A Study of Some Generalizations of Local Homology

: Tate local cohomology and Gorenstein local cohomology theory, which are important generalizations of the classical local cohomology, has been investigated. It has been found that they have such vanishing properties and long exact sequences. However, for local homology, what about the duality? In this paper we are concerned with Tate local homology and Gorenstein local homology. In the ﬁrst part of the paper we generalize local homology as Tate local homology, and study such vanishing properties, artinianness and some exact sequence of Tate local homology modules. We ﬁnd that for an artianian R -module M and a ﬁnitely generated R -module N with ﬁnite Gorenstein projective dimension, the Tate local homology module of M and N with respect to an ideal I is also an artinian module. In the second part of the paper we consider Gorenstein local homology modules as Gorenstein version. We discuss vanishing properties and some exact sequences of Gorenstein local homology modules and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. Finally, as an applicaton of the exact sequence connecting these local homology modules, we ﬁnd that for ﬁnitely generated R -modules with ﬁnite projective dimension and admitting Gorenstein projective proper resolution respectively, Gorenstein local homology coincides with generalized local homology in certain cases.


Introduction
Local cohomology functors were introduced by Grothendieck [11] in local algebra originally. Let I be an ideal of R. It is defined as the right derived functors of the torsion functor Γ I (−) which is naturally equivalent to the functor lim − → t Hom R (R/I n , −). It is denoted by H i I (−) [3]. The local homology functor were first studied by Matlis [17,18] for ideal I generated by a regular sequence. Then the work of Greenlees and May [10], and Tarrio and coauthers [22] show the strong connection between local homology and local cohomlogy. In detail, consider the I-adic completion functor Λ I (−) = lim ← − t (R/I n ⊗ R −). Left deriving Λ I (−) we can obtain the so-called local homology functors denoted H I i (−). Later, Herzog and Zamani [13] introduced the definition of the generalized local cohomology which is an extension of local cohomology of Grothendieck. Let M, N be R-modules, then was investigated as a dual of the generalized local cohomology module of M, N with respect to I [19]. Some properties of them such as vanishing properties, artianness, noetherianness are investigated, see [9,19]. Recently, the local cohomology and local duality to Notherian connected cochain DG algebras are investigated and they found that the functor can be used to detect the Gorensteinness of a homologically smooth DG algebra [16]. Avramov and Martsinkovsky [2] studied the Tate cohmology theory in the subcategory of modules of finite Gdimension and study the interaction of the absolute, the relative and the Tate cohomology theories. Parallel to the theory of Tate cohomology, they also noted Tate homology. That is for a right R-module M admitting a complete projective for each R-module N and each n ∈ Z. Later, Asadollahi and Salarian [1] made an intensive study of the relative and Tate cohomology of modules of finite Gorenstein injective dimension. Let N be an R-module with finite Gorenstein projective dimension. The Tate torsion functors are defined by means of a complete projective dimension of N : if V → P → N is a complete projective resolution for each i ∈ Z, let Tor [15]. The balance of Tate homology was discussed. Christensen and Jorgensen [5,6] extended Tate homology to complexes. They established a depth formula that holds for every pair of Tate Tor-independent modules over a Gorenstein local ring which subsumes previous generalizations of Auslander's formula.
Furthermore, Asadollahi and Salarian [1] presented a theory of Gorenstein local cohomology theory, using the Gorenstein injective version of the relative cohomology. This is a natural generalization of a notion introduced in the article [13] which generalizes the classical local cohomology. They also introduced the Tate local cohomology and studied properties of these cohomology theories. We pay attention to the dual version of these cohomology theories-Tate local homology and Gorenstein local homology. We will study some properties of them and show that these two variations of local homology are tightly connected to the generalized local homology modules introduced by Nam [

Preliminaries and Basic Facts
Throughout this paper, R is a noetherian commutative ring with non-zero identity. We first review some basic facts. For terminology we follow [2,7].
Gorenstein projective module. [7] An R-module M is said to be Gorenstein projective if there is a Hom R (−, P roj) exact exact sequence · · · → P 1 → P 0 → P 0 → P 1 → · · · of projective modules such that M = Ker(P 0 → P 1 ). The class of all Gorenstein projective modules is denoted by GP.
Complete projective resolution. [2] A complex T is acyclic if it is exact, with modules of T is projective, and Hom R (T, Q) is exact for any projective R-module Q. A complete projective T is a totally acyclic complex of projective R-modules, π is a projective resolution of M , τ is a morphism of complexes and τ n is bijective for n 0. Gorenstein projective dimension. [7] A resolution P → M is called a GP-resolution if P i belongs to GP for all i ∈ Z. A module M has finite Gorenstein projective dimension if there is a Gorenstein projective resolution of M of the form If n is the least with this property then we set Let GP denote the full subcategory of the category of R-modules whose objects are the modules admitting some GP-proper resolution. Every module of finite Gorenstein projective dimension has a GP-proper resolution.

