ON GENERALIZED HARTSHORNE’S CONJECTURE AND LOCAL
COHOMOLOGY MODULES

arXiv:1803.07546v1 [] 20 Mar 2018

T. H. FREITAS, V. H. JORGE PÉREZ, AND L. C. MERIGHE*
Abstract. Let a denote an ideal of a commutative Noetherian ring R. Let M and N be
two R-modules. In this paper, we give partial answers on the extension of Hartshorne’s
conjecture about the cofiniteness of torsion and extension functors. For this purpose, we
study the cofiniteness of the generalized local cohomology module Hia (M, N ) for a new class
of modules, called a-weakly finite modules, in the local and non-local case. Furthermore, we
derive some results on attached primes of top generalized local cohomology modules.

1. Introduction
Throughout this paper, let a and b be ideals of a commutative Noetherian ring with
nonzero identity R. If M and N are two R-modules and i ∈ Z, consider
Hia (M, N) := lim ExtiR (M/an M, N),
−→
n

the ith generalized local cohomology module with respect to a and R-modules M and N,
introduced by Herzog [19]. If M = R the definition of generalized local cohomology modules
reduces to the notion introduced by Grothendieck of local cohomology modules Hia (N) [6].
One of the interesting open questions within the local cohomology theory is the following:
Conjecture: When is ExtiR (R/a, Hja (N)) finite for all integer i and j?
This question was proposed by Hartshorne [18] who, in turn, was motivated by the conjecture made by Grothendieck [16] about the finiteness of HomR (R/a, Hja (N)). Also, Hartshorne
introduced an interesting class of modules called a-cofinite modules. Recall that an R-module
M is said to be a-cofinite if Supp(M) ⊆ V (a) and ExtiR (R/a, M) is finite for all i.
In this sense, as a generalization of Hartshorne’s conjecture, we have a natural question.
Question 1: When is Hia (M, N) a-cofinite for all i?
Concerning this question, several results were obtained and in most of them, the finiteness
assumption on the modules is crucial in the proof (for example [8], [15] and [27]).
Date: March 21, 2018.
2010 Mathematics Subject Classification. 13D45, 13H10, 13H15, 13E10.
Key words and phrases. Torsion functors, Local Cohomology, cofinite module, Artinian module, Attached
prime.
The first and second author was partially supported CNPq-Brazil 421440/2016-3.
The third author was partially supported by CNPq-Brazil - Grants 142376/2016-7.
1

In this paper, we will introduce a new class of modules, called a-weakly finite modules
over M. This class is an extension of the class of weakly finite modules introduced in [3]
(which contains the class of finitely generated modules, big Cohen-Macaulay modules and
a-cofinite modules). In the local and non-local case we will show the following two results,
that improves [9, Theorem 2.2] and [14, Theorem 2.2] respectively.
Theorem 1.1. Suppose that a ⊆ b and dimR R/b = 0. Let M be a finitely generated
R-module and N be an a-weakly finite R-module over M. Then Hib (M, N) is an Artinian
R-module and a and b-cofinite, for all i ∈ N.
Theorem 1.2. Let (R, m) be a local ring. Consider M, N two R-modules such that M is
finitely generated of finite projective dimension (pd M = d < ∞), N is b-weakly finite over
M and dimR N = n < ∞. Suppose that a ⊆ b. Then Hd+n
(M, N) is Artinian and b-cofinite.
a
In particular, if a = b, then Hd+n
(M,
N)
is
Artinian
and
a-cofinite.
a
Another aim of the present paper is to show some results concerning to attached primes of
generalized local cohomology modules Hia (M, N). More precisely, we will show an extension
of the main result in [10].
Theorem 1.3. Let M be a finitely generated R-module such that pd M < ∞. Let N be an
R-module such that dimR N = dimR R = n. Then Hd+n
(M, N) has secondary representation
a
and
(M, N)) ⊆ {p ∈ AssR (N) | cd(a, M, R/p) = d + n}.
AttR (Hd+n
a
Returning to Question 1, Melkersson [26, Theorem 2.1] has shown that ExtjR (R/a, M) is
finite for all j if and only if TorR
j (R/a, M) is finite for all j. We can analyze the Question 1
on a more general point of view.
Question 2: When are ExtiR (M, N) and TorR
i (M, N) a-cofinite (or finite length, a-cominimax,
or a-weakly cofinite) for all (or for some) integer i ?
Note that a-cominimax [2] and a-weakly cofinite modules ([11] and [12]) are extensions of
the definition of a-cofinite modules. We may also ask, as a particular case of the above
i
question, when ExtjR (L, Hia (M, N)) and TorR
j (L, Ha (M, N)) are a-cofinite (or finite length,
or a-cominimax, a-weakly cofinite) for all (or for some) integers i and j, and some R-module
L. In this sense, we show the following result.
Theorem 1.4. Let H and M be R-modules such that H is Artinian and a-cofinite and M
is minimax. Then, for each i ≥ 0, the module ExtiR (M, H) is minimax and b
a-cofinite over
b
R.

