1. Introduction
In [
1], the notion of local cohomology functors for categories of modules is introduced, alongside complexes that have garnered significant attention from numerous scholars, as evidenced in [
1,
2,
3]. Over the past six decades, the theory of local cohomology has flourished, emerging as a pivotal tool in the domains of algebraic geometry and commutative algebra. Nevertheless, its counterpart, the theory of local homology, has remained relatively underdeveloped. Initial investigations into local homology functors were conducted by Matlis in [
4,
5], focusing on ideals generated by regular sequences. Subsequently, Greenlees and May’s work in [
6], along with Tarrĺo, López and Lipman’s contributions in [
7], highlighted a profound connection between local homology and local cohomology. This linkage becomes particularly apparent (as shown in [
7]) when formulated within the derived category
of the
R-module category, specifically, before delving into homology.
The concept of support serves as a cornerstone, offering a geometric framework for delving into diverse algebraic formations. Benson, Iyengar and Krause [
8] introduced the concept of support for entities within any compactly generated triangulated category that permits small coproducts. Their methodology is grounded in the establishment of local cohomology functors “
” within triangulated categories, relative to a central ring of operators. Since then, numerous academics have pursued their vision through various endeavors (see [
9,
10,
11,
12]). For instance, Asadollahi, Salarian and Sazeedeh [
13] allocated to each entity in a category a distinct subset of
, also known as the Support (big support). Their investigation into this support revealed that it adheres to principles such as exactness, orthogonality and separation. Using this support, they analyzed the dynamics of local cohomology functors and demonstrated that these triangulated functors adhere to boundedness. Liu and coauthors [
14] introduced the depth concept for objects within such triangulated categories and discovered that when
is a graded commutative Noetherian local ring, the depth of any cohomologically bounded and cohomologically finite object does not exceed its dimension.
To classify colocalized subcategories, Benson, Iyengar and Krause [
15] developed the local homology functor “
” and a cosupport for triangulated categories, which is based on their work on local cohomology and support. Since then, many scholars have begun to study the cosupport and related theories (see [
16,
17,
18,
19]). Motivated by these studies, the aim of this paper is to study the invariants (see Definitions 1 and 2)
for some object
X in
. Clearly,
for any
. We give some basic properties and the following boundness of these two invariants for an ideal
.
Theorem 1. Let be a homogeneous ideal of Noetherian graded ring R. For any , the following hold: Theorem 2. For any non-zero object X of
let be a homogeneous ideal of Noetherian graded ring R. Then Let be a homogeneous ideal of Noetherian -graded ring . Then 2. Preliminaries
In this section, we revisit some fundamental concepts and terminology pertinent to the current paper. Our terminology will adhere to the standards set forth in [
8,
13,
15,
20].
Compact generation. Let be a triangulated category, and the suspension on . An object C in is considered if it admits set-indexed coproducts and the functor preserves all coproducts. We write for the full subcategory of compact objects in . A set of objects of is called a for if for each non-zero object there exists an object G in such that . The category is if it is generated by a set of compact objects.
For any two objects X and Y in , is defined as the graded abelian group of morphisms given by . Additionally, forms a graded ring, and serves as a right and left bimodule.
Assume that R is a graded commutative Noetherian ring, satisfying the property that for any homogeneous elements in R, the multiplication follows the rule , where denote the degree of r. We refer to a triangulated category as R-linear if there exists a homomorphism of graded rings , where represents the graded center of . Consequently, for each object X, there arises a homomorphism of graded rings , such that, for all objects , the R-module structures on induced by and agree, up to the usual sign rule.
We denote the set of homogeneous prime ideals of
R as Spec
R. Given a point
that belongs to
, we represent the homogeneous localization of
R with respect to
as
; this localization is a graded local ring, as defined by Bruns and Herzog [
21] (1.5.13), with maximal ideal
. We set
Given a homogeneous ideal
in
R, we set
Let
be a subset of Spec
R. The
of
is the set
The subset
is
-
if
. Note that the subsets
and
are specialization-closed.
Throughout this paper, we shall assume that R is a graded commutative Noetherian ring and is a compactly generated R-linear triangulated category that possesses set-indexed coproducts.
Koszul object. Let
r be a homogeneous element of
R, and let
X be an object in the category
. We refer to any object that arises in an exact triangle
as a
of
r acting on
X, denoted by
. This notation indicates a well-defined module (non-unique) isomorphism. For a given homogeneous ideal
in
R, we denote
as any Koszul object obtained by repeatedly applying the aforementioned construction using a finite sequence of generators for
. It is worth noting that this object may vary depending on the choice of the minimal generating sequence for
.
Local cohomology and homology. An exact functor
is referred to as
when there exists a morphism
that satisfies the condition that the composition
is invertible and also fulfills the equality
. Given a specialization-closed subset
of Spec
R,
is the corresponding localization functor. According to [
8] (Definition 3.2), there arises an exact functor
on
, which for each object
X induces an exact triangle
We refer to
as the
of
X supported on
. According to [
8] (Corollary 6.5), the functors
and
on
preserve coproducts, thereby possessing right adjoints due to Brown representability. The right adjoints of
and
are denoated as
and
, respectively. They induce a functorial exact triangle
This triangle leads to the definition of an exact
functor
.
