ON ARITHMETIC MACAULAYFICATION OF LOCAL RINGS

TAKESI KAWASAKI

19 August 1999

Abstract. The Rees algebra is an important object of Algebraic Geometry and Commutative Algebra. We are interested in the Cohen-Macaulay property of Rees algebras. The present article says that an unmixed local ring has a Cohen-Macaulay Rees algebra if all the formal fibers are Cohen-Macaulay.

1. Introduction

Let A be a commutative ring with identity and 𝔟 an ideal of A. The Rees algebra of 𝔟 is the graded ring

R(𝔟) = ⨁_{n≥0}(𝔟t)^{n},

where t is an indeterminate. We often regard R(𝔟) as an A-subalgebra A[𝔟t] of the polynomial ring A[t]. The Rees algebra is an important object of Algebraic Geometry and Commutative Algebra because the canonical morphism Proj R(𝔟) → Spec A is the blowing-up of Spec A along the closed subscheme Spec A/𝔟.

In the present paper, we consider the existence of Cohen-Macaulay Rees algebras. We say that a Noetherian ring A has an arithmetic Macaulayfication if there exists an ideal 𝔟 in A of positive height such that R(𝔟) is a Cohen-Macaulay ring. The main theorem of this paper is the following.

Theorem 1.1. Let A be a Noetherian local ring of positive dimension. Then the following statements are equivalent:

(A) A has an arithmetic Macaulayfication; (B) A is unmixed and all the formal fibers of A are Cohen-Macaulay.

1991 Mathematics Subject Classification. Primary: 13A30, Secondary: 13D45, 13H10.

Key words and phrases. arithmetic Macaulayfication, Cohen-Macaulay rings, formal fibers, local cohomology, Macaulayfication, Rees algebra, unmixed local ring.

The author is partially supported by Grant-in-Aid for Co-Operative Research.

Here a Noetherian local ring A is said to be unmixed if dim Â/𝔭 = dim Â for every associated prime 𝔭 of the completion Â. The formal fibers of A are the fiber rings of the natural homomorphism A → Â.

The studies in the Cohen-Macaulay property of Rees algebras started from Barshay’s paper [2]. He gave the defining ideal of R(𝔟) and its free resolution in the case where 𝔟 is generated by a regular sequence. He also showed that R(𝔟) is Cohen-Macaulay if A is also and if 𝔟 is generated by a regular sequence. Therefore a Cohen-Macaulay ring has an arithmetic Macaulayfication. Around 1980, Goto and Shimoda studied several properties of R(𝔟) in the case where A is a Buchsbaum local ring and 𝔟 a parameter ideal. See [6], [7], [8] and [24]. Summarizing these investigations, Goto and Yamagishi [11] established the theory of unconditioned strong d-sequences. Their theory contains the existence of an arithmetic Macaulayfication in the case where A is unmixed and Spec Â is Cohen-Macaulay except for the closed point. See also Brodmann [3] and Schenzel [21]. Recently Kurano [17] proved that a Noetherian local ring A containing a finite field has an arithmetic Macaulayfication if the non F -rational locus of A is of dimension 1. Independently this was also done by Aberbach [1]. Motivated by Kurano’s work, the author [16] also gave some sufficient condition for A to have an arithmetic Macaulayfication. Theorem 1.1 gives a necessary and sufficient condition to exist an arithmetic Macaulayfication.

If the Rees algebra R(𝔟) is a Cohen-Macaulay ring, then the projective scheme Proj R(𝔟) is Cohen-Macaulay. However the converse is not true in general. The author [15] gave an ideal 𝔟 such that Proj R(𝔟) is a Cohen-Macaulay scheme for fairly general Noetherian local rings. Fortunately, in this case, we can show that R(𝔟^{n}) is a Cohen-Macaulay ring for a sufficiently large integer n. Such a 𝔟 is generated by monomials of a certain system of parameters, named a p-standard system of parameters. Sections 2 and 3 are devoted to discussing the existence and properties of a p-standard system of parameters. Theorems 2.5 and 3.6 are improvements of Theorems 2.7 and 3.1 of [15], respectively. We give a proof of Theorem 1.1 in Section 4. In our proof, the theory of multigraded Rees algebras, which was introduced by Herrmann, Hyry and Ribbe [13], plays a key role. Our ideal 𝔟 is very complicated. However their theory makes the proof of Theorem 1.1 simple.

Finally we should mention a consequence of Theorem 1.1. The theorem gives the following.

Corollary 1.2. An unmixed local ring is a homomorphic image of a Cohen-Macaulay local ring if all the formal fibers of it are Cohen-Macaulay.

However there exists no analogy in Gorenstein property. In fact, Ogoma [19, Example 1] gave an example of an acceptable, unmixed local ring which is not a homomorphic image of a Gorenstein ring.

Throughout this paper, A denotes a Noetherian local ring with maximal ideal 𝔪. We assume that the dimension of A is positive. We refer the reader to [12] and [18] for unexplained terminology.

2. A p-standard System of Parameters, I

In this section, we give the definition of a p-standard system of parameters and discuss the existence of it. For a finitely generated A-module M, let 𝔞^{p}(M) denote the annihilator of the p^{th} local cohomology module H_{𝔪}^{p}(M) of M and let 𝔞(M) = ∏_{p<dim M} 𝔞^{p}(M).

Definition 2.1. Let M be a finitely generated A-module of dimension d > 0, x_{1}, …, x_{d} a system of parameters for M and s an integer such that 0 ≤ s < d. We say that x_{1}, …, x_{d} is a p-standard system of parameters of type s for M if

(1) x_{s+1}, …, x_{d} ∈ 𝔞(M);

(2) x_{i} ∈ 𝔞(M/(x_{i+1}, …, x_{d})M) for 1 ≤ i ≤ s.

This notion was given by N. T. Cuong [4]. He showed that there exists a pstandard system of parameters of type d−1 for M whenever A possesses a dualizing complex. We improve his result. For a finitely generated A-module M, let NCM(M) denote the non-Cohen-Macaulay locus of M, that is, NCM(M) = {𝔭 ∈ Spec A | M_{𝔭} is not a Cohen-Macaulay A_{𝔭}-module}. By modifying the proof of [23, Theorem 3.3], we obtain the following lemma.

Lemma 2.2. Let B and C be Noetherian rings and B → C a faithfully flat ring homomorphism. We assume that C_{𝔭}/𝔭C_{𝔭} is a Cohen-Macaulay ring for every prime ideal 𝔭 of B. Let M be a finitely generated B-module. If there exists an ideal 𝔠 of C such that NCM(M ⊗_{B} C) = V (𝔠), then NCM(M) = V (𝔠 ∩ B).

We need the following propositions to choose a p-standard system of parameters.

Proposition 2.3. Assume that A is quasi-unmixed and that all the formal fibers of A are Cohen-Macaulay. Let M be a finitely generated A-module of dimension d > 0. If M is equidimensional, then NCM(M) = V (𝔞(M)). In particular dim A/𝔞(M) < d.

Proof. If A has a dualizing complex, then the assertion was given by Schenzel [20, p. 52]. Assume that A has no dualizing complex. The completion Â of A has a dualizing complex and is a faithfully flat A-algebra. Since A is quasi-unmixed, M ⊗ Â is also equidimensional [18, Theorem 31.6]. Therefore the non-Cohen-Macaulay locus of M ⊗ Â is

V (𝔞(M ⊗ Â)) = V (𝔞^{0}(M ⊗ Â) ∩ ⋯ ∩ 𝔞^{d−1}(M ⊗ Â)).

By using Lemma 2.2, we find that the non-Cohen-Macaulay locus of M is

V (𝔞^{0}(M ⊗ Â) ∩ ⋯ ∩ 𝔞^{d−1}(M ⊗ Â) ∩ A) = V (𝔞^{0}(M) ∩ ⋯ ∩ 𝔞^{d−1}(M)).

The right hand side of the equation above is equal to V (𝔞(M)). Since NCM(M) contains no minimal prime of M, dim A/𝔞(M) = dim NCM(M) < d.

Corollary 2.4. Assume that A is quasi-unmixed and that all the formal fibers of A are Cohen-Macaulay. Let M be a finitely generated A-module of dimension d > 0. If dim A/𝔭 = d for every associated prime ideal 𝔭 of M, then dim A/𝔞(M) < d − 1.

Proof. Let 𝔭 be a prime ideal of A such that dim A/𝔭 = d−1 and M_{𝔭} ≠ 0. Then the one-dimensional A_{𝔭}-module M_{𝔭} is Cohen-Macaulay because M_{𝔭} has no embedded prime. Therefore dim A/𝔞(M) = dim NCM(M) < d − 1.

The following theorem assures us the existence of p-standard system of parameters.

