# Cofiniteness of generalized local cohomology modules with respect to a
system of ideals

Moharram Aghapournahr1 1Department of Mathematics, Faculty of Science, Arak
University, Arak, 38156-8-8349, Iran<EMAIL_ADDRESS>,  TRAN TUAN
NAM2 2Department of Mathematics-Informatics, Ho Chi Minh University of
Pedagogy, Ho Chi Minh city, Vietnam<EMAIL_ADDRESS><EMAIL_ADDRESS>,  NGUYEN THANH NAM3 3Faculty of Mathematics & Computer
Science, University of Science, VNU-HCM, Ho Chi Minh City, Vietnam
<EMAIL_ADDRESS>and  NGUYEN MINH TRI∗4,5 4Vietnam National University, Ho
Chi Minh City, Vietnam  5Department of Mathematics and Physics, University of
Information Technology, Ho Chi Minh City, Vietnam<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and
$M,X$ two $R$-modules. In this paper, we study the Artinianness and
cofiniteness of the module $H^{i}_{\Phi}(M,X)$ which is an extension of the
generalized local cohomology modules of Herzog.

###### Key words and phrases:

Local cohomology, ${\rm FD_{\leq n}}$ modules, cofinite modules,
$ETH$-cofinite modules, minimax modules

###### 2010 Mathematics Subject Classification:

13D45, 13E05, 14B15.

∗ Corresponding author

The last three authors are funded by Vietnam National Foundation for Science
and Technology Development (NAFOSTED) under Grant No. 101.04-2021.03

##  1\. Introduction

Throughout this paper, $R$ is a commutative noetherian ring and $M$ is a
finitely generated $R$-module. Let $\Phi$ be a non-empty set of ideals of $R.$
We call $\Phi$ a system of ideals of $R$ if
$\mathfrak{a},\mathfrak{b}\in\Phi,$ then there is an ideal
$\mathfrak{c}\in\Phi$ such that
$\mathfrak{c}\subseteq\mathfrak{a}\mathfrak{b}$ (see [9]) For every $R$-module
$X,$ one can define

$\Gamma_{\Phi}(X)=\\{x\in X\mid\mathfrak{a}x=0\text{ for some
}\mathfrak{a}\in\Phi\\}.$

Then $\Gamma_{\Phi}(-)$ is an additive, covariant, $R$-linear functor from the
category of $R$-modules to itself. Moreover, the functor $\Gamma_{\Phi}(-)$ is
left exact. The functor $\Gamma_{\Phi}(-)$ is called the general local
cohomology functor with respect to the system $\Phi$. For each $i\geq 0,$ the
$i$th right derived functor of $\Gamma_{\Phi}(-)$ is denoted by
$\operatorname{H}^{i}_{\Phi}(-).$ For an ideal $\mathfrak{a}$ of $R,$ if
$\Phi=\\{\mathfrak{a}^{n}\mid n\in\mathbb{N}_{0}\\},$ then the functor
$\operatorname{H}^{i}_{\Phi}(-)$ coincides with the ordinary local cohomology
functor $\operatorname{H}^{i}_{\mathfrak{a}}(-).$ Some basic properties of the
module $\operatorname{H}^{i}_{\Phi}(X)$ were shown in [2], [4], [9], [10] and
[20].

In [8], the author introduced the bifunctor $\operatorname{H}^{i}_{\Phi}(-,-)$
as follows: Let $M,X$ be two $R$-modules, the module
$\operatorname{H}_{\Phi}^{i}(M,X)$ is defined as

$\operatorname{H}_{\Phi}^{i}(M,X)\cong\mathop{\lim}\limits_{\begin{subarray}{c}\longrightarrow\\\
\mathfrak{a}\in\Phi\end{subarray}}\operatorname{Ext}_{R}^{i}(M/{\mathfrak{a}}M,X).$

Then $\operatorname{H}^{i}_{\Phi}(-,-)$ is an additive, $R$-linear functor
which is contravariant in the first variable and covariant in the second
variable. If $\Phi=\\{\mathfrak{a}^{n}\mid n\in\mathbb{N}_{0}\\},$ then the
bifunctor $\operatorname{H}^{i}_{\Phi}(-,-)$ is naturally equivalent to the
bifunctor $\operatorname{H}^{i}_{\mathfrak{a}}(-,-)$ of Herzog in [21].

The cofiniteness and Artinianness of local cohomology modules are two of the
most interesting properties which have been studied by many authors. In the
case of local cohomology modules with respect to a system of ideals, the
$\mathfrak{a}$-cofinite modules of Hartshorne was modified and called
$\Phi$-cofinite modules in [4]. We can see some conditions such that the
module $H^{i}_{\Phi}(X)$ is $\Phi$-cofinite in [2] and [4, 4.8, 4.14, 4.15].
As an special case of [29, Definition 2.1] and generalization of FSF modules
(see [24, Definition 2.1]), in [3, Definition 2.1] the first author of present
paper and Bahmanpour introduced the class of ${\rm FD_{\leq n}}$ modules. A
module $M$ is said to be ${\rm FD_{\leq n}}$ module, if there exists a
finitely generated submodule $N$ of $M$ such that $\dim M/N\leq n$. Note that
the class of ${\rm FD_{\leq-1}}$ is the same as finitely generated
$R$-modules. Recall that an $R$-module $M$ is called _weakly Laskerian_ if
$\operatorname{Ass}_{R}(M/N)$ is a finite set for each submodule $N$ of $M$.
The class of weakly Laskerian modules is introduced in [19]. Recall also that
a module $M$ is a _minimax_ module if there is a finitely generated submodule
$N$ of $M$ such that the quotient module $M/N$ is Artinian. Minimax modules
have been studied by Zöschinger in [30].

In this paper, we introduce the concept of $\Phi$-cofiniteness with respect to
module $\operatorname{H}^{i}_{\Phi}(M,X).$ In Section 2, we extend some
results on the cofiniteness of the generalized local cohomology modules with
respect to an ideal of Herzog. We prove in Theorem 2.4 that the module
$\operatorname{H}^{i}_{\Phi}(M,X)$ is $\Phi$-cofinite for all $i\geq 0$ if
$\mathrm{cd}(\mathfrak{a})\leq 1$ for all $\mathfrak{a}\in\Phi$ (or
$\operatorname{cd}\Phi\leq 1$). Next, we prove the following important and key
proposition:

######  Proposition 1.1.

(See Proposition 2.13) Let $R$ be a Noetherian ring, $\mathfrak{a}\in\Phi$ an
ideal of $R$ and $M$ a finitely generated $R$-module. Let $t\in\mathbb{N}_{0}$
be an integer and $X$ an $R$-module such that the $R$-modules
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ are finite for all $i\leq t+1$
and the $R$-module $\operatorname{H}^{i}_{\Phi}(M,X)$ is $\Phi$-cofinite for
all $i<t$. Then the $R$-modules ${\rm
Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$ and ${\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$ are finitely
generated.

Using the above proposition we prove the following main result:

######  Theorem 1.2.

(See Lemma 2.14) Let $R$ be a Noetherian ring, $\mathfrak{a}\in\Phi$ an ideal
of $R$ and $M$ a finitely generated $R$-module. Let $t\in\mathbb{N}_{0}$ be an
integer and $X$ an $R$-module such that the $R$-modules
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ are finitely generated for all
$i\leq t+1$ and the $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are ${\rm
FD_{\leq 1}}$ $R$-modules for all $i<t$. Then, the following conditions hold:

(i) The $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are
$\mathfrak{a}$-$ETH$-cofinite (in particular $\Phi$-cofinite) for all $i<t$.

(ii) For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of
$\operatorname{H}^{t}_{\Phi}(M,X)$, the $R$-modules

${\operatorname{Hom}}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)\,\,\,{\rm
and}\,\,\,{\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)$

are finitely generated.

As consequences of above main theorem we prove Corollaries 2.15, 2.16, 2.17,
2.18 and 2.19.

In Section 3, we study $\Phi$-cofinite Artinianness of
$\operatorname{H}^{i}_{\Phi}(M,X)$. More precisely we prove the following
theorem:

######  Theorem 1.3.

(See Theorems 3.1 and 3.5) Let $\Phi$ be a system of ideals of a Noetherin
ring $R$ and $M,X$ two finitely generated $R$-modules. Then the following
statements hold:

  * (i)

If $(R,\mathfrak{m})$ is a local ring and $\operatorname{pd}(M)<\infty$, then
$\operatorname{H}^{\mathrm{pd}(M)+\dim(R)}_{\Phi}(M,R)$ is $\Phi$-cofinite
Artinian.

  * (ii)

If $R$ is a semi-local and $\dim R/{\mathfrak{a}}\leq 0$ for all
$\mathfrak{a}\in\Phi$, then $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\Phi$-cofinite Artinian for all $i\geq 0.$

As a consequence of part (i) of above theorem, we prove the following result:

######  Theorem 1.4.

(See Theorem 3.3) Let $(R,\mathfrak{m})$ be a local ring and $M,X$ two
finitely generated $R$-modules with $\mathrm{pd}(M)<\infty.$ If $\dim(X)\leq
2$ or $\dim(M)\leq 2,$ then $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\Phi$-cofinite for all $i\geq 0.$

Throughout this paper, $R$ will always be a commutative Noetherian ring with
non-zero identity. We denote $\\{\mathfrak{p}\in{\rm
Spec}\,R:\,\mathfrak{p}\supseteq\mathfrak{a}\\}$ by $V(\mathfrak{a})$. For any
unexplained notation and terminology we refer the reader to [13], [26] and
[12].

##  2\. Cofiniteness

The following definition is a generalization of the concept
$\mathfrak{a}$-cofinite modules which was introduced by Hartshorne.

######  Definition 2.1 ([4]).

Let $\Phi$ be a system of ideals of $R$ and $X$ an $R$-module. The general
local cohomology module $\operatorname{H}^{j}_{\Phi}(X)$ is called to be
$\Phi$-cofinite if there exists an ideal $\mathfrak{a}\in\Phi$ such that
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},\operatorname{H}^{j}_{\Phi}(X))$ is
finitely generated for all $i.$

In the case of module $\operatorname{H}^{i}_{\Phi}(M,X),$ where $M$ is a
finitely generated $R$-module, we have a similar definition.

######  Definition 2.2.

Let $\Phi$ be a system of ideals of $R$. The module
$\operatorname{H}^{j}_{\Phi}(M,X)$ is called to be $\Phi$-cofinite if there
exists an ideal $\mathfrak{a}\in\Phi$ such that
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},\operatorname{H}^{j}_{\Phi}(M,X))$
is finitely generated for all $i.$

The cohomological dimension with respect to ideal $\mathfrak{a}$ is denoted by
$\mathrm{cd}(\mathfrak{a})$ and defined as:

$\mathrm{cd}(\mathfrak{a})=\sup\\{i\mid\operatorname{H}^{i}_{\mathfrak{a}}(X)\neq
0\text{ for any $R$-module }X\\}.$

######  Definition 2.3.

([2]) For any system of ideals $\Phi$ of $R$ and any $R$-module $X,$ we denote
the cohomological dimension of $X$ with respect to $\Phi$ by
$\mathrm{cd}(\Phi,X)$ and define as

$\mathrm{cd}(\Phi,X)=\sup\\{i\mid\operatorname{H}^{i}_{\Phi}(X)\neq 0\\},$

if this supremum exists, otherwise, we define it as $-\infty.$

We denote $\mathrm{cd}(\Phi,R)$ by $\mathrm{cd}(\Phi)$, therefore
$\mathrm{cd}(\Phi)\leq\sup\\{\mathrm{cd}(\mathfrak{a})\mid\mathfrak{a}\in\Phi\\}.$

The first result deals with the cohomological dimension with respect to an
ideal. It is also an extension of [25, 2.2] and [14, Theorem 1.1].

