Local (Co)homology and Čech (Co)complexes with Respect to a Pair of Ideals

Abstract

Let I and J be two ideals of a commutative ring R. We introduce the concepts of the $\check{\mathrm{C}}$ech complex and $\check{\mathrm{C}}$ech cocomplex with respect to $(I,J)$ and investigate their homological properties. In addition, we show that local cohomology and local homology with respect to $(I,J)$ are expressed by the above complexes. Moreover, we provide a proof for the Matlis–Greenless–May equivalence with respect to $(I,J)$, which is an equivalence between the category of derived $(I,J)$-torsion complexes and the category of derived $(I,J)$-completion complexes. As an application, we use local cohomology and the $\check{\mathrm{C}}$ech complex with respect to $(I,J)$ to prove Grothendieck’s local duality theorem for unbounded complexes.

1. Introduction

Local cohomology is an effective tool for studying commutative algebra and algebraic geometry, and it has become a research hotspot. Many scholars have immersed themselves in its study and have made some effort toward its development (see, for instance, [1,2,3,4,5,6,7]). Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring R, and let M be an R-module. Grothendieck [3] introduced the local cohomology module

$${\mathrm{H}}_{\mathfrak{a}}^i\left(M\right)={\mathrm{H}}_{-i}\left(\mathrm{R}{\Gamma }_{\mathfrak{a}}\left(M\right)\right)={\underrightarrow{\lim }}_n{\mathrm{Ext}}_R^i(R/{\mathfrak{a}}^n,M),$$

where the functor $\mathrm{R}{\Gamma }_{\mathfrak{a}}(-)$ is the right-derived functor of the $\mathfrak{a}$-torsion functor ${\Gamma }_{\mathfrak{a}}(-):={\underrightarrow{\lim }}_n{\mathrm{Hom}}_R(R/{\mathfrak{a}}^n,-).$ Corresponding to the $\mathfrak{a}$-torsion functor, the $\mathfrak{a}$-adic completion functor is given by ${\Lambda }_{\mathfrak{a}}(-):={\underleftarrow{\lim }}_n(R/{\mathfrak{a}}^n{\otimes }_R-)$, and the left-derived functor of the $\mathfrak{a}$-adic completion functor ${\Lambda }_{\mathfrak{a}}(-)$ is denoted by $\mathrm{L}{\Lambda }_{\mathfrak{a}}(-)$. These functors have been widely used to study local (co)homology in different disciplines, such as commutative algebra [2,7,8,9], algebraic geometry [1,3,6], category theory [10,11], representation theory [12], topology [13,14,15], and noncommutative algebra [16]. In particular, Porta, Shaul, and Yekutieli [12] used the derived functors $\mathrm{R}{\Gamma }_{\mathfrak{a}}$ and $\mathrm{L}{\Lambda }_{\mathfrak{a}}$ to prove the Matlis–Greenless–May (MGM) equivalence, which establishes an equivalence between the category of derived $\mathfrak{a}$-adically complete complexes and the category of derived $\mathfrak{a}$-torsion complexes, that is,

$$\mathrm{R}{\Gamma }_{\mathfrak{a}}:D{\left(R\right)}_{\mathfrak{a}\text{-}com}\rightleftarrows D{\left(R\right)}_{\mathfrak{a}\text{-}tor}:\mathrm{L}{\Lambda }_{\mathfrak{a}}.$$

Recently, for a commutative unital ring R, the categories of $\mathfrak{a}$-reduced complexes of R-modules and $\mathfrak{a}$-coreduced complexes of R-modules were established in the context of the MGM equivalence in [17].

Grothendieck’s local duality theorem is a very important and fundamental tool in the study of local cohomology theory. Assume that the local ring $(R,\mathfrak{m})$ is a homomorphic image of a Gorenstein local ring $(R^{\prime },{\mathfrak{m}}^{\prime })$ of dimension $n^{\prime }$, and M is a finitely generated R-module. Brodmann and Sharp [2] proved Grothendieck’s local duality theorem as follows

$${\mathrm{H}}_{\mathfrak{m}}^i\left(M\right)\cong {\mathrm{Hom}}_R({\mathrm{Ext}}_{R^{\prime }}^{n^{\prime }-i}(M^{\prime },R^{\prime }),E(R/\mathfrak{m})),$$

where $E(R/\mathfrak{m})$ is a injective hull of $R/\mathfrak{m}$. Later, Iyengar et al. [6] used the $\check{\mathrm{C}}$ech complex to prove the above theorem. Since then, Grothendieck’s local duality theorem has been extended to complexes, and it was considered in the case of unbounded complexes with finitely generated homology by Schenzel and Simon [7].

In 2009, Takahashi et al. [18] introduced the notion of a local cohomology module with respect to $(I,J)$ and discussed its connection with classical local cohomology modules. Let I and J be the ideals of a commutative Noetherian ring R, and let M be an R-module. The $(I,J)$-torsion submodule of M is defined by

$${\Gamma }_{I,J}\left(M\right)=\{x\in M|I^{n}x\subseteq Jx\mathrm{for}\mathrm{some}n\in \mathbb{N}\},$$

which is similar to the classical $\mathfrak{a}$-torsion module, proving that ${\Gamma }_{I,J}\left(M\right)\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\Gamma }_{\mathfrak{a}}\left(M\right)$, where $\tilde{W}(I,J)$ is a special set of ideals $\mathfrak{a}$ of R such that $I^n\subseteq \mathfrak{a}+J$ for some $n\in \mathbb{N}$. If $J=0$, then ${\Gamma }_{I,J}\left(M\right)={\Gamma }_I\left(M\right)$. The authors discussed the relation between the local cohomology modules with respect to $(I,J)$ and the classical local cohomology modules, ${\mathrm{H}}_{I,J}^i\left(M\right)\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\mathrm{H}}_{\mathfrak{a}}^i\left(M\right)$. Subsequently, many authors have started studying local cohomology theory with respect to a pair of ideals. For example, Chu and Wang [19,20] investigated the top local cohomology module ${\mathrm{H}}_{I,J}^{\dim M}\left(M\right)$ being Artinian, Payrovi and Parsa [21,22] showed the Artinianness and finiteness of local cohomology modules with respect to $(I,J)$, and Nguyem [23] studied the attached primes of the top local cohomology module with respect to $(I,J)$ in order to prove the Lichtenbaum–Hartshorne vanishing theorem for local cohomology modules with respect to $(I,J)$. In 2015, Jorge Perez and Tobnon [24] introduced the $(I,J)$-completion submodule of M as

$${\Lambda }_{I,J}\left(M\right)={\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\Lambda }_{\mathfrak{a}}\left(M\right)={\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underleftarrow{\lim }}_{n\in \mathbb{N}}M/{\mathfrak{a}}^{n}M,$$

and considered local homology theory with respect to a pair of ideals. Recently, Li and Yang [25] defined ${\Gamma }_{I,J}\left(X\right)$ and the ${\Lambda }_{I,J}\left(X\right)$ for the R-complex X and introduced the local cohomology complexes $\mathrm{R}{\Gamma }_{I,J}\left(X\right)$ and the local homology complexes $\mathrm{L}{\Lambda }_{I,J}\left(X\right)$. They established the generalized MGM equivalence between the category of derived $(I,J)$-torsion complexes and the category of derived $(I,J)$-completion complexes, that is,

$$\mathrm{R}{\Gamma }_{I,J}:D{\left(R\right)}_{(I,J)\text{-}com}\rightleftarrows D{\left(R\right)}_{(I,J)\text{-}tor}:\mathrm{L}{\Lambda }_{I,J}.$$