Theorem 2.5. Assume that A is quasi-unmixed and that all the formal fibers of A are Cohen-Macaulay. Let M be a finitely generated A-module of dimension d > 0. If M is equidimensional and s an integer such that dim A/𝔞(M) ≤ s < d, then there exists a p-standard system of parameters of type s for M.

Proof. Since d − dim A/𝔞(M) ≥ d − s, there exist d − s elements x_{s+1}, …, x_{d} in 𝔞(M) such that dim M/(x_{s+1}, …, x_{d})M = s. If elements x_{i+1}, …, x_{d} in A such that dim M/(x_{i+1}, …, x_{d})M = i are given, then M/(x_{s+1}, …, x_{d})M is also equidimensional. Therefore dim A/𝔞(M/(x_{i+1}, …, x_{d})M) < i and hence there exists an element x_{i} in 𝔞(M/(x_{i+1}, …, x_{d})M) such that dim M/(x_{i}, …, x_{d})M = i − 1.

3. A p-standard System of Parameters, II

In this section, we give some properties of a p-standard system of parameters. First we recall the definition of d-sequences and the one of unconditioned strong d-sequences.

Definition 3.1. Let M be an A-module. A sequence x_{1}, …, x_{d} of elements in A is said to be a d-sequence on M if

(x_{1}, …, x_{i−1})M : x_{i}x_{j} = (x_{1}, …, x_{i−1})M : x_{j}

for any 1 ≤ i ≤ j ≤ d. Here we set (x_{1}, …, x_{i−1}) = (0) if i = 1.

A sequence x_{1}, …, x_{d} of elements in A is said to be an unconditioned strong d-sequence (for short, a u.s.d-sequence) on M if x_{1}^{n_{1}}, …, x_{d}^{n_{d}} is a d-sequence on M for any positive integers n_{1}, …, n_{d} and in any order.

The following is one of important properties of d-sequences. It was first given by Goto and Shimoda [8] for the system of parameters for a Buchsbaum local ring, which is a typical example of d-sequences.

Proposition 3.2 ([11, Theorem 1.3]). Let M be an A-module and x_{1}, …, x_{d} a d-sequence on M. If we put 𝔮 = (x_{1}, …, x_{d}), then

(x_{1}, …, x_{i−1})M : x_{i} ∩ 𝔮^{n}M = (x_{1}, …, x_{i−1})𝔮^{n−1}M

for all n > 0 and 1 ≤ i ≤ d.

A p-standard system of parameters has several nice properties. The following two properties was given by [15].

Proposition 3.3 ([15, Proposition 2.8]). Let M be a finitely generated A-module of dimension d > 0 and x_{1}, …, x_{d} a p-standard system of parameters of type s for M. Then x_{s+1}, …, x_{d} is a u.s.d-sequence on M/(y_{1}, …, y_{u})M where y_{1}, …, y_{u} is a subsystem of parameters for M/(x_{s+1}, …, x_{d})M.

Proposition 3.4 ([15, Theorem 2.9]). Let M be a finitely generated A-module of dimension d > 0, x_{1}, …, x_{d} a p-standard system of parameters of type s for M and y_{1}, …, y_{u} a subsystem of parameters for M/(x_{i}, …, x_{d})M where 2 ≤ i ≤ d and 1 ≤ u < i. If y_{u} ∈ 𝔞(M) or y_{u} ∈ 𝔞(M/(x_{i}, …, x_{d})M), then

(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{v}y_{u} = (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{u}

for any 1 ≤ v ≤ u and Λ ⊆ {i, …, d}.

The next proposition is not in [15] but we need it to prove Theorem 1.1. The author is inspired by [4, Theorem 2.6].

Proposition 3.5. Let M be a finitely generated A-module of dimension d > 0, x_{1}, …, x_{d} a p-standard system of parameter of type s for M and y_{1}, …, y_{u} a subsystem of parameters for M/(x_{i}, …, x_{d})M where 1 ≤ i ≤ d and 1 ≤ u < i. Then x_{i}, …, x_{j} is a d-sequence on M/(y_{1}, …, y_{u}, x_{j+1}, …, x_{d})M for any i ≤ j ≤ d.

Proof. Let i ≤ l ≤ j be an integer. By applying Proposition 3.4 to a subsystem of parameters y_{1}, …, y_{u}, x_{i}, …, x_{l} for M/(x_{l+1}, …, x_{d})M and a subset {j + 1, …, d}, we obtain

(y_{1}, …, y_{u}, x_{i}, …, x_{k−1}, x_{j+1}, …, x_{d})M : x_{k}x_{l} = (y_{1}, …, y_{u}, x_{i}, …, x_{k−1}, x_{j+1}, …, x_{d})M : x_{l}

for any i ≤ k ≤ l

The following theorem is an improvement of [15, Theorem 3.1]. The old theorem requires that all n_{i}, …, n_{j} are positive but the new one requires only that all n_{i}, …, n_{j} are non-negative.

Theorem 3.6. Let M be a finitely generated A-module of dimension d > 0 and x_{1}, …, x_{d} a p-standard system of parameters of type s for M. We put 𝔮_{i} = (x_{i}, …, x_{d}) for all 1 ≤ i ≤ d. Then, for any integers 1 ≤ i ≤ j ≤ d and n_{i}, …, n_{j} ≥ 0, the following statements hold:

(A_{ij}): If y_{1}, …, y_{u} is a subsystem of parameters for M/𝔮_{i}M and if n_{k} > 0 for some integer i ≤ k ≤ j, then

(3.6.1) (y_{1}, …, y_{u}, x_{k}, …, x_{l−1})M : x_{l} ∩ [(y_{1}, …, y_{u})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M]

= (y_{1}, …, y_{u})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M

for arbitrary integer k ≤ l ≤ d.

(B_{ij}): If y_{1}, …, y_{u} is a subsystem of parameters for M/𝔮_{i}M and if n_{k} > 0 for some integer i ≤ k ≤ j, then

(3.6.2) [(y_{1}, …, y_{u−1})M + (x_{k}, …, x_{l})𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}

= (x_{k}, …, x_{l}){[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}}

+ (y_{1}, …, y_{u−1})M : y_{u}

for arbitrary integer k ≤ l ≤ d. In particular, by letting l = d, we have

[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}+1} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}

= 𝔮_{k}{[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}}
+ (y_{1}, …, y_{u−1})M : y_{u}.

(C_{ij}): If y_{1}, …, y_{u} is a subsystem of parameters for M/𝔮_{i}M and if n_{i} > 0, then

[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u} ⊆ (y_{1}, …, y_{u−1})M : y_{u} + 𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M. (3.6.3)

(D_{ij}): If y_{1}, …, y_{u} is a subsystem of parameters for M/𝔮_{i}M and if n_{i} > 0, then

(3.6.4) [(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}} … 𝔮_{j}^{n_{j}}M] : y_{u} ∩ x_{i}M

⊆ x_{i}{[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}} + (y_{1}, …, y_{u−1})M.

(E_{ij}): Let y_{1}, …, y_{u} is a subsystem of parameters for M/𝔮_{k}M where 2 ≤ k ≤ i and 1 ≤ u < k. If y_{u} ∈ 𝔞(M/𝔮_{k}M) and if n_{i} > 0, then

(3.6.5) [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v}y_{u}

= [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}

for any 1 ≤ v ≤ u and Λ ⊆ {k, …, i − 1}.

Proof. We work by induction on j − i. First we assume that i = j.

(A_{ii}): Since x_{i}, …, x_{d} is a d-sequence on M/(y_{1}, …, y_{u})M, (3.6.1) coincides with Proposition 3.2.

(B_{ii}): Let a be an element in the left hand side of (3.6.2) and put y_{u}a = x_{l}b + c with b ∈ 𝔮_{i}^{n_{i}}M and c ∈ (y_{1}, …, y_{u−1})M + (x_{i}, …, x_{l−1})𝔮_{i}^{n_{i}}M. By using (A_{ii}), we obtain

b ∈ (y_{1}, …, y_{u}, x_{i}, …, x_{l−1})M : x_{l} ∩ 𝔮_{i}^{n_{i}}M
⊆ (y_{1}, …, y_{u})M + (x_{i}, …, x_{l−1})𝔮_{i}^{n_{i}−1}M.

Let b = y_{u}a′ + c′ with c′ ∈ (y_{1}, …, y_{u−1})M + (x_{i}, …, x_{l−1})𝔮_{i}^{n_{i}−1}M. Then a′ ∈ [(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}}M] : y_{u} and

a − x_{l}a′ ∈ [(y_{1}, …, y_{u−1})M + (x_{i}, …, x_{l−1})𝔮_{i}^{n_{i}}M] : y_{u}.

