ULRICH MODULES — A GENERALIZATION — S. GOTO, R. TAKAHASHI, AND K. OZEKI

1. Introduction Throughout this note, let A be a Cohen-Macaulay local ring with maximal ideal m and d = dim A ≥ 0. In [BHU] B. Ulrich and other authors explored the structure of MGMCM (maximally generated maximal Cohen–Macaulay) A–modules, that is maximal Cohen–Macaulay A–modules M with e0m (M ) = µA (M ), where e0m (M ) (resp. µA (M )) denotes the multiplicity of M with respect to m (resp. the number of elements in a minimal system of generators of M ). In [HK] these modules are simply called Ulrich modules. The purpose of our note is to study Ulrich modules and ideals in a slightly generalized form. Let I be an m–primary ideal in A and assume that I contains a parameter ideal Q of A as a reduction. Definition 1.1. Our ideal I is called a Ulrich ideal of A, if (1) I ) Q, (2) I 2 = QI, and (3) I/I 2 is A/I–free. Condition (2) together with condition (1) is equivalent to saying that the associated ⊕ graded ring G(I) = n≥0 I n /I n+1 of I is a Cohen–Macaulay ring with a(G(I)) = 1 − d, so that Definition 1.1 is independent of the choice of minimal reductions Q of I. If I is a Ulrich ideal, then I/Q is a free A/I–module with rankA/I I/Q = µA (I) − d. Therefore, when A is a Gorenstein ring, Ulrich ideals are good ideals in the sense of [GIW]. Definition 1.2. Let M (̸= (0)) be a finitely generated A–module. Then we say that M is a Ulrich A–module with respect to I, if (1) M is a Cohen-Macaulay A–module with dimA M = d, (2) e0I (M ) = ℓA (M/IM ), and AMS 2000 Mathematics Subject Classification: 13H10, 13H15, 13A30.

(3) M/IM is A/I–free, where e0I (M ) denotes the multiplicity of M with respect to I and ℓA (∗) denotes the length. Ulrich modules with respect to the maximal ideal m of A are MGMCM modules in the original sense of [BHU]. Notice that condition (2) in Definition 1.2 is equivalent to saying that IM = QM , since M is a Cohen–Macaulay A–module with dimA M = dim A and the parameter ideal Q of A is a reduction of I. In this note we shall discuss some basic properties of Ulrich modules and ideals, the relation between them, and the structure of minimal free resolutions of Ulrich ideals with some applications. 2. Examples Let us note the following example. Example 2.1. Let R be a Cohen–Macaulay local ring with maximal ideal n and dim R = d. Let F = Rn for n > 0 and A=RnF the idealization of F over R. Let q be a parameter ideal of R and put I = q × F and Q = qA (= q × qF ). Then A is a d-dimensional Cohen-Macaulay local ring with maximal ideal m = n×F and I is an m–primary ideal of A which contains the parameter ideal Q of A as a reduction. It is standard to check that I is a Ulrich ideal of A. Hence this ring A contains infinitely many Ulrich ideals. We begin with the following. Theorem 2.2. Suppose that I is a Ulrich ideal in A. Then for all i ≥ d, SyziA (A/I) is a Ulrich A–module with respect to I, where SyziA (A/I) denotes the i in a minimal free resolution.

th

syzygy of A/I

Theorem 2.2 is proven by induction on d. Here we shall explain the basic technique of induction. For the moment, assume that d > 0 and let a ∈ Q \ mQ. Then a ̸∈ mI. Let A = A/(a), I = I/(a), and Q = Q/(a). We then have the following. Fact 2.3. The ideal I is a Ulrich ideal of A, if I is a Ulrich ideal of A.