In this paper, we introduce the $\check{\mathrm{C}}$ech complex and $\check{\mathrm{C}}$ech cocomplex with respect to $(I,J)$ of a commutative ring R and provide some computations for $\check{\mathrm{C}}$ech cohomology modules and $\check{\mathrm{C}}$ech homology modules. In addition, we show that local cohomology and local homology with respect to $(I,J)$ are expressed by the above complexes and prove the MGM equivalence with respect to $(I,J)$. As an application of local cohomology and the $\check{\mathrm{C}}$ech complex with respect to $(I,J)$, we prove Grothendieck’s local duality theorem for unbounded complexes.

2. Preliminaries

Throughout this paper, we assume that I and J are ideals of a commutative ring R. We fix some necessary notations and recall the main notions and facts. We refer the reader to [7,11,18] and the references therein for more details.

Complexes. X is a complex of R-modules

$$\cdots ⟶X_{n+1}\stackrel{d_{n+1}}{⟶}{X}_n\stackrel{d_n}{⟶}{X}_{n-1}\stackrel{d_{n-1}}{⟶}\cdots$$

such that $d_n{d}_{n+1}=0$. The derived category $D\left(R\right)$ of the category of R-complexes is the category of R-complexes localized at the class of all quasi-isomorphisms. If the homology groups of an R-complex X are finitely generated, then X is called the finitely generated homology. The left- and right-derived functors of $-{\otimes }_R-$ and ${\mathrm{Hom}}_R(-,-)$ are denoted by $-{\otimes }_R^{\mathrm{L}}-$ and $\mathrm{R}{\mathrm{Hom}}_R(-,-)$, respectively.

Čech complex. For a sequence of elements $\underline{x}=x_1,\dots ,x_k$, where $x_i\in R$, let $K\left(\underline{x}\right)$ (resp. $K\left(\underline{x}\right)$) denote the Koszul complex (resp. the Koszul cocomplex). For an R-complex of X, we define

$$K(\underline{x};X)=K\left(\underline{x}\right)\otimes X\mathrm{and}K(\underline{x};X)=\mathrm{Hom}(K\left(\underline{x}\right),X).$$

Denote by ${\mathrm{H}}_i(\underline{x};X)$ (resp. ${\mathrm{H}}^i(\underline{x};X)$) the homology (resp. cohomology) of the corresponding complexes. The $\check{\mathrm{C}}$ech complex on this sequence $\underline{x}$ is defined by ${\check{C}}_{\underline{x}}={\underrightarrow{\lim }}_{t}K\left({\underline{x}}^t\right)$. It is easy to see that ${\check{C}}_{\underline{x}}={\otimes }_{i=1}^k{\check{C}}_{x_i}$, where ${\check{C}}_{x_i}$ is the complex ${\check{C}}_{x_i}=0\to R\to R_{x_i}\to 0)$. The set $S=\{1,x,x^2,\dots \}$ is a multiplicatively closed subset of R, and $R_x=S^{-1}R$ represents the fractions of R with respect to S. For an element $x\in R$, let $S_{x,J}=\{x^n+j\mid n\in \mathbb{N},j\in J\}$ be a multiplicatively closed subset of R. For an R-module M, $M_{x,J}$ denotes the module of fractions of M with respect to $S_{x,J}$, i.e., $M_{x,J}=S_{x,J}^{-1}M.$ In particular, if $M=R$, then $R_{x,J}=S_{x,J}^{-1}R.$ For an element $x\in R$, the complex ${\check{C}}_{x,J}$ is defined as

$${\check{C}}_{x,J}=\left(0\to R\to R_{x,J}\to 0\right),$$

where R is sited in the 0th degree and $R_{x,J}$ in the 1st degree in the complex. For a sequence $\underline{x}=x_1,\dots ,x_k$ of elements in R, we define a complex ${\check{C}}_{\underline{x},J}$ as follows:

$${\check{C}}_{\underline{x},J}={\otimes }_{i=1}^k{\check{C}}_{x_i,J}.$$

In particular, if $J=0$, then ${\check{C}}_{\underline{x},J}={\check{C}}_{\underline{x}}$.

Fact 1.

(1) Li and Yang ([25], Lemma 2.2) proved the isomorphism ${\check{C}}_{\underline{x},J}\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\check{C}}_{\mathfrak{a}}$, where $\mathfrak{a}$ is generated by the sequence $\underline{\mathfrak{a}}={\mathfrak{a}}_1,\dots ,{\mathfrak{a}}_l$ of elements in R.

(2) In [26], the author constructed a quasi-isomorphism ${\check{L}}_{\underline{x}}\stackrel{\sim }{\to }{\check{C}}_{\underline{x}}$ to compute the complex $\mathrm{RHom}({\check{C}}_{\underline{x}},X)$, where ${\check{L}}_{\underline{x}}$ is a bounded complex of free R-modules with ${\check{L}}_{\underline{x}}^i=0$ for $i<0$ or $i>k$.

Observation 1.

For $\underline{x}=x_1,\dots ,x_k$, let $I=\underline{x}R$ and J be ideals of R, and $\underline{\mathfrak{a}}={\mathfrak{a}}_1,\dots ,{\mathfrak{a}}_l$ be generators of $\mathfrak{a}\in R$. Then, the following statements hold:

(1) According to Fact 1$\left(1\right)$ and the definition of the $\check{\mathrm{C}}$ech complex, we have

$${\check{C}}_{\underline{x},J}\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\check{C}}_{\mathfrak{a}}\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_{t}K\left({\underline{\mathfrak{a}}}^t\right).$$

(2) We also have

$$\begin{array}{cc}\hfill \mathrm{Hom}({\check{C}}_{\underline{x},J},X)& \cong \mathrm{Hom}({\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\check{C}}_{\mathfrak{a}},X)\hfill \\ & \cong {\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}\mathrm{Hom}({\check{C}}_{\mathfrak{a}},X)\hfill \\ & \cong {\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}\mathrm{Hom}({\underrightarrow{\lim }}_{t}K\left({\underline{\mathfrak{a}}}^t\right),X)\hfill \\ & \cong {\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underleftarrow{\lim }}_t\mathrm{Hom}(K\left({\underline{\mathfrak{a}}}^t\right),X)\hfill \\ & \cong {\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underleftarrow{\lim }}_{t}K({\underline{\mathfrak{a}}}^t;X),\hfill \end{array}$$

where the first isomorphism comes from Fact 1$\left(1\right)$, the third isomorphism comes from the definition of the $\check{\mathrm{C}}$ech complex, and the last isomorphism follows from the self-duality of the Koszul complex.

