} are the I : fR -corner-elements, □

(4.12) Corollary. With the notation of (4.11), let J = (f_{1}, …, f_{n})R be a monomial ideal and let w_{j,i} be monomials such that z_{j} = w_{j,i}f_{i} (if z_{j} ∈ f_{i}R) or w_{j,i} = 1 (if z_{j} ∉ f_{i}R). Then I : J = ∩{P (w_{j,i}); j = 1, …, m and i = 1, …, n}, so the I : J -corner-elements are among the mn monomials w_{j,i}.

Proof. If w_{j,i} = 1, then P (w_{j,i}) = X, and X contains all other parameter ideals. Therefore the conclusion follows from (4.11.2) and the fact that I : J = ∩{I : f_{i}R; i = 1, …, n}, □

To prove the next theorem, which gives an irredundant parametric decomposition of the ideal generated by the k-th powers of the monomial generators of a monomial ideal, we need the following definition and lemma.

(4.13) Definition. If J = (f_{1}, …, f_{n})R is a monomial ideal and k is a positive integer, then J^{[k]} denotes the ideal (f_{1}^{k}, f_{2}^{k}, …, f_{n}^{k})R.

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 19

(4.14) Lemma. Let J be a monomial ideal, let g be a monomial, and let k be a positive integer. Then g ∈ J if and only if g^{k} ∈ J^{[k]}.

Proof. It is clear that g ∈ J implies that g^{k} ∈ J^{[k]}.

For the converse assume that g^{k} ∈ J^{[k]}. Let J = (f_{1}, …, f_{n})R, where each f_{i} is a monomial. Then the hypothesis and (2.2.1) imply that g^{k} ∈ f_{i}^{k}R for some i = 1, …, n. Now g^{k} and f_{i}^{k} are monomials in the R -sequence x_{1}^{k}, …, x_{d}^{k}, so by (2.2.2) there exists a monomial s in x_{1}^{k}, …, x_{d}^{k} such that g^{k} = sf_{i}^{k}. Then it is clear that there exists a monomial t in x_{1}, …, x_{d} such that t^{k} = s, so g^{k} = t^{k}f_{i}^{k}, hence g = tf_{i} ∈ J, as desired, □

(4.15) Theorem. Let I be a monomial ideal such that Rad(I) = Rad(X), let z_{1}, …, z_{m} be the I -corner-elements, and let k be a positive integer. Then I^{[k]} = ∩{(P (z_{j}))^{[k]}; j = 1, …, m} is a decomposition of I^{[k]} as an irredundant intersection of parameter ideals.

Proof. Let I = (f_{1}, …, f_{n})R and note that x_{1}^{k}, …, x_{d}^{k} is an R -sequence. Therefore since each f_{i} is a monomial (in x_{1}, …, x_{d}) and since I^{[k]} = (f_{1}^{k}, …, f_{n}^{k})R, it follows that I^{[k]} is generated by monomials in x_{1}^{k}, …, x_{d}^{k}, and Rad(I^{[k]}) = Rad(X^{[k]}) (since Rad(I) = Rad(X)). Also, for each j = 1, …, m it holds that z_{j} is a monomial in x_{1}, …, x_{d} such that z_{j} ∉ I and z_{j}x_{i} ∈ I for i = 1, …, d, so it follows that z_{j}^{k} is a monomial in x_{1}^{k}, …, x_{d}^{k} such that z_{j}^{k} ∉ I^{[k]} (by (4.14)) and z_{j}^{k}x_{i}^{k} ∈ I^{[k]} for i = 1, …, d. Therefore the m elements z_{1}^{k}, …, z_{m}^{k} are among the I^{[k]} -corner-elements (for the R -sequence x_{1}^{k}, …, x_{d}^{k}).

Now let z^{∗} be an I^{[k]} -corner-element (for the R -sequence x_{1}^{k}, …, x_{d}^{k}), so z^{∗} is a monomial in x_{1}^{k}, …, x_{d}^{k} such that z^{∗} ∉ I^{[k]} and z^{∗}x_{i}^{k} ∈ I^{[k]} for i = 1, …, d. Then it is clear that there exists a monomial z in x_{1}, …, x_{d} such that z^{k} = z^{∗}, so z ∉ I (since z^{k} ∉ I^{[k]}) and zx_{i} ∈ I (by (4.14), since z^{k}x_{i}^{k} ∈ I^{[k]}). Therefore z is an I -corner-element, so z = z_{p} for some p = 1, …, m, so z^{∗} = z^{k} = z_{p}^{k}. Therefore it follows that z_{1}^{k}, …, z_{m}^{k} are all the I^{[k]} -corner-elements (for x_{1}^{k}, …, x_{d}^{k}) so it follows from (4.1) that I^{[k]} = ∩{P (z_{j}^{k}); j = 1, …, m}.

