Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules
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irk-123456789-1654872020-02-14T01:28:33Z Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules Kianezhad, A. Taherizadeh, A.J. Статті 2013 Article Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules / A. Kianezhad, A.J. Taherizadeh // Український математичний журнал. — 2013. — Т. 65, № 6. — С. 796–801. — Бібліогр.: 13 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165487 512.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Kianezhad, A. Taherizadeh, A.J. Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules Український математичний журнал |
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Kianezhad, A. Taherizadeh, A.J. |
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Kianezhad, A. |
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Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules |
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Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules |
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Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules |
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Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules |
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Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules |
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finiteness properties of minimax and α-minimax generalized local cohomology modules |
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Інститут математики НАН України |
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2013 |
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Finiteness Properties of Minimax and α-Minimax Generalized Local Cohomology Modules / A. Kianezhad, A.J. Taherizadeh // Український математичний журнал. — 2013. — Т. 65, № 6. — С. 796–801. — Бібліогр.: 13 назв. — англ. |
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Український математичний журнал |
| work_keys_str_mv |
AT kianezhada finitenesspropertiesofminimaxandaminimaxgeneralizedlocalcohomologymodules AT taherizadehaj finitenesspropertiesofminimaxandaminimaxgeneralizedlocalcohomologymodules |
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2025-07-14T18:41:04Z |
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UDC 512.5
A. Kianezhad (Islamic Azad Univ., Tehran, Iran),
A. J. Taherizadeh (Kharazmi Univ., Tehran, Iran)
FINITENESS PROPERTIES OF MINIMAX AND a-MINIMAX
GENERALIZED LOCAL COHOMOLOGY MODULES
ВЛАСТИВОСТI ФIНIТНОСТI МIНIМАКСНИХ ТА a-МIНIМАКСНИХ
УЗАГАЛЬНЕНИХ ЛОКАЛЬНИХ КОГОМОЛОГIЧНИХ МОДУЛIВ
Let R be a commutative Noetherian ring with nonzero identity, a be an ideal of R, and M and N be two (finitely generated)
R-modules. We prove that Hi
a(M,N) is а minimax a-cofinite R-module for all i < t, t ∈ N0, if and only if Hi
a(M,N)p
is a minimax Rp-module for all i < t. We also show that, under some conditions, HomR
(
R
a
, Ht
a(M,N)
)
is minimax
(t ∈ N0). Finally, we investigate a necessary condition for Hi
a(M,N) to be a-minimax.
Нехай R — комутативне нетерове кiльце з ненульовою одиницею, a — iдеал кiльця R, а M та N — два (скiнчен-
нопороджених) R-модулi. Доведено, що Hi
a(M,N) є мiнiмаксним a-кофiнiтним R-модулем для всiх i < t, t ∈ N0,
тодi i тiльки тодi, коли Hi
a(M,N)p є мiнiмаксним Rp-модулем для всiх i < t. Показано також, що за деяких умов
HomR
(
R
a
, Ht
a(M,N)
)
є мiнiмаксним (t ∈ N0). Дослiджено необхiднi умови a-мiнiмаксностi Hi
a(M,N).
1. Introduction. Let R be a commutative Noetherian ring with nonzero identity, a be an ideal
of R and M, N be two R-modules. The generalized local cohomology was first introduced in the
local case by Herzog [8] and in the general case by Bijan – Zadeh [4]. The i th generalized local
cohomology module H i
a(M,N) is defined by
H i
a(M,N) = lim−→
n∈N
ExtiR
(
M
anM
,N
)
for all i ∈ N0, where we use N0 (resp. N) to denote the set of nonnegative (resp. positive) integers.
With M = R, one clearly obtains the ordinary local cohomology modules H i
a(N) of N with respect
to a, which was introduced by Grothendieck, see, for example, [6].
It is well known that the generalized local cohomology modules have some similar properties as
ordinary local cohomology modules. We recall some properties of the generalized local cohomology
modules which will be needed in this paper.
I) Let 0 −→ N −→ M −→ L −→ 0 be an exact sequence of R-modules. Then we have two
long exact sequences (K is an arbitrary R-module):
0 −→ H0
a (K,N) −→ H0
a (K,M) −→ H0
a (K,L) −→ H1
a (K,N) −→ . . .
and
0 −→ H0
a (L,K) −→ H0
a (M,K) −→ H0
a (N,K) −→ H1
a (L,K) −→ . . .
of generalized local cohomology modules.
II) If N is an a-torsion R-module, then for all i ∈ N0, we have H i
a(M,N) ∼= ExtiR(M,N).