(3) By Fact 1$\left(2\right)$, there exists a quasi-isomorphism ${\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\check{L}}_{\mathfrak{a}}\stackrel{\sim }{\to }{\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\check{C}}_{\mathfrak{a}}$. Putting ${\check{L}}_{\underline{x},J}:={\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\check{L}}_{\mathfrak{a}}$, we can obtain that ${\check{L}}_{\underline{x},J}\stackrel{\sim }{\to }{\check{C}}_{\underline{x},J}$ is a bounded free resolution of ${\check{C}}_{\underline{x},J}$.

3. $\check{\mathrm{C}}$ech (Co)homology with Respect to (I,J)

In this section, we introduce $\check{\mathrm{C}}$ech complexes and $\check{\mathrm{C}}$ech cocomplexes with respect to a pair of ideals and study their homological properties.

Definition 1.

Let $\underline{x}=x_1,\dots ,x_k$ be a sequence of elements in R, and let J be an ideal of R. For an R-complex X, the complex ${\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X$ is called the $\check{\mathrm{C}}$ech complex with respect to a pair of ideals. Due to the flatness of the $\check{\mathrm{C}}$ech complex ${\check{C}}_{\underline{x},J}$, we have the following two isomorphisms

$${\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\cong {\check{C}}_{\underline{x},J}{\otimes }_{R}X\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_{t}K({\underline{\mathfrak{a}}}^t;X).$$

The complex $\mathrm{RHom}({\check{C}}_{\underline{x},J},X)$ is called the $\check{\mathrm{C}}$ech cocomplex with respect to a pair of ideals, which satisfies the following isomorphic relations

$$\mathrm{Hom}({\check{C}}_{\underline{x},J},X^{\prime })\simeq \mathrm{Hom}({\check{L}}_{\underline{x},J},X^{\prime })\simeq \mathrm{Hom}({\check{L}}_{\underline{x},J},X^{\prime }),$$

where $X\stackrel{\simeq }{\to }{X}^{\prime }$ is a semi-injective resolution of X.

Remark 1.

Note that the $\check{\mathrm{C}}$ech complex ${\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X$ and the $\check{\mathrm{C}}$ech cocomplex $\mathrm{RHom}({\check{C}}_{\underline{x},J},X)$ can be thought of in terms of their homology. So, we regard the modules ${\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)$ as $\check{\mathrm{C}}$ech cohomology modules and the modules ${\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\right)$ as $\check{\mathrm{C}}$ech homology modules. Therefore, we have an adjointness formula

$$\mathrm{RHom}({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X,X^{\prime })\cong \mathrm{RHom}(X,\mathrm{RHom}({\check{C}}_{\underline{x},J},X^{\prime })).$$

Let $E=\prod E_R(R/\mathfrak{m})$ be an injective cogenerator of the category of R-modules. We denote a general Matlis duality functor by ${(-)}^{\vee }:={\mathrm{Hom}}_R(-,E)$. By setting $X^{\prime }=E$ in the above, we obtain the isomorphism

$${\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)}^{\vee }\simeq \mathrm{RHom}({\check{C}}_{\underline{x},J},X^{\vee }),$$

which induces isomorphisms of the modules

$${\left({\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)\right)}^{\vee }\cong {\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X^{\vee })\right),$$

for all $i\in \mathbb{Z}$.

The following theorem discusses some homological properties of the $\check{\mathrm{C}}$ech cohomology and $\check{\mathrm{C}}$ech homology with respect to a pair of ideals.

Theorem 1.

Let $\underline{x}=x_1,\dots ,x_k$; the ideals $I=\underline{x}R$ and $J\in R$; and $\underline{\mathfrak{a}}={\mathfrak{a}}_1,\dots ,{\mathfrak{a}}_l$ be generators of $\mathfrak{a}\in R.$ Then, for an R-complex $X,$ the following statements hold:

$\left(1\right)$ ${\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_{R}X\right)\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_t{\mathrm{H}}^i({\underline{\mathfrak{a}}}^t;X);$

$\left(2\right)$ If ${\mathrm{Supp}}_R{\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_{R}X\right)\subseteq W(I,J),$ then ${\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_{R}X\right)$ is an $(I,J)$-torsion R-module, and $\mathrm{Hom}(R/I,{\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_{R}X\right))\ne 0$ when ${\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_{R}X\right)\ne 0;$

$\left(3\right)$ ${\mathrm{Supp}}_R{\check{C}}_{\underline{x},J}:={\cup }_i{\mathrm{Supp}}_R{\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}\right)=W(I,J);$

$\left(4\right)$ ${\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\right)\cong {\mathrm{H}}_i\left(\mathrm{Hom}({\check{C}}_{\underline{x},J},X^{\prime })\right)\cong {\mathrm{H}}_i\left(\mathrm{Hom}({\check{L}}_{\underline{x},J},X)\right);$

$\left(5\right)$$0\to \underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{{\underleftarrow{\lim }}^1}{\mathrm{H}}_{i+1}({\underline{\mathfrak{a}}}^t;X)\to {\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\right)\to \underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{\underleftarrow{\lim }}{\mathrm{H}}_i({\underline{\mathfrak{a}}}^t;X)\to 0$ is a short exact sequence;

$\left(6\right)$ If ${\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\right)\ne 0,$ then $R/\mathfrak{a}{\otimes }_R{\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\right)\ne 0.$

Proof. 

$\left(1\right)$ According to the definition of the $\check{\mathrm{C}}$ech complex, we have

$${\check{C}}_{\underline{x},J}{\otimes }_{R}X\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_{t}K({\underline{\mathfrak{a}}}^t;X),$$

which, together with the exactness of $\underrightarrow{\lim }$, implies the following isomorphisms

$$\begin{array}{cc}\hfill {\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_{R}X\right)& \cong {\mathrm{H}}^i\left({\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_{t}K({\underline{\mathfrak{a}}}^t;X)\right)\hfill \\ & \cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_t{\mathrm{H}}^i({\underline{\mathfrak{a}}}^t;X).\hfill \end{array}$$

$\left(2\right)$ Let $\mathfrak{p}\notin W(I,J)$; we find that ${\mathrm{H}}^i{\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)}_{\mathfrak{p}}=0$. Note that

$${\check{C}}_{\underline{x},J}{\otimes }_{R}X\otimes R_{\mathfrak{p}}\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_{t}K({\underline{\mathfrak{a}}}^t;X)\otimes R_{\mathfrak{p}}$$

is an exact complex. Hence, by ([18], Proposition 1.7), ${\mathrm{Supp}}_R{\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)\subseteq W(I,J)$ if and only if ${\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)$ is an $(I,J)$-torsion R-module.

