PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 102(116) (2017), 115–120 DOI: https://doi.org/10.2298/PIM1716115V

COHOMOLOGICAL DIMENSIONS WITH RESPECT TO SUM AND INTERSECTION OF IDEALS

Alireza Vahidi

Abstract. Let R be a commutative Noetherian ring with non-zero identity, 𝔞 and 𝔟 proper ideals of R, M a finitely generated R-module with finite projective dimension, and X a finitely generated R-module. We study the cohomological dimensions of M and X with respect to 𝔞 + 𝔟 and 𝔞 ∩ 𝔟. We show that the inequality cd_{𝔞+𝔟} (M, X) ⩽ cd_{𝔞}(M, X) + cd_{𝔟}(X) holds true and we find an equivalent condition for it to be equality.

1. Introduction

Throughout R is a commutative Noetherian ring with non-zero identity, 𝔞 and 𝔟 are proper ideals of R, M is a finite R-module (i.e., finitely generated) and X is an arbitrary R-module which is not necessarily finite. For the basic results, notations and terminology not given in this paper, the reader is referred to [3, 4].

The ith generalized local cohomology module

H_{𝔞}^{i}(M, X) ≅ lim_{\overrightarrow{n∈ℕ}} Ext_{R}^{i}(M/𝔞^{n}M, X),

which is a generalization of the ith ordinary local cohomology module

H_{𝔞}^{i}(X) ≅ lim_{\overrightarrow{n∈ℕ}} Ext_{R}^{i}(R/𝔞^{n}, X),

was introduced by Herzog in his Habilitation [7] and then continued by Suzuki [8], Bijan-Zadeh [2], Yassemi [9] and some other authors. In the study of vanishing of generalized local cohomology modules, Amjadi and Naghipour in [1] extended the cohomological dimension of X with respect to 𝔞,

cd_{𝔞}(X) := sup{i : H_{𝔞}^{i}(X) ≠ 0},

to the cohomological dimension of M and X with respect to 𝔞,

cd_{𝔞}(M, X) := sup{i : H_{𝔞}^{i}(M, X) ≠ 0},

2010 Mathematics Subject Classification: Primary 13D05; Secondary 13D45.

Key words and phrases: cohomological dimensions; generalized local cohomology modules; local cohomology modules.

Communicated by Zoran Petrović.

and proved some basic results of this notion.

Our main ideas come from [6] where, for a finite R-module X, Dibaei and the author proved the inequalities

(1.1)  cd_{𝔞 +𝔟} (X) ⩽ cd_{𝔞}(X) + cd_{𝔟}(X),
cd_{𝔞 ∩𝔟} (X) ⩽ cd_{𝔞}(X) + cd_{𝔟}(X),

and found some equivalent conditions for (1.1) to be equality [6, Corollaries 2.2 and 2.6] (see also [5, Corollary 4.2]). Here, we first present some technical results (Lemmas 2.1, 2.2, and 2.3) to generalize the above inequalities. More precisely, in Theorem 2.1 and Corollary 2.1, we show that, for a finite R-module M with finite projective dimension and for a finite R-module X, the inequality

(1.2)  cd_{𝔞+𝔟} (M, X) ⩽ cd_{𝔞}(M, X) + cd_{𝔟}(X),

and one of the inequalities

cd_{𝔞∩𝔟} (M, X) ⩽ cd_{𝔞}(M, X) + cd_{𝔟}(X),
cd_{𝔞∩𝔟} (M, X) ⩽ cd_{𝔞}(X) + cd_{𝔟}(M, X),

holds true. Then, in Theorem 2.2, we show that there exists an isomorphism between the R-modules H_{𝔞+𝔟}^{cd_{𝔞}(M,X)+cd_{𝔟}(X)} (M, X) and H_{𝔞}^{cd_{𝔞}(M,X)} (M, H_{𝔟}^{cd_{𝔟}(X)} (X)). Finally, in Corollary 2.2, we find an equivalent condition for (1.2) to be equality.

Even though we can show some of our results by using spectral sequences, we are avoiding the use of this technique completely in this work and we provide more elementary proofs for the results.

2. Main results

Our method to prove some results of the paper is based on the induction argument and we need the following useful lemma for the base cases and the inductive steps. Note that, for all i, we have the isomorphism

H_{𝔞}^{i}(M, X) ≅ H^{i}(Γ_{𝔞}(Hom_{R}(M, E^{•}))),

where E^{•} is an injective resolution of X.

Lemma 2.1. Let M be a finite R-module, X an arbitrary R-module, and Y a 𝔟-torsion R-module. Then the following statements hold true.