######  Theorem 2.4.

Let $M,X$ be two finitely generated $R$-modules. Assume that
$\mathrm{cd}(\mathfrak{a})\leq 1$ for all $\mathfrak{a}\in\Phi$ (or
$\operatorname{cd}(\Phi)\leq 1$). Then $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\Phi$-cofinite for all $i\geq 0.$

###### Proof.

Let $F=\Gamma_{\Phi}(-)$ and $G=\operatorname{Hom}_{R}(M,-)$ be functors from
the category of $R$-modules to itself. For any $R$-module $K,$ by [20, 2.8]

$FG(K)=\Gamma_{\Phi}(\operatorname{Hom}_{R}(M,K))=\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(K))=\Gamma_{\Phi}(M,K).$

If $E$ is an injective $R$-module, then so is $\Gamma_{\Phi}(E)$ by [4, 4.4].
Let $\textbf{F}_{\bullet}$ be a free resolution of $M$

$\textbf{F}_{\bullet}:\cdots\to F_{2}\to F_{1}\to F_{0}\to M\to 0,$

where $F_{i}$ is finitely generated free for all $i\geq 0.$ By applying the
functor $\operatorname{Hom}_{R}(-,\Gamma_{\Phi}(E))$ to the free resolution of
$M$, there is an exact sequence

$0\to\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(E))\to\operatorname{Hom}_{R}(F_{0},\Gamma_{\Phi}(E))\to\operatorname{Hom}_{R}(F_{1},\Gamma_{\Phi}(E))\to\cdots$

and $\operatorname{Hom}_{R}(F_{i},\Gamma_{\Phi}(E))$ is injective for all
$i\geq 0.$ Since $M$ is finitely generated, the exact sequence can be
rewritten

$0\to\Gamma_{\Phi}(\operatorname{Hom}_{R}(M,E))\to\Gamma_{\Phi}(\operatorname{Hom}_{R}(F_{0},E))\to\Gamma_{\Phi}(\operatorname{Hom}_{R}(F_{1},E))\to\cdots.$

Note that $\operatorname{Hom}_{R}(\textbf{F}_{\bullet},E)$ is an injective
resolution of $\operatorname{Hom}_{R}(M,E).$ Then
$R^{i}F(G(E))=\operatorname{H}^{i}_{\Phi}(\operatorname{Hom}_{R}(M,E))=0$ for
all $i>0.$ By [28, 10.47] there is a Grothendieck spectral sequence

$E_{2}^{p,q}=\operatorname{H}^{p}_{\Phi}(\operatorname{Ext}^{q}_{R}(M,X))\underset{p}{\Rightarrow}\operatorname{H}^{p+q}_{\Phi}(M,X).$

Since $\mathrm{cd}(\mathfrak{a})\leq 1$ for all $\mathfrak{a}\in\Phi$ (or
$\operatorname{cd}(\Phi)\leq 1$), we see that $E_{2}^{p,q}=0$ for all $q\geq
0,~{}p>1.$ Let $n$ be a non-negative integer, there is a filtration $\phi$ of
submodules of $\operatorname{H}^{n}=\operatorname{H}^{n}_{\Phi}(M,X)$

$0=\phi^{n+1}\operatorname{H}^{n}\subseteq\phi^{n}\operatorname{H}^{n}\subseteq\ldots\subseteq\phi^{1}\operatorname{H}^{n}\subseteq\phi^{0}\operatorname{H}^{n}=\operatorname{H}^{n}$

such that

$E_{\infty}^{i,n-i}\cong\phi^{i}\operatorname{H}^{n}/\phi^{i+1}\operatorname{H}^{n}$

for all $i\leq n.$ Since $E_{\infty}^{p,q}$ is a sub-quotient of
$E_{2}^{p,q},$ it follows that $E_{\infty}^{p,q}=0$ for all $p>1.$
Consequently,

$\phi^{2}\operatorname{H}^{n}=\phi^{3}\operatorname{H}^{n}=\ldots=\phi^{n+1}\operatorname{H}^{n}=0.$

Furthermore,
$E_{2}^{1,n-1}=E_{\infty}^{1,n-1}\cong\phi^{1}\operatorname{H}^{n}/\phi^{2}\operatorname{H}^{n}=\phi^{1}\operatorname{H}^{n}$
and
$E_{2}^{0,n}=E_{\infty}^{0,n}\cong\phi^{0}\operatorname{H}^{n}/\phi^{1}\operatorname{H}^{n}.$
Hence, there is a short exact sequence

$0\to\operatorname{H}^{1}_{\Phi}(\operatorname{Ext}_{R}^{n-1}(M,X))\to\operatorname{H}^{n}_{\Phi}(M,X)\to\operatorname{H}^{0}_{\Phi}(\operatorname{Ext}_{R}^{n}(M,X))\to
0.$

By [2, 2.15], the module
$\operatorname{H}^{1}_{\Phi}(\operatorname{Ext}_{R}^{n-1}(M,X))$ is
$\Phi$-cofinite. Moreover,
$\operatorname{H}^{0}_{\Phi}(\operatorname{Ext}_{R}^{n}(M,X))$ is finitely
generated, this implies that $\operatorname{H}^{n}_{\Phi}(M,X)$ is
$\Phi$-cofinite. ∎

######  Corollary 2.5.

Let $M,X$ be two finitely generated $R$-modules. Assume that $\dim(R)\leq 1.$
Then $\operatorname{H}^{i}_{\Phi}(M,X)$ is $\Phi$-cofinite for all $i\geq 0.$

###### Proof.

Under the conditions stated above, $\operatorname{H}^{i}_{\mathfrak{a}}(X)=0$
for all $i>1$ and for all finitely generated $R$-module $X$ and
$\mathfrak{a}\in\Phi.$ Hence $\mathrm{cd}(\mathfrak{a})\leq 1$ for all
$\mathfrak{a}\in\Phi.$ The assertion follows from 2.4. ∎

The following two theorems are needed in the proof of the next main results of
this section.

######  Theorem 2.6.

Let $\Phi$ be a system of ideals of $R$ and $\mathfrak{a}\in\Phi$. Let $M$ be
a finitely generated $R$-module, $X$ an arbitrary $R$-module, and $t$ a non-
negative integer such that

  * (i)

$\operatorname{Ext}_{R}^{t-i}(\operatorname{Tor}_{i}^{R}(R/\mathfrak{a},M),X)$
is finitely generated for all $i\leq t$, and

  * (ii)

$\operatorname{Ext}_{R}^{t+1-i}(R/\mathfrak{a},\operatorname{H}^{i}_{\Phi}(M,X))$
is finitely generated for all $i\leq t$.

Then $\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$
is finitely generated.

###### Proof.

Let

$\operatorname{E}^{\bullet}=0\rightarrow
X\rightarrow\operatorname{E}^{0}\rightarrow...\rightarrow\operatorname{E}^{i}\rightarrow...$

be an injective resolution of $X$ and apply
$\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(-))$ to its deletion
$\operatorname{E}^{\bullet X}$ to get the complex

$\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(\operatorname{E}^{\bullet
X}))=0\rightarrow\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(\operatorname{E}^{0}))\rightarrow...\rightarrow\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(\operatorname{E}^{i}))\rightarrow...$

Let

$0\rightarrow\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(\operatorname{E}^{\bullet
X}))\rightarrow\operatorname{T}^{\bullet,0}\rightarrow\operatorname{T}^{\bullet,1}\rightarrow...\rightarrow\operatorname{T}^{\bullet,i}\rightarrow...$

be a Cartan-Eilenberg injective resolution of
$\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(\operatorname{E}^{\bullet X}))$, which
exists by [28, Theorem 10.45] and consider the third quadrant bicomplex
$\mathcal{T}=\\{\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{T}^{p,q})\\}$.
We denoted the total complex of $\mathcal{T}$ by Tot($\mathcal{T}$).

The first filtration has ${}^{I}\operatorname{E}_{2}$ term the iterated
homology $\operatorname{H}^{\prime p}\operatorname{H}^{\prime\prime
p,q}(\mathcal{T})$. We have

$\displaystyle\operatorname{H}^{\prime\prime p,q}(\mathcal{T})$
$\displaystyle\cong\operatorname{H}^{q}(\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{T}^{p,\bullet}))$
$\displaystyle\cong\operatorname{Ext}^{q}_{R}(R/\mathfrak{a},\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(\operatorname{E}^{p}))$
$\displaystyle\cong\operatorname{Hom}_{R}(\operatorname{Tor}_{q}^{R}(R/\mathfrak{a},M),\Gamma_{\Phi}(\operatorname{E}^{p}))$

from [4, Lemma 4.4] and [28, Theorem 10.63] therefore

${}^{I}\operatorname{E}_{2}^{p,q}$ $\displaystyle\cong\operatorname{H}^{\prime
p}\operatorname{H}^{\prime\prime p,q}(\mathcal{T})$
$\displaystyle\cong\operatorname{H}^{p}(\operatorname{Hom}_{R}(\operatorname{Tor}_{q}^{R}(R/\mathfrak{a},M),\Gamma_{\Phi}(\operatorname{E}^{\bullet
X}))$
$\displaystyle\cong\operatorname{H}_{\Phi}^{p}(\operatorname{Tor}_{q}^{R}(R/\mathfrak{a},M),X)$

which yields, by [20, Lemma 2.8 (ii)], the third quadrant spectral sequence

${}^{I}\operatorname{E}_{2}^{p,q}:=\operatorname{Ext}^{p}_{R}(\operatorname{Tor}_{q}^{R}(R/\mathfrak{a},M),X)\underset{p}{\Rightarrow}\operatorname{H}^{p+q}(\operatorname{Tot}(\mathcal{T})).$

For all $i\leq t$, we have
${}^{I}\operatorname{E}_{\infty}^{t-i,i}=^{I}\operatorname{E}_{t+2}^{t-i,i}$
because, for all $j\geq
t+2,^{I}\operatorname{E}_{j}^{t-i-j,i+j-1}=0=~{}^{I}\operatorname{E}_{j}^{t-i+j,i+1-j}$;
so that ${}^{I}\operatorname{E}_{\infty}^{t-i,i}$ is finitely generated from
the fact that ${}^{I}\operatorname{E}_{t+2}^{t-i,i}$ is a subquotient of
${}^{I}\operatorname{E}_{2}^{t-i,i}$ which is finitely generated by assumption
(i). There exists a finite filtration

$0=\phi^{t+1}\operatorname{H}^{t}\subseteq\phi^{t}\operatorname{H}^{t}\subseteq...\subseteq\phi^{1}\operatorname{H}^{t}\subseteq\phi^{0}\operatorname{H}^{t}=\operatorname{H}^{t}(\operatorname{Tot}(\mathcal{T}))$

such that
${}^{I}\operatorname{E}_{\infty}^{t-i,i}\cong\phi^{t-i}\operatorname{H}^{t}/\phi^{t-i+1}\operatorname{H}^{t}$
for all $i\leq t$. Now the exact sequences

$0\rightarrow\phi^{t-i+1}\operatorname{H}^{t}\rightarrow\phi^{t}\operatorname{H}^{t-i}\rightarrow~{}^{I}\operatorname{E}_{\infty}^{t-i,i}\rightarrow
0,$

for all $i\leq t$, show that
$\operatorname{H}^{t}(\operatorname{Tot}(\mathcal{T}))$ is finitely generated.

On the other hand, the second filtration has ${}^{II}\operatorname{E}_{2}$
term the iterated homology $\operatorname{H}^{\prime\prime
p}\operatorname{H}^{{}^{\prime}q,p}(\mathcal{T})$. Note that every short exact
sequence of injective modules splits and so it remains split after applying
the functor $\operatorname{Hom}_{R}(R/\mathfrak{a},-)$. By using this fact and
the fact that $\operatorname{T}^{\bullet,\bullet}$ is a Cartan-Eilenberg
injective resolution of
$\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(\operatorname{E}^{\bullet X}))$, we
get

$\displaystyle\operatorname{H}^{\prime q,p}(\mathcal{T})$
$\displaystyle\cong\operatorname{H}^{q}(\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{T}^{\bullet,p}))$
$\displaystyle\cong\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{q}(\operatorname{T}^{\bullet,p}))$
$\displaystyle\cong\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{q,p}).$

Therefore

${}^{II}\operatorname{E}_{2}^{p,q}$
$\displaystyle\cong\operatorname{H}^{\prime\prime p}\operatorname{H}^{\prime
q,p}(\mathcal{T})$
$\displaystyle\cong\operatorname{H}^{p}(\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{q,\bullet}))$
$\displaystyle\cong\operatorname{Ext}_{R}^{p}(R/\mathfrak{a},\operatorname{H}^{q}_{\Phi}(M,X))$

which gives the third quadrant spectral sequence

${}^{II}\operatorname{E}_{2}^{p,q}:=\operatorname{Ext}_{R}^{p}(R/\mathfrak{a},\operatorname{H}^{q}_{\Phi}(M,X))\underset{p}{\Rightarrow}\operatorname{H}^{p+q}(\operatorname{Tot}(\mathcal{T})).$

There exists a finite filtration

$0=\psi^{t+1}\operatorname{H}^{t}\subseteq\psi^{t}\operatorname{H}^{t}\subseteq...\subseteq\psi^{1}\operatorname{H}^{t}\subseteq\psi^{0}\operatorname{H}^{t}=\operatorname{H}^{t}(\operatorname{Tot}(\mathcal{T}))$

such that
${}^{II}\operatorname{E}_{\infty}^{t-i,i}\cong\psi^{t-i}\operatorname{H}^{t}/\psi^{t-i+1}\operatorname{H}^{t}$
for all $i\leq t$. Since
$\operatorname{H}^{t}(\operatorname{Tot}(\mathcal{T}))$ is finitely generated,
${}^{II}\operatorname{E}_{\infty}^{0,t}\cong\psi^{0}\operatorname{H}^{t}/\psi^{1}\operatorname{H}^{t}$
is finitely generated. Therefore ${}^{II}\operatorname{E}_{t+2}^{0,t}$ is
finitely generated because
${}^{II}\operatorname{E}_{t+2}^{0,t}=^{II}\operatorname{E}_{\infty}^{0,t}$
from the fact that
${}^{II}\operatorname{E}_{j}^{-j,t+j-1}=0=^{II}\operatorname{E}_{j}^{j,t+1-j}$
for all $j\geq t+2$. For all $r\geq 2$, let
${}^{II}Z^{0,t}_{r}=\operatorname{Ker}(^{II}\operatorname{E}_{r}^{0,t}\rightarrow^{II}\operatorname{E}_{r}^{r,t+1-r})$
and
${}^{II}B^{0,t}_{r}=\operatorname{Im}(^{II}\operatorname{E}_{r}^{-r,t+r-1}\rightarrow^{II}\operatorname{E}_{r}^{0,t})$.
We have the exact sequences

$0\rightarrow^{II}Z^{0,t}_{r}\rightarrow^{II}\operatorname{E}^{0,t}_{r}\rightarrow^{II}\operatorname{E}^{0,t}_{r}/^{II}Z^{0,t}_{r}\rightarrow
0$

and

$0\rightarrow^{II}B^{0,t}_{r}\rightarrow\rightarrow^{II}Z^{0,t}_{r}\rightarrow^{II}\operatorname{E}^{0,t}_{r+1}\rightarrow
0$.