By induction on l, we find that a is in the right hand side of (3.6.2). The opposite inclusion is obvious.

(C_{ii}): By using (B_{ii}) repeatedly, we have

[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}}M] : y_{u} = (y_{1}, …, y_{u−1})M : y_{u}
+ 𝔮_{i}^{n_{i}−1}{[(y_{1}, …, y_{u−1})M + 𝔮_{i}M] : y_{u}}
⊆ (y_{1}, …, y_{u−1})M : y_{u} + 𝔮_{i}^{n_{i}−1}M.

(D_{ii}): If n_{i} = 1, then the right hand side of (3.6.4) equals to (y_{1}, …, y_{u−1}, x_{i})M and hence contains the left hand side.

Assume that n_{i} > 1. Let a be an element in M such that x_{i}a is in the left hand side of (3.6.4). Then

y_{u}x_{i}a ∈ [(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}}M] ∩ (y_{1}, …, y_{u−1}, x_{i})M
= (y_{1}, …, y_{u−1})M + x_{i}𝔮_{i}^{n_{i}−1}M

because of (A_{ii}). Hence

x_{i}a ∈ [(y_{1}, …, y_{u−1})M + x_{i}𝔮_{i}^{n_{i}−1}M] : y_{u}
= (y_{1}, …, y_{u−1})M : y_{u} + x_{i}{[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}−1}M] : y_{u}}.

Here we used (B_{ii}). By applying Proposition 3.4 to a subsystem of parameters y_{1}, …, y_{u}, x_{i} for M/𝔮_{i+1}M, we have

(y_{1}, …, y_{u−1})M : y_{u}x_{i} = (y_{1}, …, y_{u−1})M : x_{i}

and hence

\begin{aligned}(y_{1}, …, y_{u−1})M : y_{u} ∩ x_{i}M & = x_{i}[(y_{1}, …, y_{u−1})M : y_{u}x_{i}] \\ & ⊆ (y_{1}, …, y_{u−1})M. \end{aligned} (3.6.6)

Therefore

x_{i}a ∈ x_{i}{[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}−1}M] : y_{u}} + (y_{1}, …, y_{u−1})M : y_{u} ∩ x_{i}M
⊆ x_{i}{[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}−1}M] : y_{u}} + (y_{1}, …, y_{u−1})M.

(E_{ii}): Because of (B_{ii}),

[(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}^{n_{i}}M] : y_{v}y_{u}
= (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{v}y_{u}
+ 𝔮_{i}^{n_{i}−1}{[(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}M] : y_{v}y_{u}}.

Applying Proposition 3.4 to a subsystem of parameters y_{1}, …, y_{u} for M/𝔮_{k}M and subsets of {k, …, d}: Λ and Λ ∪ {i, …, d}, we obtain

(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{v}y_{u}
+ 𝔮_{i}^{n_{i}−1}{[(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}M] : y_{v}y_{u}}
= (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{u}
+ 𝔮_{i}^{n_{i}−1}{[(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}M] : y_{u}}
= [(y_{1}, …, y_{u−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}^{n_{i}}M] : y_{u}.

Thus (3.6.5) is shown.

Next we assume that j > i and prove (A_{ij})–(E_{ij}). If n_{i} = 0, then (A_{ij}) and (B_{ij}) are contained in (A_{i+1,j}) and (B_{i+1,j}), respectively. Therefore we may assume that n_{i} > 0. Similarly we may also assume that n_{j} > 0.

(A_{ij}): Let a be an element in the left hand side of (3.6.1). If k = l = i, then

a ∈ (y_{1}, …, y_{u})M : x_{i} ∩ (y_{1}, …, y_{u}, x_{i}, …, x_{d})M = (y_{1}, …, y_{u})M.

Otherwise, by using (A_{i+1,j}), we have

a ∈ (y_{1}, …, y_{u}, x_{i}, x_{k}, …, x_{l−1})M : x_{l} ∩ [(y_{1}, …, y_{u}, x_{i})M + 𝔮_{i+1}^{n_{i}+n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M]

= \begin{cases}(y_{1}, …, y_{u}, x_{i})M + (x_{i+1}, …, x_{l−1})𝔮_{i+1}^{n_{i}+n_{i+1}−1} ⋯ 𝔮_{j}^{n_{j}}M & if k = i or i + 1 \\ (y_{1}, …, y_{u}, x_{i})M + (x_{k}, …, x_{l−1})𝔮_{i+1}^{n_{i}+n_{i+1}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M & if k > i + 1 \end{cases}

= (y_{1}, …, y_{u}, x_{i})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M.

Taking intersection with (y_{1}, …, y_{u})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M, we obtain

a ∈ (y_{1}, …, y_{u})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M
+ x_{i}M ∩ [(y_{1}, …, y_{u})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M].

Because of (C_{i+1,j}),

x_{i}M ∩ [(y_{1}, …, y_{u})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M]
= x_{i}𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M
+ x_{i}M ∩ [(y_{1}, …, y_{u})M + 𝔮_{i+1}^{n_{i}+n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M]
= x_{i}𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M + x_{i}{[(y_{1}, …, y_{u})M + 𝔮_{i+1}^{n_{i}+n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : x_{i}}
⊆ x_{i}𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M + x_{i}[(y_{1}, …, y_{u})M : x_{i} + 𝔮_{i+1}^{n_{i}+n_{i+1}−1} ⋯ 𝔮_{j}^{n_{j}}M]
⊆ (y_{1}, …, y_{u})M + x_{i}𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M.

Therefore

a ∈ (y_{1}, …, y_{u})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M
+ x_{i}𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M.

If k = i, then the proof is completed. If k > i, then we work by induction on n_{i}. Let a = x_{i}b + c with b ∈ 𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M and

c ∈ (y_{1}, …, y_{u})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M.

If we apply Proposition 3.4 to a subsystem of parameters y_{1}, …, y_{u}, x_{k}, …, x_{l−1}, x_{i}, x_{l} for M/𝔮_{l+1}M, then we have

b ∈ (y_{1}, …, y_{u}, x_{k}, …, x_{l−1})M : x_{i}x_{l} = (y_{1}, …, y_{u}, x_{k}, …, x_{l−1})M : x_{l}.

If n_{i} = 1, then (A_{i+1,j}) says that

b ∈ (y_{1}, …, y_{u}, x_{k}, …, x_{l−1})M : x_{l} ∩ 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M
⊆ (y_{1}, …, y_{u})M + (x_{k}, …, x_{l−1})𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M

and hence a = x_{i}b + c is in the right hand side of (3.6.1). If n_{i} > 1, then we obtain

b ∈ (y_{1}, …, y_{u}, x_{k}, …, x_{l−1})M : x_{l} ∩ 𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M
⊆ (y_{1}, …, y_{u})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M

by the induction hypothesis. Thus a = x_{i}b+c is also in the right hand side of (3.6.1).

(B_{ij}): Let a be an element in the left hand side of (3.6.2) and put y_{u}a = x_{l}b + c with b ∈ 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M and c ∈ (y_{1}, …, y_{u−1})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M. Then

b ∈ (y_{1}, …, y_{u}, x_{k}, …, x_{l−1})M : x_{l} ∩ 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M
⊆ (y_{1}, …, y_{u})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M.

Here we used (A_{ij}). If we put b = y_{u}a′ + c′ with

c′ ∈ (y_{1}, …, y_{u−1})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{j}^{n_{j}}M,

then a′ ∈ [(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u} and

a − x_{l}a′ ∈ [(y_{1}, …, y_{u−1})M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}.

By induction on l, we find that a is in the right hand side of (3.6.2). The opposite inclusion is obvious.

(C_{ij}): We first show that

(y_{1}, …, y_{u−1}, x_{i})M : y_{u} ∩ (y_{1}, …, y_{u−1}, x_{i}, …, x_{l})M = (y_{1}, …, y_{u−1}, x_{i})M (3.6.7)

for all i ≤ l ≤ d. We work by induction on l. If l = i, then there exists nothing to prove. Assume that l > i and let a be an element in the left hand side of (3.6.7). If we put a = x_{l}b + c with c ∈ (y_{1}, …, y_{u−1}, x_{i}, …, x_{l−1})M, then

b ∈ (y_{1}, …, y_{u−1}, x_{i}, …, x_{l−1})M : y_{u}x_{l} = (y_{1}, …, y_{u−1}, x_{i}, …, x_{l−1})M : x_{l}.

Here we applied Proposition 3.4 to a subsystem of parameters y_{1}, …, y_{u−1}, x_{i}, …, x_{l−1}, y_{u}, x_{l} for M/𝔮_{l+1}M. Thus we obtain

a = x_{l}b + c ∈ (y_{1}, …, y_{u−1}, x_{i})M : y_{u} ∩ (y_{1}, …, y_{u−1}, x_{i}, …, x_{l−1})M
= (y_{1}, …, y_{u−1}, x_{i})M

by the induction hypothesis.