(i) Γ_{𝔞}(M, X) ≅ Hom_{R}(M, Γ_{𝔞}(X)). (ii) Γ_{𝔞+𝔟} (M, X) ≅ Γ_{𝔞}(M, Γ_{𝔟}(X)). (iii) H_{𝔞+𝔟}^{i} (M, Y) ≅ H_{𝔞}^{i}(M, Y) for all i ⩾ 0.

Proof. This is easy and left to the reader. □

In the next lemma, for a non-negative integer t, we find out when the generalized local cohomology module H_{𝔞+𝔟}^{t} (M, X) vanishes.

Lemma 2.2. Let M be a finite R-module, X an arbitrary R-module, and t a non-negative integer such that H_{𝔞}^{t−i} (M, H_{𝔟}^{i}(X)) is zero for all i, 0 ⩽ i ⩽ t. Then H_{𝔞+𝔟}^{t} (M, X) is also zero.

COHOMOLOGICAL DIMENSIONS W.R.T SUM AND INTERSECTION OF IDEALS 117

Proof. We prove by using induction on t. The case t = 0 is clear from Lemma 2.1 (ii). Suppose that t > 0 and that t − 1 is settled. Let X̄ = X/Γ_{𝔟}(X) and L = E(X̄)/X̄ where E(X̄) is an injective hull of X̄. Since Γ_{𝔟}(X̄) = 0 = Γ_{𝔞+𝔟} (X̄), we have Γ_{𝔟}(E(X̄)) = 0 = Γ_{𝔞+𝔟} (E(X̄)) and so, by Lemma 2.1 (i), Γ_{𝔟}(E(X̄)) = 0 = Γ_{𝔞+𝔟} (M, E(X̄)). Applying the derived functors of Γ_{𝔟}(−) and Γ_{𝔞+𝔟} (M, −) to the short exact sequence

0 −→ X̄ −→ E(X̄) −→ L −→ 0,

we obtain the isomorphisms

H_{𝔟}^{i}(L) ≅ H_{𝔟}^{i+1} (X̄) ≅ H_{𝔟}^{i+1} (X), (2.1)

H_{𝔞+𝔟}^{i} (M, L) ≅ H_{𝔞+𝔟}^{i+1} (M, X̄) (2.2)

for all i ⩾ 0. From the isomorphisms (2.1), for all i, 0 ⩽ i ⩽ t − 1, we have

H_{𝔞}^{(t−1)−i} (M, H_{𝔟}^{i}(L)) ≅ H_{𝔞}^{t−(i+1)} (M, H_{𝔟}^{i+1} (X))

which is zero by the assumptions. Thus, from the induction hypothesis on L, we have H_{𝔞+𝔟}^{t−1} (M, L) = 0. Therefore H_{𝔞+𝔟}^{t} (M, X̄) = 0 by the isomorphisms (2.2). Now, by the short exact sequence

0 −→ Γ_{𝔟}(X) −→ X −→ X̄ −→ 0,

we get the long exact sequence

⋯−→ H_{𝔞+𝔟}^{t} (M, Γ_{𝔟}(X)) −→ H_{𝔞+𝔟}^{t} (M, X) −→ H_{𝔞+𝔟}^{t} (M, X̄) −→ · · ·.

The above long exact sequence in conjunction with Lemma 2.1 (iii) and the assumptions, show that H_{𝔞+𝔟}^{t} (M, X) = 0. □

The following lemma is a generalization of [1, Theorem B] and is crucial in the rest of the paper.

Lemma 2.3. Suppose that M is a finite R-module with finite projective dimension, X is a finite R-module, and Y is an arbitrary R-module such that Supp_{R}(Y) ⊆ Supp_{R}(X). Then

cd_{𝔞}(M, Y) ⩽ cd_{𝔞}(M, X).

Proof. Since generalized local cohomology commutes with direct limit and Y is the direct limit of its finite submodules, the assertion follows from [1, Theorem B]. □

The following theorem is one of our main results which generalizes [6, Corollary 2.2 (i)] (see also [5, Corollary 4.2]).

Theorem 2.1. Let M be a finite R-module with finite projective dimension and X a finite R-module. Then we have

cd_{𝔞+𝔟} (M, X) ⩽ cd_{𝔞}(M, X) + cd_{𝔟}(X).