III) Let R′ be a second commutative Noetherian ring with identity and let f : R −→ R′ be a flat
ring homomorphism. Then there is an isomorphism (i ∈ N0)
c© A. KIANEZHAD, A. J. TAHERIZADEH, 2013
796 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
FINITENESS PROPERTIES OF MINIMAX AND a-MINIMAX GENERALIZED . . . 797
H i
a(M,N)⊗R R
′ ∼= H i
aR′(M ⊗R R
′, N ⊗R R
′).
The organization of the paper is an follows:
In Section 2, we study the minimax generalized local cohomology modules (Theorem 2.2 and
Propositions 2.1 and 2.2). Also the minimaxness of HomR
(
R
a
,
H i
a(M,N)
K
)
, t ∈ N0, and
HomR
(
R
a
,
Ht
a(M,N)
K
)
will be considered whenever M,N are finitely generated R-modules and
K is a submodule of Ht
a(M,N) (Theorem 2.2 and Corollary 2.1) which generalize [9] (Theorem
2.2).
In Section 3, we study the a-minimax generalized local cohomology modules. In Theorem 2.1, we
show that whenever M, N are finitely generated and a-minimax R-modules such that AssR(N) ⊆
⊆ V (a) and pd(M) <∞, then Ht
a(M,N) is a-minimax for all i ≥ 0.
Throughout the paper R is a commutative Noetherian ring with nonzero identity.
2. Minimax generalized local cohomology modules. In this section we prove the minimaxness
of some generalized local cohomology modules.
Definition 2.1. Let N be an R-module. Then N is said to be a minimax module if there is a
finitely generated submodule L of N such that
N
L
is Artinian.
The class of minimax modules includes all finite and all Artinian module. Moreover it is closed
under taking submodules, quotients and extensions, i.e., it is a serre subcategory of the category of
R-modules, cf. [12] and [13]. Of course this class is strictly larger than the class of all finite modules
and Artinian modules as well, cf. [3] (Theorem 12). Also we note that a minimax R-module has only
finitely many associated primes.
Lemma 2.1. Let N
f−→M
g−→ L be an exact sequence of R-modules such that N and L are
minimax. Then M is minimax.
Proof. The result follows from the fact that the class of minimax R-modules is a serre subcate-
gory of the category of R-modules, and the exact sequence
0 −→ f(N)
⊆−→M
g−→ g(M) −→ 0.
Lemma 2.2. Let M, N be two R-modules such that M is projective. Then H i
a(M,N) ∼=
∼= H i
a(HomR(M,N)) for all i ∈ N0.
Proof. For all i ∈ N0
H i
a(M,N) = lim−→
n∈N
ExtiR
(
M
anM
,N
)
∼= lim−→
n∈N
ExtiR
(
R
an
⊗R M,N
)
∼=
∼= lim−→
n∈N
ExtiR
(
R
an
,Hom(M,N)
)
(by [11], exercise 9.21) ∼= H i
a(Hom(M,N)).
The next theorem generalize [1] (Theorem 2.8).
Theorem 2.1. Suppose that t ∈ N0 and M, N are two finitely generated R-modules such that
pd(M) <∞ and H i
a(N) is minimax for all i < t . Then the following are equivalent:
(i) H i
a(M,N) is minimax for all i < t,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
798 A. KIANEZHAD, A. J. TAHERIZADEH
(ii) H i
a(M,N) is minimax and a-cofinite for all i < t,
(iii) H i
a(M,N)p is minimax for all i < t and all p ∈ Spec (R),
(iv) H i
a(M,N)m is minimax for all i < t and all m ∈ max (R).
Proof. (i) ⇒ (ii). By induction on n := pd(M). If n = 0, then the result follows from Lemma
2.3 and [2] (Theorem 2.8).
Next let n > 0. We have
H0
a (M,N) = lim−→
n∈N
HomR
(
M
anM
,N
)
∼= lim−→
n∈N
HomR
(
M ⊗R
R
an
, N
)
∼=
∼= lim−→
n∈N
HomR
(
R
an
,Hom(M,N)
)
∼= H0
a (Hom(M,N))
and so the result follows. Now suppose i > 0. From the short exact sequence 0 −→ L −→
−→
⊕n
j=1R −→M −→ 0 we get the exact sequence
H i
a(N)n−→H i
a(L,N) −→ H i+1
a (M,N), i ∈ N,
according to our assumption and Lemma 2.1 we deduce that H i
a(L,N) is minimax for all i < t− 1.
By induction hypothesis H i
a(L,N) is a-cofinite for all i < t− 1. But from the exact sequence
H i−1
a (L,N) −→ H i
a(M,N) −→ H i
a(N)n.
and [2] (Theorem 2.8) and [10] (Corollary 4.4) we get the result.