Next we show (3.6.3). By using (B_{ij}), we may assume that n_{i} = 1. Let a be an element in the left hand side of (3.6.3). Then

a ∈ [(y_{1}, …, y_{u−1}, x_{i})M + 𝔮_{i+1}^{n_{i+1}+1} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}
⊆ (y_{1}, …, y_{u−1}, x_{i})M : y_{u} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M

because of (C_{i+1,j}). On the other hand, since n_{j} > 0, we obtain

a ∈ [(y_{1}, …, y_{u−1})M + 𝔮_{i}^{2}M] : y_{u}
⊆ (y_{1}, …, y_{u−1})M : y_{u} + 𝔮_{i}M.

Here we used (C_{ii}). Hence

a ∈ [(y_{1}, …, y_{u−1}, x_{i})M : y_{u} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] ∩ [(y_{1}, …, y_{u−1})M : y_{u} + 𝔮_{i}M]
= (y_{1}, …, y_{u−1})M : y_{u} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M + (y_{1}, …, y_{u−1}, x_{i})M : y_{u} ∩ 𝔮_{i}M
= (y_{1}, …, y_{u−1})M : y_{u} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M + x_{i}M.

Here we used (3.6.7). Taking intersection with [(y_{1}, …, y_{u−1})M+𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}, we obtain

a ∈ (y_{1}, …, y_{u−1})M : y_{u} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M
+ x_{i}{[(y_{1}, …, y_{u−1})M + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}x_{i}}.

By applying (E_{i+1,j}) to a subsystem of parameters y_{1}, …, y_{u}, x_{i} for M/𝔮_{i+1}M, we have

[(y_{1}, …, y_{u−1})M + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}x_{i} = [(y_{1}, …, y_{u−1})M + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : x_{i}.

Therefore a ∈ (y_{1}, …, y_{u−1})M : y_{u} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M.

(D_{ij}): Let a be an element in M such that x_{i}a is in the left hand side of (3.6.4). Then

y_{u}x_{i}a ∈ x_{i}M ∩ [(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{j}^{n_{j}}M]
⊆ (y_{1}, …, y_{u−1})M + x_{i}𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M.

Here we used (A_{ij}). We put y_{u}x_{i}a = x_{i}b + c with b ∈ 𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M and c ∈ (y_{1}, …, y_{u−1})M. Then

b ∈ (y_{1}, …, y_{u})M : x_{i} ∩ 𝔮_{j}M
⊆ (y_{1}, …, y_{u})M : x_{i} ∩ 𝔮_{i}M
⊆ (y_{1}, …, y_{u})M

because n_{j} > 0 and x_{i}, …, x_{d} is a d-sequence on M/(y_{1}, …, y_{u})M. If we put b = y_{u}a′ + c′ with c′ ∈ (y_{1}, …, y_{u−1})M, then

a′ ∈ [(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}

and

x_{i}(a − a′) ∈ (y_{1}, …, y_{u−1})M : y_{u} ∩ x_{i}M
⊆ (y_{1}, …, y_{u−1})M.

Here we used (3.6.6) again. Therefore

x_{i}a ∈ (y_{1}, …, y_{u−1})M + x_{i}{[(y_{1}, …, y_{u−1})M + 𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}}.

(E_{ij}): We may assume that n_{i} = 1 in the same way as the proof of (E_{ii}). We divide the proof into two cases.

First we assume that n_{i+1} + ⋯ + n_{j} = 1, that is, n_{i+1} = ⋯ = n_{j−1} = 0 and n_{j} = 1. We show that

(3.6.8) [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + (x_{i}, …, x_{l−1}, x_{j}, …, x_{d})𝔮_{i}M] : y_{v}y_{u} = [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + (x_{i}, …, x_{l−1}, x_{j}, …, x_{d})𝔮_{i}M] : y_{u}

for all i ≤ l ≤ j by descending induction on l. If l = j, then (3.6.8) coincides with (E_{ii}). Assume that l < j and let a be an element in the left hand side of (3.6.8). The induction hypothesis says that

a ∈ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + (x_{i}, …, x_{l}, x_{j}, …, x_{d})𝔮_{i}M] : y_{u}.

We put y_{u}a = x_{l}b + c with b ∈ 𝔮_{i}M and

c ∈ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + (x_{i}, …, x_{l−1}, x_{j}, …, x_{d})𝔮_{i}M.

On the other hand, Proposition 3.4 says that

a ∈ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, x_{i}, …, x_{l−1}, x_{j}, …, x_{d})M : y_{v}y_{u}
= (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, x_{i}, …, x_{l−1}, x_{j}, …, x_{d})M : y_{u}.

Hence

b ∈ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, x_{i}, …, x_{l−1}, x_{j}, …, x_{d})M : x_{l} ∩ 𝔮_{i}M
⊆ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, x_{i}, …, x_{l−1}, x_{j}, …, x_{d})M

because x_{i}, …, x_{j−1} is a d-sequence on

M/(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, x_{j}, …, x_{d})M.

Therefore

y_{u}a = x_{l}b + c ∈ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + (x_{i}, …, x_{l−1}, x_{j}, …, x_{d})𝔮_{i}M.

Thus (3.6.8) is proved. If we put l = i, then we obtain

[(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}𝔮_{j}M] : y_{v}y_{u} = [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}𝔮_{j}M] : y_{u}.

Next we assume that n_{i+1} + ⋯ + n_{j} > 1. Let

a ∈ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v}y_{u}.

Then (E_{i+1,j}) says that

a ∈ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, x_{i})M + 𝔮_{i+1}^{n_{i+1}+1} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v}y_{u}
= [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, x_{i})M + 𝔮_{i+1}^{n_{i+1}+1} ⋯ 𝔮_{j}^{n_{j}}M] : y_{u}.

Therefore

y_{u}a ∈ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v}
∩ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, x_{i})M + 𝔮_{i+1}^{n_{i+1}+1} ⋯ 𝔮_{j}^{n_{j}}M]
= (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i+1}^{n_{i+1}+1} ⋯ 𝔮_{j}^{n_{j}}M
+ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v} ∩ x_{i}M
= (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i+1}^{n_{i+1}+1} ⋯ 𝔮_{j}^{n_{j}}M
+ x_{i}{[(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v}}.

Here we used (D_{ij}) to show the second equation. We put y_{u}a = x_{i}b + c with

b ∈ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v} (3.6.9)

and

c ∈ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i+1}^{n_{i+1}+1} ⋯ 𝔮_{j}^{n_{j}}M.

By applying (C_{i+1,j}) to a subsystem of parameters y_{1}, …, y_{v−1}, y_{u}, {x_{λ} | λ ∈ Λ}, x_{i} for M/𝔮_{i+1}M, we obtain

\begin{aligned}b & ∈ [(y_{1}, …, y_{v−1}, y_{u}, \{x_{λ} | λ ∈ Λ\})M + 𝔮_{i+1}^{n_{i+1}+1} ⋯ 𝔮_{j}^{n_{j}}M] : x_{i} \\ & ⊆ (y_{1}, …, y_{v−1}, y_{u}, \{x_{λ} | λ ∈ Λ\})M : x_{i} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M. \end{aligned} (3.6.10)

On the other hand, since n_{i+1} + ⋯ + n_{j} > 1, we have

\begin{aligned}b & ∈ [(y_{1}, …, y_{v−1}, \{x_{λ} | λ ∈ Λ\})M + 𝔮_{i+1}^{2}M] : y_{v} \\ & ⊆ [(y_{1}, …, y_{v−1}, \{x_{λ} | λ ∈ Λ\})M : y_{v} + 𝔮_{i+1}M \end{aligned} (3.6.11)

by using (C_{i+1,i+1}).

Furthermore, by applying Proposition 3.4 to a subsystem of parameters y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ}, y_{v}, x_{i} for M/𝔮_{i+1}M, we obtain

\begin{aligned}(y_{1}, …, y_{v−1}, \{x_{λ} | λ ∈ Λ\})M : y_{v} & ⊆ (y_{1}, …, y_{v−1}, \{x_{λ} | λ ∈ Λ\})M : y_{v}x_{i} \\ & = (y_{1}, …, y_{v−1}, \{x_{λ} | λ ∈ Λ\})M : x_{i}. \end{aligned} (3.6.12)

Hence, by taking intersection of (3.6.10) and (3.6.11), we have

b ∈ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{v} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M
+ (y_{1}, …, y_{v−1}, y_{u}, {x_{λ} | λ ∈ Λ})M : x_{i} ∩ 𝔮_{i+1}M
⊆ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{v} + y_{u}M + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M.