Proof. Let t be an integer bigger than cd_{𝔞}(M, X) + cd_{𝔟}(X). We prove that H_{𝔞+𝔟}^{t} (M, X) = 0. From Lemma 2.2, it is enough to show that H_{𝔞}^{t−i} (M, H_{𝔟}^{i}(X)) = 0 for all i, 0 ⩽ i ⩽ t. By the definition of cohomological dimension of X with respect to 𝔟, H_{𝔞}^{t−i} (M, H_{𝔟}^{i}(X)) = 0 if i > cd_{𝔟}(X). Otherwise, t − i > cd_{𝔞}(M, X). Since Supp_{R}(X) ⊇ Supp_{R}(H_{𝔟}^{i}(X)), cd_{𝔞}(M, X) ⩾ cd_{𝔞}(M, H_{𝔟}^{i}(X)) from Lemma 2.3. Therefore H_{𝔞}^{t−i} (M, H_{𝔟}^{i}(X)) = 0 by the definition of cohomological dimension of M and H_{𝔟}^{i}(X) with respect to 𝔞. □

The following corollary is an immediate application of the above theorem which generalizes [6, Corollary 2.2 (ii)].

Corollary 2.1. Let M be a finite R-module with finite projective dimension and X a finite R-module. Then one of the following inequalities holds true.

(i) cd_{𝔞∩𝔟} (M, X) ⩽ cd_{𝔞}(M, X) + cd_{𝔟}(X) or (ii) cd_{𝔞∩𝔟} (M, X) ⩽ cd_{𝔞}(X) + cd_{𝔟}(M, X).

Proof. By considering the Mayer–Vietoris exact sequence

0 −→ Γ_{𝔞+𝔟} (M, X) −→ Γ_{𝔞}(M, X) ⊕ Γ_{𝔟}(M, X)
−→ Γ_{𝔞∩𝔟} (M, X) −→ H_{𝔞+𝔟}^{1} (M, X) −→ ⋯
−→ H_{𝔞+𝔟}^{i} (M, X) −→ H_{𝔞}^{i}(M, X) ⊕ H_{𝔟}^{i}(M, X)
−→ H_{𝔞∩𝔟}^{i} (M, X) −→ H_{𝔞+𝔟}^{i+1} (M, X) −→ ⋯

and Theorem 2.1, we get the inequality cd_{𝔞∩𝔟} (M, X) ⩽ cd_{𝔞}(M, X) + cd_{𝔟}(X) if cd_{𝔞}(M, X) ⩾ cd_{𝔟}(M, X), and the inequality cd_{𝔞∩𝔟} (M, X) ⩽ cd_{𝔞}(X) + cd_{𝔟}(M, X) if cd_{𝔞}(M, X) ⩽ cd_{𝔟}(M, X). □

In the following lemma, for non-negative integers s and t, we find some sufficient conditions for validity of the isomorphism H_{𝔞+𝔟}^{s+t} (M, X) ≅ H_{𝔞}^{s}(M, H_{𝔟}^{t}(X)).

Lemma 2.4. Let M be a finite R-module, X an arbitrary R-module, and s, t non-negative integers such that

(i) H_{𝔞}^{s+t−i} (M, H_{𝔟}^{i}(X)) = 0 for all i ≠ t,

(ii) H_{𝔞}^{s+t+1−i} (M, H_{𝔟}^{i}(X)) = 0 for all i < t,
(iii) H_{𝔞}^{s+t−1−i} (M, H_{𝔟}^{i}(X)) = 0 for all i > t.

Then we have

H_{𝔞+𝔟}^{s+t} (M, X) ≅ H_{𝔞}^{s}(M, H_{𝔟}^{t}(X)).

Proof. Let X̄ = X/Γ_{𝔟}(X) and L = E(X̄)/X̄ where E(X̄) is an injective hull of X̄. We prove by using induction on t. In the case that t = 0, we have H_{𝔞+𝔟}^{s−1} (M, X̄) = 0 = H_{𝔞+𝔟}^{s} (M, X̄) from hypothesis (iii) and (i), and Lemma 2.2. Now, the assertion follows by the exact sequence

H_{𝔞+𝔟}^{s−1} (M, X̄) −→ H_{𝔞+𝔟}^{s} (M, Γ_{𝔟}(X)) −→ H_{𝔞+𝔟}^{s} (M, X) −→ H_{𝔞+𝔟}^{s} (M, X̄),

obtained from the short exact sequence

0 −→ Γ_{𝔟}(X) −→ X −→ X̄ −→ 0,

and Lemma 2.1 (iii).