One of the main goals of this paper is to generalizes the results showed by Naguipour,
Bahmanpour and Khalili [4]. In this direction we show the following theorem:

Theorem 1.5. Let N be a nonzero a-cominimax R-module and M be a finitely generated
R-module.
(i) If dimR M = 1, then the R-module TorR
i (M, N) is Artinian and a-cofinite for all
i ≥ 0.
2

(ii) If dimR M = 2, then the R-module TorR
i (M, N) is a-cofinite for all i ≥ 0.
For larger dimensions we obtain the following result.
Theorem 1.6. Let (R, m) be a local ring and a an R-ideal. Let N be a nonzero a-cominimax
R-module and M be a finitely generated R-module. Then the R-module TorR
i (M, N) is aweakly cofinite for all i ≥ 0 when one of the following cases holds:
(i) dimR N ≤ 2.
(ii) dimR M = 3.
This paper is organized as follows. In Section 2, we discuss Question 1, define the class
of a-weakly finite modules over M and with the purpose to show Theorem 1.1 and Theorem
1.2.
In Section 3, we obtain some results on attached primes of generalized local cohomology
in order to prove our second main result (Theorem 1.3).
In order to understand the cofiniteness behavior of extension and torsion functors, in
Section 4 we show some preliminary results that will be useful for this purpose. Also, we
recall the definitions of minimax, weakly Laskerian modules, cominimax and weakly cofinite
modules. This notions improves the definitions of Noetherian (and Artinian) modules and
cofinite modules.
In last section, we show some cases where the torsion functor TorR
i (M, N) is cofinite or
weakly cofinite module for all i. As applications of the results showed in the Section 2, we
i
study the behavior of ExtjR (L, Hia (M, N)) and TorR
j (L, Ha (M, N)), for some R-module L.
For conventions of the notation, basic results, and terminology not given in this paper,
the reader should consult the books Brodmann-Sharp [6] and Matsumura [22].
2. Cofiniteness and Artinianess of Generalized Local Cohomology Modules
In this section, we fix our notation and list several results for the convenience of the reader.
Let R be a commutative Noetherian ring. Throughout this paper, unless otherwise noted,
the R-module N is not necessarily finitely generated and we understand the R-dimension of
N, denoted by dimR N, by dimR N := sup{dimR R/p | p ∈ SuppR (N)}.
Definition 2.1. Let S be the largest class of R-modules satisfying the following four properties:
1. If N ∈ S, then HomR (R/a, N) and HomR (M/aM, N) are finitely generated, where
M is an R-module.
2. If N is a non-zero element of S and x is a regular element, then N/xN ∈ S is
non-zero and dimR N/xN = dimR N − 1.
3. If N ∈ S, then |AssR (N)| < ∞.
4. If N ∈ S, then N/Γa (N) ∈ S.
We say an R-module N is a-weakly finite over M, if it belongs to S.
If M = R, we say that the R-module is a-weakly finite. If M = R, (R, m) is a local ring
and a = m we just say that the R-module is weakly finite (see [3, Definition 2.1]).
Note that if M is a finitely generated R-module, then M is a weakly finite R-module.
Another important class of modules are the cofinite modules. Recall that a module N is
said to be a-cofinite if SuppR (N) ⊆ V (a) and ExtiR (R/a, N) is a finite module for all i ∈ Z
3

[18]. With this notion, Bagheri ([3, Lemma 2.1]) has shown, in the local case, that the class
of cofinite modules is contained in the class of weakly finite modules. Our next proposition
shows a similar behaviour for a-weakly finite modules and the proof is analogous to what
was done in [3, Lemma 2.1].
Proposition 2.2. Let (R, m) be a local ring and let M be a non-zero R-module of dimension
n > 0. If M is a-cofinite, then M is a-weakly finite.
Also, Bagheri [3, Theorem 2.1] has shown in the local case that the local cohomology
module Him (N) is Artinian when N is a weakly finite R-module. Our main goal in this
section is to generalize this result in the non-local case (Theorem 2.9). To do this, we will
need some previous results.
Lemma 2.3 ([26, Corollary 4.4]). The class of Artinian a-cofinite modules is closed under
taking submodules, quotients and extensions, i.e., it is a Serre subcategory of the category of
R-modules.
Proposition 2.4. Let M be a finitely generated R-module and N ∈ S any R-module. Then
for all i ≥ 0, ExtiR (M, N) and TorR
i (M, N) are elements of S.
Proof. The proof follows from the definition of extension and torsion functors.



Lemma 2.5 ([14, Lemma 2.1(i)]). If M is a finitely generated R-module, then
Hi (M, Γa (N)) ∼
= Exti (M, Γa (N)).
R

a

Lemma 2.6 ([6, Lemma 2.1.1]). If T is an R-module such that |AssR (T )| < ∞, then T is
a-torsion free if and only if there is an element r ∈ a such that r is T -regular.
Proposition 2.7 ([26, Proposition 4.1]). Let M be a module with support in V (a). Then
M is Artinian and a-cofinite if and only if (0 :M a) has finite length. If there is an element
x ∈ a such that (0 :M x) is Artinian and a-cofinite, then M is Artinian and a-cofinite.
Lemma 2.8. Suppose that a ⊆ b and dimR R/b = 0. Let N be an a-weakly finite R-module.
Then Γb (N) is Artinian and a and b-cofinite.
Proof. Note that HomR (R/a, N) = (0 :N a) is finitely generated, since N is a-weakly finite.
By hypothesis, a ⊆ b then HomR (R/b, N) = (0 :N b) ⊆ (0 :N a) = HomR (R/a, N) and, since
R is a Noetherian ring, HomR (R/b, N) is finitely generated. Moreover, applying functors
HomR (R/a, −) and HomR (R/b, −) in the exact sequence
0 → Γb (N) → N
and using the same argument, we conclude HomR (R/a, Γb(N)) and HomR (R/b, Γb (N)) are
finitely generated.
Since dimR R/b = 0, it follows that HomR (R/b, Γb (N)) = (0 :Γb (N ) b) has finite length.
Thus by Proposition 2.7, Γb (N) is Artinian and b-cofinite. On the other hand,
Γb (N) = (0 :Γb (N ) b) ⊆ (0 :Γb (N ) a) ⊆ Γb (N)
thus (0 :Γb (N ) a) = Γb (N) = (0 :Γb (N ) b) which has finite length. Then, again by Proposition
2.7, Γb (N) is Artinian and a-cofinite. Therefore, Γb (N) is Artinian and a and b-cofinite. 
Now we are able to show the first main result of this section.
4