Here we applied Proposition 3.2 to a d-sequence x_{i}, …, x_{d} on

M/(y_{1}, …, y_{v−1}, y_{u}, {x_{λ} | λ ∈ Λ})M.

Taking intersection with (3.6.9), we obtain

b ∈ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{v} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M
+ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v} ∩ y_{u}M
= (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{v} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M
+ y_{u}{[(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M] : y_{v}y_{u}}
= (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M : y_{v} + 𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M.

Here we used (E_{i+1,j}) to show the last equation. By using (3.6.12) again, we find that

y_{u}a = x_{i}b + c ∈ (y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}M.

That is,

a ∈ [(y_{1}, …, y_{v−1}, {x_{λ} | λ ∈ Λ})M + 𝔮_{i}𝔮_{i+1}^{n_{i+1}} ⋯ 𝔮_{j}^{n_{j}}] : y_{u}.

The opposite inclusion is obvious. The proof is completed.

4. The Proof of Theorem 1.1

Before the proof of Theorem 1.1, we give some statements on ℤ^{r}-graded rings. Let R = ⨁_{n_{1},...,n_{r}≥0} R_{(n_{1},...,n_{r})} be a Noetherian ℤ^{r}-graded ring. For such a ring, let R_{+} = ⨁_{(n_{1},...,n_{r})≠(0,...,0)} R_{(n_{1},...,n_{r})} and R^{++} = ⨁_{n_{1},...,n_{r}>0} R_{(n_{1},...,n_{r})}. We assume that R_{(0,...,0)} = A and that R is generated by non-zero A-modules R_{(1,0,...,0)}, …, R_{(0,...,0,1)} over A. Let Proj R denote the set of all homogeneous prime ideals of R not containing R^{++}. It has a scheme structure as same as ℤ-graded rings. Let R^{Δ} = ⨁_{n≥0} R_{(n,...,n)} be the diagonal subring of R. The inclusion map R^{Δ} → R induces an isomorphism of schemes Proj R ≅ Proj R^{Δ}.

If M = ⨁_{n_{1},...,n_{r}∈ℤ} M_{(n_{1},...,n_{r})} is a ℤ^{r}-graded R-module, then it corresponds to a quasi-coherent sheaf M^{∼} on Proj R. Let M^{Δ} denote the R^{Δ}-module ⨁_{n∈ℤ} M_{(n,...,n)}. Then there exists an isomorphism of sheaves M^{∼} ≅ (M^{Δ})^{∼}.

Following two complementary propositions will play a key role in this section.

Proposition 4.1. Let M be a finitely generated graded R-module and 𝔟 an ideal of A. Then there exists an integer n such that

[H_{𝔟R+R_{+}}^{p}(M)]_{(n_{1},...,n_{r})} = 0 unless n_{1}, …, n_{r} < n

for all p ≥ 0.

Proof. If 𝔟 = (0), then we can prove the assertion in the same way as [22, Chapter III §3 Théorème 2]. The spectral sequence E_{2}^{pq} = H_{𝔟R}^{p}H_{R_{+}}^{q}(−) ⇒ H_{𝔟R+R_{+}}^{p+q}(−) says that the assertion holds in general.

Proposition 4.2. Let M be a finitely generated graded R-module and assume that A has a dualizing complex. Then the following are equivalent:

(1) M^{∼} is a Cohen-Macaulay, equidimensional sheaf on Proj R;

(2) If 𝔭 is a prime ideal of R not containing R^{++}, then M_{𝔭} is a Cohen-Macaulay R_{𝔭}-module and dim R/𝔭 + dim M_{𝔭} = dim M;

(3) There is an integer m such that

(R^{++})^{m}H_{𝔪R+R_{+}}^{p}(M) = 0

for all 0 ≤ p < dim M;

(4) There is an integer n such that

[H_{𝔪R+R_{+}}^{p}(M)]_{(n_{1},...,n_{r})} = 0 whenever n_{1}, …, n_{r} < n

for all 0 ≤ p < dim M.

Here a coherent sheaf is said to be equidimensional if all the irreducible components of its support have the same dimension.

Proof. Since R is a finitely generated A-algebra, R has a dualizing complex. The equivalence between (1) and (2) comes from Proposition 1.2.2 and Corollary 1.2.4 of [10]. The one between (2) and (3) comes from [5, Satz 1]. Since H_{𝔪R+R_{+}}^{p}(M) is Artinian and hence has a finite socle, (3) and (4) are equivalent.

Let φ: ℤ^{r} → ℤ^{s} be a group homomorphism satisfying φ(ℕ^{r}) ⊆ ℕ^{s}. We put

R^{φ} = ⨁_{m_{1},...,m_{s}≥0} ( ⨁_{φ(n_{1},...,n_{r})=(m_{1},...,m_{s})} R_{(n_{1},...,n_{r})}) ,

which is a ℤ^{s}-graded ring. For a graded R-module M, let

M^{φ} = ⨁_{m_{1},...,m_{s}∈ℤ} ( ⨁_{φ(n_{1},...,n_{r})=(m_{1},...,m_{s})} M_{(n_{1},...,n_{r})}) ,

which is a graded R^{φ}-module. We know that

[H_{𝔟R+R_{+}}^{p}(M)]^{φ} = H_{𝔟R^{φ}+(R^{φ})_{+}}^{p}(M^{φ})

for any ideal 𝔟 of A. See Lemma 1.1 of [13].

The following proposition is contained in the proof of [13, Lemma 2.1].

Proposition 4.3. Let M = ⨁_{n_{1},...,n_{r}≥0} M_{(n_{1},...,n_{r})} be a finitely generated graded R-module and 𝔟 an ideal of A. We put

S = ⨁_{n_{1},...,n_{r+1}≥0} R_{(n_{1},...,n_{r−1},n_{r}+n_{r+1})}

and

L = ⨁_{n_{1},...,n_{r+1}≥0} M_{(n_{1},...,n_{r−1},n_{r}+n_{r+1})}.

If

[H_{𝔟R+R_{+}}^{p}(M)]_{(n_{1},...,n_{r})} = 0 unless n_{1}, …, n_{r} < 0

for all p, then

[H_{𝔟S+S_{+}}^{p}(L)]_{(n_{1},...,n_{r+1})} = 0 unless n_{1}, …, n_{r+1} < 0

for all p.

Proof. Let

I = ⨁_{n_{1},...,n_{r}≥0,n_{r+1}>0} R_{(n_{1},...,n_{r−1},n_{r}+n_{r+1})}

and

N_{1} = ⨁_{n_{1},...,n_{r}≥0,n_{r+1}>0} M_{(n_{1},...,n_{r−1},n_{r}+n_{r+1})}.

If we put φ(n_{1}, …, n_{r}) = (n_{1}, …, n_{r}, 0), then S/I ≅ R^{φ} and L/N_{1} ≅ M^{φ}. Therefore

[H_{𝔟S+S_{+}}^{p}(L/N_{1})]_{(n_{1},...,n_{r+1})} = 0 unless n_{1}, …, n_{r} < 0, n_{r+1} = 0

for all p. Similarly, if we put

N_{2} = ⨁_{n_{1},...,n_{r−1},n_{r+1}≥0,n_{r}>0} M_{(n_{1},...,n_{r−1},n_{r}+n_{r+1})},

then

[H_{𝔟S+S_{+}}^{p}(L/N_{2})]_{(n_{1},...,n_{r+1})} = 0 unless n_{1}, …, n_{r−1}, n_{r+1} < 0, n_{r} = 0

for all p.

There is a long exact sequence of local cohomology modules

⋯ ⟶ H_{𝔟S+S_{+}}^{p−1}(L/N_{i}) ⟶ H_{𝔟S+S_{+}}^{p}(N_{i}) ⟶ H_{𝔟S+S_{+}}^{p}(L) ⟶ H_{𝔟S+S_{+}}^{p}(L/N_{i}) ⟶ ⋯

for i = 1 and 2. On the other hand N_{1} ≅ N_{2}(0, …, 0, 1, −1). If one of n_{1}, …, n_{r} is non-negative, then

[H_{𝔟S+S_{+}}^{p}(L)]_{(n_{1},...,n_{r+1})} ≅ [H_{𝔟S+S_{+}}^{p}(N_{1})]_{(n_{1},...,n_{r+1})}
≅ [H_{𝔟S+S_{+}}^{p}(N_{2})]_{(n_{1},...,n_{r−1},n_{r}+1,n_{r+1}−1)}
≅ [H_{𝔟S+S_{+}}^{p}(L)]_{(n_{1},...,n_{r−1},n_{r}+1,n_{r+1}−1)}
≅ ⋯ = 0.

Here we used Proposition 4.1. We can also prove that [H_{𝔟S+S_{+}}^{p}(L)]_{(n_{1},...,n_{r+1})} = 0 if n_{r} ≥ 0.