Suppose that t > 0 and that t − 1 is settled. Since Γ_{𝔟}(X̄) = 0 = Γ_{𝔞+𝔟} (X̄), we have Γ_{𝔟}(E(X̄)) = 0 = Γ_{𝔞+𝔟} (E(X̄)) and so, by Lemma 2.1 (i), we get Γ_{𝔟}(E(X̄)) = 0 = Γ_{𝔞+𝔟} (M, E(X̄)). Applying the derived functors of Γ_{𝔟}(−) and Γ_{𝔞+𝔟} (M, −) to the short exact sequence

0 −→ X̄ −→ E(X̄) −→ L −→ 0,

we obtain the isomorphisms

H_{𝔟}^{i}(L) ≅ H_{𝔟}^{i+1} (X̄) ≅ H_{𝔟}^{i+1} (X), (2.3)

H_{𝔞+𝔟}^{i} (M, L) ≅ H_{𝔞+𝔟}^{i+1} (M, X̄) (2.4)

for all i ⩾ 0. From isomorphisms (2.3) and the assumptions, we have • H_{𝔞}^{s+(t−1)−i} (M, H_{𝔟}^{i}(L)) = H_{𝔞}^{s+t−(i+1)} (M, H_{𝔟}^{i+1} (X)) = 0 for all i ≠ t − 1,

• H_{𝔞}^{s+(t−1)+1−i} (M, H_{𝔟}^{i}(L)) = H_{𝔞}^{s+t+1−(i+1)} (M, H_{𝔟}^{i+1} (X)) = 0 for all i < t − 1,

• H_{𝔞}^{s+(t−1)−1−i} (M, H_{𝔟}^{i}(L)) = H_{𝔞}^{s+t−1−(i+1)} (M, H_{𝔟}^{i+1} (X)) = 0 for all i > t − 1.

Thus, by the induction hypothesis on L, we get H_{𝔞+𝔟}^{s+(t−1)} (M, L) ≅ H_{𝔞}^{s}(M, H_{𝔟}^{t−1} (L)). Therefore H_{𝔞+𝔟}^{s+t} (M, X̄) ≅ H_{𝔞}^{s}(M, H_{𝔟}^{t}(X)) from isomorphisms (2.3) and (2.4). Now, by Lemma 2.1 (iii), assumptions (i) and (ii), and the exact sequence

H_{𝔞+𝔟}^{s+t} (M, Γ_{𝔟}(X)) −→ H_{𝔞+𝔟}^{s+t} (M, X) −→ H_{𝔞+𝔟}^{s+t} (M, X̄) −→ H_{𝔞+𝔟}^{s+t+1} (M, Γ_{𝔟}(X)),

obtained from the short exact sequence

0 −→ Γ_{𝔟}(X) −→ X −→ X̄ −→ 0,

we get H_{𝔞+𝔟}^{s+t} (M, X) ≅ H_{𝔞}^{s}(M, H_{𝔟}^{t}(X)) which completes the proof. □

The following theorem generalizes [6, Corollary 2.6 (i)]. Note that, for finite R-modules M and X, if (Ann_{R}(M) + 𝔞 + 𝔟)X ≠ X, then (Ann_{R}(M) + 𝔞)X ≠ X and 𝔟X ≠ X. Thus we get cd_{𝔞}(M, X) ⩾ 0 and cd_{𝔟}(X) ⩾ 0.

Theorem 2.2. Let M be a finite R-module with finite projective dimension and X a finite R-module with (Ann_{R}(M) + 𝔞 + 𝔟)X ≠ X. Then we have

H_{𝔞+𝔟}^{cd_{𝔞}(M,X)+cd_{𝔟}(X)} (M, X) ≅ H_{𝔞}^{cd_{𝔞}(M,X)} (M, H_{𝔟}^{cd_{𝔟}(X)} (X)).

Proof. Consider Lemma 2.3 and apply Lemma 2.4 with s = cd_{𝔞}(M, X) and t = cd_{𝔟}(X). □

In the following corollary, we find an equivalent condition for

cd_{𝔞+𝔟} (M, X) ⩽ cd_{𝔞}(M, X) + cd_{𝔟}(X)

to be equality. This corollary generalizes [6, Corollary 2.6 (ii)].

Corollary 2.2. Let M be a finite R-module with finite projective dimension and X a finite R-module with (Ann_{R}(M) + 𝔞 + 𝔟)X ≠ X. Then the following statements are equivalent.

(i) cd_{𝔞+𝔟} (M, X) = cd_{𝔞}(M, X) + cd_{𝔟}(X).

(ii) cd_{𝔞}(M, X) = cd_{𝔞}(M, H_{𝔟}^{cd_{𝔟}(X)} (X)).

Proof. (i)⇒ (ii). This follows from Theorem 2.2 and Lemma 2.3.

(ii)⇒ (i). It follows by Theorems 2.1 and 2.2. □

Acknowledgement. This research was in part supported by a grant from Payame Noor University.

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Department of Mathematics (Received 03 02 2016) Payame Noor University (PNU) Iran

    vahidi.ar@pnu.ac.ir