Generalization of Hartshorne’s Connectedness Theorem

In this paper, we use local cohomology theory to present some results about connectedness property of prime spectrum of modules. In particular, we generalize the Hartshorne’s connectedness theorem.


Introduction
In last decades, the connectedness of some varieties of the prime spectra of a commutative ring is investigated by many authors. Falting's connectedness theorem asserts that in an analytically irreducible local ring (R, m) of dimension n, if a ⊆ m is an ideal generated by at most n − 2 elements, then the punctured spectrum of R/a is connected (see [10]). On the other hand, Hartshorne's connectedness result (see [12]), says that Spec(R) \ V (a) is a connected subset of Spec(R) provided grade(a, R) > 1. In [8] Divaani-Aazar and Schenzel proved a generalization of these results for finitely generated modules. The reader can refer to [7,9,19,20] for more references to the subject mentioned above. In this paper, we use local cohomology to prove some connectedness results for varieties of prime spectrum of certain modules. In particular, we generalize the Hartshorne's connectedness theorem.
Over the past several decades, the theory of prime modules and prime submodules (and its related topics such as Zariski topology on the prime spectrum of modules) is investigated by many algebraist (see [1,2,6,11,13,14]). The Zariski topology on the spectrum of prime ideals of a ring is one of the main tools in algebraic geometry. In the literature, there are many different generalizations of the Zariski topology of rings to modules via prime submodules (see [3,16,18]). Here, we use the Zariski topology on the prime spectrum of modules which is considered by C. P. Lu in [16]. It is shown by Lu that if M is finitely generated, then Spec(M ), the set of all prime submodule of M when is equipped with the Zariski topology, is connected if and only if Spec(R/Ann(M )) is a connected space (see [16]). We find that, this is the only result on the connectedness of the prime spectra of modules in the previous literatures. Here, we are going to give some connectedness results for certain subspaces of prime spectrum of a module.
Throughout this paper, all rings are commutative with identity and all modules are unital. For a submodule N of an R-module M , (N : R M ) denotes the ideal {r ∈ R | rM ⊆ N } and annihilator of M , denoted by Ann R (M ), is the ideal (0 : R M ). If there is no ambiguity, we will consider (N : M ) (resp. Ann(M )) instead of (N : R M ) (resp. Ann R (M )). A submodule N of an R-module M is said to be prime if N = M and whenever rm ∈ N (where r ∈ R and m ∈ M ), then r ∈ (N : M ) or m ∈ N . If N is prime, then p = (N : M ) is a prime ideal of R. In this case, N is said to be p-prime (see [14]). The set of all prime submodules of an R-module M is called the prime spectrum of M and denoted by Spec(M ). Similarly, the collection of all p-prime submodules of an R-module M for any p ∈ Spec(R) is designated by Spec p (M ).
We remark that Spec(0) = ∅ and that Spec(M ) may be empty for some nonzero R-module M (for example see [15,18]). Let B be a nonzero finitely generated R-module. Since every proper submodule of B is contained in a maximal submodule and since every maximal submodule is prime, Spec(B) is nonempty.
Then the elements of the set Z(M ) satisfy the axioms for closed sets in a topological space Spec(M ) (see [16]). The resulting topology due to Z(M ) is called the Zariski topology relative to M . We recall that the Zariski radical of a submodule N of an R-module M , denoted by rad(N ), is the intersection of all members of V (N ), that is, rad Let N be an R-module. For an ideal I of R we recall that the i-th local cohomology module of N with respect to I is defined as The reader can refer to [4] for the basic properties of local cohomology modules. Let H be an R-module.
An element a ∈ R is said to be H-regular if ax = 0 for all 0 = x ∈ H. A sequence a 1 , . . . , a n of elements of R is an H-sequence (or an H-regular sequence) if the following two conditions hold:

Main Results
Our first main result is the following statement which is a generalization of Hartshorne's connectedness result [   Before bringing the final main result of this paper, recall that for any topological space Z and y ∈ Z, Z y is the set of all points y ′ ∈ Z whose closure contains y (see [12]).

4
Dawood Hassanzadeh-lelekaami Lemma 2.5. Let T be a connected topological space and Y a closed subspace such that for each y ∈ Y , T y \ {y} is nonempty and connected. Then T \ Y is connected.
We are now ready to state and prove the second main result which is an application of Theorem 2.1.
Theorem 2.6. Let R be a noetherian ring and M be an R-module such that X := Spec(M ) is a connected topological space. Suppose that Y is a closed subset of X such that for each P ∈ Y , M p , where p := (P : M ), is a cyclic indecomposable R p -module with depth(M p ) > 1. Then Spec(M )\Y is connected.
Proof. By Lemma 2.5, it is enough to show that for each P ∈ Y , X P \ {P } is nonempty and connected. Note that by [16, Proposition 5.2(1)], X P can be described as following: By [17, Theorem 3.7(1)], this set (as a subspace of X) is homeomorphic to Spec(M p ). More precisely, P ∈ X P is corresponded to P e ∈ Spec(M p ) (the extension of P with respect to the natural map M → M p ). Moreover, (P : R M ) p = (P e : Rp M p ) = pR p ∈ Max(R p ).
Since, M p is a cyclic R p -module, |Spec q (M p )| ≤ 1 for all q ∈ Spec(R p ) by [18,Theorem 3.5]. Therefore, P e = pM p is the unique maximal submodule of M p . Hence, X P \ {P } is homeomorphic to Recently, a sheaf on the prime spectra of modules is introduced in [13]. Let N be an R-module. Then the sheaf associated to N relative to M is denoted by A(N, M ). For exact definition and the results on this sheaf see [13]. By [  Proof. Use Theorem 2.6.