(ii) ⇒ (iii) and (iii)⇒ (iv) is obvious.
(iv) ⇒ (i) The proof is similar to that for (i)⇒ (ii).
The next result follows by an standard argument.
Proposition 2.1. Suppose that M,N are two R-modules such that N is finitely generated and
for each p of suppR(N), Ht
a
(
M,
R
p
)
be minimax. then Ht
a(M,N) is minimax (t ∈ N0).
Proposition 2.2. Let M,N,L be finitely generated R-modules such that pd(M) and dim(N)
is finite and suppR(L) ⊆ suppR(N). Suppose H i
a(M,N) is minimax for all i ≥ r. Then H i
a(M,L)
is minimax for all i ≥ r, r ∈ N0.
Proof. By [4] (Proposition 5.5) H i
a(M,N) = 0 for all i > pd(M) + dimN. The proof is by
decrasing induction on i = r, r + 1, . . . , pd(M) + dimN + 1. If i = pd(M) + dim(N) + 1 then
there is nothing to prove. So let r ≤ i ≤ pd(M) + dim(N). By Gruson’s theorem there exist a chain
0 −→ L0 ⊆ L1 ⊆ . . . ⊆ Lt = L such that the quotient
Lj
Lj−1
a homomorphic image of a direct sum
of finitely many copies of N for all j = 1, 2, . . . , t. Consider the short exact sequences
0 −→ Lj−1 −→ Lj −→
Lj
Lj−1
−→ 0, j = 1, 2, . . . , t,
we may reduce to the case t = 1. So let there exsits a short exact sequence 0 −→ K −→ Nm −→
−→ L −→ 0 in which m > 0 and K is a finitely generated R-module. This exact sequence induce
the long exact sequence
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
FINITENESS PROPERTIES OF MINIMAX AND a-MINIMAX GENERALIZED . . . 799
. . . −→ H i
a(M,Nm) −→ H i
a(M,L) −→ H i+1
a (M,K) −→ . . . . (2.1)
Since suppR(K) ⊆ SuppR(N), by induction hypothesis H i+1
a (M,K) is minimax for all i =
= r, . . . , pd(M) + dimN. Moreover H i
a(M,Nm) ∼= H i
a(M,N)m is minimax. Thus from the exact
sequence (2.1) and Lemma 2.1 the result follows.
The next two results generalized [9] (Theorem 2.2 and Corollary 2.3).
Theorem 2.2. Let M,N be two finitely generated R-modules such that H i
a(M,N) and H i
a(N)
is minimax for all i < t, t ∈ N0, and pd(M) <∞. Then HomR
(
R
a
, Ht
a(M,N)
)
are minimax and
so AssHt
a(M,N) is finite.
Proof. If t = 0, then,
Ext0R
(
R
a
,HomR(M,N)
)
∼= HomR
(
R
a
,HomR(M,N)
)
which is finitely generated. Furthermore
HomR
(
R
a
, H0
R(M,N)
)
∼= HomR
(
R
a
, H0
a (Hom(M,N)
)
∼= HomK
(
R
a
,HomR(M,N)
)
.
Next let t > 0. The proof is by induction on n := pdR(M) ≥ 0. If n = 0, then by Lemma 2.2
H i
a(M,N) ∼= H i
a(Hom(M,N)) for all i, and
HomR
(
R
a
, Ht
a(M,N)
)
∼= HomR
(
R
a
, Ht
a(Hom(M,N))
)
.
So since ExttR
(
R
a
,HomR(M,N)
)
is finitely generated and H i
a(Hom(M,N)) is minimax for all
i < t, the result will be deduced from [9] (Theorem 2.2).
Now suppose that n > 0. There is a short exact sequence
0 −→ K −→ Rn −→M −→ 0
from which we get the long exact sequcence
HomR
(
R
a
, Ht−1
a (K,N)
)
−→ HomR
(
R
a
, Ht
a(M,N)
)
−→ HomR
(
R
a
, Ht
a(N)n
)
−→ . . . .
(2.2)
Next, H i
a(K,N) is minimax for all i < t− 1, by the exact sequence
. . . −→ H i
a(N)n −→ H i
a(K,N) −→ H i+1
a (M,N) −→ . . .
the minimaxness of H i
a(N) and H i
a(M,N) for all i < t − 1 and Lemma 2.1. Thus by induction
hypothesis HomR
(
R
a
, Ht−1
a (K,N)
)
is minimax. In addition HomR
(
R
a
, Ht
a(N)
)
is minimax by
[9] (Theorem 2.2). Now we deduce the result from exact sequence (2.2) and Lemma 2.1.