If $\mathrm{Hom}(R/I,{\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right))=0$, then $0=\left(0{:}_{{\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)}I\right)=\left(0{:}_{{\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)}{I}^t\right)$, and ${\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)={\Gamma }_{I,J}\left({\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)\right)=\bigcup \left(0{:}_{{\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\right)}{I}^t\right)=0$. This produces a contradiction.

$\left(3\right)$ For all $\mathfrak{p}\in W(I,J)$, we have $I{R}_{\mathfrak{p}}\ne R_{\mathfrak{p}}$. Hence, the complex ${\check{C}}_{\underline{x},J}\otimes R_{\mathfrak{p}}$ is not exact. This together with $\left(2\right)$ means that the result holds.

$\left(4\right)$ The complex $\mathrm{RHom}({\check{C}}_{\underline{x},J},X)$ is expressed by each of the following quasi-isomorphic complexes

$$\mathrm{Hom}({\check{C}}_{\underline{x},J},X^{\prime })\stackrel{\simeq }{\to }\mathrm{Hom}({\check{L}}_{\underline{x},J},X^{\prime })\stackrel{\simeq }{\leftarrow }\mathrm{Hom}({\check{L}}_{\underline{x},J},X),$$

where $X\stackrel{\simeq }{\to }{X}^{\prime }$ is a semi-injective resolution of X.

$\left(5\right)$ By the isomorphism ${\check{C}}_{\underline{x},J}\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_{t}K\left({\underline{\mathfrak{a}}}^t\right)$, we have an exact sequence

$$0\to \underset{t}{\otimes }\left(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right)\right)\to \underset{t}{\otimes }\left(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right)\right)\to {\check{C}}_{\underline{x},J}\to 0.$$

Let $X\stackrel{\simeq }{\to }{X}^{\prime }$ be a semi-injective resolution of X. Then, we obtain the short exact sequence

$$0\to \mathrm{Hom}({\check{C}}_{\underline{x},J},X^{\prime })\to \prod _t\mathrm{Hom}(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right),X^{\prime })\to \prod _t\mathrm{Hom}(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right),X^{\prime })\to 0,$$

which, together with the kernels and cokernels of the following long exact homology sequence

$$\cdots \to \prod _t{\mathrm{H}}_{i+1}(\mathrm{Hom}(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right),X^{\prime }))\to {\mathrm{H}}_i(\mathrm{Hom}({\check{C}}_{\underline{x},J},X^{\prime }))\to$$

$$\prod _t{\mathrm{H}}_i(\mathrm{Hom}(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right),X^{\prime }))\to \cdots$$

gives the short exact sequence in $\left(5\right).$

$\left(6\right)$ Assume ${\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\right)\ne 0$. It follows from $\left(5\right)$ that at least one of the two modules $\underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{{\underleftarrow{\lim }}^1}{\mathrm{H}}_{i+1}({\underline{\mathfrak{a}}}^t;X)$ and $\underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{\underleftarrow{\lim }}{\mathrm{H}}_i({\underline{\mathfrak{a}}}^t;X)$ is not zero. However, for all $j\in \mathbb{Z}$, there is an exact sequence

$$0\to \underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{\underleftarrow{\lim }}{\mathrm{H}}_j({\underline{\mathfrak{a}}}^t;X)\to \prod _t{\mathrm{H}}_j(\mathrm{Hom}(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right),X))\to \prod _t{\mathrm{H}}_j(\mathrm{Hom}(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right),X))\to$$

$$\underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{{\underleftarrow{\lim }}^1}{\mathrm{H}}_j({\underline{\mathfrak{a}}}^t;X)\to 0.$$

Note that the modules ${\mathrm{H}}_j(\mathrm{Hom}(\underset{\mathfrak{a}\in \tilde{W}}{\underrightarrow{\lim }}K\left({\underline{\mathfrak{a}}}^t\right),X))$ can be annihilated by a power of $\mathfrak{a}$, and there is a sequence

$$R/\mathfrak{a}\otimes \underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{{\underleftarrow{\lim }}^1}{\mathrm{H}}_{i+1}({\underline{\mathfrak{a}}}^t;X)\to R/\mathfrak{a}\otimes {\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\right)\to R/\mathfrak{a}\otimes \underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{\underleftarrow{\lim }}{\mathrm{H}}_i({\underline{\mathfrak{a}}}^t;X)\to 0.$$

Therefore, it follows from ([7], 2.2.11) that we find that $R/\mathfrak{a}{\otimes }_R\underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{{\underleftarrow{\lim }}^1}{\mathrm{H}}_{i+1}({\underline{\mathfrak{a}}}^t;X)\ne 0$ if $\underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{{\underleftarrow{\lim }}^1}{\mathrm{H}}_{i+1}({\underline{\mathfrak{a}}}^t;X)\ne 0$, and $R/\mathfrak{a}{\otimes }_R\underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{\underleftarrow{\lim }}{\mathrm{H}}_i({\underline{\mathfrak{a}}}^t;X)\ne 0$ if $\underset{\mathfrak{a}\in \tilde{W}}{\underleftarrow{\lim }}\underset{t}{\underleftarrow{\lim }}{\mathrm{H}}_i({\underline{\mathfrak{a}}}^t;X)\ne 0$. This means that $\left(6\right)$ holds. □

Next, we investigate when $\mathrm{RHom}({\check{C}}_{\underline{x},J},-)$ and ${\check{C}}_{\underline{x},J}{\otimes }_R-$ vanish. Recall the class ${\mathcal{T}}_{\mathfrak{a}}$ of complexes introduced in ([7], 5.1.4), and there exist some equivalent characterizations as follows

$${\mathcal{T}}_{\mathfrak{a}}=\left\{X\right|\mathrm{T}\text{-}\mathrm{codp}(\mathfrak{a},X)=\infty \}=\left\{X\right|R/\mathfrak{a}{\otimes }_{R}X\simeq 0\}=\left\{X\right|\mathrm{RHom}(R/\mathfrak{a},X)\simeq 0\},$$

where $\mathrm{T}\text{-}\mathrm{codp}(\mathfrak{a},X):=inf\{i\in \mathbb{Z}|{\mathrm{H}}_i(R/\mathfrak{a}{\otimes }_{R}F)\ne 0\}=\infty$, $F\stackrel{\simeq }{\to }X$ is a semi-flat resolution of X.

Proposition 1.