Since ${}^{II}\operatorname{E}^{r,t+1-r}_{2}$ is finitely generated and
${}^{II}\operatorname{E}^{-r,t+r-1}_{2}=0,^{II}\operatorname{E}^{r,t+1-r}_{r}$
is finitely generated and ${}^{II}\operatorname{E}^{-r,t+r-1}_{r}=0$ and so
${}^{II}\operatorname{E}^{0,t}_{r}/^{II}Z^{0,t}_{r}$ is finitely generated and
${}^{II}B^{0,t}_{r}=0$. Hence ${}^{II}\operatorname{E}^{0,t}_{r}$ is finitely
generated when ${}^{II}\operatorname{E}^{0,t}_{r}$ is finitely generated.
Therefore
${}^{II}\operatorname{E}^{0,t}_{2}=\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$
is finitely generated. ∎

######  Theorem 2.7.

Let $\Phi$ be a system of ideals of $R$ and $\mathfrak{a}\in\Phi$. Let $M$ be
a finitely generated $R$-module, $X$ an arbitrary $R$-module, and $t$ a non-
negative integer such that

  * (i)

$\operatorname{Ext}_{R}^{t+1-i}(\operatorname{Tor}_{i}^{R}(R/\mathfrak{a},M),X)$
is finitely generated for all $i\leq t+1$, and

  * (ii)

$\operatorname{Ext}_{R}^{t+2-i}(R/\mathfrak{a},\operatorname{H}_{\Phi}^{i}(M,X))$is
finitely generated for all $i<t$.

Then
$\operatorname{Ext}_{R}^{1}(R/\mathfrak{a},\operatorname{H}_{\Phi}^{t}(M,X))$is
finitely generated.

###### Proof.

This is sufficiently similar to that of Theorem 2.6 to be omitted. We leave
the proof to the reader. ∎

######  Lemma 2.8.

Let $R$ be a Noetherian ring, $\mathfrak{a}$ an ideal of $R$, $t$ a non-
negative integer. Then for any $R$–modules $T$, the following conditions are
equivalent:

  * (i)

$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},T)$ is finitely generated for all
$0\leq i\leq t$.

  * (ii)

for any finitely generated $R$–module $N$ with
$\operatorname{Supp}_{R}(N)\subseteq\operatorname{V}(\mathfrak{a})$,
$\operatorname{Ext}^{i}_{R}(N,T)$ is finitely generated for all $0\leq i\leq
t$.

###### Proof.

See [22, Lemma 1]. ∎

As a generalization of definition of cofinite modules with respect to an ideal
([27, Definition 2.2]), it is introduced in [1, Definition 2.2] the following
definition.

######  Definition 2.9.

An $R$-module $M$ (not necessary $\mathfrak{a}$-torsion) is called
$ETH$-cofinite with respect to ideal $\mathfrak{a}$ of $R$ or
$\mathfrak{a}$-$ETH$-cofinite if
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},M)$ is a finitely generated
$R$-module for all $i$.

For more details about properties of this class see [1, Example 2.3].

######  Definition 2.10.

An $R$-module $M$ is said to be ${\rm FD_{\leq n}}$ module, if there exists a
finitely generated submodule $N$ of $M$ such that $\dim M/N\leq n$.

The following remark is some elementary properties of the class of ${\rm
FD_{\leq n}}$ modules which we shall use.

######  Remark 2.11.

There are some basic properties of the class of ${\rm FD_{\leq n}}$ modules:

  * (i)

The class of ${\rm FD_{\leq n}}$ modules contains all finitely generated,
Artinian, minimax and weakly Laskerian modules.

  * (ii)

Let $0\to L\to M\to N\to 0$ be a short exact sequence of $R$-modules. Then $M$
is a ${\rm FD_{\leq n}}$ module if and only if $L$ and $N$ are both ${\rm
FD_{\leq n}}$ module. Thus, any submodule and quotient of a ${\rm FD_{\leq
n}}$ module is ${\rm FD_{\leq n}}$.

  * (iii)

If $M$ is a finitely generated $R$-module and $L$ is a ${\rm FD_{\leq n}}$
$R$-module, then $\operatorname{Ext}^{j}_{R}(M,L)$ and
$\operatorname{Tor}^{R}_{j}(M,L)$ are ${\rm FD_{\leq n}}$ for all $j\geq 0.$

In what follows, the next lemma plays key role.

######  Lemma 2.12.

Let $R$ be a Noetherian ring and $\mathfrak{a}$ an ideal of $R$. Let $M$ be an
${\rm FD_{\leq 1}}$ $R$-module. Then $M$ is $\mathfrak{a}$-$ETH$-cofinite if
and only if $\operatorname{Hom}_{R}(R/\mathfrak{a},M)$ and
$\operatorname{Ext}^{1}_{R}(R/\mathfrak{a},M)$ are finitely generated.

###### Proof.

See [2, Theorem 3.1]. ∎

The following proposition which is one of our main results of this section is
a generalization of Theorems [17, Theorem 2.1] and [16, Theorem A].

######  Proposition 2.13.

Let $R$ be a Noetherian ring, $\mathfrak{a}\in\Phi$ an ideal of $R$ and $M$ a
finitely generated $R$-module. Let $t\in\mathbb{N}_{0}$ be an integer and $X$
an $R$-module such that the $R$-modules
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ are finitely generated for all
$i\leq t+1$ and the $R$-module $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\Phi$-cofinite for all $i<t$. Then the $R$-modules ${\rm
Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$ and ${\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$ are finitely
generated.

###### Proof.

Using Lemma 2.8, it is easy to see that the conditions (i) and (ii) of
Theorems 2.6 and 2.7 are established. Thus the $R$-modules ${\rm
Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$ and ${\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$ are finitely
generated by Theorems 2.6 and 2.7. ∎

We are now ready to state and prove the following main results of this section
(Theorem 2.14 and the Corollaries 2.17 and 2.18) which are extension of
Bahmanpour-Naghipour’s results in [5], [6], Quy’s result in [24], [15, Theorem
2.5], [11, Theorem 3.6] and [3, Theorem 3.4 and Corollaries 3.5 and 3.6] and
[14, Theorem 1.2].

######  Theorem 2.14.

Let $R$ be a Noetherian ring, $\mathfrak{a}\in\Phi$ an ideal of $R$ and $M$ a
finitely generated $R$-module. Let $t\in\mathbb{N}_{0}$ be an integer and $X$
an $R$-module such that the $R$-modules
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ are finitely generated for all
$i\leq t+1$ and the $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are ${\rm
FD_{\leq 1}}$ $R$-modules for all $i<t$. Then, the following conditions hold:

(i) The $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are
$\mathfrak{a}$-$ETH$-cofinite (in particular $\Phi$-cofinite) for all $i<t$.

(ii) For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of
$\operatorname{H}^{t}_{\Phi}(M,X)$, the $R$-modules

${\operatorname{Hom}}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)\,\,\,{\rm
and}\,\,\,{\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)$

are finitely generated.

###### Proof.

(i) We proceed by induction on $t$. For the case $t=0$ there is nothing to
prove. So, let $t>0$ and the result has been proved for smaller values of $t$.
By the inductive assumption, $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\mathfrak{a}$-$ETH$-cofinite for $i=0,1,...,t-2$. Hence by Lemma 2.13 and
assumption, ${\rm Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t-1}_{\Phi}(M,X))$
and ${\rm Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t-1}_{\Phi}(M,X))$ are
finitely generated. Therefore by Lemma 2.12,
$\operatorname{H}^{i}_{\Phi}(M,X)$ is $\mathfrak{a}$-$ETH$-cofinite for all
$i<t$. This completes the inductive step.
(ii) In view of (i) and Lemma 2.13, ${\rm
Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))\,\,\,{\rm
and}\,\,\,{\rm Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$
are finitely generated. On the other hand, according to [2, Theorem 2.7], $N$
is $\mathfrak{a}$-$ETH$-cofinite. Now, the exact sequence

$0\longrightarrow N\longrightarrow H^{t}_{\Phi}(M,X)\longrightarrow
H^{t}_{\Phi}(M,X)/N\longrightarrow 0$

induces the following exact sequence,

${\rm
Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))\longrightarrow{\rm
Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)\longrightarrow{\rm
Ext}^{1}_{R}(R/\mathfrak{a},N)\longrightarrow$ ${\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))\longrightarrow{\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)\longrightarrow{\rm
Ext}^{2}_{R}(R/\mathfrak{a},N).$

Consequently

${\rm Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)\,\,\,{\rm
and}\,\,\,{\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)$

are finitely generated, as required.∎

The following theorem illustrates a relationship between the minimaxness and
the $\Phi$-cofiniteness. In [14, 2.3], if
$\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is minimax for all $i<t,$ then
$\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is $\mathfrak{a}$-cofinite for all
$i<t.$ Now we extend this property in the case
$\operatorname{H}^{i}_{\Phi}(M,X).$

######  Corollary 2.15.

Let $R$ be a Noetherian ring, $\mathfrak{a}\in\Phi$ an ideal of $R$ and $M$ a
finitely generated $R$-module. Let $t\in\mathbb{N}_{0}$ be an integer and $X$
an $R$-module such that the $R$-modules
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ are finitely generated for all
$i\leq t+1$ and the $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are minimax
for all $i<t.$ Then, the following conditions hold:

(i) The $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are
$\mathfrak{a}$-$ETH$-cofinite (in particular $\Phi$-cofinite) for all $i<t$.

(ii) For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of
$\operatorname{H}^{t}_{\Phi}(M,X)$, the $R$-modules

${\operatorname{Hom}}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)\,\,\,{\rm
and}\,\,\,{\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)$

are finitely generated.

###### Proof.

Use Theorem 2.14 and note that the category of minimax modules is contained in
the category of ${\rm FD_{\leq 1}}$ modules. ∎

######  Corollary 2.16.

Let $R$ be a Noetherian ring, $\mathfrak{a}\in\Phi$ an ideal of $R$ and $M$ a
finitely generated $R$-module. Let $t\in\mathbb{N}_{0}$ be an integer and $X$
an $R$-module such that the $R$-modules
$\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ are finitely generated for all
$i\leq t+1$. Assume that $\operatorname{H}^{i}_{\Phi}(X)$ is ${\rm FD_{\leq
1}}$ module for all $i<t.$ Then $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\Phi$-cofinite for all $i<t$ and
$\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))~{}\text{and}~{}{\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X))$ are finitely
generated.

###### Proof.

Let $F=\operatorname{Hom}_{R}(M,-),G=\Gamma_{\Phi}(-)$ be functors from the
category of $R$-modules to itself and $\mathfrak{a}\in\Phi.$ For any
$R$-module $K,$ by [20, Lemma 2.8]

$FG(K)=\operatorname{Hom}_{R}(M,\Gamma_{\Phi}(K))\cong\operatorname{H}^{0}_{\Phi}(M,K).$

Let $\operatorname{E}$ be an injective $R$-module, then
$\Gamma_{\Phi}(\operatorname{E})$ is injective by [4, Lemma 4.4]. Hence
$R^{i}F(G(\operatorname{E}))=0$ for all $i>0.$ By [28, 10.47], there is a
Grothendieck spectral sequence

$E_{2}^{p,q}=\operatorname{Ext}_{R}^{p}(M,\operatorname{H}^{q}_{\Phi}(X))\underset{p}{\Rightarrow}\operatorname{H}^{p+q}_{\Phi}(M,X).$

Combining the hypothesis with 2.11(iii), $E_{2}^{p,q}$ is ${\rm FD_{\leq 1}}$
module for all $p\geq 0,~{}q<t.$ Let $k<t,$ there is a filtration $\phi$ of
submodules of $\operatorname{H}^{k}=\operatorname{H}^{k}_{\Phi}(M,X)$ such
that

$E_{\infty}^{i,k-i}\cong\phi^{i}\operatorname{H}^{k}/\phi^{i+1}\operatorname{H}^{k}$

for all $i\leq k.$ The module $E_{\infty}^{i,k-i}$ is ${\rm FD_{\leq 1}}$
module for all $0\leq i\leq k$ by 2.11(ii) as $E_{\infty}^{i,k-i}$ is a sub-
quotient of $E_{2}^{i,k-i}.$ Therefore $\operatorname{H}^{k}_{\Phi}(M,X)$ is
${\rm FD_{\leq 1}}$ module for all $k<t$ and the assertion follows from
Theorem 2.14. ∎

######  Corollary 2.17.

Let $R$ be a Noetherian ring, $\mathfrak{a}\in\Phi$ an ideal of $R$ and $M$ a
finitely generated $R$-module. Let $X$ be an $R$-module such that the
$R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are ${\rm FD_{\leq 1}}$ (or
weakly Laskerian) $R$-modules for all $i$. Then,

(i) The $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are
$\mathfrak{a}$-$ETH$-cofinite (in particular $\Phi$-cofinite) for all $i$.