Corollary 2.1. With the same assumptions as in (2.2), let K be an R-submodule of Ht
a(M,N)
such that Ext1R
(
R
a
,K
)
is minimax. Then HomR
(
R
a
,
Ht
a(M,N)
K
)
is a minimax module. In par-
ticular Ass
(
Ht
a(M,N)/K)
)
is finite.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
800 A. KIANEZHAD, A. J. TAHERIZADEH
3. a-Minimax modules and generalized local cohomology modules. Recall that for an R-
module M, the Goldie dimension of M is defined as the cardinal of the set of indecompos-
able submodules of E(M), which appear in a decomposition of E(M) into direct sum of in-
decomposable submodules. The notation G. dim(M) is used for Goldie dimension of M. For a
prime ideal p of R µ0(p,M) denotes the 0 th Bass number of M with respect to p. It is known
that µ0(p,M) > 0 if and only if p ∈ AssR(M). Therefore by definition of Goldie dimension,
GdimM =
∑
p∈Ass(M)
µ0(p,M). Also for an ideal a of R and R-module M the a-relative Goldie
dimension of M is defined as
G dimaM :=
∑
p∈V (a)
µ0(p,M).
The a-relative Goldie dimension of an R-module has been introduced and studied in [7].
Definition 3.1. Let a be an ideal of R. An R-module M is said to be minimax with respect to
a or a- minimax if the a-relative Goldie dimension of any quotient module of M is finite, i.e., for any
submodule N of M, G dima
(
M
N
)
<∞.
The concept of a-minimax modules was introduced and studied in [2]. By [2] (Proposition 2.6)
if M is an a-minimax R-module such that AssM ⊆ V (a), then H i
a(M) is a-minimax for all i ≥ 0.
Now we intend to generalize the result to obtain another similar result for generalized local
cohomology modules.
Theorem 3.1. Let M and N be two R-modules such that M is finitely generated and N is
a-minimax with AssR(N) ⊆ V (a) and p(M) <∞. Then H i
a(M,N)is a-minimax for all i ≥ 0.
Proof. By induction on n := pdM. If n = 0, then H i
a(M,N) ∼= H i
a(Hom(M,N)) by Lemma
2.2 and HomR(M,N) is a-minimax by [2] ( Corollary 2.5). Also Ass (HomR(M,N)) = AssR(N)∩
∩ SuppR(M) ⊆ V (a), by [5] (Ch. 4, § 2.1, Proposition 10) the result follows from [2] (Proposition
2.6). Next, let n > 0 and suppose that the result is true for n − 1. From the short exact sequence
0 −→ L −→ Rk −→ M −→ 0 (in which L is a finitely generated R-module) we get the following
long exact sequence:
. . . −→ H i−1
a (L,N) −→ H i
a(M,N) −→ H i
a(N)k −→ . . . . (3.1)
By induction hypothesis H i−1
a (L,N) is a-minimax for all i ≥ 1. Moreover H i
a(N)k is a-minimax
by [2] (Proposition 2.6 and Corollary 2.4). Next we deduce from (3.1) and [2] (Proposition 2.3) that
H i
a(M,N) is a- minimax for all i ≥ 1. For i = 0, we have H0
a (M,N) ∼= H0
a (HomR(M,N)) which
is a-minimax by [2] (Corollary 2.5 and Proposition 2.3).
Proposition 3.1. Let M be a finitely generated R-module and N an arbitrary R-module such
that HomR(M,N) is a-minimax and suppR(M) ⊆ V (a). Furthermore let t ∈ N0 and for each
i 6= t, H i
a(M,N) be a-minimax. Then Ht
a(M,N) is a-minimax.
Proof. By induction on t ≥ 0. If t = 0, then H0
a (M,N) ∼= H0
a (HomR(M,N)), and
HomR(M,N) is a-minimax by our assumption and
AssR(HomR(M,N)) = AssR(N) ∩ SuppR(M) ⊆ V (a),
the result follows from [2] (Proposition 2.6).
Next let t > 0 and assume that the result holds for t− 1. Set E := E(N), L :=
E
N
and consider
the short exact sequence 0 −→ N −→ E −→ L −→ 0. Since H i
a(M,E) = 0 for all i > 0, we
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
FINITENESS PROPERTIES OF MINIMAX AND a-MINIMAX GENERALIZED . . . 801
deduce that
H i
a(M,L) ∼= H i−1
a (M,N).
Thus it follows from the hypothesis that for all i 6= t−1 H i
a(M,L) is a-minimax. Hence by induction
hypothesis Ht−1
a (M,L) is a-minimax and consequently by (2.1) Ht
a(M,N) is a-minimax as well.
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Received 11.03.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
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