Theorem 2.9. Suppose that a ⊆ b and dimR R/b = 0. Let M be a finitely generated Rmodule and let N be a a-weakly finite R-module over M. Then Hib (M, N) is an Artinian
R-module and a and b-cofinite, for all i ∈ N.
Proof. We use induction on i. Assume that i = 0. We claim H0b (M, N) is Artinian and a
and b-cofinite. Since H0b (M, N) ∼
= HomR (M, Γb (N)) the assertion is trivial by Lemma 2.8
and Proposition 2.4.
Now, assume i > 0. By induction, Hjb (M, N) is Artinian and a and b-cofinite, for j < i.
Consider the exact sequence
Hib (M, Γb (N)) → Hib (M, N) → Hib (M, N/Γb (N)) .
By Lemma 2.8, Γb (N) is Artinian and a and b-cofinite. Since M is finitely generated, by
Proposition 2.4 and Lemma 2.5, Hib (M, Γb (N)) is Artinian and a and b-cofinite. Therefore,
Hib (M, N) is Artinian and a and b-cofinite if and only if Hib (M, N/Γb (N)) is Artinian and
a and b-cofinite. Hence we may assume that Γb (N) = 0 and so, by Lemma 2.6, there is an
element r ∈ b which is N-regular.
The exact sequence
·r

0 → N → N → N/rN → 0
induces the following exact sequence
·r

Hi−1
(M, N/rN) −→ Hib (M, N) −→ Hib (M, N).
b
Since N/rN is a-weakly finite, by induction Hi−1
(M, N/rN) is Artinian and a and b-cofinite.
b
Thus, by Lemma 2.3, (0 :Hib (M,N ) r) is Artinian and a and b-cofinite. Therefore, by Proposition 2.7, Hib (M, N) is Artinian and a and b-cofinite.

Corollary 2.10. Let (R, m) be a local ring. Let M be a finitely generated R-module and N
be a weakly finite R-module over M. Then Him (M, N) is Artinian for all i ∈ N.
Corollary 2.11 ([3, Theorem 2.1]). Let (R, m) be a local ring and let N be a weakly finite
R-module. Then Him (N) is Artinian for all i ∈ N.
Corollary 2.12. Suppose that a ⊆ b and dimR R/b = 0. Let M be a finitely generated
R-module and N be a a-weakly finite R-module over M. Then ExtjR (R/a, Hib (M, N)) and
ExtjR (R/b, Hib(M, N)) have finite length for all i and j in N.
Proof. By Theorem 2.9, Hib (M, N) is Artinian and a and b-cofinite. So, ExtjR (R/a, Hib (M, N))
and ExtjR (R/b, Hib (M, N)) are finitely generated and, since R/a and R/b are finitely generated, by Proposition 2.4, ExtjR (R/a, Hib (M, N)) and ExtjR (R/b, Hib (M, N)) are Artinian.
Therefore, ExtjR (R/a, Hib (M, N)) and ExtjR (R/b, Hib (M, N)) have finite length.

As a consequence of this corollary, we have the following result.
Corollary 2.13. Let (R, m) be a local ring. Suppose a ⊆ b such that dimR R/b = 0. Let
M be a finitely generated R-module and N be a a-weakly finite R-module over M. Then
AttR (ExtjR (R/a, Hib (M, N))) = {m} = AttR (ExtjR (R/b, Hib (M, N))).
5

Proof. We show that AttR (ExtjR (R/a, Hib (M, N))) = {m}, the other one is analogous. By
Corollary 2.12, ExtjR (R/a, Hib (M, N)) has finite length, so SuppR (ExtjR (R/a, Hib (M, N))) =
{m}.
Now, by [28, Proposition 2.9 (2) and (3)], AttR (ExtjR (R/a, Hib(M, N))) ⊆ {m}. Therefore
AttR (ExtjR (R/a, Hib (M, N))) = {m}, since AttR (ExtjR (R/a, Hib(M, N))) 6= ∅.

The local case. From now on, in this section, we will assume that our ring R is local with
maximal ideal m.
Theorem 2.14. Let M be a finitely generated R-module such that pd M = d < ∞ and let
N be a b-weakly finite R-module over M such that dimR N = n < ∞. Suppose a ⊆ b. Then
Hd+n
(M, N) is Artinian and b-cofinite.
a
In particular, if a = b, Hd+n
(M, N) is Artinian and a-cofinite.
a
Proof. We may choose x ∈ m \ a. By [13, Lemma 3.1], there is an exact sequence
d+n
Hd+n
(M, N) −→ Hd+n
a+xR (M, N) −→ Ha
aRx (Mx , Nx ),

where Nx is the localization of N at {xi | i ≥ 0}. Note that dimRx (Nx ) < n, so Hd+n
aRx (Mx , Nx ) =
d+n
d+n
0. Thus there is an epimorphism Ha+xR (M, N) −→ Ha (M, N). Now the assertion follows
by assuming m = a + (x1 , . . . , xr ) and applying the argument for finite steps, since R is
Noetherian ring. Therefore
φ

d+n
0 −→ ker φ −→ Hd+n
(M, N) −→ 0
m (M, N) −→ Ha

is a short exact sequence. Now, from Theorem 2.9, Hd+n
m (M, N) is Artinian and b-cofinite.
Therefore Hd+n
(M,
N)
is
Artinian
and
b-cofinite.