(ii) For any $i\geq 0$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule
$N$ of $\operatorname{H}^{i}_{\Phi}(M,X)$, the $R$-module
$\operatorname{H}^{i}_{\Phi}(M,X)/N$ is $\mathfrak{a}$-$ETH$-cofinite (in
particular $\Phi$-cofinite).

###### Proof.

(i) Follows by Theorem 2.14 (i).
(ii) In view of (i) the $R$-module $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\mathfrak{a}$-$ETH$-cofinite for all $i$. Hence the $R$-module ${\rm
Hom}_{R}(R/\mathfrak{a},N)$ is finitely generated, and so it follows from [2,
Theorem 2.7] that $N$ is $\mathfrak{a}$-$ETH$-cofinite. Now, the exact
sequence

$0\longrightarrow
N\longrightarrow\operatorname{H}^{i}_{\Phi}(M,X)\longrightarrow\operatorname{H}^{i}_{\Phi}(M,X)/N\longrightarrow
0$

and [2, Lemma 2.4] implies that the $R$-module
$\operatorname{H}^{i}_{\Phi}(M,X)/N$ is $\mathfrak{a}$-$ETH$-cofinite.∎

######  Corollary 2.18.

Let $\mathfrak{a}$ be an ideal of a Noetherian ring $R$, $M$ a finitely
generated $R$-module and $X$ an $\mathfrak{a}$-$ETH$-cofinite $R$-module. Let
$t\in\mathbb{N}_{0}$ such that the $R$-modules
$\operatorname{H}^{i}_{\Phi}(M,X)$ are weakly Laskerian for all $i<t$. Then,
the following conditions hold:

(i) The $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are
$\mathfrak{a}$-$ETH$-cofinite (in particular $\Phi$-cofinite) for all $i<t$.

(ii) For all ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of
$\operatorname{H}^{t}_{\Phi}(M,X)$, the $R$-modules

${\rm Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)\,\,\,{\rm
and}\,\,\,{\rm
Ext}^{1}_{R}(R/\mathfrak{a},\operatorname{H}^{t}_{\Phi}(M,X)/N)$

are finitely generated.

###### Proof.

Use Theorem 2.14 and note that the category of weakly Laskerian modules is
contained in the category of ${\rm FD_{\leq 1}}$ modules.∎

The following corollary is a generalization of [5, Corollary 2.7], [18,
Theorem 2.5] and [14, Theorem 1.2].

######  Corollary 2.19.

Let $R$ be a Noetherian ring, $\mathfrak{a}\in\Phi$ an ideal of $R$ and $M$ a
finitely generated $R$-module. Let $X$ be an $\mathfrak{a}$-$ETH$-cofinite
$R$-module such that $\dim X/{\mathfrak{a}}X\leq 1$ (e.g., $\dim
R/\mathfrak{a}\leq 1$). Then,

  * (i)

the $R$-modules $\operatorname{H}^{i}_{\Phi}(M,X)$ are
$\mathfrak{a}$-$ETH$-cofinite (in particular $\Phi$-cofinite) for all $i$.

  * (ii)

for any $i\geq 0$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$
of $\operatorname{H}^{i}_{\Phi}(M,X)$, the $R$-module
$\operatorname{H}^{i}_{\Phi}(M,X)/N$ is $\mathfrak{a}$-$ETH$-cofinite (in
particular $\Phi$-cofinite).

###### Proof.

(i) Since by [8, Lemma 2.1],

$\operatorname{H}^{i}_{\Phi}(M,X)\cong\underset{\mathfrak{a}\in\Phi}{\varinjlim}\operatorname{H}^{i}_{\mathfrak{a}}(M,X),$

it is easy to see that
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M,X))\subseteq\underset{\mathfrak{a}\in\Phi}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M,X))$
and therefore

$\operatorname{dimSupp}\operatorname{H}^{i}_{\Phi}(M,X)\leq\sup\\{\operatorname{dimSupp}\operatorname{H}^{i}_{\mathfrak{a}}(M,X)|\mathfrak{a}\in\Phi\\}\leq
1,$

thus $\operatorname{H}^{i}_{\Phi}(M,X)$ is ${\rm FD_{\leq 1}}$ $R$-module and
the assertion follows by Corollary 2.17 (i).
(ii) Proof is the same as 2.17 (ii). ∎

######  Corollary 2.20.

Let $\Phi$ be a system of ideals of a Noetherian ring $R$, $M$ a finitely
generated $R$-module and $X$ an $\mathfrak{a}$-$ETH$-cofinite $R$-module such
that $\operatorname{H}^{i}_{\Phi}(M,X)$ are ${\rm FD_{\leq 1}}$(or weakly
Laskerian) $R$-modules for all $i\geq 0$. Then for each finitely generated
$R$-module $N$, the $R$-modules ${\rm
Ext}^{j}_{R}(N,\operatorname{H}^{i}_{\Phi}(M,X))$ and ${\rm
Tor}^{R}_{j}(N,\operatorname{H}^{i}_{\Phi}(M,X))$ are
$\mathfrak{a}$-$ETH$-cofinite (in particular $\Phi$-cofinite) and ${\rm
FD_{\leq 1}}$ modules for all $i\geq 0$ and $j\geq 0$.

###### Proof.

Follows by [1, Corollary 2.11]. ∎

##  3\. Artinianess and cofiniteness

It is well-known that $\operatorname{H}^{\dim(X)}_{\mathfrak{a}}(X)$ is an
$\mathfrak{a}$-cofinite Artinian (see [27, Proposition 5.1]) $R$-module and
$\operatorname{H}^{\mathrm{pd}(M)+\dim(X)}_{\mathfrak{a}}(M,N)$ is an
$\mathfrak{a}$-cofinite Artinian $R$-module (see [15, Proposition 2.7 and
Theorem 2.8]). Now, we consider the top generalized local cohomology module
with respect to a system of ideals.

######  Theorem 3.1.

Let $(R,\mathfrak{m})$ be a local ring and $M$ a finitely generated $R$-module
with $\mathrm{pd}(M)<\infty.$ Then
$\operatorname{H}^{\mathrm{pd}(M)+\dim(R)}_{\Phi}(M,R)$ is $\Phi$-cofinite
Artinian.

###### Proof.

Let $p=\mathrm{pd}(M)$ and $d=\dim(R).$ We consider the spectral sequence

$E_{2}^{p,q}=\operatorname{Ext}_{R}^{p}(M,\operatorname{H}^{q}_{\Phi}(R))\underset{p}{\Rightarrow}\operatorname{H}^{p+q}_{\Phi}(M,R).$

There is a filtration $\phi$ of submodules of $H^{p+d}=H^{p+d}_{\Phi}(M,R)$
such that

$E_{\infty}^{i,p+d-i}\cong\phi^{i}\operatorname{H}^{p+d}/\phi^{i+1}\operatorname{H}^{p+d}$

for all $i\leq p+d.$ By [9, 2.7], $E_{2}^{i,j}=0$ for all $i>p$ or $j>d.$
Since $E_{\infty}^{i,p+d-i}$ is a sub-quotient of $E_{2}^{i,p+d-i}$, we can
assert that $E_{\infty}^{i,p+d-i}=0$ for all $i\neq p.$ Consequently,

$\phi^{0}\operatorname{H}^{p+d}=\ldots=\phi^{p}\operatorname{H}^{p+d}\text{
and
}\phi^{p+1}\operatorname{H}^{p+d}=\ldots=\phi^{p+d+1}\operatorname{H}^{p+d}=0.$

The homomorphisms of the spectral sequence

$0=E_{2}^{p-2,d+1}\to E_{2}^{p,d}\to E_{2}^{p+2,d-1}=0$

induces that $E_{2}^{p,d}=E_{\infty}^{p,d}.$ Hence, there is an isomorphism

$\operatorname{Ext}_{R}^{p}(M,\operatorname{H}^{d}_{\Phi}(R))\cong\phi^{p}\operatorname{H}^{p+d}/\phi^{p+1}\operatorname{H}^{p+d}=\operatorname{H}^{p+d}_{\Phi}(M,R).$

The Artianness of $\operatorname{H}^{p+d}_{\Phi}(M,R)$ is followed from [10,
Theorem 3.1].

Our next goal is to prove the cofiniteness by induction on $d.$ When $d=0,$
$R$ is an Artinian $R$-module and
$\operatorname{Supp}(R)\subseteq\\{\mathfrak{m}\\}\subseteq\bigcup_{\mathfrak{a}\in\Phi}V(\mathfrak{a}).$
Hence $R$ is $\Phi$-torsion by [23, Lemma 2.4]. It follows from [20, Lemma
2.8] that

$H^{p}_{\Phi}(M,R)\cong\operatorname{Ext}_{R}^{p}(M,R).$

Therefore, $\operatorname{H}^{p}_{\Phi}(M,R)$ is finitely generated and then
it is $\Phi$-cofinite.

Assume that $d>0.$ Let $\overline{R}=R/\Gamma_{\Phi}(R)$ which is
$\Phi$-torsion-free by [20, Lemma 2.4]. Using again [20, Lemma 2.4],
$\overline{R}$ is $\mathfrak{a}$-torsion free for all $\mathfrak{a}\in\Phi.$
We can take an element $x\in\mathfrak{a}$ which is regular on $\overline{R}.$
The short exact sequence

$0\to\overline{R}\to\overline{R}\to\overline{R}/x\overline{R}\to 0$

deduces the following exact sequence

$\cdots\overset{f}{\to}\operatorname{H}^{p+d-1}_{\Phi}(M,\overline{R}/x\overline{R})\overset{g}{\to}\operatorname{H}^{p+d}_{\Phi}(M,\overline{R})\overset{.x}{\to}\operatorname{H}^{p+d}_{\Phi}(M,\overline{R})\to
0.$

Since $\dim(\overline{R}/x\overline{R})\leq d-1,$ by the inductive hypothesis
$\operatorname{H}^{p+d-1}_{\Phi}(M,\overline{R}/x\overline{R})$ is
$\Phi$-cofinite Artinian. Consequently,
$\operatorname{Hom}_{R}(R/\mathfrak{a},\mathrm{Im}f)$ is finitely generated as
it is a submodule of

$\operatorname{Hom}_{R}(R/\mathfrak{a},H^{p+d-1}_{\Phi}(M,\overline{R}/x\overline{R})).$

Moreover, $\mathrm{Im}f$ is an Artinian $R$-module. It follows from [2, Lemma
2.5 and Definition 2.1] that
$\operatorname{Ext}_{R}^{i}(R/\mathfrak{a},\mathrm{Im}f)$ is finitely
generated for all $i\geq 0.$ Therefore,
$\operatorname{Ext}_{R}^{i}(R/\mathfrak{a},\mathrm{Im}g)$ is finitely
generated for all $i\geq 0.$ In particularly,
$\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{Im}g)$ is finitely
generated and

$\displaystyle\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{Im}g)$
$\displaystyle=\operatorname{Hom}_{R}(R/\mathfrak{a},0:_{\operatorname{H}^{p+d}_{\Phi}(M,\overline{R})}x)$
$\displaystyle\cong\operatorname{Hom}_{R}(R/\mathfrak{a},\operatorname{H}^{p+d}_{\Phi}(M,\overline{R})).$

The proof above gives that $\operatorname{H}^{p+d}_{\Phi}(M,\overline{R})$ is
Artinian as $\dim{\overline{R}}\leq d.$ By [2, Lemma 2.5 and Definition 2.1],
$\operatorname{H}^{p+d}_{\Phi}(M,\overline{R})$ is $\Phi$-cofinite. The short
exact sequence

$0\to\Gamma_{\Phi}(R)\to R\to\overline{R}\to 0$

induces a long exact sequence

$\cdots\to\operatorname{H}^{p+d}_{\Phi}(M,\Gamma_{\Phi}(R))\to\operatorname{H}^{p+d}_{\Phi}(M,R)\to\operatorname{H}^{p+d}_{\Phi}(M,\overline{R})\to
0.$

We get by [20, Lemma 2.8] that
$\operatorname{H}^{i}_{\Phi}(M,\Gamma_{\Phi}(R))\cong\operatorname{Ext}_{R}^{i}(M,\Gamma_{\Phi}(R))$
for all $i\geq 0$ and then $\operatorname{H}^{i}_{\Phi}(M,\Gamma_{\Phi}(R))$
is finitely generated for all $i\geq 0.$ Therefore
$\operatorname{H}^{p+d}_{\Phi}(M,R)$ is $\Phi$-cofinite and the proof is
complete. ∎

######  Corollary 3.2.

Let $(R,\mathfrak{m})$ be a local ring. Then $H^{\dim(R)}_{\Phi}(R)$ is
$\Phi$-cofinite Artinian.

When $\dim(M)\leq 2$ or $\dim(X)\leq 2,$ the cofiniteness of $H^{i}_{I}(M,X)$
was shown in [25, Theorem 2.9] or [14, Theorem 1.3]. We also see in [2,
Corollary 2.15] that if $(R,\mathfrak{m})$ is a local ring with $\dim(R)\leq
2,$ then $\operatorname{H}^{i}_{\Phi}(X)$ is $\Phi$-cofinite for all $i\geq
0.$ The following result might be considered as a generalization of these
facts.