Finally, fix j ∈ {1, …, m} and let z_{j} = x_{1}^{a_{1}} ⋯ x_{d}^{a_{d}}. Then z_{j}^{k} = x_{1}^{ka_{1}} ⋯ x_{d}^{ka_{d}} = (x_{1}^{k})^{a_{1}} ⋯ (x_{d}^{k})^{a_{d}}, so it follows from (2.1.2) that P (z_{j}^{k}) = ((x_{1}^{k})^{a_{1}+1}, …, (x_{d}^{k})^{a_{d}+1})R and that P (z_{j}) = (x_{1}^{a_{1}+1}, …, x_{d}^{a_{d}+1})R, so it follows that P (z_{j}^{k}) = (P (z_{j}))^{[k]}.

Therefore it follows from the preceding paragraph that I^{[k]} = ∩{(P (z_{j}))^{[k]}); j = 1, …, m}, □

The final result in this section shows that the I -corner-elements determine the I^{[k]} -corner-elements. (4.16) Corollary. With the notation of (4.15), c(I) = c(I^{[k]}) (see (3.9)). More specifically, if z_{1}, …, z_{m} are the I -corner-elements, and if z_{j} = x_{1}^{a_{j,1}} ⋯ x_{d}^{a_{j,d}}, then the I^{[k]} -corner-elements are the m monomials z_{j}^{(k)} = x_{1}^{ka_{j,1}+k−1} ⋯ x_{d}^{ka_{j,d}+k−1}. Proof. This follows immediately from (4.15) and (4.8), □

5. A Related Irredundant Irreducible Decomposition. Let I be a monomial ideal in a local ring (R, M) such that Rad(I) = Rad(X). Then the main result in this section, (5.1), gives a decomposition of I + M(I : X) that is closely related to (4.1). (5.1) Theorem. Assume that R is local with maximal ideal M, let I be a monomial ideal such that Rad(I) = Rad(X), let z_{1}, …, z_{m} be the I -corner-elements, and for j = 1, …, m let Q_{j} be maximal in S_{j} = {Q; Q is an ideal in R, P (z_{j}) ⊆ Q, and z_{j} ∉ Q}. Then each Q_{j} is irreducible, ∩^{m}_{j=1} Q_{j} is an irredundant intersection, and

( ∩^{m}_{j=1} Q_{j}) ∩ (I : X) = I + M(I : X).

Proof. Fix j ∈ {1, …, m}. Then P (z_{j}) ∈ S_{j}, by (2.3), so S_{j} is not empty, so there exists an ideal Q_{j} that is maximal with respect to being in S_{j}. Then each ideal that properly contains Q_{j} must contain z_{j}, so Q_{j} is irreducible.

Also, z_{j} ∈ P (z_{i}) ⊆ Q_{i} for i ∈ {1, …, j − 1, j + 1, …, m} (by (3.3)) and z_{j} ∉ Q_{j}, so ∩^{m}_{j=1} Q_{j} is an irredundant intersection.

Further, since z_{j}M ⊂ z_{j}R, it follows that z_{j}M ⊆ Q_{j}, and I + (z_{1}, …, z_{j−1}, z_{j+1}, … z_{m})R ⊆ P (z_{j}) ⊆ Q_{j}, by (3.3) and (4.1), so it follows from (3.7) that I + M(I : X) ⊆ I + (z_{1}, …, z_{j−1}, z_{j+1}, … z_{m})R + z_{j}M ⊆ Q_{j}. Therefore it follows

that I + M(I : X) ⊆ ( ∩^{m}_{j=1} Q_{j}) ∩ (I : M).