Let $\underline{x}=x_1,\dots ,x_k$; the ideals $I=\underline{x}R$ and $J\in R$; and $\mathfrak{a}\in \tilde{W}(I,J)$ be generated by the sequence $\underline{\mathfrak{a}}={\mathfrak{a}}_1,\dots ,{\mathfrak{a}}_l.$ Then, for any R-complex $X,$ the following conditions are equivalent:

$\left(1\right)$ ${\check{C}}_{\underline{x},J}{\otimes }_{R}X$ is exact;

$\left(2\right)$ $X\in {\mathcal{T}}_{\mathfrak{a}};$

$\left(3\right)$ $\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\simeq 0;$

$\left(4\right)$ $\mathrm{Hom}({\check{L}}_{\underline{x},J},X)$ is exact.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ We have $R/\mathfrak{a}{\otimes }_{R}X\simeq R/\mathfrak{a}{\otimes }_R{\check{C}}_{\underline{x},J}{\otimes }_{R}X\simeq 0.$

$\left(2\right)\Rightarrow \left(1\right)$ Let $X\in {\mathcal{T}}_{\mathfrak{a}}$. It follows from ([7], 5.3.5) that ${\mathrm{H}}^i({\underline{\mathfrak{a}}}^t;X)=0={\mathrm{H}}_i({\underline{\mathfrak{a}}}^t;X)$. Hence,

$${\mathrm{H}}^i\left({\check{C}}_{\underline{x},J}{\otimes }_{R}X\right)\cong {\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}{\underrightarrow{\lim }}_t{\mathrm{H}}^i({\underline{\mathfrak{a}}}^t;X)=0.$$

$\left(2\right)\Rightarrow \left(3\right)$ According to Conclusion (5) of Theorem 1, we have ${\mathrm{H}}_i\left(\mathrm{RHom}({\check{C}}_{\underline{x},J},X)\right)=0.$

$\left(3\right)\Rightarrow \left(2\right)$ We have the following isomorphisms

$$\mathrm{RHom}(R/\mathfrak{a},X)\cong \mathrm{RHom}(R/\mathfrak{a}{\otimes }_R{\check{C}}_{\underline{x},J},X)\cong \mathrm{RHom}(R/\mathfrak{a},\mathrm{RHom}({\check{C}}_{\underline{x},J},X))=0.$$

$\left(3\right)\iff \left(4\right)$ Since ${\check{L}}_{\underline{x},J}\stackrel{\simeq }{\to }{\check{C}}_{\underline{x},J}$ is a free resolution of ${\check{C}}_{\underline{x},J}$, $\mathrm{RHom}({\check{C}}_{\underline{x},J},X)$ can be expressed as $\mathrm{Hom}({\check{L}}_{\underline{x},J},X)$ in $D\left(R\right)$. The equivalence is clearly established. □

4. Local (Co)homology with Respect to (I,J)

In this section, we show that local cohomology and local homology with respect to $(I,J)$ are expressed by the $\check{\mathrm{C}}$ech complex and the $\check{\mathrm{C}}$ech cocomplex. And then, we provide a proof for the MGM equivalence with respect to $(I,J)$, which is an equivalence between the category of derived $(I,J)$-torsion complexes and the category of derived $(I,J)$-completion complexes.

Definition 2.

(1) The functor ${\Gamma }_{I,J}$ has a right-derived functor (local cohomology functor)

$$\mathrm{R}{\Gamma }_{I,J}(-)={\underrightarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}\mathrm{R}{\Gamma }_{\mathfrak{a}}(-):D\left(R\right)\to D\left(R\right).$$

Let X be an R-complex. According to ([25], 3.2), we show that

$$\mathrm{R}{\Gamma }_{I,J}\left(X\right)\cong {\check{C}}_{\underline{x},J}{\otimes }_R^{\mathrm{L}}X\cong {\check{C}}_{\underline{x},J}{\otimes }_{R}X.$$

In particular, if $X=R$, then $\mathrm{R}{\Gamma }_{I,J}\left(R\right)\cong {\check{C}}_{\underline{x},J}{\otimes }_{R}R\cong {\check{C}}_{\underline{x},J}$. For an integer n, ${\mathrm{H}}_{I,J}^n(-)$ denotes the nth cohomology of $\mathrm{R}{\Gamma }_{I,J}(-)$.

(2) The functor ${\Lambda }_{I,J}$ has a left-derived functor (local homology functor)

$$\mathrm{L}{\Lambda }_{I,J}(-)={\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}\mathrm{L}{\Lambda }_{\mathfrak{a}}(-):D\left(R\right)\to D\left(R\right).$$

Let X be an R-complex. Then, we have the following isomorphisms

$$\begin{array}{cc}\hfill \mathrm{L}{\Lambda }_{I,J}\left(X\right)& \cong {\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}\mathrm{L}{\Lambda }_{\mathfrak{a}}\left(X\right)\hfill \\ & \cong {\underleftarrow{\lim }}_{\mathfrak{a}\in \tilde{W}(I,J)}\mathrm{RHom}(\mathrm{R}{\Gamma }_{\mathfrak{a}}\left(R\right),X)\hfill \\ & \cong \mathrm{RHom}(\mathrm{R}{\Gamma }_{I,J}\left(R\right),X)\hfill \\ & \cong \mathrm{RHom}({\check{C}}_{\underline{x},J},X)\hfill \\ & \cong \mathrm{Hom}({\check{L}}_{\underline{x},J},X).\hfill \end{array}$$

Lemma 1.

Let $\underline{x}=x_1,\dots ,x_k$, and the ideals $I=\underline{x}R$ and $J\in R.$ Then, for an R-complex X:

$\left(1\right)$ $\mathrm{Hom}({\check{L}}_{\underline{x},J},{\check{C}}_{\underline{x},J}{\otimes }_{R}X)\simeq \mathrm{Hom}({\check{L}}_{\underline{x},J},X);$

$\left(2\right)$ ${\check{C}}_{\underline{x},J}{\otimes }_{R}X\simeq {\check{C}}_{\underline{x},J}{\otimes }_R\mathrm{Hom}({\check{L}}_{\underline{x},J},X).$

Proof. 