######  Theorem 3.3.

Let $(R,\mathfrak{m})$ be a local ring and $M,X$ two finitely generated
$R$-modules with $\mathrm{pd}(M)<\infty.$ If $\dim(X)\leq 2$ or $\dim(M)\leq
2,$ then $\operatorname{H}^{i}_{\Phi}(M,X)$ is $\Phi$-cofinite for all $i\geq
0.$

###### Proof.

There is a Grothendieck spectral sequence

$E_{2}^{p,q}=\operatorname{H}^{p}_{\Phi}(\operatorname{Ext}^{q}_{R}(M,X))\underset{p}{\Rightarrow}\operatorname{H}^{p+q}_{\Phi}(M,X).$

The assumption shows that $\dim(\operatorname{Ext}_{R}^{i}(M,X))\leq 2$ and
$\operatorname{Ext}_{R}^{i}(M,X)$ is finitely generated for all $i\geq 0.$

By Corollary 3.2,
$\operatorname{H}^{2}_{\Phi}(\operatorname{Ext}^{q}_{R}(M,X))$ is
$\Phi$-cofinite for all $q\geq 0.$ On the other hand
$\operatorname{H}^{p}_{\Phi}(\operatorname{Ext}^{q}_{R}(M,X))=0$ for all
$p>2,q\geq 0$ by [9, 2.7]. Let $q\geq 0,$ the homomorphisms of spectral
sequence

$0=E_{2}^{i-2,q-i+1}\to E_{2}^{i,q-i}\to E_{2}^{i+2,q-i-1}=0$

induce

$E_{2}^{i,q-i}=E_{\infty}^{i,q-i}$

where $i=0,1.$ There is a filtration $\phi$ of submodules of
$\operatorname{H}^{q}=\operatorname{H}^{q}_{\Phi}(M,X)$ such that

$E_{\infty}^{i,q-i}\cong\phi^{i}\operatorname{H}^{q}/\phi^{i+1}\operatorname{H}^{q}$

for all $0\leq i\leq q.$ Since $E_{\infty}^{i,q-i}$ is a sub-quotient of
$E_{2}^{i,q-i},$ we claim that $E_{\infty}^{i,q-i}=0$ for all $i>2.$ Thus,

$\phi^{3}\operatorname{H}^{q}=\phi^{4}\operatorname{H}^{q}=\ldots=\phi^{q+1}\operatorname{H}^{q}=0.$

The homomorphisms of spectral sequence

$0\to
E_{2}^{0,q-1}\overset{d_{2}^{0,q-1}}{\to}E_{2}^{2,q-2}\overset{d_{2}^{2,q-2}}{\to}E_{2}^{4,q-3}=0$

turn out

$E_{\infty}^{2,q-2}=E_{3}^{2,q-2}=\operatorname{Ker}d_{2}^{2,q-2}/\operatorname{Im}d_{2}^{0,q-1}=E_{2}^{2,q-2}/\operatorname{Im}d_{2}^{0,q-1}.$

The short exact sequence

$0\to\operatorname{Im}d_{2}^{0,q-1}\to E_{2}^{2,q-2}\to E_{\infty}^{2,q-2}\to
0$

induces the following exact sequence

$\operatorname{Hom}_{R}(R/\mathfrak{a},E_{2}^{2,q-2})\longrightarrow\operatorname{Hom}_{R}(R/\mathfrak{a},E_{\infty}^{2,q-2})\longrightarrow\operatorname{Ext}_{R}^{1}(R/\mathfrak{a},\operatorname{Im}d_{2}^{0,q-1}).$

Since $E_{2}^{2,q-2}$ is $\Phi$-cofinite Artinian by Corollary 3.2,
$\operatorname{Hom}_{R}(R/\mathfrak{a},E_{2}^{2,q-2})$ is finitely generated.
Furthermore, $\operatorname{Im}d_{2}^{0,q-1}$ is finitely generated.
Therefore, $\operatorname{Hom}_{R}(R/\mathfrak{a},E_{\infty}^{2,q-2})$ is
finitely generated. Note that $E_{\infty}^{2,q-2}$ is Artinian as it is a sub-
quotient of the Artinian $R$-module
$E_{2}^{2,q-2}=\operatorname{H}^{2}_{\Phi}(\operatorname{Ext}_{R}^{q-2}(M,X))$
by Corollary 3.2. It follows from [2, Lemma 2.5 and Definition 2.1] that
$\operatorname{Ext}_{R}^{i}(R/\mathfrak{a},E_{\infty}^{2,q-2})$ is finitely
generated for all $i\geq 0.$

The equality $E_{\infty}^{1,q-1}=E_{2}^{1,q-1}$ shows that
$\phi^{1}\operatorname{H}^{q}/\phi^{2}\operatorname{H}^{q}$ is $\Phi$-cofinite
by [2, Corolary 2.15]. On the other hand, we have
$\phi^{2}\operatorname{H}^{q}\cong E_{\infty}^{2,q-2}$ and then
$\operatorname{Ext}_{R}^{i}(R/\mathfrak{a},\phi^{1}\operatorname{H}^{q})$ is
finitely generated for all $i\geq 0.$

Note that $\phi^{0}\operatorname{H}^{q}/\phi^{1}\operatorname{H}^{q}\cong
E_{\infty}^{0,q}$ is finitely generated since $E_{\infty}^{0,q}$ is a sub-
quotient of $E_{2}^{0,q}$ which is finitely generated. Thus
$\operatorname{Ext}_{R}^{i}(R/\mathfrak{a},\phi^{0}\operatorname{H}^{q})$ is
finitely generated for all $i\geq 0.$ This means that the module
$\phi^{0}\operatorname{H}^{q}=\operatorname{H}^{q}_{\Phi}(M,X)$ is
$\Phi$-cofinite. ∎

######  Corollary 3.4.

Let $(R,\mathfrak{m})$ be a local ring with $\dim(R)\leq 2.$ Then
$\operatorname{H}^{i}_{\Phi}(M,X)$ is $\Phi$-cofinite for all $i\geq 0.$

The following theorem is a generalization of [7, Theorem 3.6].

######  Theorem 3.5.

Let $R$ be a semi-local Noetherian ring and $\Phi$ a system of ideals of $R$.
Let $M$ and $X$ be two $\mathfrak{a}$-$ETH$-cofinite $R$-module for all
$\mathfrak{a}\in\Phi$ (or finitely generated). If $\dim R/{\mathfrak{a}}\leq
0$ for all $\mathfrak{a}\in\Phi$, then $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\Phi$-cofinite Artinian for all $i\geq 0.$

###### Proof.

Since by [8, Lemma 2.1],

$\operatorname{H}^{i}_{\Phi}(M)\cong\underset{\mathfrak{a}\in\Phi}{\varinjlim}\operatorname{H}^{i}_{\mathfrak{a}}(M),$

it is easy to see that
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\underset{\mathfrak{a}\in\Phi}{\bigcup}\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\mathfrak{a}}(M))$
and therefore

$\operatorname{dimSupp}\operatorname{H}^{i}_{\Phi}(M)\leq\sup\\{\operatorname{dimSupp}\operatorname{H}^{i}_{\mathfrak{a}}(M)|\mathfrak{a}\in\Phi\\}\leq
0,$

thus
$\operatorname{Supp}_{R}(\operatorname{H}^{i}_{\Phi}(M))\subseteq\operatorname{Max}(R)$.
Now, it follows from [7, Corolary 3.4] that $\operatorname{H}^{i}_{\Phi}(M)$
is an Artinian $R$-module for all $i\geq 0$. Therefore Remark 2.11 (i) and
Corollary 2.16 imply that $\operatorname{H}^{i}_{\Phi}(M,X)$ is
$\Phi$-cofinite for all $i\geq 0$. ∎

Acknowledgments

The authors are deeply grateful to the referee for careful reading of the
manuscript and for the helpful suggestions.

## References

  * [1] M. Aghapournahr, Cofiniteness of local cohomology modules for a pair of ideals for small dimensions, J. Algebra Appl., 17 (2018), 1850020.
  * [2] M. Aghapournahr and K. Bahmanpour, Cofiniteness of general local cohomology modules for small dimensions, Bull. Korean Math. Soc., 53 (2016), 1341-1352.
  * [3] M. Aghapournahr and K. Bahmanpour, Cofiniteness of weakly Laskerian local cohomology modules, Bull. Math. Soc. Sci. Math. Roumanie, 105 (2014), 347-356.
  * [4] J. Asadollahi, K. Khashyarmanesh and Sh. Salarian, _A generalization of the cofiniteness problem in local cohomology modules_ , J. Aust. Math. Soc., 75 (2003), 313–324.
  * [5] K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra, 321 (2009), 1997–2011.
  * [6] K. Bahmanpour and R. Naghipour, On the cofiniteness of local cohomology modules, Proc. Amer. Math. Soc., 136 (2008), 2359-2363.
  * [7] M. Behrouzian and M. Aghapournahr, Lower bounds of certain general local cohomology modules, Comm. Alg., 48 (2020), 2406–2417.
  * [8] M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J., 21(1980), 173-181.
  * [9] M. H. Bijan-Zadeh, Torsion theory and local cohomology over commutative Noetherian ring, J. London Math. Soc., 19(1979), 402-410.
  * [10] M. H. Bijan-Zadeh, On the Artinian property of certain general local cohomology, J. London Math. Soc., 35(1985), 399-403.
  * [11] K. Borna Lorestani, P. Sahandi and S. Yassemi, Cofiniteness of generalized local cohomology modules, Bull. Aust. Math. Soc., 83 (2011), 382-388.
  * [12] M. P. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge. Univ. Press, 1998.
  * [13] W. Bruns and J. Herzog, Cohen Macaulay Rings, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge Univ. Press, Cambridge, UK, 1993.
  * [14] N. T. Cuong, S. Goto and N. V. Hoang, _On the cofiniteness of generalized local cohomology modules_ , Kyoto J. Math., 55 (2015), 169-185
  * [15] L. Chu, Cofiniteness and finiteness of generalized local cohomology modules, Bull. Aust. Math. Soc., 83 (2009), 244-250.
  * [16] M. T. Dibaei and S. Yassemi, Finiteness of extention functors of local cohomology modules, Comm. Algebra, 34 (2006), 3097–3101.
  * [17] M. T. Dibaei and S. Yassemi, Associated primes and cofiniteness of local cohomology modules, Manuscripta Math., 117 (2005), 199-205.
  * [18] K. Divaani-Aazar and A. Hajkarimi, _Cofiniteness of generalized local cohomology modules for one-dimensional ideals_ , Canad. Math. Bull. 55(2012), 81–87.
  * [19] K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc., 133 (2005), 655-660.
  * [20] A. Hajikarimi, Cofiniteness with respect to a Serre subcategory, Math. Notes, 89(1) (2011) 121-130.
  * [21] J. Herzog, Komplexe, Auflösungen und dualität in der localen Algebra, Habilitationsschrift, Universität Regensburg, 1970.
  * [22] K.-I. Kawasaki, _On the finiteness of Bass numbers of local cohomology modules_ , Proc. Amer. Math. Soc., 124 (1996), 3275–3279.
  * [23] K. Khashyarmanesh and Sh. Salarian, General local cohomology modules and Koszul homology modules, Colloq. Math., 77(2) (1998) 305-313.
  * [24] P. H. Quy, On the finiteness of associated primes of local cohomology modules, Proc. Amer. Math. Soc., 138 (2010), 1965-1968.
  * [25] A. Mafi and H. Saremi,  Cofinite modules and generalized local cohomology,, Houston J. Math., 35(4) (2009) 1013-1018.
  * [26] H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, UK, 1986.
  * [27] L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285 (2005), 649–668.
  * [28] J. Rotman, An introduction to Homological Algebra, 2nd edn., Springer, 2009.
  * [29] T. Yoshizawa, Subcategories of extension modules by subcategories, Proc. Amer. Math. Soc., 140 (2012), 2293-2305.
  * [30] H. Zöschinger, _Minimax Moduln_ , J. Algebra, 102 (1986), 1–32.


# Deep and Wide Photometry of Two Open Clusters NGC 1245 and NGC 2506: CCD
Observation and Physical Properties

S. H. Lee,1 Y.-W. Kang1 and H. B. Ann2
1Korea Astronomy and Space Science Institute, Daejeon 305-348, Korea
2Department of Earth Sciences, Pusan National University, Busan 609-735, Korea
<EMAIL_ADDRESS>to whom any correspondence should be
addressed<EMAIL_ADDRESS>
(Accepted 2012 June 26. Received 2012 June 26; in original form 2012 March 1)

###### Abstract

We have conducted $VI$ CCD photometry of the two open clusters NGC 1245 and
NGC 2506 using the CFH12K CCD camera. Our photometry covers a sky area of
$84^{\prime}\times 82^{\prime}$ and $42^{\prime}\times 81^{\prime}$ for the
two clusters, respectively, and reaches down to $V\approx 23$. We derived the
physical parameters using detailed theoretical isochrone fittings using
$\chi^{2}$ minimization. The derived cluster parameters are $E(B-V)=0.24\pm
0.05$ and $0.03\pm 0.04$, $(V-M_{V})_{0}=12.25\pm 0.12$ and $12.47\pm 0.08$,
$age(Gyr)=1.08\pm 0.09$ and $2.31\pm 0.16$, and [Fe/H]= $-0.08\pm 0.06$ and
$-0.24\pm 0.06$, respectively for NGC 1245 and NGC 2506, We present the
luminosity functions (LFs) of the two clusters, which reach down to
$M_{V}\approx 10$, and derive mass functions (MFs) with slopes of
$\Gamma=-1.29$ for NGC 1245 and $\Gamma=-1.26$ for NGC 2506. The slopes are
slightly shallower than that of the solar neighbourhood, implying the
existence of dynamical evolution that drives the evaporation of the low-mass
stars in the clusters.