Tate Local Homology
Definition 3.1. Let I be an ideal of the ring R and N be an R-module with Gpd R N < ∞. Consider the complete projective resolution T → P → N of N . For any i ∈ Z, the i-th Tate local homology module of M and N with respect to I is defined by the formula where M is an arbitrary R-module. These modules will be denoted Nam [21] introduced the category of R-modules, denoted by Proof Since N is a finitely generated R-module with Gpd R N < ∞, then we can take the complete projective resolution T → P → N of N such that each P i and T i are finitely generated free. Hence M/I t M ⊗ T is degreewise artinian for all t > 0. It should be noted that the inverse limit lim ← − t is exact on artinian R-modules by [12, 9.1]. Therefore it commutes with homology functor H i and the proof is complete.
Theorem 3.1. Let N be a finitely generated R-module of finite Gorenstein projective dimension. Let 0 → M → M → M → 0 be a short exact sequence of R-modules. If M is artianian, then there exists a long exact sequence of Tate local homology groups Proof By assumption, there exists a complete projective resolution T → P → N of N such that each T i is finitely generated free. So the functor − ⊗ T i is exact. Hence we obtain an exact sequence of complexes  [12, 9.1]. We get the following exact sequence of complexes It induces a long exact sequence of homology groups as desired by Proposition 3.2.
is exact for all t > 0. Hence we get Therefore we obtain the long exact sequence of Tate local homology modules. The proof is complete. Now we list some properties of Tate local homology.  As M/m r M has finite length and N is finitely generated, Tor

Gorenstein Local Homology Modules
The main purpose of this part is to introduce and study Gorenstein local homology.
which is independent (up to isomorphism) of the choice of GP-proper resolution of N by standard facts of homological algebra, will be denoted by GH I i (M, N ) and will be called the ith Gorenstein local homology module of M and N with respect to I.  N ). Especially, if M is an artinian module, then as the proof of Proposition 3.2, for any i ∈ Z, there is an isomorphism where Tor GP i (M/I t M, N ) is the Gorenstein torsion functor computed by the GP-proper resolution of N , that is, Proof There is an exact sequence of flat modules such that M ∼ = Im(F 0 → F −1 ). Set L 0 := M and L i := Im(F i → F i−1 ) for i < 0. Then for each i < 0, the exact sequence yields the following long exact sequence of local homology modules Note that H I i (F ) = 0 for any flat R-module F by [17, (1) It is easy to see that a resolution of Λ Iacyclic R-modules can be used to computed local homology. So if every flat R-module has finite projective dimension, then by [14,Proposition 3.4], every Gorenstein projective Rmodule also is Gorenstein flat. Hence by Lemma 4.1 one can compute local homology modules using a GP-proper . Hence our definition is in fact a generalization of usual local homology functor.
(2) There is a generalization of local homology given by Nam [19]. The following theorem provides a tight connection between Gorenstein, Tate and generalized local homology.
Theorem 4.1. Let N be a finitely generated R-module with Gpd R N = d < ∞. For each artinian R-module M there is an exact sequence So it induces an exact sequence of complexes. Its homology exact sequence has the form Since P → N is a projective resolution, we have The right exactness of M/I t M ⊗ R − yields  Theorem 4.2. Let M and N be finitely generated R-modules such that pd R M < ∞ and N ∈ GP. If every flat R-module has finite projective dimension, then for any i ∈ Z, N ).
Proof There exists a GP-proper exact sequence of R- where P is Gorenstein projective. Hence Gorenstein flat and L ∈ GP. So by Lemma 4.2, we obtain a commutative diagram

Conclusion
In this paper we mainly study some generalizations of local homology as the duality of local cohomology. Firstly, Tate local homology is introduced. Such vanishing properties, artinianness and some exact sequence of Tate local homology modules are obtained. Then we consider Gorenstein local homology modules as Gorenstein version. We present vanishing properties and some exact sequences of Gorenstein local homology models and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. As an application of the exact sequence, we obtain when Gorenstein local homology coincides with generalized local homology. However the vanishing of Tate homology is a sufficient condition implying the depth formula to hold for some modules [4], we may further give a new sufficient condition implying the depth formula to hold for certain modules by vanishing of Tate local homology introduced here.