2

φ

Proof. The exact sequence 0 → [I 2 + (a)]/I 2 → I/I 2 → I/I → 0 is split, since [I 2 + (a)]/I 2 ∼ = (a)/[(a) ∩ I 2 ] ∼ = (a)/aI ∼ = A/I and since the homomorphism φ sends 1 to a = a + I 2 which is a part of an A/I–free 2 basis of I/I 2 . Thus I/I is A/I–free.  Fact 2.4 (Vasconcelos [V]). Suppose that I/I 2 is A/I–free. Then (A/I) ⊕ SyziA (A/I) SyziA (A/I)/a·SyziA (A/I) ∼ = Syzi−1 A for all i ≥ 1. Proof. It is enough to show that the exact sequence b a

0 → A/I → I/aI → I → 0 is split. We write I = (a) + (x1 , x2 , . . . , xℓ ) with ℓ = µA (I) − 1. Then I/aI = Aa +

n ∑

Axi

i=1

where a and xi denote the images of a and xi in I/aI, respectively. We claim that this ∑ sum is direct. Assume that ca + ni=1 ci xi = 0 in I/aI with c, ci ∈ A. Then, because I/I 2 is a homomorphic image of I/aI, we still have that ca +

n ∑

ci xi = 0

i=1

in I/I 2 (here a and xi denote the images of a and xi in I/I 2 , respectively). Since {a, xi ∈ I/I 2 (1 ≤ i ≤ n)} forms a free A/I-basis of I/I 2 , we get c ∈ I. Thus ∑ ca = ni=1 ci xi = 0 in I/aI, so that I/aI ∼  = A/I ⊕ I. Theorem 2.2 now readily follows by induction on d. Remember that when d = 0, we get I 2 = (0) and I ∼ = (A/I)n (n = µA (I) > 0). Hence i SyziA (A/I) ∼ = (A/I)n

for all i ≥ 0. Remark 2.5. Fact 2.4 was known by W. V. Vasconcelos [V]. Using this, he proved the famous result that an ideal I (̸= A) in a Noetherian local ring (A, m) is generated by an A–regular sequence, if I has finite projective dimension and if the A/I–module I/I 2 is free. Hence A is a RLR, once m has finite projective dimension.

3. Relation between Ulrich modules and Ulrich ideals The converse of Theorem 2.2 is also true. Namely we have the following. Theorem 3.1. The following conditions are equivalent. (1) I is a Ulrich ideal of A. (2) SyziA (A/I) is a Ulrich A–module with respect to I for all i ≥ d. (3) There exists an exact sequence 0→X→F →Y →0 of finitely generated A–modules such that (a) F is free, (b) X ⊆ mF , and (c) both X and Y are Ulrich A–modules with respect to I. When d > 0, one can add the following. (4) µA (I) > d, I/I 2 is A/I–free, and SyziA (A/I) is a Ulrich A–module with respect to I for some i ≥ d. The implication (3) ⇒ (1) of Theorem 3.1 is based on the following. Proposition 3.2. Let 0→X→F →Y →0 be an exact sequence of finitely generated A-modules and assume that (i) F is a free A-module, (ii) X ⊆ mF , and (iii) Y is a Ulrich A-module with respect to I. Then X is a Ulrich A-module with respect to I if and only if I is a Ulrich ideal of A. Proof. Let us note the proof of the only if part. We may assume that the field A/m is infinite. Suppose that X is a Ulrich A–module with respect to I and look at the exact sequence 0 → X/QX → F/QF → Y /QY → 0; hence X/QX = Syz1A/Q (Y /QY ). Then, because Y /QY = Y /IY ∼ = (A/I)r (r = rankA F > 0), X/QX ∼ = (I/Q)r , so that I ) Q and I 2 ⊆ Q, because X ̸= (0) and QX = IX. Besides, I/Q is a free A/I–module, since X/IX ∼ = (I/Q)r and X/IX