###### keywords:

(Galaxy:) open clusters and associations: individual: – methods: observational
– techniques: photometric

††pagerange: Deep and Wide Photometry of Two Open Clusters NGC 1245 and NGC
2506: CCD Observation and Physical Properties–LABEL:lastpage††pubyear: 2012
Figure 1: Observed fields ($84^{\prime}\times 82^{\prime}$) of NGC 1245. We
combined 6 V-filter images, where one CCD image has a size of
$42^{\prime}\times 28^{\prime}$, to make the mosaic image. North is up and
east is to the left.

##  1 Introduction

Open clusters are located in the Galactic disc as Population I, and it is
important to investigate their properties and the spatial distribution of the
Galaxy to understand the Galaxy structure and evolution, especially the
formation and dynamical evolution of the Galactic disc. Because stars in the
Galactic disc are originated in evaporated stars from open clusters, the
luminosity function of open clusters is an important parameter for
understanding their formation and evolution. We can easily obtain the LF from
CCD observations, but an analysis of the LF is not easy because the observed
LF of open clusters reflects the dynamical evolution which depends on the
cluster’s age as well as the initial mass function (IMF).

It is necessary to have a deep photometry over the entire cluster field to
understand the LF of old open clusters down to the magnitudes of the
evaporating stars because they are much fainter than the turn-off stars and
are supposed to be located around the tidal radius of the clusters. But there
are only a few open clusters of which photometry is deep and wide enough to
study the effect of dynamical evolution on the structure of open clusters.
Thus, we selected open clusters which are old enough for the low mass stars to
evaporate from the clusters by dynamical evolution and rich enough suitable
for a detailed analysis of the spatial distribution of low mass stars. We also
took into account the distance of open clusters to ensure that our photometry
reaches up to $M_{V}\sim 10$.

The old open cluster NGC 1245 is located in the Perseus arm at a Galactic
latitude of $-8^{\circ}.93$. Using photographic photometry data from the
Schmidt telescope, Peruanskii & Ryadchenko (1978) discussed the radial density
distribution of stars with a limiting magnitude of $B=17$ and a radius of
$r\approx 83^{\prime}$. They presented cluster boundaries with a nucleus
($r<13^{\prime}$) and corona ($13^{\prime}<r<61^{\prime}$). Carraro & Patat
(1994) and Subramaniam (2003) found a shallower slope of the mass function of
NGC 1245 than Salpeter (1955). Using $BV$ CCD photometry, Subramaniam (2003)
presented the surface number density with a limiting magnitude of $V=17$ and a
radius of $r\sim 6^{\prime}.5$. They also indicated a lack of stars in the
cluster centre. Burke et al. (2004) derived physical parameters and a core
radius of $R_{c}=3^{\prime}.1$ from $BVI$ CCD photometry. They determined an
optical absorption of $A_{V}=0.68$ magnitude and a distance modulus
$(V-M_{V})_{0}=12.27$; they also reported that the cluster has no differential
reddening, in contrast to previous studies such as Carraro & Patat (1994) and
Wee & Lee (1996).

The old open cluster NGC 2506 also has rich members located at a Galactic
latitude of $+9^{\circ}.94$. A radial density profile of the proper motion
membership stars obtained by Chiu & van Altena (1981) indicated that bright
stars are more concentrated than faint stars in this cluster. Marconi et al.
(1997) suggested that more than 20% of the main sequence stars are binary.
From a photometric study by Henderson et al. (2007), the distance of this
cluster is known to be 3.4 kpc and its age is 1.79 Gyr.

This paper presents the deep $VI$ CCD photometry of the two old open clusters
NGC 1245 and NGC 2506, covering the entire cluster fields including the
surrounding field regions for field star correction, along with the derivation
of the cluster parameters such as age, metallicity, distance, and interstellar
reddening. We will discuss the dynamical structures of the open clusters in a
forthcoming paper (Lee, Kang $\&$ Ann, in preparation), but here we present
the physical properties, luminosity functions, and mass functions of the two
open clusters along with their physical properties.

The present paper is organized as follows. In §2, we describe the observations
and data reductions. We present the physical parameters of the clusters in §3,
and describe the luminosity functions and mass functions in §4 and §5,
respectively. A summary is given in the final section.

Figure 2:  Observed fields ($42^{\prime}\times 81^{\prime}$) of NGC 2506. We
combined 3 V-filter images, where one CCD image has a size of
$42^{\prime}\times 28^{\prime}$, to make the mosaic image. North is up and
east is to the left.

##  2 Observation

We selected target clusters best suited for studying the dynamical properties
of open clusters which is the subject of the second paper (lee12). The two
target clusters NGC 1245 and NGC 2506 are located at a relatively high
Galactic latitude to reduce field star contamination and are old enough for
the evaporation of low mass stars to be prevalent due to dynamical evolution.
Also, they are at proper distances for optimum sky coverage and the deep
photometry to reach up to $M_{V}\sim 10$. The basic known properties of the
selected clusters are listed in Table 1.

Table 1: Basic data of the selected clusters, extracted from the WEBDA web site (http://obswww.unige.ch/webda/) Cluster | $\alpha$ | $\delta$ | $l$ | $b$ | Distance | $E(B-V)$ | log(age)
---|---|---|---|---|---|---|---
name | J2000.0 | J2000.0 |  |  | (kpc) |  |
NGC 1245 | $03^{h}14^{m}42^{s}$ | +47 14 12 | $146^{\circ}.65$ | $-8^{\circ}.93$ | 2.88 | 0.300 | 8.704
NGC 2506 | $08^{h}00^{m}01^{s}$ | -10 46 12 | $230^{\circ}.56$ | $+9^{\circ}.94$ | 3.46 | 0.081 | 9.045

###  2.1 Observation

We conducted deep $VI$ CCD photometry using the CFH12K mosaic CCD camera
mounted at the prime focus of the Canada-France-Hawaii Telescope (CFHT). The
effective aperture of the CFHT is 3.6 m, and the CFH12K mosaic CCD camera
consists of 12 CCDs of $2048\times 4096$ pixels. The CFH12K field of view is
$42^{\prime}\times 28^{\prime}$ with pixel size of $0.^{\prime\prime}206$.

For better sky coverage, we observed 6 regions for NGC 1245 and 3 regions for
NGC 2506. The field of view of the combined images was $84^{\prime}\times
82^{\prime}$ for NGC 1245 and $42^{\prime}\times 81^{\prime}$ for NGC 2506
(see Fig. 1 and Fig. 2). The observations were performed on the night of
December 6, 2001, at CFHT. The observation log is given in Table 2.

Table 2: Log of observations Cluster | Region |  $V$ Filter |  $I$ Filter | Seeing
---|---|---|---|---
NGC 1245 | North-East |  $140^{s}\times$ 4, $5^{s}\times$ 4 |  $90^{s}\times$ 4, $5^{s}\times$ 4 | $1.^{\prime\prime}6$
| East |  $140^{s}\times$ 4, $5^{s}\times$ 4 |  $90^{s}\times$ 4, $5^{s}\times$ 4 | $1.^{\prime\prime}2$
| South-East |  $140^{s}\times$ 4, $5^{s}\times$ 4 |  $90^{s}\times$ 4, $5^{s}\times$ 4 | $1.^{\prime\prime}1$
| North-West |  $140^{s}\times$ 4, $5^{s}\times$ 4 |  $90^{s}\times$ 4, $5^{s}\times$ 4 | $1.^{\prime\prime}3$
| West |  $140^{s}\times$ 4, $5^{s}\times$ 4 |  $90^{s}\times$ 4, $5^{s}\times$ 4 | $1.^{\prime\prime}5$
| South-West |  $140^{s}\times$ 4, $5^{s}\times$ 4 |  $90^{s}\times$ 4, $5^{s}\times$ 4 | $1.^{\prime\prime}3$
NGC 2506 | North |  $140^{s}\times$ 3, $5^{s}\times$ 3 |  $90^{s}\times$ 4, $5^{s}\times$ 3 | $1.^{\prime\prime}2$
| Centre |  $140^{s}\times$ 3, $5^{s}\times$ 3 |  $90^{s}\times$ 4, $5^{s}\times$ 3 | $1.^{\prime\prime}0$
| South |  $140^{s}\times$ 3 |  $90^{s}\times$ 4 | $1.^{\prime\prime}2$
Figure 3:  Photometric residuals of the transformation to the standard system.
We used standard stars in SA 98 (Landolt, 1992). Standard deviations of the
$VI$ photometry are $\sigma_{\Delta V}=0.018$ and
$\sigma_{\Delta(V-I)}=0.028$.  Figure 4:  Incompleteness correction factor
$\Lambda$ as a function of $V$ magnitude.

###  2.2 Data Reduction

We followed the standard data reduction procedures, including bias
subtraction, overscan correction, and trimming and flat-fielding using
IRAF/CCDRED (Stetson, 1987). We used the twilight sky flat fields obtained
before and after the observations for flat-fielding. In order to increase the
signal-to-noise ratio and to reach low limiting magnitudes, we took multiple
exposures of each region. These images were aligned and median-combined into a
master image for each region using images of both the long and short
exposures. For the photometry calibration, we obtained calibration images for
each chip of the CFH12K. We used 20–40 isolated bright stars to derive the
point-spread function (PSF) of each calibration image, and we performed PSF
photometry (Stetson, 1987) using IRAF/DAOPHOT to obtain the instrumental
magnitudes.

For standard star transformation in each band, we observed the SA 98 region
(Landolt, 1992) on the same night. The standard star observations were
carefully designed to ensure that the maximum number of stars were imaged in
each chip of the CFH12K mosaic CCD. The number of observed standard stars was
8–19 for each chip. We derived the standard magnitudes of the programme stars
by using the following equations given on the CFHT web page:

$V=v-0.12(X-1)+0.014(V-I)+z_{v}$ (1) $I=i-0.04(X-1)+0.050(V-I)+z_{i}$ (2)

where $V$ and $I$ are the standard magnitudes and $v$ and $i$ are the
instrumental magnitudes. The terms $z_{v}$ and $z_{i}$ are the zero points of
the photometry that were determined from the photometry of the standard stars
in SA 98, and $X$ is the airmass. Fig. 3 shows the residuals of the standard
star transformation in the $V$ and $I$ bands.

Figure 5: CMD of the central region ($r<5^{\prime}$) of NGC 1245 with
isochrones fitted by eyeball (red line) and $\chi^{2}$ minimum (green line).
Black lines indicate the boundaries of main-sequence stars and giant box.
Figure 6: CMD of the central region ($r<5^{\prime}$) of NGC 2506 with
isochrones fitted by eyeball (red line) and $\chi^{2}$ minimum (green line).
Black lines indicate the boundaries of main-sequence stars and giant box.
Figure 7: Confidence regions of the joint variation in age, metallicity,
distance modulus and reddening for NGC 1245. The three colors red, green, and
blue represent the confidence limits of 2, 4, 6 $\sigma$, respectively,
corresponding to $\Delta\chi^{2}=4,16,36$ Figure 8: Confidence regions of the
joint variation in age, metallicity, distance modulus and reddening for NGC
2506. The three colors red, green, and blue represent the confidence limits of
2, 4, 6 $\sigma$, respectively, corresponding to $\Delta\chi^{2}=4,16,36$

To correct the incompleteness of the photometry, we generated artificial stars
on the CCD frames using the addstar task in the IRAF/DAOPHOT utility and
performed reduction in the same manner as the observed frames. The additional
artificial stars were randomly generated in each unit-magnitude bin on all
original images. To prevent crowding in the images, the number of artificial
stars were controlled to be less than 10% of the detected stars, and we
repeated this for 40 trials. The incompleteness factor ($\Lambda$) of the
photometric data was determined in the central and outer regions of each
cluster. Finally, the incompleteness correction factor was obtained from
$\Lambda=N_{r}/N_{a}$, where $N_{a}$ is the number of added stars and $N_{r}$
is the number of recovered stars. Fig. 4 shows the relationship between
$\Lambda$ and $V$ magnitude.

##  3 Physical Parameters

We determined the basic physical parameters of the two open clusters by
analyzing the color magnitude diagrams (CMDs). We used Yonsei-Yale ($Y^{2}$)
isochrones (Demarque et al., 2004) to fit to the observed CMDs. We assumed
$E(V-I)/E(B-V)=1.25$ and a total-to-selective extinction ratio $R_{V}$ of 3.2
(Cardelli et al., 1989). Initially, we adjusted the distance, age,
metallicity, and interstellar reddening by eyeball fitting (Fig. 5 and Fig. 6)
but we determined the best fit parameters by applying $\chi^{2}$ minimization
algorithm as described below.