Finally, if y ∈ ( ∩^{m}_{j=1} Q_{j})∩(I : M), then y = ∑^{n} ; i=1 s_{i}f_{i} + ∑^{m} ; j=1 r_{j}z_{j} (by (3.7), where I = (f_{1}, …, f_{n})R) and y is in each Q_{j}. However, since I+(z_{1}, …, z_{j−1}, z_{j+1}, …, z_{m})R+ z_{j}M ⊆ Q_{j} and z_{j} ∉ Q_{j}, it follows that r_{j}z_{j} ∈ Q_{j}, hence r_{j} ∈ Q_{j} : z_{j}R = M. Since this holds for each j = 1, …, m it follows that y ∈ I + M(I : X), □

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 21

(5.2) Remark. It is readily checked that the following are equivalent for two ideals J and Y in a local ring (R, M): (a) M(J : Y) ⊆ J; (b) J : Y = J : M; (c) J : (J : Y) = M. If any of (a) - (c) hold with I and X in place of J and Y, then X = M and R is a regular local ring.

Proof. It follows from [T, Theorem 6] that I : (I : X) is generated by monomials, so if any of (a) - (c) hold, then in particular (c) holds, so M is generated by monomials. But every ideal generated by monomials is contained in X, so M = X is generated by an R -sequence, hence R is a regular local ring, □

(5.3) Corollary. With the notation of (5.1) assume that R is a regular local ring with maximal ideal M = X. Then I = ∩{Q_{j}; j = 1, …, m} is an irredundant irreducible decomposition of I and Q_{j} = P (z_{j}) for j = 1, …, m.

Proof. Since X = M, M(I : X) ⊆ I, so (5.1) shows that ( ∩^{m}_{j=1} Q_{j}) ∩ (I : M) = I, hence ( ∩^{m}_{j=1} (Q_{j}/I)) ∩ ((I : M)/I) = I/I. Also, (I : X)/I = (I : M)/I is the socle of R/I, and it is shown in [HRS3, (3.3.2)] that ( ∩^{m}_{j=1} (Q_{j}/I)) ∩ ((I : M)/I) = (0) if and only if ∩^{m}_{j=1} (Q_{j}/I) = (0). It therefore follows that I = ∩{Q_{j}; j = 1, …, m}, and (5.1) shows that this is an irredundant intersection of irreducible ideals.

To see that Q_{j} = P (z_{j}) for j = 1, …, m, fix j ∈ {1, …, m} and note that it is shown in (4.1) that I = ∩{P (z_{j}); j = 1, …, m}. Since R is regular, it follows that each parameter ideal P (z_{j}) is irreducible. Also, P (z_{j}) ⊆ Q_{j}, by construction (see (5.1)). Further, it is shown in [HRS3, (3.6)] that there are no containment relations among the ideals in IC(I) = {q; there exists an irredundant irreducible decomposition of I with q as a factor}. Therefore it follows that Q_{j} = P (z_{j}) for j = 1, …, m, □

(5.4) Remark. If either M ≠ X or if R is a Gorenstein local ring, but not regular, in (5.3), then the parameter ideal P (z_{j}) is irreducible and every monomial ideal that contains P (z_{j}) must contain z_{j} (by (3.16), since z_{j} is the unique P (z_{j}) -corner-element, by (4.3)). However, there are ideals that contain P (z_{j}) that are not monomial ideals, so the unique cover of P (z_{j}) is properly contained in (P (z_{j}), z_{j})R, and hence P (z_{j}) ⊂ Q_{j} in (5.1).

6. Parametric and Irreducible ideals. In this section we prove a few additional

22 WILLIAM HEINZER, L. J. RATLIFF, JR. AND KISHOR SHAH

results concerning parameter ideals and their relation to irreducible ideals.

(6.1) Proposition. Consider the following statements about a monomial ideal I such that Rad(I) = Rad(X):

(6.1.1) (6.1.1) (6.1.1) I has exactly one corner-element.

(6.1.2) (6.1.2) (6.1.2) I is a parameter ideal.

(6.1.3) (6.1.3) (6.1.3) I is irreducible.

Then (6.1.3) ⇒ (6.1.1) ⇔ (6.1.2), and all three statements are equivalent when R is Cohen-Macaulay and Rad(X) = P is a prime ideal such that R_{P} is a Gorenstein local ring of altitude d.

Proof. Since Rad(I) = Rad(X), (4.1) shows that I = ∩{P (z_{j}); j = 1, …, m}, where z_{1}, …, z_{m} are the I -corner-elements. Therefore if I is irreducible, then m = 1, so (6.1.3) ⇒(6.1.1).

(4.1) shows that (6.1.1) ⇒ (6.1.2).