Let 𝔟_{1}, …, 𝔟_{r} be ideals of A. The multigraded Rees algebra of A (for short, the mult-Rees algebra) with respect to them is defined to be

R(𝔟_{1}, …, 𝔟_{r}) = A[𝔟_{1}t_{1}, …, 𝔟_{r}t_{r}],

where t_{1}, …, t_{r} are indeterminates. If 𝔟_{1}, …, 𝔟_{r} are of positive height, then dim R(𝔟_{1}, …, 𝔟_{r}) = dim A + r. See Proposition 1.17 of [13]. For an A-module M, let R_{M}(𝔟_{1}, …, 𝔟_{r}) denote the R(𝔟_{1}, …, 𝔟_{r})-module

⨁_{n_{1},...,n_{r}≥0}(𝔟_{1}t_{1})^{n_{1}} ⋯ (𝔟_{r}t_{r})^{n_{r}}M.

As stated above, there exist the isomorphism of schemes

Proj R(𝔟_{1}, …, 𝔟_{r}) ≅ Proj R(𝔟_{1} ⋯ 𝔟_{r})

and the one of sheaves

R_{M}(𝔟_{1}, …, 𝔟_{r})^{∼} ≅ R_{M}(𝔟_{1} ⋯ 𝔟_{r})^{∼}.

Recently Hyry gives the following theorem.

Theorem 4.4 ([14, Corollary 2.10]). Let 𝔟_{1}, …, 𝔟_{r} be ideals in A of positive height. If the multi-Rees algebra R(𝔟_{1}, …, 𝔟_{r}) is Cohen-Macaulay, then the ordinary Rees algebra R(𝔟_{1} ⋯ 𝔟_{r}) is also Cohen-Macaulay.

We start to prove Theorem 1.1. If A has a p-standard system of parameters for A, then there exists an ideal 𝔟 of positive height making Proj R(𝔟) Cohen-Macaulay and equidimensional [15, Proposition 3.1]. In fact, let x_{1}, …, x_{d} be a p-standard system of parameters of type s for A and put 𝔮_{i} = (x_{i}, …, x_{d}) for all i ≤ s + 1. Then Proj R(𝔮_{1} ⋯ 𝔮_{s+1}) is Cohen-Macaulay and equidimensional and hence Proj R(𝔮_{1}, …, 𝔮_{s+1}) is also. We show that the multi-Rees algebra R = R(𝔮_{1}^{n+1}, …, 𝔮_{s+1}^{n+1}) is a Cohen-Macaulay ring for a sufficiently large n. Then Hyry’s theorem says that R((𝔮_{1} ⋯ 𝔮_{s+1})^{n+1}) is a Cohen-Macaulay ring. First we show that

[H_{𝔪R+R_{+}}^{p}(R)]_{(n_{1},...,n_{s+1})} = 0 unless n_{1}, …, n_{s+1} < 0

for all p.

Proposition 4.5. Let M be a finitely generated A-module and x_{i}, …, x_{d} elements in A. Fix an integer s such that i ≤ s + 1 < d and put 𝔮_{j} = (x_{j}, …, x_{d}) for all i ≤ j ≤ s + 1. We put S = A[𝔮_{i}t_{i}, …, 𝔮_{s+1}t_{s+1}] and L = R_{M}(𝔮_{i}, …, 𝔮_{s+1}). If x_{i}, …, x_{d} satisfies the following five conditions:

(1) the sequence x_{j}, …, x_{d} is a d-sequence on M/(x_{λ} | λ ∈ Λ)M for any

integer i ≤ j ≤ s + 1 and Λ ⊆ {i, …, j − 1};

(2) the sequence x_{j}, …, x_{d−1} is a d-sequence on M/({x_{λ} | λ ∈ Λ}, x_{d})M for

any integer i ≤ j ≤ s + 1 and Λ ⊆ {i, …, j − 1};

(3) the sequence x_{s+1}, …, x_{d} is a u.s.d-sequence on M/(x_{λ} | λ ∈ Λ)M for

any Λ ⊆ {i, …, s};

(4) the equality

({x_{λ} | λ ∈ Λ}, x_{k}, …, x_{l−1})M : x_{l} ∩ [(x_{λ} | λ ∈ Λ)M + 𝔮_{j}^{n_{j}} ⋯ 𝔮_{s+1}^{n_{s+1}}M]

= (x_{λ} | λ ∈ Λ)M + (x_{k}, …, x_{l−1})𝔮_{j}^{n_{j}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{s+1}^{n_{s+1}}M

holds for any integers i ≤ j ≤ k ≤ s + 1, k ≤ l ≤ d, n_{j}, …, n_{k−1}, n_{k+1}, …, n_{s+1} ≥ 0, n_{k} > 0 and Λ ⊆ {i, …, j − 1};

(5) 0 :_{M} x_{i} = 0 :_{M} x_{d}, then H_{𝔮_{i}S+S_{+}}^{0}(L) = [H_{𝔮_{i}S+S_{+}}^{0}(L)]_{(0,...,0)} = 0 :_{M} x_{i}, H_{𝔮_{i}S+S_{+}}^{1}(L) = H_{𝔮_{i}S+S_{+}}^{2}(L) = 0 and

[H_{𝔮_{i}S+S_{+}}^{p}(L)]_{(n_{i},...,n_{s+1})} = 0 unless n_{i}, …, n_{s+1} < 0

for p > 2.

Proof. First we remark that d − s ≥ 2 because of the assumption. We work by descending induction on i. If i = s + 1, then 𝔮_{s+1} is generated by a u.s.d-sequence on M. In this case, the assertion is contained in [11, Theorem 4.1].

Assume that i < s+1. Without loss of generality, we may assume that 0 :_{M} x_{i} = 0. Indeed, 0 :_{M} x_{i} ⊆ 0 :_{M} x_{l} for all i ≤ l ≤ d because of (1). Therefore

[0 :_{M} x_{i} + ({x_{λ} | λ ∈ Λ}, x_{k}, …, x_{l−1})M] : x_{l} ⊆ ({x_{λ} | λ ∈ Λ}, x_{k}, …, x_{l−1})M : x_{l}^{2}.

The assumption (1) also says that

({x_{λ} | λ ∈ Λ}, x_{k}, …, x_{l−1})M : x_{l}^{2} = ({x_{λ} | λ ∈ Λ}, x_{k}, …, x_{l−1})M : x_{l}.

Therefore

[0 :_{M} x_{i} + ({x_{λ} | λ ∈ Λ}, x_{k}, …, x_{l−1})M] : x_{l}

∩ [0 :_{M} x_{i} + (x_{λ} | λ ∈ Λ)M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}}M]

= ({x_{λ} | λ ∈ Λ}, x_{k}, …, x_{l−1})M : x_{l} ∩ [(x_{λ} | λ ∈ Λ)M + 𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}}M]

+ 0 :_{M} x_{i}

= (x_{λ} | λ ∈ Λ)M + (x_{k}, …, x_{l−1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{k}^{n_{k}−1} ⋯ 𝔮_{s+1}^{n_{s+1}}M + 0 :_{M} x_{i}.

Thus x_{i}, …, x_{d} satisfies (4) on \overline{M} = M/0 :_{M} x_{i}. Similarly we can show that the sequence x_{i}, …, x_{d} satisfies (1)–(3) on \overline{M}. Naturally 0 :\overline{M} x_{i} = 0 :\overline{M} x_{d} = 0.

On the other hand, since x_{i}, …, x_{d} is a d-sequence on M, we obtain 0 :_{M} x_{i} ∩ 𝔮_{i}M = 0. Therefore there exists a short exact sequence

0 ⟶ 0 :_{M} x_{i} ⟶ L ⟶ R\overline{M}(𝔮_{i}, …, 𝔮_{s+1}) ⟶ 0.

Since 𝔮_{i}[0 :_{M} x_{i}] = 0, we have an exact seuqnece

0 ⟶ 0 :_{M} x_{i} ⟶ H_{𝔮_{i}S+S_{+}}^{0}(L) ⟶ H_{𝔮_{i}S+S_{+}}^{0}(R\overline{M}(𝔮_{i}, …, 𝔮_{s+1})) ⟶ 0

and isomorphisms

H_{𝔮_{i}S+S_{+}}^{p}(L) ≅ H_{𝔮_{i}S+S_{+}}^{p}(R\overline{M}(𝔮_{i}, … 𝔮_{s+1}))

for p > 0. Thus if the assertion holds for \overline{M}, then the one also holds for M.