is a free A/I–module. With the condition that for every minimal reduction Q of I, (1) I 2 ⊆ Q and (2) I/Q is a free A/I–module, one can deduce that I/I 2 is A/I–free and that I 2 = QI as well, which we leave to the reader.  We are now in a position to finish the proof of Theorem 3.1. Proof of Theorem 3.1. (4) ⇒ (1) Let a ∈ Q \ mQ and put A = A/(a), I = I/(a), Q = Q/(a). Then by Lemma 2.4 SyziA (A/I)/a·SyziA (A/I) ∼ (A/I) ⊕ SyziA (A/I). = Syzi−1 A i−1 Therefore, because both SyzA (A/I) and SyziA (A/I) are Ulrich modules with respect to I, thanks to the implication (3) ⇒ (1) I is a Ulrich ideal of A. Hence I 2 ⊆ Q, which yields I 2 = QI, because I/I 2 is A/I–free. 

Remark 3.3. Let k[[X]] be the formal power series ring over a filed k and put A = k[[X]]/(X 3 ). We look at the exact sequence x

0 → m2 → A → A → A/m → 0 of A–modules, where x denotes the image of X in A and m = (x) the maximal ideal in A. Then, since m3 = (0), the A–module m2 = Syz2A (A/m) is a Ulrich module with respect to m, but m not a Ulrich ideal of A, since m2 ̸= (0). This example shows that the implication (4) ⇒ (1) is not true in general, unless d > 0. Question 3.4. It seems interesting to explore how many Ulrich ideals are contained in a given Cohen–Macaulay local ring. For example, let k[[t]] be the formal power series ring over a field k and let A = k[[ta1 , ta2 , . . . , taℓ ]] ⊆ k[[t]] be a numerical semigroup ring, where 0 XAg

<

a1 , a2 , . . . , aℓ

∈

Z such that

GCD(a1 , a2 , . . . , aℓ ) = 1. Let be the set of Ulrich ideals in A which are generated by monomials in t. It is then not difficult to check that XAg is finite and for example, we have the following. g (1) Xk[[t 3 ,t4 ,t5 ]] = {m}. g (2) Xk[[t4 ,t5 ,t6 ]] = {(t4 , t6 )}. g (3) Xk[[t a ,ta+1 ,...,t2a−2 ]] = ∅, if a ≥ 5.

g (4) Let 1 < a < b be integers such that GCD(a, b) = 1. Then Xk[[t a ,tb ]] ̸= ∅ if and only if a or b is even. (5) Let A = k[[t4 , t6 , t4ℓ−1 ]] (ℓ ≥ 2). Then ♯XAg = 2ℓ − 2.

4. Minimal free resolutions of Ulrich ideals We now explore minimal free resolutions of Ulrich ideals. Let I be a Ulrich ideal of A which contains a parameter ideal Q of A as a reduction. Let ∂

∂

· · · → Fi →i Fi−1 → · · · → F1 →1 F0 → A/I → 0 be a minimal free resolution of A/I and let βi = βiA (A/I) (i ≥ 0) be the i-th Betti number of A/I. We put n = µA (I) = β1 . We then have the following, which is proven by induction on d. Theorem 4.1. The following assertions hold true. (1) A/I ⊗A ∂i = 0 for all i ≥ 1. (2)  i−d d  (n (d)− d) ·(n − d + 1) (i ≥ d), βi = + (n − d)βi−1 (1 ≤ i ≤ d),  i 1 (i = 0) ( d) for i ≥ 0. Hence βi = i + (n − d)βi−1 for all i ≥ 1. What Theorem 4.1 (3) says is the following. Corollary 4.2. The minimal free resolution of I is obtained by the direct sum of those of Q and (A/I)n−d . i n−d ∼ Corollary 4.3. Syzi+1 for all i ≥ d. Hence A (A/I) = [SyzA (A/I)] i ∼ Syzi+1 A (A/I) = SyzA (A/I)

for all i ≥ d, if A is a Gorenstein ring. This result shows we can expect, in some sense, only one Ulrich module arising from syzygies. We furthermore have the following. Let I1 (∂i ) (i ≥ 1) be the ideal of A generated by the entries of the matrix ∂i . Theorem 4.4. I1 (∂i ) = I for all i ≥ 1.