Assume that (6.1.2) holds and let I = (x_{1}^{a_{1}}, …, x_{d}^{a_{d}})R. Then x_{1}^{a_{1}−1} ⋯ x_{d}^{a_{d}−1} is the unique I -corner-element (since I : X = x_{1}^{a_{1}−1} ⋯ x_{d}^{a_{d}−1}R), so (6.1.2) ⇒ (6.1.1).

Finally, if R is Cohen-Macaulay and Rad(X) = P is a prime ideal such that R_{P} is a Gorenstein local ring, then it readily follows from (4.3) that (6.1.1) ⇒ (6.1.3), □

(6.2) Remark. (6.1) provides an alternate proof that open monomial ideals are finite intersections of parameter ideals in Gorenstein local rings. Specifically, let Q be such an ideal. If Q is irreducible, then Q is a parameter ideal by (6.1). If Q is not irreducible, then Q is the intersection of two monomial ideals that properly contain it. (For if Q = (f_{1}, …, f_{n})R and f_{i} = x_{1}^{e_{i,1}} ⋯ x_{d}^{e_{i,d}} is such that e_{i,j} ≥ 1 and e_{i,k} ≥ 1, then Q = Q_{1} ∩ Q_{2}, where Q_{1} = (f_{1}, …, f_{i−1}, x_{1}^{e_{i,1}} ⋯ x_{j}^{e_{i,j}−1} ⋯ x_{d}^{e_{i,d}}, f_{i+1}, …, f_{n})R and Q_{2} = (f_{1}, …, f_{i−1}, x_{1}^{e_{i,1}} ⋯ x_{k}^{e_{i,k}−1} ⋯ x_{d}^{e_{i,d}}, f_{i+1}, …, f_{n})R.) Therefore by induction on the (finite) number of monomial ideals between X and Q it follows that the open monomial ideals Q_{1} and Q_{2} are finite intersections of parameter ideals, so Q is.

(6.3) characterizes the parameter ideals that are minimal with respect to containing a given monomial ideal I.

PARAMETRIC DECOMPOSITION OF MONOMIAL IDEALS (I) 23

(6.3) Proposition. Let I be a monomial ideal such that Rad(I) = Rad(X) and let Q be an ideal that is minimal in {q; I ⊆ q and q is a parameter ideal}. Then Q = P (z) for some I -corner-element z.

Proof. By (4.1) and (6.1), Q = P (w) for the unique Q -corner-element w. Then w ∉ Q, so w ∉ I, hence fw is an I -corner-element for some monomial f by (3.15). Then I ⊆ P (fw) ⊆ P (w) = Q, and P (fw) is a parameter ideal. Therefore the definition of Q shows that P (fw) = P (w), so w = fw is an I -corner-element, □

(6.4) Corollary. Let I ⊆ J be monomial ideals such that Rad(I) = Rad(X), and let z_{1}, … z_{m} (resp., w_{1}, …, w_{n}) be the I (resp. J) -corner-elements. Then each P (w_{i}) contains some P (z_{j}) and then z_{j} ∈ w_{i}R.

Proof. Let P(I) = {q; I ⊆ q and q is a parameter ideal}. Then I ⊆ J ⊆ P (w_{i}) for i = 1, …, n, by (4.1), so each P (w_{i}) ∈ P(I). Fix i ∈ {1, …, n}. Then P (w_{i}) ∈ P(I), so P (w_{i}) contains an ideal q that is minimal in P(I). Then q = P (z_{j}) for some I -corner-element z_{j}, by (6.3), so P (z_{j}) ⊆ P (w_{i}), and it is readily checked that this implies that z_{j} ∈ w_{i}R, □

In our final result, by “Q is an irreducible component of I” we mean that there exists a decomposition ∩{Q_{j}; j = 1, …, m} of I as an irredundant finite intersection of irreducible ideals Q_{j} such that Q = Q_{j} for some j = 1, …, m.

(6.5) Corollary. Let I be a monomial ideal such that Rad(I) = Rad(X) and let Q be minimal in {q; I ⊆ q and q is an irreducible monomial ideal in R}. If R is a Gorenstein local ring, then Q is an irreducible component of I.

Proof. If R is a Gorenstein local ring, and if Q is an irreducible monomial ideal, then Q is a parameter ideal, by (6.1), so this follows immediately from (6.3) and (4.1), □

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Department of Mathematics, Purdue University, W. Lafayette, IN 47907 Department of Mathematics, University of California, Riverside, CA 92521 Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804