From now on, we assume that 0 :_{M} x_{i} = 0. Then 0 :_{M} x_{i+1} = ⋯ = 0 :_{M} x_{d} = 0 and

x_{i}M : x_{i+1} = x_{i}M : x_{d}. (4.5.1)

In fact, if a ∈ x_{i}M : x_{d} and we put x_{d}a = x_{i}b, then b ∈ x_{d}M : x_{i} ⊆ x_{d}M : x_{i+1} because of (2). If we put x_{i+1}b = x_{d}c, then x_{i+1}x_{d}a = x_{i}x_{i+1}b = x_{i}x_{d}c, that is, x_{i+1}a − x_{i}c ∈ 0 :_{M} x_{d} = 0. Therefore a ∈ x_{i}M : x_{i+1}. The opposite inclusion comes from (1).

We first show that x_{s}t_{s}, x_{s+1}t_{s+1}, x_{s+2} is a regular sequence on L. Since x_{s} is regular on M, x_{s}t_{s} is regular on L.

Let at_{i}^{n_{i}} ⋯ t_{s+1}^{n_{s+1}} ∈ x_{s}t_{s}L: x_{s+1}t_{s+1}. If n_{s} = 0, then x_{s+1}a = 0 and hence a = 0. If n_{s} > 0, then

a ∈ x_{s}M : x_{s+1} ∩ 𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}}M = x_{s}𝔮_{i}^{n_{i}} ⋯ 𝔮_{s}^{n_{s}−1}𝔮_{s+1}^{n_{s+1}}M.

Hence at_{i}^{n_{i}} ⋯ t_{s+1}^{n_{s+1}} ∈ x_{s}t_{s}L.

Let at_{i}^{n_{i}} ⋯ t_{s+1}^{n_{s+1}} ∈ (x_{s}t_{s}, x_{s+1}t_{s+1})L: x_{s+2}. If n_{s} = n_{s+1} = 0, then a = 0 because x_{s+2} is regular on M. If n_{s} > 0 and n_{s+1} = 0, then a ∈ x_{s}M : x_{s+2}. Because of (3), we have x_{s}M : x_{s+1} = x_{s}M : x_{s+2}. Hence

a ∈ x_{s}M : x_{s+1} ∩ 𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}}M = x_{s}𝔮_{i}^{n_{i}} ⋯ 𝔮_{s}^{n_{s}−1}𝔮_{s+1}^{n_{s+1}}M,

that is, at_{i}^{n_{i}} ⋯ t_{s+1}^{n_{s+1}} ∈ x_{s}t_{s}L. If n_{s} = 0 and n_{s+1} > 0, then

a ∈ x_{s+1}M : x_{s+2} ∩ 𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}}M = x_{s+1}𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}−1}M

and hence at_{i}^{n_{i}} ⋯ t_{s+1}^{n_{s+1}} ∈ x_{s+1}t_{s+1}L. If n_{s}, n_{s+1} > 0, then

a ∈ (x_{s}, x_{s+1})M : x_{s+2} ∩ 𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}}M
= (x_{s}, x_{s+1})𝔮_{i}^{n_{i}} ⋯ 𝔮_{s}^{n_{s}−1}𝔮_{s+1}^{n_{s+1}}M
⊆ x_{s}𝔮_{i}^{n_{i}} ⋯ 𝔮_{s}^{n_{s}−1}𝔮_{s+1}^{n_{s+1}}M + x_{s+1}𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}−1}M.

Therefore at_{i}^{n_{i}} ⋯ t_{s+1}^{n_{s+1}} ∈ (x_{s}t_{s}, x_{s+1}t_{s+1})L. Just we have shown that H_{𝔮_{i}S+S_{+}}^{p}(L) = 0 for p ≤ 2.

Let M^{∗} = M/x_{i}M and M^{∗∗} = M/x_{i}M : x_{i+1}. Then x_{i+1}, …, x_{d} satisfies (1)– (5) on M^{∗} and hence on M^{∗∗}. In fact, it is obvious that x_{i+1}, …, x_{d} satisfies (1)–(4) on M^{∗}. Furthermore (4.5.1) says that x_{i+1}, …, x_{d} satisfies (5) on M^{∗}.

Let S′ = A[𝔮_{i+1}t_{i+1}, …, 𝔮_{s+1}t_{s+1}]. Then the induction hypothesis says that

[H_{𝔮_{s+1}S′+S′_{+}}^{p}(R_{M^{∗∗}}(𝔮_{i+1}, …, 𝔮_{s+1}))]_{(n_{i+1},...,n_{s+1})} = 0 unless n_{i+1}, …, n_{s+1} < 0

for all p. Let S′′ = A[𝔮_{i+1}t_{i}, 𝔮_{i+1}t_{i+1}, …, 𝔮_{s+1}t_{s+1}]. Proposition 4.3 says that

[H_{𝔮_{i+1}S′′+S′′_{+}}^{p}(R_{M^{∗∗}}(𝔮_{i+1}, 𝔮_{i+1}, …, 𝔮_{s+1}))]_{(n_{i},...,n_{s+1})} = 0 unless n_{i}, …, n_{s+1} < 0

for all p. Since S′′ is an A-subalgebra of S, we can regard R_{M^{∗∗}}(𝔮_{i}, …, 𝔮_{s+1}) as an S′′-module and there exists an S′′-isomorphism

R_{M^{∗∗}}(𝔮_{i+1}, 𝔮_{i+1}, …, 𝔮_{s+1}) ≅ R_{M^{∗∗}}(𝔮_{i}, …, 𝔮_{s+1}).

ON ARITHMETIC MACAULAYFICATION OF LOCAL RINGS 19

Since (x_{i}, x_{i}t_{i})R_{M^{∗∗}}(𝔮_{i}, …, 𝔮_{s+1}) = 0, we obtain an S′′-isomorphism

H_{𝔮_{i}S+S_{+}}^{p}(R_{M^{∗∗}}(𝔮_{i}, …, 𝔮_{s+1})) ≅ H_{𝔮_{i+1}S′′+S′′_{+}}^{p}(R_{M^{∗∗}}(𝔮_{i}, …, 𝔮_{s+1})). Therefore

[H_{𝔮_{i}S+S_{+}}^{p}(R_{M^{∗∗}}(𝔮_{i}, …, 𝔮_{s+1}))]_{(n_{i},...,n_{s+1})} = 0 unless n_{i}, …, n_{s+1} < 0 for all p. Since there exists an exact sequence

0 ⟶ 0 :_{M^{∗}} x_{i} ⟶ R_{M^{∗}}(𝔮_{i}, …, 𝔮_{s+1}) ⟶ R_{M^{∗∗}}(𝔮_{i}, …, 𝔮_{s+1}) ⟶ 0,

we find that

[H_{𝔮_{i}S+S_{+}}^{p}(R_{M^{∗}}(𝔮_{i}, …, 𝔮_{s+1}))]_{(n_{i},...,n_{s+1})} = 0 unless n_{i}, …, n_{s+1} < 0 for all p > 0.

On the other hand, by regarding R_{M^{∗}}(𝔮_{i+1}, …, 𝔮_{s+1}) as an S-module via the isomorphism S′ ≅ S/𝔮_{i}t_{i}S, we have

[H_{𝔮_{i+1}S+S_{+}}^{p}(R_{M^{∗}}(𝔮_{i+1}, …, 𝔮_{s+1}))]_{(n_{i},...,n_{s+1})} = 0

unless n_{i} = 0, n_{i+1}, …, n_{s+1} < 0 for all p > 0 and

H_{𝔮_{i+1}S+S_{+}}^{0}(R_{M^{∗}}(𝔮_{i+1}, …, 𝔮_{s+1})) ≅ 0 :_{M^{∗}} x_{i+1}.

The spectral sequence

E_{2}^{pq} = H_{x_{i}}^{p}H_{𝔮_{i+1}S+S_{+}}^{q}(−) ⇒ H_{𝔮_{i}S+S_{+}}^{p+q}(−)

gives an exact sequence

0 ⟶ H_{x_{i}}^{1}H_{𝔮_{i+1}S+S_{+}}^{p−1}(−) ⟶ H_{𝔮_{i}S+S_{+}}^{p}(−) ⟶ H_{x_{i}}^{0}H_{𝔮_{i+1}S+S_{+}}^{p}(−) ⟶ 0. By using it, we obtain that [H_{𝔮_{i}S+S_{+}}^{p}(R_{M^{∗}}(𝔮_{i+1}, …, 𝔮_{s+1}))]_{(n_{i},...,n_{s+1})} = 0 unless n_{i} = 0, n_{i+1}, …, n_{s+1} < 0 for all p > 0. Since x_{i+1}, …, x_{d} satisfies the assumptions (1)–(5) on M itself, we obtain that [H_{𝔮_{i}S+S_{+}}^{p}(R_{M}(𝔮_{i+1}, …, 𝔮_{s+1}))]_{(n_{i},...,n_{s+1})} = 0 unless n_{i} = 0, n_{i+1}, …, n_{s+1} < 0 for all p ≥ 0 in the same way as above.