a
Corollary 2.15. Let N be a b-weakly finite R-module such that dimR N = n < ∞. Suppose
a ⊆ b. Then Hna (N) is Artinian and b-cofinite.
In particular, if a = b, Hna (N) is Artinian and a-cofinite.
The next result give us an important isomorphism using a a-weakly finite module. By
Corollary 2.15, the proof follows analogously to [17, Proposition 2.2].
Proposition 2.16. Let M and N be R-modules such that M is finitely generated and N is
a-weakly finite over M. Assume pd M = d < ∞ and dimR N = n < ∞. Also, assume n ∈ N
and x1 , . . . , xn is an a -filter regular sequence on N. Then
1. Hd+n
(M, N) ∼
= ExtdR (M, Hna (N)).
a
2. AttR (Hd+n
(M, N)) ⊆ AttR (Hna (N)).
a
Proposition 2.17. Let M and N be R-modules such that M is finitely generated and N
is weakly finite. Assume pd M = d < ∞, dimR R = dimR N = n and 0 < n < ∞. If
Hd+n
m (M, N) 6= 0, then it is not finitely generated.
d+n
Proof. As Hd+n
m (M, N) 6= 0, then AttR (Hm (M, N)) 6= ∅. By Proposition 2.16 (2) and (3.1),
n
AttR (Hd+n
m (M, N)) ⊆ AttR (Hm (N)) ⊆ {p ∈ AssR (N) | dimR R/p = n}.
d+n
Since n > 0, we have AttR (Hd+n
m (M, N)) * {m}. Since Hm (M, N) is Artinian, it follows
d+n
that Hm (M, N) is not finitely generated by [6, Corollary 7.2.12].


6

3. Attached primes of the top generalized local cohomology modules
In this section, we assume that (R, m) is a commutative Noetherian local ring with maximal
ideal m. In [10, Theorem B], M. T. Dibaei and S. Yassemi have shown that, if dimR N =
dimR R = n, then Hna (N) has a secondary representation and
AttR (Hna (N)) ⊆ {p ∈ AssR (N) | cd(a, R/p) = n}.

(3.1)

The main purpose of this section is to generalize this result for generalized local cohomology
modules. In order to do this, we give some preliminaries result.
Lemma 3.1. Let M be finitely generated R-module and N = lim Nj , where {Nj | j ∈ J} is
−→
j

a family of finitely generated R-modules. Then
Hia (M, N) = lim Hia (M, Nj ).
−→
j

Proof. Since M is finitely generated, M/an M is finitely generated. Then, by [7, Theorem 1],
it is possible to commute direct limit and the functor ExtiR (M/an M, −). Therefore
Hia (M, N) = lim ExtiR (M/an M, N)
−→
n

= lim ExtiR (M/an M, lim Nj )
−→
−→
n

=

j

lim(lim ExtiR (M/an M, Nj ))
−→ −→
n
j

= lim Hia (M, Nj ).
−→
j


Lemma 3.2 ([30, Lemma 1.2]). If M is an R-module with AttR (M) = {p} where p is a
minimal prime of R, then M is secondary.
Proposition 3.3. Let M be a finitely generated R-module such that pd M = d < ∞ and
dimR N = n < ∞. Consider L ⊆ N be an arbitrary R-submodule. If AssR (L) is a finite set
and AttR (Hd+n
(M, L)) ⊆ AssR (L), then Hd+n
(M, L) has a secondary representation, for all
a
a
R-submodule L ⊆ N , in particular for L = N.
Proof. Let AssR (N) = {p1 , . . . , pl }. We proceed by induction on l.
If l = 1, then ∅ 6= AttR (Hd+n
(M, N)) ⊆ {p1 }. Therefore Hd+n
(M, N) is a p1 -secondary
a
a
R-module, by Lemma 3.2.
Now assume l > 1. By [5, Proposition 2 - p.263], for each i, 1 ≤ i ≤ l, there exist Li ⊆ N
submodule such that AssR (Li ) = {pi } and AssR (N/Li ) = AssR (N) \ {pi }. By hypothesis,
AttR (Hd+n
(M, Li )) ⊆ AssR (Li ) = {pi }
a

and

AttR (Hd+n
(M, N/Li )) ⊆ AssR (N) \ {pi }.
a

Thus AttR (Hd+n
(M, Li ) is pi -secondary or zero and AttR (Hd+n
(M, N/Li )) has secondary
a
a
representation by induction.
By the exact sequence
ϕi

AttR (Hd+n
(M, Li )) −→ AttR (Hd+n
(M, N)) −→ AttR (Hd+n
(M, N/Li )) −→ 0,
a
a
a
7

we obtain that ϕi (AttR (Hd+n
(M, Li ))) is pi -secondary or zero. If ϕi (AttR (Hd+n
(M, Li ))) = 0
a
a
d+n
d+n
for some i, then AttR (Ha (M, N)) ∼
Att
(H
(M,
N/L
))
which
has
secondary
represen=
R
i
a
d+n
tation. So we may assume ϕi (AttR (Ha (M, Li ))) 6= 0 for all i = 1, . . . , l. Hence
AttR (Hd+n
(M, N))/
a

l
X

ϕi (AttR (Hd+n
(M, Li )))
a

⊆

i=1

l
\

AttR (Hd+n
(M, N)/ϕi (Hd+n
(M, Li )))
a
a

i=1

=

l
\

AttR (Hd+n
(M, Li )) = ∅.
a

i=1

Therefore
Hd+n
(M, N) =
a

l
X

ϕi (Hd+n
(M, Li ))
a

i=1

which is secondary representation for Hd+n
(M, N).
a



The next theorem is the main result of this section.
Theorem 3.4. Let M be a finitely generated R-module such that pd M = d < ∞. Let
dimR N = dimR R = n. Then Hd+n
(M, N) has secondary representation and
a
AttR (Hd+n
(M, N)) ⊆ {p ∈ AssR (N) | cd(a, M, R/p) = d + n}.
a
Proof. Firstly we will show the inclusion of the set of attached primes. We can assume
Hd+n
(M, N) 6= 0.
a
Let X = {p ∈ AssR (N) | cd(a, M, R/p) = d + n}. By [5, Proposition 2 - p.263], there
exist L ⊆ N submodule such that AssR (N/L) = X and AssR (L) = AssR (N) \ X.
Consider the exact sequence
Hd+n
(M, L) −→ Hd+n
(M, N) −→ Hd+n
(M, N/L) −→ Had+n+1 (M, L).
a
a
a
Then Had+n+1 (M, L) = 0, since n + 1 > dimR L. On the other hand, by Lemma 3.1,
Hd+n
(M, L) = lim Hd+n
(M, Lj ), where {Lj | j ∈ I} is a family of finite submodules of
a
−→ a
j