$\left(1\right)$ Let ${\check{D}}_{\underline{x},J}$ be the kernel of the morphism ${\check{C}}_{\underline{x},J}\to R$. By applying the functor $\mathrm{Hom}({\check{L}}_{\underline{x},J},-)$ to the following short exact sequence of complexes

$$0\to {\check{D}}_{\underline{x},J}{\otimes }_{R}X\to {\check{C}}_{\underline{x},J}{\otimes }_{R}X\to X\to 0,$$

we can obtain the short exact sequence

$$0\to \mathrm{Hom}({\check{L}}_{\underline{x},J},{\check{D}}_{\underline{x},J}{\otimes }_{R}X)\to \mathrm{Hom}({\check{L}}_{\underline{x},J},{\check{C}}_{\underline{x},J}{\otimes }_{R}X)\to \mathrm{Hom}({\check{L}}_{\underline{x},J},X)\to 0.$$

Therefore, it is sufficient to show that the first complex in the previous sequence is exact. There exists a short exact sequence $0\to R/\mathfrak{a}{\otimes }_R{\check{D}}_{\underline{x},J}\to R/\mathfrak{a}{\otimes }_R{\check{C}}_{\underline{x},J}\stackrel{\simeq }{\to }R/\mathfrak{a}\to 0$ and a quasi-isomorphism ${\check{D}}_{\underline{x},J}{\otimes }_{R}F\stackrel{\simeq }{\to }{\check{D}}_{\underline{x},J}{\otimes }_{R}X$ such that $R/\mathfrak{a}{\otimes }_R{\check{D}}_{\underline{x},J}{\otimes }_{R}F\simeq 0$, where $F\stackrel{\simeq }{\to }X$ is a semi-flat resolution of X. And we further obtain ${\check{D}}_{\underline{x},J}{\otimes }_{R}X\in {\mathcal{T}}_{\mathfrak{a}}$. Thus, by Proposition 1, $\mathrm{Hom}({\check{L}}_{\underline{x},J},{\check{D}}_{\underline{x},J}{\otimes }_{R}X)$ is exact.

$\left(2\right)$ Let ${\check{K}}_{\underline{x},J}$ be the kernel of the morphism ${\check{L}}_{\underline{x},J}\to R$, which is a sequence of bounded free complexes and produces a short exact sequence

$$0\to X\to \mathrm{Hom}({\check{L}}_{\underline{x},J},X)\to \mathrm{Hom}({\check{K}}_{\underline{x},J},X)\to 0.$$

By applying ${\check{C}}_{\underline{x},J}{\otimes }_R-$ to the above sequence, we can obtain the following short exact sequence

$$0\to {\check{C}}_{\underline{x},J}{\otimes }_{R}X\to {\check{C}}_{\underline{x},J}{\otimes }_R\mathrm{Hom}({\check{L}}_{\underline{x},J},X)\to {\check{C}}_{\underline{x},J}{\otimes }_R\mathrm{Hom}({\check{K}}_{\underline{x},J},X)\to 0.$$

We also need to prove that the last complex in the sequence is exact. It is easy to see that the quasi-isomorphism ${\check{K}}_{\underline{x},J}\to {\check{D}}_{\underline{x},J}$, and let $X\to X^{\prime }$ be a semi-injective resolution. It follows from $R/\mathfrak{a}{\otimes }_R{\check{D}}_{\underline{x},J}\cong 0$ that $\mathrm{Hom}(R/\mathfrak{a},\mathrm{Hom}({\check{D}}_{\underline{x},J},X^{\prime }))\cong \mathrm{Hom}(R/\mathfrak{a}{\otimes }_R{\check{D}}_{\underline{x},J},X^{\prime })$. Then, we have

$$\mathrm{Hom}({\check{D}}_{\underline{x},J},X)\stackrel{\simeq }{\to }\mathrm{Hom}({\check{K}}_{\underline{x},J},X^{\prime })\in {\mathcal{T}}_{\mathfrak{a}}.$$

Which, together with Proposition 1, implies that the complex ${\check{C}}_{\underline{x},J}{\otimes }_R\mathrm{Hom}({\check{K}}_{\underline{x},J},X)$ is exact. □

Remark 2.

As ${\check{L}}_{\underline{x},J}\stackrel{\simeq }{\to }{\check{C}}_{\underline{x},J}$ is a free resolution of ${\check{C}}_{\underline{x},J}$, we have the two following quasi-isomorphisms: $\mathrm{Hom}({\check{L}}_{\underline{x},J},{\check{L}}_{\underline{x},J}{\otimes }_{R}X)\simeq \mathrm{Hom}({\check{L}}_{\underline{x},J},X)$ and ${\check{L}}_{\underline{x},J}{\otimes }_{R}X\simeq {\check{L}}_{\underline{x},J}{\otimes }_R\mathrm{Hom}({\check{L}}_{\underline{x},J},X)$.

Lemma 2.

Let $\underline{x}=x_1,\dots ,x_k,$ and the ideals $I=\underline{x}R$ and $J\in R.$ Then, for an R-complex X, we have the following statements:

$\left(1\right)$ $\mathrm{R}{\Gamma }_{I,J}\left(\mathrm{L}{\Lambda }_{I,J}\left(X\right)\right)\simeq \mathrm{R}{\Gamma }_{I,J}\left(X\right),$

$\left(2\right)$ $\mathrm{L}{\Lambda }_{I,J}\left(\mathrm{R}{\Gamma }_{I,J}\left(X\right)\right)\simeq \mathrm{L}{\Lambda }_{I,J}\left(X\right).$

Proof. 

(1) According to the definition of the derived functors and Lemma 1, we can easily obtain

$$\begin{array}{cc}\hfill \mathrm{R}{\Gamma }_{I,J}\left(\mathrm{L}{\Lambda }_{I,J}\left(X\right)\right)& \simeq {\check{C}}_{\underline{x},J}{\otimes }_R\mathrm{Hom}({\check{L}}_{\underline{x},J},X)\hfill \\ & \simeq {\check{L}}_{\underline{x},J}{\otimes }_R\mathrm{Hom}({\check{L}}_{\underline{x},J},X)\hfill \\ & \simeq {\check{L}}_{\underline{x},J}{\otimes }_{R}X\simeq \mathrm{R}{\Gamma }_{I,J}\left(X\right),\hfill \end{array}$$

as desired.

(2) According to the definition of the derived functors and Lemma 1, we can easily obtain $\mathrm{L}{\Lambda }_{I,J}\left(\mathrm{R}{\Gamma }_{I,J}\left(X\right)\right)\simeq \mathrm{Hom}({\check{L}}_{\underline{x},J},{\check{L}}_{\underline{x},J}{\otimes }_{R}X)\simeq \mathrm{Hom}({\check{L}}_{\underline{x},J},X)\simeq \mathrm{L}{\Lambda }_{I,J}\left(X\right)$, as desired. □

Next, we define the two derived complexes and prove the MGM equivalence with respect to $(I,J)$.

Definition 3.

The R-complex X is called derived $(I,J)$-torsion if the morphism $\mathrm{R}{\Gamma }_{I,J}\left(X\right)\to X$ is an isomorphism. The R-complex X is called derived $(I,J)$-complete if the morphism $X\to \mathrm{L}{\Lambda }_{I,J}\left(X\right)$ is an isomorphism. We denote by $D{\left(R\right)}_{(I,J)\text{-}tor}$ and $D{\left(R\right)}_{(I,J)\text{-}com}$ the full subcategories of $D\left(R\right)$, representing categories that are derived $(I,J)$-torsion complexes and derived $(I,J)$-complete complexes, respectively.

Theorem 2.