In order to reduce the effects of field star contamination, we constructed
cluster CMDs using stars in the central region ($R<5^{\prime}$) of the
cluster. We also defined the boundaries of main-sequence stars, which are the
blue and red edge along main-sequence stars. These boundaries are tightly
selected in order to exclude binary sequence. According to evolution tracks,
the blue edge of giant clump stars are more massive and more evolved than the
red edge of the giant clump. And the giant branch of $Y^{2}$ isochrone
represents the red edge of giant stars. So we defined a region of red edge of
the giant clump. In Fig. 5 and Fig. 6, we showed the boundaries, and the red
edge regions. The $Y^{2}$ isochrones provide an interpolation scheme to
calculate isochrones for an arbitrary age and metallicity.

Following Burke et al. (2004), for a given set of fitting parameters (age,
metallicity, distance modulus and interstellar reddening), we defined the
$\chi^{2}$ as

$\chi^{2}=\sum_{i}\chi_{i}^{2}$ (3)

where the sum is over all stars selected for isochrone fitting and
$\chi_{i}^{2}$ is the contribution from star $i$ which is defined as

$\chi_{i}^{2}\equiv[V_{m}-V_{o}]_{i}^{2}+[(V-I)_{m}-(V-I)_{o}]_{i}^{2}.$ (4)

Here, $V$ and $V-I)$ are magnitude and color of stars and the subscripts $m$
and $o$ stand for model and observation, respectively. We assumed an equal
weight for all stars used in the calculation of $\chi^{2}$ in order to avoid
overemphasizing the main-sequence turn-off where the systematic difference
between theoretical isochrones are more pronounced (Grocholski & Sarajedini,
2003; Burke et al., 2004) although their photometric errors are the smallest.
We performed a grid search over age, metallicity, distance modulus and
reddening to find the best-fit parameters which minimize the $\chi^{2}$. In
table 3, we presents the ranges and intervals of parameters used for NGC 1245
and NGC 2506.

The confidence limits on the derived parameters are obtained by scaling the
resulting $\chi^{2}$ statistic (Burke et al., 2004)

$\Delta\chi^{2}=\frac{\chi^{2}-\chi_{min}^{2}}{\chi_{min}^{2}/\nu}$ (5)

where $\chi^{2}_{min}$ is the minimum $\chi^{2}$ and $\nu$ is the number of
degrees of freedom. In Fig. 7 and Fig. 8, we plotted the confidence regions
for the joint variation in the isochrone parameters representing the
$\Delta\chi^{2}=4,16,36$ (red, green, blue), which correspond to 2, 4, and 6
$\sigma$ error respectively. As shown in Fig. 7 and Fig. 8, the local minimum
of $\Delta\chi^{2}$ is fairly well defined and the $\Delta\chi^{2}$ surface
can be approximated by a paraboloid near minimum. In this approximation, the
statistical errors (1 $\sigma$ errors) are the projection of the
$\Delta\chi^{2}=1.0$ extent of the paraboloid on the parameter axes. The best-
fit parameters and their statistical errors ($\sigma_{stat}$) are given in
Table 4. As can be seen in Fig. 7 and Fig. 8, the statistical errors in the
best-fit parameters are smaller in NGC 2506 than NGC 1245. The larger errors
in NGC 1245 is caused by the broad main-sequence near the turn-off in NGC
1245.

Table 3: Isochrone grid parameters for NGC 1245 and NGC 2506 Cluster | NGC 1245 | NGC 2506
---|---|---
| Range | Interval | Range | Interval
age(Gyr) | $1.02\sim 1.18$ | 0.005 | $2.19\sim 2.41$ | 0.005
[Fe/H] | $-0.24\sim 0.0$ | 0.005 | $-0.29\sim-0.16$ | 0.005
$(V-M_{V})_{0}$ | $12.13\sim 12.36$ | 0.005 | $12.41\sim 12.53$ | 0.002
$E(B-V)$ | $0.2\sim 0.3$ | 0.005 | $0.015\sim 0.070$ | 0.002
Table 4: Physical parameters of NGC 1245 and NGC 2506

Cluster |  | $E(B-V)$ | $(V-M_{V})_{0}$ | age(Gyr) | [Fe/H]
---|---|---|---|---|---
NGC 1245 | best $\chi^{2}_{min}$ | $0.24$ | $12.25$ | $1.08$ | $-0.08$
| $zeropoint+\sigma_{z}$ | 0.30 | 12.20 | 1.06 | -0.12
| $zeropoint-\sigma_{z}$ | 0.21 | 12.32 | 1.08 | -0.09
| $R_{V}=3.0$ | 0.26 | 12.33 | 1.06 | -0.12
| $R_{V}=3.4$ | 0.25 | 12.18 | 1.06 | -0.11
| $V_{lim}=18.0$ | 0.27 | 12.33 | 0.9 | -0.09
| $V_{lim}=21.0$ | 0.225 | 12.23 | 1.08 | 0.00
| Padova (Eyeball) | 0.12 | 12.45 | 1.00 | 0.07
|  $Y^{2}$ (Eyeball) | $0.22$ | $12.35$ | $1.0$ | $0.0$
| $\sigma_{stat}$ | $\pm 0.01$ | $\pm 0.02$ | $\pm 0.01$ | $\pm 0.02$
| $\sigma_{sys}$ | $\pm 0.05$ | $\pm 0.12$ | $\pm 0.09$ | $\pm 0.06$
| $\sigma_{tot}$ | $\pm 0.05$ | $\pm 0.12$ | $\pm 0.09$ | $\pm 0.06$
| Previous | $0.26^{2},0.28^{8},0.28^{3}$ | $13.2^{2},12.0^{3},12.4^{5}$ | $0.8^{2},1.1^{3}$ | $-0.04^{3},-0.05^{6}$
|  | $0.29^{5},0.21^{6}$ | $12.27^{6},12.1^{8}$ | $0.9^{5},1.05^{6}$ | $+0.06^{8},+0.14^{9}$
NGC 2506 | best $\chi^{2}_{min}$ | $0.035$ | $12.47$ | $2.31$ | $-0.24$
| $zeropoint+\sigma_{z}$ | 0.08 | 12.38 | 2.30 | -0.24
| $zeropoint-\sigma_{z}$ | 0.00 | 12.53 | 2.32 | -0.24
| $R_{V}=3.0$ | 0.035 | 12.47 | 2.33 | -0.24
| $R_{V}=3.4$ | 0.035 | 12.46 | 2.32 | -0.24
| $V_{lim}=18.0$ | 0.02 | 12.33 | 2.48 | -0.22
| $V_{lim}=21.0$ | 0.03 | 12.32 | 2.47 | -0.23
| Padova (Eyeball) | 0.02 | 12.4 | 2.5 | -0.32
|  $Y^{2}$ (Eyeball) | $0.02$ | $12.4$ | $2.19$ | $-0.2$
| $\sigma_{stat}$ | $\pm 0.004$ | $\pm 0.01$ | $\pm 0.01$ | $\pm 0.01$
| $\sigma_{sys}$ | $\pm 0.04$ | $\pm 0.08$ | $\pm 0.16$ | $\pm 0.06$
| $\sigma_{tot}$ | $\pm 0.04$ | $\pm 0.08$ | $\pm 0.16$ | $\pm 0.06$
| Previous | $0.1^{1},0\sim 0.07^{4}$ | $11.7^{1},12.6^{4},12.95^{7}$ | $1.6\sim 2.0^{4},1.8^{7}$ | $-0.32\sim 0.07^{4},-0.32^{7}$
|  | $0.10^{8},$ | $11.8^{8}$ |  | $-0.41^{8}$

1 Purgathofer (1964), 2 Carraro & Patat (1994), 3 Wee & Lee (1996), 4 Marconi
et al. (1997), 5 Subramaniam (2003), 6 Burke et al. (2004), 7 Henderson et al.
(2007) , 8 Janes (1979), 9 Lyngå (1987)

The statistical errors presented in Table 4 are very small. However, the
uncertainties of the physical parameters can be larger than these errors
because they merely represent the statistical errors associated with the
$\chi^{2}$ minimization. There are other sources of errors that affect the
accuracy of the derived physical parameters systematically. Possible causes of
systematic errors are the uncertainties related to the photometric
calibration, assumed parameters such as the total-to-selective extinction
ratio $R_{V}$, limiting magnitudes of observed data $V_{lim}$ and the
theoretical models for isochrone fittings. In order to know the systematic
errors in the derived physical parameters, we performed some experiments by
varying the assumed parameters and theoretical models. We present the results
in Table 4.

From equation (1) and (2), systematic errors associated with the calibration
of the photometry can be caused by uncertainties in the extinction and
transformation coefficients and zero points. But we examined only the effect
of the uncertainties in the zero points ($\sigma_{z}(V)$ and $\sigma_{z}$)
because the calibration coefficients which are provided by CHFT web site are
assumed to be well determined. We derived physical parameters from the data
with the zero point shifts of $\pm\sigma_{z}(V)=0.05$ and
$\pm\sigma_{z}(V-I)=0.10$ and found that the systematic errors in $E(B-V)$ and
$(V-M_{V})_{0}$ are relatively large.

Since systematic uncertainties in $R_{V}$ affect the cluster parameters,
variations in $R_{V}$ alter the $(V-I)$ color at fixed $A_{V}$ (Burke et al.,
2004), we performed $\chi^{2}$ minimization procedures to derive cluster
parameters with $R_{V}=3.0$ and $R_{V}=3.4$ using the same grid search. We
found that the systematic errors associated with the $R_{V}$ variation are
similar to or less than the statistical errors, except for those in the
distance modulus of NGC 1245.

In Fig. 5 and Fig. 6, there is a non-negligible discrepancy between the
observations and the theoretical isochrones for the lower main-sequence stars
($V>19$). This effect is most serious for stars in $19<V<21$ of NGC 2506. Due
to this discrepancy, the derived cluster parameters can be changed if we
assume different limiting magnitude for isochrone fittings. To estimate the
uncertainties related to the discrepancy between the models and the
observations, we performed the same procedures again to derive the cluster
parameters with limiting magnitudes of $V_{lim}=18.0$ and $V_{lim}=21.0$. The
systematic errors related to the $V_{lim}$ are generally larger than the
statistical errors and they are larger in NGC 1245. But, the systematic errors
in $(V-M_{V})_{0}$ are larger in NGC 2506. The reason for the larger
uncertainties in the parameters of NGC 1245 is partly due to the broad
distribution of stars near the turn-off and upper main-sequence of NGC 1245.
The large systematic error in $(V-M_{V})_{0}$ in NGC 2506 seems to be caused
by the maximum discrepancy between the models and observations at $19<V<21$ in
NGC 2506.

Since the cluster parameters derived from isochrone fittings significantly
depend on the theoretical models, we used the Padova isochrones (Girardi et
al., 2000) for comparison models. However, we did not apply $\chi^{2}$
minimization procedures because we do not have interpolation programs for the
Padova isochrones. Rather, we applied eye-ball fittings for both the $Y^{2}$
isochrones and the Padova isochrones and compared the derived cluster
parameters. As can be seen in Table 4, the systematic errors resulting from
the theoretical models are much larger than the statistical errors but
comparable to the systematic errors associated with the assumed parameters. In
general, the systematic errors associated with the theoretical models are
larger in NCC 1245 than NGC 2506. The largest difference is observed in the
age of NGC 2506.

We calculated the systematic errors of cluster parameters from the derived
values in Table 4 by assuming that we explored typical values of $R_{V}$,
$V_{lim}$ and zero point shifts in the derivation of cluster parameters.
Actually, we considered the standard deviations of the cluster parameters
resulting from different sets of assumed parameters (zero points, $R_{V}$ and
$V_{lim}$) and theoretical models as the systematic errors of the cluster
parameters. Then, the total systematic error ($\sigma_{sys}$) can be
determined by a quadratic sum of the systematic errors as

${\sigma_{sys}}^{2}={\sigma_{z}}^{2}+{\sigma(R_{V})}^{2}+{\sigma(V_{lim})}^{2}+{\sigma(iso)}^{2}$
(6)

here $\sigma(R_{V})$ and $\sigma(V_{lim})$ are systematic errors associated
with the parameters $R_{V}$ and $V_{lim}$, respectively, and $\sigma(iso)$ is
the systematic error related to the theoretical models. Then, the total error
($\sigma_{tot}$) can be written as

${\sigma_{tot}}^{2}={\sigma_{stat}}^{2}+{\sigma_{sys}}^{2}$ (7)

where $\sigma_{stat}$ is the statistical error related to the $\chi^{2}$
minimization. In Table 4, we present the total uncertainties in each cluster
along with the statistical errors and the systematic errors. However, the
actual uncertainties in the derived cluster parameters can be larger than
these values since we did not consider the errors associated with the
unresolved binaries and the field star contamination. Our estimates of the
systematic errors are similar to those of Burke et al. (2004) who derived the
cluster parameters by the same method as ours and the total uncertainties are
comparable to those of Janes & Hoq (2011) who derived the cluster parameters
using synthetic CMDs.