Cosupport. The concept of the
of an object
X in
is introduced in [
15] (
Section 4) as follows:
One has the following statements by [
15] (Theorem 4.5) and [
15] (Proposition 4.7t):
;
;
.
We refer to the exact triangles in the subsequent text as the
–
triangles corresponding to
and
, which is the dual of [
8] (Theorem 7.5) and is a useful tool in this article.
Let
be specialization-closed subsets within
R. Consider the following exact triangles:
Apply the functors
and
to the above triangles, respectively. By [
15] (4.2), one has natural morphisms:
Lemma 1. Given that and are specialization-closed subsets of , there exist natural exact triangles for each X in : Arithmetic rank and dimension. Let
be a graded ideal of
R. The
of
, abbreviated as
, is defined as follows:
Note that
.
Let X be an object of . The dimension of X is defined as Obviously, for any object X of , the equality holds true.
For any object
C belonging to
and
X in
, we define
. We denote
An object
X in
is said to be cohomologically bounded above (resp., bounded below) if, for any compact object
C, there exists a positive integer
such that
(resp.,
).
X is termed cohomologically bounded if it satisfies both conditions. We denote by
(resp.,
,
) the full subcategory of
, consisting of all cohomologically bounded above (resp., bounded below) objects.
The following lemma is used very frequently in this article, which recovers part of [
13] (Lemma 4.5) and [
13] (Lemma 4.8).
Lemma 2. Let r be a homogeneous element from the set R, and let C represent a compact object of . Then, for any ,
, if ; , if .
, if ; , if .
Proof. By [
13] (Lemma 4.5) and [
13] (Lemma 4.8), we know that
and
for any
. We only need to prove
. To demonstrate the desired inequality
, we shall assume the contrary:
, where
.
Consider the exact triangle
This induces the exact sequence
Clearly,
.
For any compact object
,
induces the long exact sequence
Clearly,
. If
, then
. If
, assume that
, then we have
since
, which is impossible. Therefore,
. □
3. Width in Triangulated Categories
In this section, we define the width for objects relative to a homogeneous ideal utilizing the local homology functors on triangulated categories. Additionally, we provide a bound and discuss some fundamental properties of it.
Definition 1. Let represent a homogeneous ideal in the ring R and let X be an object belonging to the triangulated category . We define the - of X as follows:We set whenever . Remark 1. If , then
if ;
For any , by Lemma 2, one has that For any , the following theorem gives an upper bound and a lower bound.
Theorem 3. For any object , the following hold: Proof. The second inequality is simpler; we just have to prove
for any compact object
C. We validate this assertion for the specific case of
, and by a straightforward inductive argument, we derive the generalized result. If
, then
by Lemma 2 and [
15] (Corollary 4.8). Therefore, according to [
19] (Proposition 2.2(1)), it follows that
. If
, then
by Lemma 2 and [
15] (Corollary 4.8). Hence [
19] (Proposition 2.2(2)) implies that
.
For the first inequality, if
, then
by [
15] (Theorem 4.5), so we have
by [
15] (Lemma 4.12). The conclusion follows from Remark 1(2).
Assume
; we can employ induction on the number of generators of
for any compact object
C in
to demonstrate that
. To initiate this, let us assume
. The triangle
induces the triangle
If
, then in light of Lemma 2, we can establish the following equivalences:
Here, the second equality follows from
by [
15] (Lemma 4.12) and [
15] (Corollary 4.8). Therefore,
.
If
, the (§) induces the following exact sequence of
-modules:
Obviously,
for all
. If
, we find that the morphism
is injective. Utilizing [
15] (Lemma 6.1), we infer that any element of
is annihilated by a power of
r, leading to the conclusion that
, which contradicts our previous assumption. Therefore,
. Finally, considering Lemma 2, we deduce that
.
In the case where
with
, the exact triangle
yields the following exact sequence of
-modules:
Clearly,
for all
. If
, we get that
since
, which is a contradiction. Thus
. Lastly, in view of Lemma 2, we obtain
Therefore, the desired conclusion holds in this case. Now, let us assume inductively that the result has been established for all values less than
n. Given
. let us define
. Based on the inductive step, we obtain the following inequalities:
This complete the proof. □
The subsequent proposition encapsulates the manner in which width behaves in exact triangles.
Proposition 1. Given a homogeneous ideal of R, for any exact triangle in , the following hold:
Proof. The triangle
yields the induced triangle
For any compact object
, by applying the functor
to the above triangle, we obtain the long exact sequence:
It is noteworthy that
if
and
both vanish. Consequently, the first inequality arises from the definition of the width of an object in triangulated categories. The same reasoning applies to the other inequalities. □
Proposition 2. Given a homogeneous ideal of R and an object X of , the following holds: Proof. According to the definition of width, we can establish the following series of equalities:
This complete the proof. □
4. Local Homology Dimension at a Homogeneous Ideal
Given a homogeneous ideal of R and an object X in , this section introduces an invariant denoted as , which is defined as . The main results of this section, Theorems 4 and 5, establish the relationship between the local homology dimension and the arithmetic rank of and .