Let $\underline{x}=x_1,\dots ,x_k,$ and the ideals $I=\underline{x}R$ and $J\in R.$ Then, for an R-complex $X,$ the functors $\mathrm{R}{\Gamma }_{I,J}:D{\left(R\right)}_{(I,J)\text{-}com}\rightleftarrows D{\left(R\right)}_{(I,J)\text{-}tor}:\mathrm{L}{\Lambda }_{I,J}$ form an equivalence.

Proof. 

By Lemma 2(1), there are functorial isomorphisms

$$X\cong \mathrm{R}{\Gamma }_{I,J}\left(X\right)\cong \mathrm{R}{\Gamma }_{I,J}\left(\mathrm{L}{\Lambda }_{I,J}\left(X\right)\right),$$

for $X\in D{\left(R\right)}_{(I,J)\text{-}tor}$. It follows from Lemma 2(2) that there are functorial isomorphisms

$$Y\cong \mathrm{L}{\Lambda }_{I,J}\left(X^{\prime }\right)\cong \mathrm{L}{\Lambda }_{I,J}\left(\mathrm{R}{\Gamma }_{I,J}\left(X^{\prime }\right)\right),$$

for $X^{\prime }\in D{\left(R\right)}_{(I,J)\text{-}com}$. These isomorphisms establish the desired equivalence. □

5. Application

In this section, we use local cohomology and the $\check{\mathrm{C}}$ech complex with respect to $(I,J)$ to prove Grothendieck’s local duality theorem for unbounded complexes. The following definition comes from [7].

Definition 4.

Let R be a Noetherian ring, and let Z be a finitely generated homology-bounded complex of injective R-modules. We call Z a dualizing complex if the natural morphism

$$X\to {\mathrm{Hom}}_R({\mathrm{Hom}}_R(X,Z),Z)$$

is a quasi-isomorphism for any R-complex X with finitely generated homology. In particular, assume that R is a local Noetherian ring. Then, a dualizing complex Z is said to be normalized if $\sup \left(Z\right)=\dim \left(R\right)$ , where $\sup \left(Z\right)$ denotes the supremum of non-zero homology groups of Z.

Lemma 3.

Let $\underline{x}=x_1,\dots ,x_k$ be a sequence of elements in a Noetherian ring R. Consider the ideals $I=\underline{x}R$ and $J\in R,$ and let M and N be two bounded complexes of injective R-modules. Assume that the homology group of M is finitely generated. Then, the following evaluation morphisms

$${\mathrm{Hom}}_R(M,N){\otimes }_R{\check{C}}_{\underline{x},J}\to {\mathrm{Hom}}_R(M,N{\otimes }_R{\check{C}}_{\underline{x},J})$$

and

$${\mathrm{Hom}}_R(M,N){\otimes }_R{\check{L}}_{\underline{x},J}\to {\mathrm{Hom}}_R(M,N{\otimes }_R{\check{L}}_{\underline{x},J})$$

are quasi-isomorphisms of bounded complexes of flat R-modules.

Proof. 

It follows from R being a Noetherian ring that $N{\otimes }_R{\check{C}}_{\underline{x},J}$ and $N{\otimes }_R{\check{L}}_{\underline{x},J}$ are bounded complexes of injective R-modules. There exists a semi-free resolution $F\stackrel{\simeq }{\to }M$, where F is a right-bounded complex of finitely generated R-modules. Thie yields a commutative diagram

$$\begin{array}{ccc}{\mathrm{Hom}}_R(M,N){\otimes }_R{\check{C}}_{\underline{x},J}& \to & {\mathrm{Hom}}_R(M,N{\otimes }_R{\check{C}}_{\underline{x},J})\\ \downarrow & & \downarrow \\ {\mathrm{Hom}}_R(F,N){\otimes }_R{\check{C}}_{\underline{x},J}& \to & {\mathrm{Hom}}_R(F,N{\otimes }_R{\check{C}}_{\underline{x},J}),\end{array}$$

where the horizontal arrows are given by the evaluation. In the above diagram, since N and $N{\otimes }_R{\check{C}}_{\underline{x},J}$ are bounded complexes of injective R-modules, the vertical arrows are quasi-isomorphisms. Note that the bottom horizontal arrow is an isomorphism by ([7], 11.1.2); hence, we have the quasi-isomorphisms

$${\mathrm{Hom}}_R(M,N){\otimes }_R{\check{C}}_{\underline{x},J}\to {\mathrm{Hom}}_R(M,N{\otimes }_R{\check{C}}_{\underline{x},J}).$$

□

Lemma 4.

Let R be a Noetherian ring admitting a dualizing complex Z; $\underline{x}=x_1,\dots ,x_k$ be a sequence of elements of R; and the ideals $I=\underline{x}R$ and $J\in R.$ Suppose that X is an R-complex. Then, there exists the following morphism

$${\check{C}}_{\underline{x},J}{\otimes }_{R}X\to {\mathrm{Hom}}_R({\mathrm{Hom}}_R(X,Z),{\check{C}}_{\underline{x},J}{\otimes }_{R}Z).$$

It is a quasi-isomorphism if the homology groups of the R-complex X are finitely generated. It induces a quasi-isomorphism

$$\mathrm{R}{\Gamma }_{I,J}\left(X\right)\simeq {\mathrm{RHom}}_R({\mathrm{RHom}}_R(X,Z),\mathrm{R}{\Gamma }_{I,J}\left(Z\right)).$$

Moreover, $\mathrm{R}{\Gamma }_{I,J}\left(X\right)$ is expressed by the complex ${\mathrm{Hom}}_R({\mathrm{Hom}}_R(X,Z),{\Gamma }_{I,J}\left(Z\right)).$

Proof. 

It follows from the definition of a dualizing complex that there exists a quasi-isomorphism $R\to {\mathrm{Hom}}_R(Z,Z)$ between bounded complexes of flat modules, which, together with Lemma 3, induces quasi-isomorphisms of bounded flat complexes

$${\check{C}}_{\underline{x},J}\to {\mathrm{Hom}}_R(Z,Z){\otimes }_R{\check{C}}_{\underline{x},J}\to {\mathrm{Hom}}_R(Z,Z{\otimes }_R{\check{C}}_{\underline{x},J}).$$

By applying $X{\otimes }_R-$ to the above sequence, we can obtain the quasi-isomorphism

$$X{\otimes }_R{\check{C}}_{\underline{x},J}\to X{\otimes }_R{\mathrm{Hom}}_R(Z,Z{\otimes }_R{\check{C}}_{\underline{x},J}).$$

And we also have the evaluation morphism

$$X{\otimes }_R{\mathrm{Hom}}_R(Z,Z{\otimes }_R{\check{C}}_{\underline{x},J})\to {\mathrm{Hom}}_R({\mathrm{Hom}}_R(X,Z),Z{\otimes }_R{\check{C}}_{\underline{x},J}).$$

It is a quasi-isomorphism when the homology groups of the R-complex X are finitely generated, as established in ([7], 11.1.5). As the complex $Z{\otimes }_R{\check{C}}_{\underline{x},J}$ is a bounded complex of injective R-modules, by composition, the first statement is obtained.