Let X be the kernal of the natural epimorphism L → R_{M^{∗}}(𝔮_{i}, …, 𝔮_{s+1}). We consider the following exact sequence 0 ⟶ X ⟶ L ⟶ R_{M^{∗}}(𝔮_{i}, …, 𝔮_{s+1}) ⟶ 0. (4.5.2) If p > 1, then it gives an isomorphism

[H_{𝔮_{i}S+S_{+}}^{p}(X)]_{(n_{i},...,n_{s+1})} ≅ [H_{𝔮_{i}S+S_{+}}^{p}(L)]_{(n_{i},...,n_{s+1})} unless n_{i}, …, n_{s+1} < 0. Since x_{i}M ∩ 𝔮_{i}^{n_{i}} ⋯ 𝔮_{s+1}^{n_{s+1}}M = x_{i}𝔮_{i}^{n_{i}−1} ⋯ 𝔮_{s+1}^{n_{s+1}}M if n_{i} > 0, we obtain

⨁_{n_{i}>0} X_{(n_{i},...,n_{s+1})} = x_{i}t_{i}L.

Taking quotient of the each term of (4.5.2) by x_{i}t_{i}L, 𝔮_{i}t_{i}L or 𝔮_{i}t_{i}R_{M^{∗}}(𝔮_{i}, …, 𝔮_{s+1}), respectively, we obtain an exact sequence

0 ⟶ X/x_{i}t_{i}L ⟶ R_{M}(𝔮_{i+1}, …, 𝔮_{s+1}) ⟶ R_{M^{∗}}(𝔮_{i+1}, …, 𝔮_{s+1}) ⟶ 0. (4.5.3)

Moreover

0 ⟶ L ⟶^{x_{i}t_{i}} X(1, 0, …, 0) ⟶ (X/x_{i}t_{i}L)(1, 0, …, 0) ⟶ 0 (4.5.4)

is also exact. Taking local cohomology of (4.5.3) and (4.5.4), we find that

[H_{𝔮_{i}S+S_{+}}^{p}(X/x_{i}t_{i}L)(1, 0, …, 0)]_{(n_{i},...,n_{s+1})} = 0 unless n_{i}, …, n_{s+1} < 0

for all p > 1 and hence

[H_{𝔮_{i}S+S_{+}}^{p}(L)]_{(n_{i},...,n_{s+1})} ≅ [H_{𝔮_{i}S+S_{+}}^{p}(X)]_{(n_{i}+1,...,n_{s+1})} unless n_{i}, …, n_{s+1} < 0

for all p > 2. Therefore if one of n_{i}, …, n_{s+1} is non-negative, then

[H_{𝔮_{i}S+S_{+}}^{p}(L)]_{(n_{i},...,n_{s+1})} ≅ [H_{𝔮_{i}S+S_{+}}^{p}(X)]_{(n_{i}+1,...,n_{s+1})}
≅ [H_{𝔮_{i}S+S_{+}}^{p}(L)]_{(n_{i}+1,...,n_{s+1})}
≅ ⋯ = 0

for all p > 2. Here we used Proposition 4.1. The proof is completed.

Proof of Theorem 1.1 and Corollary 1.2. Let A be a Noetherian local ring of dimension d > 0. First we prove that (B) implies (A). Assume that A satisfies (B). If d = 1, then A is Cohen-Macaulay because A has no embedded prime. Let a be a system of parameters for A. Then R(aA) is a polynomial ring over A and hence Cohen-Macaulay.

Assume that d ≥ 2. Since A is unmixed, dim A/𝔭 = d for any associated prime 𝔭 of A. Thus s = dim A/𝔞(A) < d − 1 because of Corollary 2.4. Theorem 2.5 assures us that there exists a p-standard system of parameters x_{1}, …, x_{d} of type s for A. Let 𝔮_{i} = (x_{i}, …, x_{d}) for all 1 ≤ i ≤ s + 1. We show that R((𝔮_{1} ⋯ 𝔮_{s+1})^{n+1}) is Cohen-Macaulay for a sufficiently large n. Since 𝔞^{p}(M)Â ⊆ 𝔞^{p}(M ⊗ Â) for any finitely generated A-module M, the sequence x_{1}, …, x_{d} is also a p-standard system of parameters for Â. On the other hand, the Cohen-Macaulay property of R((𝔮_{1} ⋯ 𝔮_{s+1})^{n+1}Â) = R((𝔮_{1} ⋯ 𝔮_{s+1})^{n+1}) ⊗ Â implies the one of R((𝔮_{1} ⋯ 𝔮_{s+1})^{n+1}). Therefore we may assume that A is complete and hence has a dualizing complex. Let R = R(𝔮_{1}, …, 𝔮_{s+1}). Propositions 3.3–3.6 say that x_{1}, …, x_{d} satisfies the assumptions (1)–(4) of Proposition 4.5. Furthermore x_{1}, …, x_{d} are non-zero divisors on A because A is unmixed. Therefore

[H_{𝔪R+R_{+}}^{p}(R)]_{(n_{1},...,n_{s+1})} = 0 unless n_{1}, …, n_{s+1} < 0

for all p.

On the other hand, we know that Proj R is a Cohen-Macaulay, equidimensional scheme. Proposition 4.2 gives an integer n such that

[H_{𝔪R+R_{+}}^{p}(R)]_{(n_{1},...,n_{s+1})} = 0 whenever n_{1}, …, n_{s+1} < n

for p < dim R. Therefore the Veronese subring R(𝔮_{1}^{n+1}, …, 𝔮_{s+1}^{n+1}) of R is Cohen-Macaulay. See [9, Theorem 3.1.1] or [13, Lemma 1.14]. Hyry’s theorem says that R((𝔮_{1} ⋯ 𝔮_{s+1})^{n+1}) is Cohen-Macaulay.

Next we show that (A) implies (B). Let 𝔟 be an ideal in A of positive height such that R = A[𝔟t] is a Cohen-Macaulay ring. Then A is a homomorphic image of a Cohen-Macaulay local ring R_{𝔪R+R_{+}} and hence all the formal fibers of A are Cohen-Macaulay. Just we also finish to prove Corollary 1.2. Next we show that A is unmixed. By passing through the completion, we may assume that A is complete. Since 𝔟 is of positive height, dim R = d + 1. See [25, Corollary 1.6]. Let 𝔭_{1}, …, 𝔭_{s} be the associated primes of A. Then

𝔭_{i}^{∗} = 𝔭_{i}A[t] ∩ R where i = 1, …, s

are the associated primes of R. Since R is a Cohen-Macaulay ring of dimension d+1, dim R/𝔭_{i}^{∗} = d+1 and hence dim A/𝔭_{i} = d, see [25, Corollary 1.6] again, for all i.

Finally we give an example.

Example 4.6. Let k be a field, B an affine semigroup ring

k[a, b, c, d, e^{2}, e^{3}, ade, bde, cde, d^{2}e]

and 𝔫 the homogeneous maximal ideal of B. Then A = B_{𝔫} is a Noetherian local ring of dimension 5. The sequence x_{1} = a^{4}, x_{2} = b^{4}, x_{3} = c^{4}, x_{4} = d^{4}, x_{5} = e^{4} is a p-standard system of parameter of type 3 for A. See [15, Appendix B].

Let 𝔮_{i} = (x_{i}, …, x_{d}) for i = 1, …, 4. Then the proof of Theorem 1.1 says that R(𝔮_{1}^{n}, …, 𝔮_{4}^{n}) is a Cohen-Macaulay ring for a sufficiently large n. However we can verify that R(𝔮_{1}, …, 𝔮_{4}) = A[𝔮_{1}t_{1}, …, 𝔮_{4}t_{4}] is already a Cohen-Macaulay ring by using a computer. Indeed the sequence x_{1}, x_{1}t_{1} + x_{2}, x_{2}t_{1} + x_{3}, x_{2}t_{2} + x_{3}t_{1} + x_{4}, x_{3}t_{2} + x_{4}t_{1} + x_{5}, x_{3}t_{3} + x_{4}t_{2} + x_{5}t_{1}, x_{4}t_{3} + x_{5}t_{2}, x_{4}t_{4} + x_{5}t_{3}, x_{5}t_{4} is a regular sequence on A[𝔮_{1}t_{1}, …, 𝔮_{4}t_{4}].

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Department of Mathematics, Tokyo Metropolitan University, Hachioji Minami-Ohsawa 1-1, Tokyo 192-0397, Japan

E-mail address: kawasaki@comp.metro-u.ac.jp

URL: http://www.comp.metro-u.ac.jp/^{∼}kawasaki/