N.
We claim that Hd+n
(M, Lj ) = 0, for all j ∈ I. If dimR Lj = lj < n, then d + n > d + lj
a
d+n
and therefore Ha (M, Lj ) = 0. Now, if dimR Lj = n, then
AttR (Hd+n
(M, Lj )) = {p ∈ AssR (Lj ) | cd(a, M, R/p) = d + n}.
a
On the other hand, as Lj is a submodule of L and AssR (L) = AssR (N/X), we have AssR (Lj )∩
X = ∅. Then, Hd+n
(M, Lj ) = 0 and we can conclude Hd+n
(M, L) = 0.
a
a
d+n
d+n
∼
Therefore, Ha (M, N) = Ha (M, N/L) and we can consider N/L instead of N. Thus
AssR (N) = AssR (N/L) = X = {p ∈ AssR (N) | cd(a, M, R/p) = n + d}.
[
Now, we claim AttR (Hd+n
(M, N)) ⊆ AssR (N). If r ∈
/
p, then the short exact
a
p∈Ass(N )

sequence
·r

0 −→ N −→ N −→ N/rN −→ 0
induces the exact sequence
·r

0 −→ Hd+n
(M, N) −→ Hd+n
(M, N) −→ Hd+n
(M, N/rN).
a
a
a
8

Since cd(a, M, N/rN) < n[
+ d, we have Hd+n
(M, N/rN)
rHd+n
(M, N) =
a
a
[ = 0. Then [
d+n
Ha (M, N) and r ∈
/
p. Therefore
p⊆
p.
p∈AttR (Had+n (M,N ))

p∈AttR (Had+n (M,N ))

p∈AssR (N )

Note that |AssR (N)| < ∞, since in this case, AssR (N) ⊆ AssR (R), and so if p ∈ AssR (N)
we have that
n + d = cd(a, M, R/p) ≤ pd(M) + dimR R/p ≤ pd(M) + dimR R = n + d,
which implies, dimR R/p = n. Thus p is minimal over the ideal 0 and p ∈ AssR (R), which is
a finite set, since R is Noetherian.
In this way, if p ∈ AttR (Hd+n
(M, N)), then p ⊆ q, for any q ∈ AssR (N). On the other
a
hand, since
cd(a, M, R/p) ≤ pd(M) + dimR R/p ≤ pd(M) + dimR R = d + n
and
n + d = cd(a, M, R/q) ≤ cd(a, M, R/p) ≤ n + d
we have p = q. Therefore, AttR (Hd+n
(M, N)) ⊆ AssR (N).
a
The secondary representation of AttR (Hd+n
(M, N)) follows immediately using Proposition
a
3.3 and the first part of this proof.

The inclusion showed in the previous theorem is strict in general (see [10, Example 6]).
But, if M and N are finitely generated R-modules such that pd M = d < ∞ and dimR N =
n < ∞, the equality holds [17, Theorem 2.3].
4. Torsion and Extension Functors: Some Preparatory Results
Now, we are interested on the cofiniteness of torsion and extension functors. Firstly we
will recall some definitions that will be used in for the rest of the paper.
(i) An R-module M is called minimax, if there is a finitely generated submodule N of
M such that M/N is Artinian [32].
(ii) An R-module M is said to be a-weakly cofinite if Supp(M) ⊆ V (a) and ExtjR (R/a, M)
is minimax for all j ([11] and [12]).
(iii) An R-module M is said to be weakly Laskerian, if the set of associated primes of any
quotient module of M is finite ([11] and [12]).
(iv) An R-module M is said to be a-cominimax if Supp(M) ⊆ V (a) and ExtjR (R/a, M)
is weakly Laskerian for all j [2].
Now we will recall some facts of minimax modules.
Remark 4.1. The following statements hold:
(i) Noetherian modules are minimax, as are Artinian modules.
(ii) The set of associated primes of any minimax R-module is finite.
For our first results, we will use the Serre subcategory of the category of R-modules, which
is a class of R-modules closed under taking submodules, quotients and extensions. We denote
by S for a Serre subcategory of a category of R-modules. Recall that the class of Noetherian
modules, Artinian modules, minimax modules, weakly Laskerian and matlis reflexive are
examples of Serre subcategories.
9

Lemma 4.2. Let M be a finitely generated R-module and N be an arbitray module. Let t
R
be a non-negative integer such that TorR
i (M, N) ∈ S for all i ≤ t. Then Tori (L, N) ∈ S for
all i ≤ t, when L is a finitely generated R-module such that SuppR (L) ⊆ SuppR (M).
Proof. Since SuppR (L) ⊆ SuppR (M), there exists a chain of R-modules
0 = L0 ⊂ L1 ⊂ · · · ⊂ Lk = L
such that the factors Lj /Lj−1 are homomorphic images of a direct sum of finitely many
copies of M (by Grusons Theorem, [31, Theorem 4.1]).
Consider the exact sequences
0 → K → M n → L1 → 0
0 → L1 → L2 → L2 /L1 → 0
..
.
0 → Lk1 → Lk → Lk /Lk−1 → 0
for some positive integer n and some finitely generated R-module K.
Now, from the long exact sequence
R
R
R
· · · → TorR
i+1 (Lj /Lj−1 , N) → Tori (Lj−1 , N) → Tori (Lj , N) → Tori (Lj /Lj−1 , N) → · · ·
R
and using the properties of Serre subcategory, TorR
i (Lj , N) ∈ S if, and only if, Tori (Lj−1 , N) ∈
S. In this sense, using an easy induction on k, it suffices to prove the case when k = 1.
So, consider the exact sequence mentioned above

0 → K → M n → L → 0.