Since $\mathrm{R}{\Gamma }_{I,J}\left(X\right)\cong {\check{C}}_{\underline{x},J}{\otimes }_{R}X$, ${\check{C}}_{\underline{x},J}{\otimes }_{R}Z$ and ${\Gamma }_{I,J}\left(Z\right)$ are bounded complexes of injective R-modules, and the last statement is followed by the first. □

Dually, the local homology for R-complexes also yields similar results to the local cohomology for R-complexes.

Proposition 2.

Let R be a Noetherian ring admitting a dualizing complex Z; $\underline{x}=x_1,\dots ,x_k$ be a sequence of elements of R; and the ideals $I=\underline{x}R$ and $J\in R.$ Suppose that X is an R-complex. Then, there exists the following morphism

$$\mathrm{Hom}({\check{L}}_{\underline{x},J},X)\to {\mathrm{Hom}}_R({\mathrm{Hom}}_R(X,Z),\mathrm{Hom}({\check{L}}_{\underline{x},J},Z).$$

It is a quasi-isomorphism if the homology groups of the R-complex X are finitely generated. Moreover, it induces a quasi-isomorphism

$$\mathrm{L}{\Lambda }_{I,J}\left(X\right)\simeq {\mathrm{RHom}}_R({\mathrm{RHom}}_R(X,Z),\mathrm{L}{\Lambda }_{I,J}\left(Z\right)).$$

Proof. 

By Definition 4, there is a morphism

$$X\to {\mathrm{Hom}}_R({\mathrm{Hom}}_R(X,Z),Z),$$

which is a quasi-isomorphism for an R-complex X with finitely generated homology. Since ${\check{L}}_{\underline{x},J}$ is a bounded complex of free R-modules, it can induce a morphism

$${\mathrm{Hom}}_R({\check{L}}_{\underline{x},J},X)\to {\mathrm{Hom}}_R({\check{L}}_{\underline{x},J},{\mathrm{Hom}}_R({\mathrm{Hom}}_R(X,Z),Z)),$$

which is a quasi-isomorphism if the homology groups of the R-complex X are finitely generated. Together with adjointness and commutativity, the second complex is isomorphic to

$${\mathrm{Hom}}_R({\mathrm{Hom}}_R(X,Z),{\mathrm{Hom}}_R({\check{L}}_{\underline{x},J},Z)),$$

which proves the first part of the claim.

Since $\mathrm{L}{\Lambda }_{I,J}\left(X\right)\cong {\mathrm{Hom}}_R({\check{L}}_{\underline{x},J},X)$ and ${\mathrm{Hom}}_R({\check{L}}_{\underline{x},J},Z)$ is a bounded complex of injective R-modules, we also prove the second statement. □

As an application of local cohomology and the $\check{\mathrm{C}}$ech complex with respect to $(I,J)$, the following theorem provides a proof of Grothendieck’s local duality theorem for unbounded complexes.

Theorem 3.

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring of dimension d admitting a normalized dualizing complex $Z,$ and let the homology groups of the R-complex X be finitely generated. Then, there exists the following quasi-isomorphism

$$\mathrm{R}{\Gamma }_{\mathfrak{m},J}\left(X\right)\simeq {\mathrm{RHom}}_R({\mathrm{RHom}}_R(X,Z),E_R{\left(k\right)}^{-d}).$$

In particular, there exist natural isomorphisms

$${\mathrm{H}}_{\mathfrak{m},J}^i\left(X\right)\cong {\mathrm{Hom}}_R({\mathrm{Ext}}_R^{d-i}(X,Z),E_R\left(k\right)),$$

for all $i\in \mathbb{Z}.$

Proof. 

Putting $I=\mathfrak{m}$ into Proposition 5.3 yields the following quasi-isomorphism

$$\mathrm{R}{\Gamma }_{\mathfrak{m},J}\left(X\right)\simeq {\mathrm{RHom}}_R({\mathrm{RHom}}_R(X,Z),\mathrm{R}{\Gamma }_{\mathfrak{m},J}\left(Z\right)).$$

And by ([7], 11.4.9), there exists a quasi-isomorphism

$${\Gamma }_{\mathfrak{m},J}\left(Z\right)\to E_R{\left(k\right)}^{-d}.$$

Thus, there exists a quasi-isomorphism

$$\mathrm{R}{\Gamma }_{\mathfrak{m},J}\left(X\right)\simeq {\mathrm{RHom}}_R({\mathrm{RHom}}_R(X,Z),E_R{\left(k\right)}^{-d}).$$

This completes the first part of the statement. Next, we prove the second one. We have the following isomorphisms

$$\begin{array}{cc}\hfill {\mathrm{H}}_{\mathfrak{m},J}^i\left(X\right)& \cong {\mathrm{H}}^i\left(\mathrm{R}{\Gamma }_{\mathfrak{m},J}\left(X\right)\right)\hfill \\ & \cong {\mathrm{H}}^i\left({\mathrm{RHom}}_R({\mathrm{RHom}}_R(X,Z),E_R{\left(k\right)}^{-d})\right)\hfill \\ & \cong {\mathrm{Hom}}_R\left({\mathrm{H}}^{d-i}({\mathrm{RHom}}_R(X,Z),E_R{\left(k\right)}^{-d})\right)\hfill \\ & \cong {\mathrm{Hom}}_R({\mathrm{Ext}}_R^{d-i}(X,Z),E_R\left(k\right)),\hfill \end{array}$$

where the first one comes from the definition of local cohomology, and the second one comes from the first part of the statement. The last one is due to the isomorphism ${\mathrm{Ext}}_R^{d-i}(X,Z)\cong {\mathrm{H}}^{d-i}\left({\mathrm{RHom}}_R(X,Z)\right)$. □

6. Conclusions

In this paper, we introduce the concepts of the $\check{\mathrm{C}}$ech complex and $\check{\mathrm{C}}$ech cocomplex with respect to $(I,J)$ and discuss some homological properties of the $\check{\mathrm{C}}$ech cohomology and $\check{\mathrm{C}}$ech homology with respect to $(I,J)$. In addition, we use the $\check{\mathrm{C}}$ech complex and $\check{\mathrm{C}}$ech cocomplex with respect to $(I,J)$ to express the local cohomology and local homology with respect to $(I,J)$, and then we provide a proof for the Matlis–Greenless–May equivalence with respect to $(I,J)$, which is an equivalence between the category of derived $(I,J)$-torsion complexes and the category of derived $(I,J)$-completion complexes. As an application of local cohomology and the $\check{\mathrm{C}}$ech complex with respect to $(I,J)$, we prove Grothendieck’s local duality theorem for unbounded complexes.