Definition 2. Given a homogeneous ideal of R and an object X in , we define the homological dimension of X with respect to , denoted as , as
We set whenever . Remark 2. Let X be an object in . By the definitions of width and homological dimension of X, we have
Lemma 3 ([
19] (Proposition 2.5))
. Let be a homogeneous element and X an object in . If , then for any ;
If , then for any .
The above lemma gives the formula for computing
, which recovers part of [
22] (Theorem 4.1). It is very useful for proving the following proposition.
Proposition 3. Let be homogeneous elements in R. For any object X in , we observe thatMore specifically, for any homogeneous ideals of R and any , one finds that . Proof. First, assume that
. If
, by Lemma 2, we find that
. Hence, we have following equalities by Lemma 3,
If
, by Lemma 2, we find that
, and then
Now, assume that
. If
, by Lemma 2, we find that
. Hence, we have following equalities by Lemma 3,
If
, by Lemma 2, we find that
, and then
This complete the proof. □
Corollary 1. Given a local ring and a homogeneous ideal of R, for any object X in , there exists an inequality stating that .
Lemma 4. Let be a graded ideal of R and be an object of . If , then .
Proof. By [
13] (Corollary 4.4), we have that if
, then
for any
. So, we just have to prove that
for any
. To validate the assertion for
, a straightforward iterative approach leads to the generalized conclusion.
Utilizing [
20] (Lemma 2.6) and [
8] (Corallary 5.7), we deduce that
. Consequently, let us consider the exact triangle
If
, then
. Consequently, Lemma 3 yields that
On the other hand, if
, then
. Therefore, Lemma 3 implies that
This complete the proof. □
Let be a homogeneous ideal of R. The following theorem gives the relationship between the homology dimension with respect to and the arithmetic rank of .
Theorem 4. Given a homogeneous ideal of R and an object of , we find that .
Proof. Suppose . We shall employ induction on n to establish the desired result.
If
,
as
by [
15] (Corollary 4.8), then
. Therefore, the conclusion holds.
Assume that
and
. By [
15] (Corollary 4.8) and [
15] (Lemma 4.12), we have that
. So, we have the triangle
If
, consider the induced long exact sequence of
-modules of (‡):
Due to Lemma 2, we know that
for all
. Therefore, the homomorphism
is injective for all
. This, in turn, implies that
for all
, since any element of
is annihilated by some power of
r according to [
15] (Lemma 6.1). Hence,
If
, where
, consider the induced long exact sequence of
-modules of (‡):
As stated in Lemma 2,
for all
. Consequently, the homomorphism
is injective for all
. This injection guarantees that
for all
, due to the fact that any element of
is annihilated by some power of
r. Therefore,
.
Assuming inductively that the integer
and the desired result holds for all ideals with an arithmetic rank less than
n, let us take an ideal
defined as
, where
are homogeneous elements of
R. Next, we consider the ideal
, which is given by
. Now, turning our attention to the Mayer–Vietoris triangle
In light of the fact that
and
, we obtain the following triangle
For any
, this induces a long exact sequence
since
; hence, if
, we find that
and this implies that
for all
. Hence,
. □
In the following section of the paper, we assume that is a Noetherian -graded ring.
Proposition 4. Let be a homogeneous ideal of R, with r being a homogeneous element belonging to , and let C be a compact object of . Then, for any , Proof. By Lemma 2, we only show that
if
. Considering the exact triangle
applying the exact functor
to the above triangle, we find that
For any
, this leads to the exact sequence
If
, then the morphism
has to be injective, which is impossible as
is
-torsion by [
15] (Lemma 6.1); then, any element of
is annihilated by a certain power of
r. Consequently,
This complete the proof. □
Theorem 5. Given a homogeneous ideal of R and an object of , we get the inequality .
Proof. If , the conclusion clearly holds.
If , the result is evident since for any object X of , and so the result follows from Lemma 4.
If
, let us assume there exists
such that
. Arrange the minimal prime ideals in
in a manner such that
for
and
for
. Choose
. Define
and
. Clearly,
and
. Utilizing Lemma 1, we investigate the Mayer–Vietoris triangle
In view of the facts that
and
, we get the following triangle
For any
, this induces a long exact sequence
Since
, we have
by [
15] (Corollary 4.8), we then find that
Hence, we obtain the following equivalence relationship:
Let
be a homogeneous element. Therefore,
. If
, then since
, we find that
, and so
. Hence the result follows from Theorem 4. Otherwise, we may find
such that
. Continuing in this way, after
d steps, we may deduce that
. Therefore,
. Consequently, according to Lemma 4 and Proposition 4, for any compact object
C of
T, we obtain
This implies that
. Now, considering the long exact sequence of cohomology modules arising from the exact triangle
we infer that
. This is due to the fact that
and any element of
vanishes by a power of
. Following back from this argument gives us
. In other words:
The proof is complete because
C was arbitrary. □