(4.1)

We now will use induction in t.
If t = 0, we have that L ⊗R N is a submodule of M n ⊗R N which belongs to S. Then
L ⊗R N ∈ S.
′
′
Now, lets assume t > 0 and TorR
j (L , N) ∈ S for every finitely generated R-module L
with SuppR (L′ ) ⊆ SuppR (M) and all j < t. The exact sequence (4.1) induces the long exact
sequence
R
R
n
· · · → TorR
i (M , N) → Tori (L, N) → Tori−1 (K, N) → · · ·
so that, by the inductive hypothesis, TorR
i−1 (K, N) ∈ S for all i ≤ t. On the other hand,
n
M
n
∼
TorR
TorR
i (M, N) ∈ S.
i (M , N) =
Therefore, TorR
i (L, N) for all i ≤ t, and the proof is complete.



Lemma 4.3. Let M be a finitely generated R-module and N be an arbitray module. Let t
be a non-negative integer such that ExtiR (M, N) ∈ S for all i ≤ t. Then ExtiR (L, N) ∈ S for
all i ≤ t, when L is a finitely generated R-module such that SuppR (L) ⊆ SuppR (M).
Proof. The proof follows by a similar way to what was done in the previous lemma.



The next theorem is a key ingredient for the rest of the paper.
Theorem 4.4. Let t be a non-negative integer. Then, for an arbitrary R-module N, the
following conditions are equivalent:
(i) TorR
i (R/a, N) ∈ S for all i ≤ t.
10

(ii) For any finitely generated R-module M with SuppR (M) ⊆ V (a), TorR
i (M, N) ∈ S
for all i ≤ t.
(iii) For any R-ideal b with a ⊆ b, TorR
i (R/b, N) ∈ S for all i ≤ t.
(iv) For any minimal prime p over a, TorR
i (R/p, N) ∈ S for all i ≤ t.
(v) ExtiR (R/a, N) ∈ S for all i ≤ t.
(vi) For any finitely generated R-module M with SuppR (M) ⊆ V (a), ExtiR (M, N) ∈ S
for all i ≤ t.
(vii) For any R-ideal b with a ⊆ b, ExtiR (R/b, N) ∈ S for all i ≤ t.
(viii) For any minimal prime p over a, ExtiR (R/p, N) ∈ S for all i ≤ t.
Proof. (i)⇒(ii) It follows from Lemma 4.2, since SuppR (R/a) = V (a).
(ii)⇒(iii) Take M = R/b and observe that SuppR (R/b) = V (b) ⊆ V (a).
(iii)⇒(iv) Immediate.
(iv)⇒(i) Let p1 , . . . , pn be the minimal
primes of a. Then, by assumption,
TorR
i (R/pj , N) ∈
L
Ln
n
R
R
∼
S for all j = 1, . . . , n. Hence
j=1 R/pj , N) ∈ S. Since
j=1 Tori (R/pj , N) = Tori (
Ln
SuppR (R/a) = SuppR ( j=1 R/pj ), it follows by Lemma 4.2 that TorR
i (R/a, N) ∈ S for all
i ≤ t, as required.
(v)⇒(vi)⇒(vii)⇒(viii)⇒(v) It follows in a similar way to what was done previously, using
Lemma 4.3.
(i)⇔(v) The proof is the same as that in [26, Theorem 2.1]

In particular, for the class of minimax modules we obtain the following result.
Corollary 4.5. Let a be an ideal of R such that dimR R/a = 1 and let t be a non-negative
integer. Then, for an arbitrary R-module N, the following conditions are equivalent:
(i) ExtiR (R/a, N) is minimax for all i ≤ t.
(ii) TorR
i (R/a, N) is minimax for all i ≤ t.
(iii) The Bass number µi (p, N) is finite for all p ∈ V (a) for all i ≤ t.
(iv) The Betti number β i (p, N) is finite for all p ∈ V (a) for all i ≤ t.
(v) Hai (N) is a-cominimax, for all integer i.
Proof. The proof follows by Theorem 4.4, [20, Theorem 3.3] and [1, Corollary 2.5].



Lemma 4.6. Let M be a minimax R-module. If there is n ∈ N and m1 , . . . , ms maximal
ideals of R such that m1 m2 · · · ms M = 0, then M has finite length.
Proof. The proof follows by the short exact sequence 0 → L → M → K → 0, where L is
finitely generated and K is an Artinian R-module, since M is minimax.

5. Cofiniteness of torsion and Extension functors
Our purpose in this section is to give partial answers about Question 2 made in the
introduction. The reader can compare the next result with [21, Theorem 2.3].
Theorem 5.1. Let H and N be R-modules such that H is Artinian and a-cofinite and N is
b
minimax. Then, for each i ≥ 0, the module ExtiR (N, H) is minimax and b
a-cofinite over R.

Proof. Since N is a minimax module, there exist L ⊆ N submodule such that L is finitely
generated and N/L is Artinian. Fix such L and consider the exact sequence
0 → L → N → N/L → 0.
11

Thus, we have a long exact sequence
· · · → ExtiR (N/L, H) → ExtiR (N, H) → ExtiR (L, H) → · · · .
By Proposition 2.4, since Artinian and a-cofinite modules are a Serre subcategory, we
obtain that ExtiR (L, H) is Artinian and a-cofinite. Then is also Artinian and b
a-cofinite over
i
b Thus Ext (L, H) is minimax and b
b
R.
a-cofinite over R.
R
i
b Thus
On the other hand, by [21, Corollary 2.3], ExtR (N/L, H) is a Noetherian over R.
b
ExtiR (N/L, H) is minimax and b
a-cofinite over R.
Therefore, since the class of b
a-cofinite minimax modules is a Serre subcategory [26, Corolb
lary 4.4], ExtiR (N, H) is minimax and b
a-cofinite over R.