##  4 Luminosity Function

###  4.1 Construction of Luminosity Function

Our observed area was wide enough to cover the field region as well as the
cluster region. To derive the LFs of the open clusters, we first determined
the cluster extent $R_{cl}$ by examining the surface density distribution of
the observed area. The radius at which the surface number density radial
profiles of the stars becomes constant was taken to be $R_{cl}$. The field
regions can then be defined as the region $R>2R_{cl}$. From this, we derived
the field star population, which was required to correct the open cluster LFs.
The cluster radii were $R_{cl}=16^{\prime}$ and $19^{\prime}$ for NGC 1245 and
NGC 2506, respectively. We derived the LFs by counting the number of stars in
each bin of magnitude $\Delta m=1$ for the $V$-band photometry of the cluster
region with the correction of field stars as follows:

$N=\frac{N_{cl}}{\Lambda}-\frac{N_{f}}{\Lambda}\frac{A_{cl}}{A_{f}}$ (8)

where $N_{cl}$ is the number of stars for each magnitude bin in the cluster
region, $N_{f}$ is that of the field region, and $A_{cl}$ and $A_{f}$ are the
areas of the cluster and field regions, respectively. Here, we assumed that
all the field regions have the same LF, but this is not usually true because
there are fluctuations in the surface number densities of the field stars as
well as inhomogeneous field star populations.

Figure 9:  Luminosity functions of four field regions around NGC 1245. Figure 10:  Luminosity functions of four field regions around NGC 2506.  Table 5: Dispersions in the LFs of field regions of NGC 1245 and NGC 2506 | Dispersion (stars/(arcmin)2)
---|---
$M_{V}$ | NGC 1245 | NGC 2506
-1.0 | 0.007 | 0.036
0.0 | 0.010 | 0.046
1.0 | 0.004 | 0.041
2.0 | 0.022 | 0.090
3.0 | 0.033 | 0.081
4.0 | 0.039 | 0.117
5.0 | 0.024 | 0.100
6.0 | 0.079 | 0.193
7.0 | 0.137 | 0.166
8.0 | 0.083 | 0.128
9.0 | 0.142 | 0.164
10.0 | 0.459 | 0.135
11.0 | 0.192 | 0.082

In order to take into account the uncertainties arising from the fluctuations
in different directions, we constructed LFs of the field regions surrounding
the clusters (see Fig. 9 and Fig. 10 for NGC 1245 and NGC 2506, respectively).
The LFs of the NGC 1245 field regions were very similar to each other except
for the large difference at $M_{V}\sim 10$. This large difference was caused
partly by the different limiting magnitudes of the individually observed
regions and partly by large photometric errors at the faintest magnitudes.
While the four field LFs were nearly the same in NGC 1245, the field LFs in
the south of NGC 2506 were much different from those in the north of NGC 2506.
More specifically, the number of stars in the southern fields of NGC 2506 was
larger than that in the northern fields. Since NGC 2506 is located in the
north of the Galactic plane, the difference seen in Fig. 10 was due to the
density gradient along the Galactic latitude. Because the LFs of the open
clusters were affected by those of the field regions, the dispersions in the
field LFs are listed in Table 5 as a measure of the uncertainties caused by
the correction. If we consider the errors of the cluster LFs as the
dispersions in the field LFs scaled to the area of the cluster, they become
larger than the Poisson errors, but this seems to be more a meaningful way to
represent the errors in the cluster LFs (Phelps & Janes, 1993).

Figure 11:  Luminosity function of NGC 1245. Error bars represent the
logarithmic dispersion of field LFs scaled to the cluster area.  Figure 12:
Luminosity function of NGC 2506. Error bars represent the logarithmic
dispersion of field LFs scaled to the cluster area.  Figure 13:  LFs compared
with that of the solar neighbourhood. All LFs have been normalized to the
value at $M_{V}$ =4.0

###  4.2 Characteristics of Luminosity Functions

The LFs of NGC 1245 and NGC 2506 are shown in Fig. 11 and Fig. 12\. The
logarithmic error bar was scaled by area using Table 5 as follows:

$e_{+}=log(N+\Delta\times A_{cl})-log(N)\\\ $ (9)
$e_{-}=log(N)-log(N-\Delta\times A_{cl})\\\ $ (10)

Here, $e_{+}$ and $e_{-}$ are the positive and negative errors, respectively,
and $\Delta$ is the LF dispersion of the field region as listed in Table 5.
Our observing limit was $M_{V}\approx 10$, but near $M_{V}\approx 10$ for NGC
1245, the uncertainties increased dramatically because of the different
limiting magnitudes of each observing region.

Stars brighter than $M_{V}\approx 1$ are thought to be turn-off and giant
stars in NGC 1245 and NGC 2506. Comparing the bright parts of the LFs of NGC
1245 and NGC 2506, the number of bright stars in NGC 1245 is larger than that
in NGC 2506. The reason for the larger number of brighter stars in NGC 1245 is
the younger age of NGC 1245. The age of NGC 2506 is nearly twice that of NGC
1245. And the giant clump stars of $M_{V}\approx 0$ are more developed in NGC
2506.

The LF of NGC 1245 is nearly flat. The flat LF of NGC 1245 could be caused by
the dynamical evolution which makes the low-mass stars evaporate from the
cluster because the LF of the cluster is expected to increase monotonically
toward the faint magnitudes to at least $M_{V}\approx 10$. The LF of NGC 2506
is similar to that of NGC 1245, but they do differ. While the LF of NGC 1245
is relatively flat between $M_{V}=3$ and $M_{V}=10$, the LF of NGC 2506 shows
a gradual increase toward fainter magnitudes. It seems unlikely that this
difference is due to the errors in the field star correction because the
dispersions in the field LFs are not large enough to erase the rising tendency
toward the faint magnitudes. It seems worthy to note that the LF of NGC 2506
is derived from a slightly larger area than that of NGC 1245, which may
include regions where evaporating stars are likely to be located, at least
temporarily. The radial dependencies of the LFs of NGC 1245 and NGC 2506 show
that the peak at $M_{V}=10$ in the LF of NGC 2506 is most pronounced in the LF
derived from the outer regions.

In Fig. 13, we compare the LFs of NGC 1245 and NGC 2506 with that of the solar
neighbourhood (Binney & Merrifield, 1998). Because turn off magnitudes are
about $M_{V}\approx 2.0$ and 3.5 for NGC 1245 and NGC 2506, respectively, we
normalized the LFs at $M_{V}=4.0$ to avoid evolutionary effects. In this
figure, we can see a similar global pattern between the solar neighbourhood
and the open clusters, but the number of bright stars in the LFs of NGC 1245
and NGC 2506 is slightly larger than that of the solar neighbourhood, while
the opposite is true for the number of faint stars. We believe this feature is
caused by the evaporation of low-mass stars in NGC 1245 and NGC 2506 because
they are old enough to expect this kind of dynamical evolution and mass
segregation.

Figure 14:  Mass function of NGC 1245. Error bars represent the logarithmic
dispersion of field MFs scaled to the cluster area. The fitting range is
$log(M/M_{\odot})=-0.2\sim 0.3$.  Figure 15:  Mass function of NGC 2506. Error
bars represent the logarithmic dispersion of field MFs scaled to the cluster
area. The fitting range is $log(M/M_{\odot})=-0.2\sim 0.1$.

##  5 Mass Function

The mass function is a useful tool for studying the IMF and to understand the
star formation process in general as well as the structure and evolution of a
stellar system. In order to construct the mass function of NGC 1245 and NGC
2506, we used the mass-luminosity relation of Schaller et al. (1992) for stars
with $M_{V}<7.0$ and that of Henry & McCarthy (1993) for $M_{V}=7$–$17.6$.
Fig. 14 and Fig. 15 show the mass functions of NGC 1245 and NGC 2506 derived
by counting the stars in the mass interval $\Delta log(M/M_{\odot})=0.1$
inside $R_{cl}$. The uncertainties in the mass function were derived from the
dispersions of the field region mass functions, similar to the uncertainties
of the LFs. We derived the slope of mass function

$\Gamma=\frac{dlogN(log(M/M_{\odot}))}{dlog(M/M_{\odot})}.$ (11)

by a least-squares fit, weighted by dispersions in the MFs of field regions.
We obtained $\Gamma=-1.35\pm 0.04$ for NGC 1245 and $\Gamma=-1.26\pm 0.07$ for
NGC 2506 with fitting ranges of $log(M/M_{\odot})=-0.2\sim 0.3$ and
$log(M/M_{\odot})=-0.2\sim 0.1$, respectively for NGC 1245 and NGC 2506.

The derived $\Gamma$ values of the two clusters were slightly shallower than
the slope of the solar neighbourhood IMF: $\Gamma=-1.35$ (Salpeter, 1955). In
the case of NGC 1245, the present $\Gamma$ value is similar to those reported
by Carraro & Patat (1994) and Subramaniam (2003) who derived a similar but
shallower slope than the Salpeter value. The slopes of the mass functions of
NGC 1245 and NGC 2506 were thought not to be the same as the slope of the
respective IMFs because of their age; evaporation of the low-mass stars
results in a shallower slope.

##  6 Summary

We conducted deep and wide $VI$ photometry of old open clusters NGC 1245 and
NGC 2506 based on CCD observations using the CFHT. The present photometry is
suitable for investigating dynamical evolution in open clusters because it is
deep enough to reach down to $M_{V}\approx 10$ and wide enough to cover the
entire regions of the two clusters including the field regions for an
effective correction of the field star contamination.

We derived the physical parameters of the two clusters using detailed
isochrone fittings based on $\chi^{2}$ minimization: $E(B-V)=0.24\pm 0.05$ and
$0.03\pm 0.04$, $(V-M_{V})_{0}=12.25\pm 0.12$ and $12.47\pm 0.08$,
$age(Gyr)=1.08\pm 0.09$ and $2.31\pm 0.16$, and [Fe/H]= $-0.08\pm 0.06$ and
$-0.24\pm 0.06$ for NGC 1245 and NGC 2506, respectively. The quoted errors are
the total errors which are quadratic sum of statistical errors associated with
the $\chi^{2}_{min}$ and systematic errors related to the assumed parameters
and theoretical models. The present estimates of the physical parameters of
two clusters are in good agreement with the previous estimates.

We also derived the LFs of NGC 1245 and NGC 2506. The LF of NGC 1245 shows a
flat profile between $M_{V}=3$ and $M_{V}=10$, whereas the LF of NGC 2506
displays a slight rise. This difference seems to be not due to the errors
related to the field star correction but due to the dynamical structures,
which can differ because dynamical evolution can be different for clusters of
a similar age as it is affected by the environment as well as by the internal
properties of the cluster. We derived the mass functions of NGC 1245 and NGC
2506, which show slightly shallower slopes than that of the solar
neighbourhood IMF. The shallower slope is understandable if we consider that
the derived mass function is the present day mass function, which is thought
to be different from the IMF of the clusters due to dynamical evolution, since
the ages of NGC 1245 and NGC 2506 are old enough for mass segregation and
evaporation of low-mass stars. We will discuss the dynamical properties,
structures and halo of these two clusters in a forthcoming paper.

## Acknowledgments

This work was supported for two years by Pusan National University Research
Grant. YWK was supported in part by the research grant from KASI (Korea
Astronomy and Space Science Institute).

## References

  * Binney & Merrifield (1998) Binney, J., Merrifield, M. 1998, Galactic Astronomy (Princeton : Princeton Univ. Press)
  * Burke et al. (2004) Burke C. J., Gaudi B. S., DePoy D. L., Pogge R. W., Pinsonneault M. H., 2004, AJ, 127, 2382
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
  * Carraro & Patat (1994) Carraro G., Patat F., 1994, A&A, 289, 397
  * Chiu & van Altena (1981) Chiu, L.-T. G., van Altena, W. F., 1981, ApJ, 243, 841
  * Demarque et al. (2004) Demarque, P., Woo, J.-H., Kim, Y.-C., Yi, S. K., 2004, ApJS,155, 667
  * Girardi et al. (2000) Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000, A&A, 141, 371
  * Grocholski & Sarajedini (2003) Grocholski, A. J., Sarajedini, A. 2003, MNRAS, 345, 1015
  * Henderson et al. (2007) Henderson C., B., Deliyannis C. P., Hughto J., Simmons A., Croxall K., Sarajedini A., Platais I., 2007, AAS, 211, 5819
  * Henry & McCarthy (1993) Henry, T, J., McCarthy, D, W., Jr., 1993, AJ, 106, 773
  * Janes (1979) Janes, K. A., 1979, ApJS, 39, 135
  * Janes & Hoq (2011) Janes, K. A., Hoq, S. 2011, AJ, 141, 92
  * Landolt (1992) Landolt A. U., 1992, AJ, 104, 340
  * Lyngå (1987) Lyngå, G., 1987, Catalogue of Open CLuster Data (5th ed.; Strasbourg: CDS)
  * Marconi et al. (1997) Marconi G., Hamilton D., Tosi M., Bragaglia A., 1997, MNRAS, 291, 763
  * Peruanskii & Ryadchenko (1978) Peruanskii S. S., Ryadchenko V. P., 1978, SvA, 22, 34
  * Phelps & Janes (1993) Phelps, R. L., Janes, K. A., 1993, AJ, 106, 1870
  * Purgathofer (1964) Purgathofer A., 1964, Zeitschrift fu”r Astrophysik, 59, 79
  * Salpeter (1955) Salpeter, E. E., 1955, ApJ, 121, 161
  * Schaller et al. (1992) Schaller, G., Schaerer, D., Meynet, G., Maeder, A, 1992, A&AS, 96, 269
  * Stetson (1987) Stetson P. B., 1987, PASP, 99, 191
  * Subramaniam (2003) Subramaniam A., 2003, BASI, 31, 49
  * Wee & Lee (1996) Wee S.-O., Lee M. G., 1996, JKAS, 29, 181