Proof. By induction on d, we have I1 (∂i ) + Q = I for all i ≥ 1, while by Corollary 4.2 I1 (∂i ) ⊇ Q for 1 ≤ i ≤ d. Therefore, since by Corollary 4.2 I1 (∂i+1 ) = I1 (∂i ) if i ≥ d, the result readily follows.  Corollary 4.5. Let I and J be Ulrich ideals of A. Then I = J if and only if SyziA (A/I) ∼ = SyziA (A/J) for some i ≥ 0. Let XA = {I | I is a Ulrich ideal of A}. We have the following answer to Question 3.4. Theorem 4.6. Suppose that A is of finite C-M representation type. Then XA is a finite set. Proof. Let YA = {[SyzdA (A/I)] | I ∈ XA }, where [SyzdA (A/I)] denotes the isomorphic class of SyzdA (A/I). Let I ∈ XA and n = µA (I). Then, because I/Q ∼ = (A/I)n−d , we have n − d ≤ (n − d)·rA (A/I) = rA (I/Q) ≤ r(A), where rA (∗) denotes the Cohen–Macaulay type. Hence µA (SyzdA (A/I)) = βdA (A/I) = (n − d + 1)d ≤ (r(A) + 1)d ≪ ∞ by Theorem 4.1. Therefore, since A is of finite C-M representation type, the set YA is finite, so that XA is also finite, because XA ⊆ YA by Proposition 4.5.  Let us explore one example. Example 4.7. Let A = k[[X, Y, Z]]/(Z 2 − XY ), where k[[X, Y, Z]] is the formal power series ring over a field k. Then XA = {m}. Proof. Let x, y, and z be the images of X, Y , and Z in A, respectively. Then the indecomposable maximal Cohen-Macaulay A–modules (up to isomorphisms) are A and p = (x, z). Since m2 = (x, y)m, we get m ∈ XA . Let I ∈ XA . Then µA (I) = 3. We put X = Syz2A (I). Then, because µA (X) = 4 and rankA X = 2, we see X∼ =p⊕p∼ = Syz2A (A/m), so that I = m by Corollary 4.5.



For one-dimensional Cohen-Macaulay local rings possessing finite C-M representation type, we have the following, where k[[X, Y ]] and k[[t]] are the formal power series rings over a field k and x, y denote the images of X, Y in the corresponding ring, respectively. Example 4.8. The following assertions hold true. (1) (2) (3) (4)

Xk[[t3 ,t4 ]] = {(t4 , t6 )}. Xk[[t3 ,t5 ]] = ∅. Xk[[X,Y ]]/(Y (X 2 −Y 2ℓ+1 )) = {(x, y 2ℓ+1 ), (x2 , y)}, where ℓ ≥ 1. Xk[[X,Y ]]/(Y (Y 2 −X 3 )) = {(x3 , y)}.

(5) Xk[[X,Y ]]/(X 2 −Y 2ℓ ) = {(x2 , y), (x − y ℓ , y(x + y ℓ )), (x + y ℓ , y(x − y ℓ ))}, where ℓ ≥ 1.

References [BHU] J. Brennan, J. Herzog, and B. Ulrich, Maximally generated Cohen-Macaulay modules, Math. Scand., 61 (1987), 181–203. [GIW] S. Goto, S. Iai, and K. Watanabe, Good ideals in Gorenstein local rings, Trans. Amer. Math. Soc., 353 (2000), 2309–2346. [HK] J. Herzog and M. K¨ uhl, Maximal Cohen-Macaulay modules over Gorenstein rings and Bourbaki sequences, Commutative Algebra and Combinatorics, Adv. Stud. Pure Math., 11 (1987), 65–92. [V] W. V. Vasconcelos, Ideals generated by regular sequences, J. Algebra, 6 (1967), 309-316. Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan E-mail address: [email protected] Department of Mathematical Sciences, Faculty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan E-mail address: [email protected] Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan E-mail address: [email protected]