The next two results partially answer a generalization of Hartshorne’s conjecture.

Corollary 5.2. Let R be a complete ring and a is an R-ideal such that dimR R/a = 0. Let
M be a finitely generated R-module and let N be an a-weakly finite R-module over M. Then,
for each i ≥ 0 and j ≥ 0, ExtiR (L, Hja (M, N)) is minimax and a-cofinite R-module, for any
minimax module L.
Proof. The proof follows by Theorem 2.9 and Theorem 5.1.



Corollary 5.3. Let (R, m) be a complete local ring. Let M be a finitely generated R-module
such that pd M = d < ∞ and let N be an a-weakly finite R-module over M such that
dimR N = n < ∞. Then, for each i ≥ 0, ExtiR (L, Hd+n
(M, N)) is minimax and a-cofinite
a
R-module, for any minimax module L.
Proof. The proof follows by Theorem 2.14 and Theorem 5.1.



Lemma 5.4. Let N be a nonzero a-cominimax R-module. Then for any nonzero R-module
M of finite length, the R-module TorR
i (M, N) has finite length for all i ≥ 0.
Proof. Firstly, note that SuppR (M) is a finite non-empty subset of the set of all maximal
ideals of R (denoted by Max(R)), since M has finite length. So, consider SuppR (M) :=
{p1 , . . . , pr } and b = p1 p2 . . . pr . By Lemma 4.2, it is sufficient to show that TorR
i (R/b, N) has
R
∼
finite
length
for
all
i
≥
0,
because
Supp
(M)
=
V
(b).
By
the
isomorphism
Tor
R
i (R/b, N) =
Lr
R
i=1 Tori (R/pi , N), we may assume without loss of generality that r = 1, and so b = p1 .
Let i ≥ 0 be an integer such that TorR
i (R/p1 , N) 6= 0. Note that p1 ∈ SuppR (N), and since
N is a-cominimax, SuppR (N) ⊆ V (a) and ExtiR (R/a, N) is minimax for all i ≥ 0. Hence
ExtiR (R/pi , N) is minimax and has finite length by Theorem 4.4 and Lemma 4.6. Therefore,
applying again Theorem 4.4, we obtain that TorR
i (R/p1 , N) is minimax and has finite length
for all i ≥ 0, and so we obtain the statement.

Now we are able to show the main result of this section.
Theorem 5.5. Let N be a nonzero a-cominimax R-module and M be a finitely generated
R-module.
(i) If dimR M = 1, then the R-module TorR
i (M, N) is Artinian and a-cofinite for all
i ≥ 0.
(ii) If dimR M = 2, then the R-module TorR
i (M, N) is a-cofinite for all i ≥ 0.
12

Proof. (i) Since N is cominimax and SuppR (Γa (M)) ⊆ V (a), we obtain that TorR
i (Γa (M), N)
is minimax for all i ≥ 0 by Theorem 4.4. Now, by the short exact sequence
0 → Γa (M) → M → M/Γa (M) → 0,
we can deduce the following long exact sequence all i ≥ 0
R
R
R
· · · → TorR
i (Γa (M), N) → Tori (M, N) → Tori (M/Γa (M), N) → Tori−1 (Γa (M), N) → · · · .

So it is sufficient to show that, for all i ≥ 0, TorR
i (M/Γa (M), N) is Artinian and a-cofinite.
Hence, we may assume that Γa (M) = 0 and therefore ensure the existence of x 6∈ a\∪p∈AssR (M )
p by [6, Lemma 2.2.1]. From the short exact sequence
x

0 → M → M → M/xM → 0,
we obtain following long exact sequence
x

R
R
R
· · · → TorR
i (M/xM, N) → Tori−1 (M, N) → Tori−1 (M, N) → Tori−1 (M/xM, N) → · · · .

By Lemma 5.4, the R-module TorR
i (M/xM, N) is of finite length i ≥ 0, because M/xM has
finite length. So, by previous long exact sequence, (0 :TorRi (M,N ) x) has finite length for all i ≥
0 and therefore (0 :TorRi (M,N ) I) has also finite length. Finally, since SuppR (TorR
i (M, N)) ⊆
R
SuppR (N) ⊆ V (a), we can conclude that Tori (M, N) is a-torsion. Therefore for all i ≥ 0,
R
TorR
i (M, N) is an Artinian R-module by [25, Theorem 1.3]. The cofiniteness of Tori (M, N)
follows by [26, Theorem 4.3].
(ii) Proceeding similarly to the proof of item (i), we may assume that assume that Γa (M) =
0, and so take x 6∈ a\ ∪p∈AssR (M ) p. The short exact sequence
x

0 → M → M → M/xM → 0,
induces following long exact sequence
x

R
R
R
· · · → TorR
i (M/xM, N) → Tori−1 (M, N) → Tori−1 (M, N) → Tori−1 (M/xM, N) → · · · .
R
Therefore (0 :TorRi (M,N ) I) and TorR
i (M, N)/x Tori (M, N) are Artinian R-modules by item
(i) and Lemma 5.4 and a-cofinite by [26, Corollary 4.4] for all i ≥ 0. Therefore TorR
i (M, N)
is a-cofinite by [26, Corollary 3.4]


Corollary 5.6. Let a be an R-ideal such that dimR R/a = 0. Let L, M be finitely generated
R-modules and let N be a a-weakly finite R-module over M.
j
(i) If dimR L = 1, then TorR
i (L, Ha (M, N)) is Artinian and a-cofinite for all i, j ≥ 0.
j
(ii) If dimR L = 2, then TorR
i (L, Ha (M, N)) is a-cofinite for all i, j ≥ 0.
Proof. The result follows by Theorem 2.9 and Theorem 5.5.



Proposition 5.7. Let N be a nonzero a-cominimax R-module and M be a finitely generated
R-module. If that dimR N ≤ 1, then the R-module TorR
i (M, N) is a-cominimax for all i ≥ 0.
Proof. Let F• be a resolution of M consisting of finite free R-modules. Since TorR
i (M, N) =
Hi (F• ⊗R N)) is a subquotient of a finite direct sum of copies of N. Now the result follows
by the fact that the category of a-cominimax modules with dimension less than or equal to
1 is an Abelian category [20, Theorem 2.5].

13

Corollary 5.8. Let N be a nonzero a-cofinite R-module and M be a finitely generated Rmodule. If dimR N ≤ 1, then the R-module TorR
i (M, N) is a-cofinite for all i ≥ 0.
Now we will investigate the behavior of torsion functors for larger dimensions. For this
purpose, weakly Laskerian modules are key ingredients.
Remark 5.9. Note that if R is a Noetherian ring, a is an ideal of R and M an Rmodule, then TorR
i (R/a, M) is a weakly Laskerian R-module for all i ≥ 0 if, and only if,
i
ExtR (R/a, M) is weakly Laskerian R-module for all i ≥ 0.
The proof follows by Theorem 4.4 and the fact that the class of weakly Laskerian modules
is a Serre subcategory.
Theorem 5.10. Let (R, m) be a local ring and a an R-ideal. Let N be a nonzero a-cominimax
R-module and M be a finitely generated R-module. Then the R-module TorR
i (M, N) is aweakly cofinite for all i ≥ 0 when one of the following cases holds:
(i) dimR N ≤ 2.
(ii) dimR M = 3.
Proof. Note that, in the light of Remark 5.9, it is sufficient to show that the R-modules
R
TorR
j (R/a, Tori (M, N)) are weakly Laskerian for all i ≥ 0 and j ≥ 0. For this purpose,
R
′
consider the set Λ = {TorR
j (R/a, Tori (M, N)) | i ≥ 0 and j ≥ 0}. Let K ∈ Λ and let K
′
be a submodule of K. The proof is complete if we show that the set AssR (K/K ) is finite.
Note that we may assume that R is complete by [22, Ex 7.7] and [23, Lemma 2.1]. Suppose
that AssR (K/K ′) is an infinite set. So, we can consider {ps }∞
s=1 the countable infinite subset
of non-maximal elements of AssR (K/K ′). Further m 6⊆ ∪∞
s=1 ps by [24, Lemma 3.2]. Define
S := R\ ∪∞
p
.
Now,
we
will
analyze
the
cases
(i)
and
(ii).
s=1 s
If dimR N ≤ 2, then we can conclude that S −1 N is a S −1 a-cominimax S −1 R-module of
dimension at most one by [22, Ex 7.7] and [29, Lemma 3.4]. Therefore TorS −1 R (S −1 M, S −1 N)
is S −1 a-cominimax by Proposition 5.7.
Therefore S −1 K/S −1 K ′ is a minimax S −1 R-module and so, AssS −1 R (S −1 K/S −1 K ′ ) is finite
−1
by Remark 4.1. However, for each pi ∈ {ps }∞
pi ∈ AssS −1 R (S −1 K/S −1 K ′ ),
s=1 , we have that S
and so we obtain a contradiction. This completes the proof.
In case dimR M = 3, we obtain that TorS −1 R (S −1 M, S −1 N) is S −1 a-cominimax, by Lemma
5.4 and Theorem 5.5. Now the proof follows similarly to the one previously made.

Now we give some applications of the results showed in this section.
Corollary 5.11. Let N be a nonzero minimax R-module and M be a finitely generated
R-module.
(i) If dimR Hia (N) ≤ 1 (e.g. dimR N ≤ 1 or dimR R/a = 1), then the R-module
i
TorR
j (M, Ha (N)) is a-cominimax for all i ≥ 0 and and j ≥ 0. Also, for for all
i
i ≥ 1 and and j ≥ 0, the R-module TorR
j (M, Ha (N)) is a-cofinite.
(ii) If (R, m) a is local ring and dimR Hia (M) ≤ 2 (e.g. dimR R/I ≤ 2 , then the R-module
i
TorR
j (M, Ha (N)) is a-weakly cofinite for all i ≥ 0 and j ≥ 0.
(iii) Let L and M be finitely generated R-modules such that pd M = d < ∞ and dimR L =
3, and let N be an a-weakly finite R-module over M such that dimR N = n < ∞.
Then, for each i ≥ 0, ToriR (L, Hd+n
(M, N)) is a-weakly cofinite R-module.
a
14

Proof. (i) First note that Hia (N) is an a-cominimax R-module for all i ≥ 0, by [1, Theorem
i
2.2]. Now the first statement follows by Proposition 5.7. The cofiniteness of TorR
j (M, Ha (N))
j
follows by the fact that ExtR (M, Hia (N)) is finite for all i ≥ 1 and j ≥ 0 by [1, Theorem 2.2]
and Corollary 5.8.
(ii) The proof follows analogously to Theorem 5.10 (i), using the previous item.
(iii) Apply Theorem 2.14 and Theorem 5.10 (ii).

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Universidade Tecnológica Federal do Paraná, 85053-525, Guarapuava-PR, Brazil
E-mail address: freitas.thf@gmail.com
Universidade de São Paulo - ICMC, Caixa Postal 668, 13560-970, São Carlos-SP, Brazil
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Universidade de São Paulo - ICMC, Caixa Postal 668, 13560-970, São Carlos-SP, Brazil
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