Resolving resolution dimensions in triangulated categories
-
Xin Ma
Abstract
Let
1 Introduction
Approximation theory is the main part of relative homological algebra and representation theory of algebras, and its starting point is to approximate arbitrary objects by a class of suitable subcategories. In particular, resolving subcategories play important roles in approximation theory (e.g., [1–3]). As an important example of resolving subcategories, Auslander and Buchweitz [4] studied the approximation theory of the subcategory consisting of maximal Cohen-Macaulay modules over an artin algebra, and Hernández et al. [5] developed an analogous theory for triangulated categories. Using the approximation triangles established by Hernández et al. [5, Theorem 5.4], Di and Wang [6] constructed additive functors (adjoint pairs) between additive quotient categories. On the other hand, Zhu [7] studied the resolution dimension with respect to a resolving subcategory in an abelian category, and Huang [8] introduced relative preresolving subcategories in an abelian category and defined homological dimensions relative to these subcategories, which generalized many known results (see [4,9,10]).
In analogy to relative homological algebra in abelian categories, Beligiannis [11] developed a relative version of homological algebra in a triangulated category
In Section 2, we give some terminology and some preliminary results.
In Section 3, some homological properties of resolving subcategories are obtained. In particular, we obtain Auslander-Buchweitz approximation triangles (see Proposition 3.10) for objects having finite resolving resolution dimensions. Our main result is the following:
Theorem
Let
There are two triangles
and
in
In Section 4, we will further study objects having finite resolution dimensions with respect to a resolving subcategory
As an application, in Section 5, given a resolving subcategory
Throughout this paper, all subcategories are full, additive, and closed under isomorphisms.
2 Preliminaries
Let
An object of
A morphism in
commutes.
A triangulated category is a triple
The following result is well known, which is an efficient tool in studying triangulated categories.
Remark 2.1
[11, Proposition 2.1] Let
Octahedral axiom. For any two morphisms
in which all rows and the third column are triangles in
Base change. For any triangle
in which all rows and columns are triangles in
Cobase change. For any triangle
in which all rows and columns are triangles in
Throughout this paper,
Definition 2.2
[11] A triangle
is called split if it is isomorphic to the triangle
We use
Definition 2.3
[11] Let
in
is in
in
is in
is in
Definition 2.4
[11] A class
Throughout this paper, we always assume that
Definition 2.5
[11] Let
be a triangle in
We use
in
Definition 2.6
(see [11, 4.1]) An object
is exact. We use
We say that
Remark 2.7
Definition 2.8
Let
A triangle
in
is exact.
[13] A
(2.1)in
(2.2)in
Asadollahi and Salarian [13] introduced the notion of
Definition 2.9
[13, Definition 3.6] Let
in
Throughout this paper, we always assume that
Let
Remark 2.10
Let
be a triangle in
of “
of “
Following Remark 2.10, we usually use the strategy of “dimension shifting,” which is an important tool in relative homological theory of triangulated categories.
Now, we set
For two subcategories
Taking
Definition 2.11
(cf. [18, Definition 3.1]) Let
in
Definition 2.12
(cf. [18, Definition 3.2]) Let
3 Resolution dimensions with respect to a resolving subcategory
Taking
Definition 3.1
Let
For a
with all
Lemma 3.2
Let
and
are
Proof
For
with
Consider the following triangle:
in
Similarly, we have the following
Set
and
Since
and
in
But from the following triangles in
we have that
Using the above, we can get:
Proposition 3.3
Let
Proof
Apply Lemma 3.2.□
Now we can compare resolution dimensions in a given triangle in
Proposition 3.4
Let
be a triangle in
Proof
For any
in
Assume
in
Assume
and a triangle
in
Assume
in
As direct results, we have the following closure properties for the subcategory
Remark 3.5
If
Corollary 3.6
Let
be a triangle in
(cf. [18, Proposition 3.11]) Assume that
Assume that
(cf. [18, Proposition 3.13]) Assume that
Proposition 3.7
Let
If
In particular, if
Proof
It is clear.
and
with
Now, we assume that
Set
in
in
Now we give the following definition.
Definition 3.8
Let
We need the following easy and useful observation.
Lemma 3.9
Let
If
If
Proof
Apply Remark 2.10.□
The following is an analogous theory of Auslander-Buchweitz approximations (see [4,5]).
Proposition 3.10
Let
and
in
In particular, if
Proof
We proceed by induction on
in
in
Since
is in
is in
For
in
Note that
is in
Now suppose
in
in
Note that
is in
in
For
in
Applying cobase change for the triangle (8) along the morphism
As a similar argument to that of the diagram (6), we obtain that the triangles
and
are in
In particular, suppose
Proposition 3.11
Keep the notion as Proposition 3.10. Assume
If
In particular, if
If
in
Let
If
Suppose that
Proof
Suppose
Moreover, applying Corollary 3.6(1) to the triangle (3) implies
The last assertion follows from the above argument and Proposition 3.10.
Since
(
(
in
Suppose
in
Suppose
(10)in
in
One can see that the triangle (11)is in
Lemma 3.12
Let
Proof
Clearly,
Conversely, let
Set
is split. It follows that
in
Proposition 3.13
Let
Proof
Clearly,
Now, let
On the other hand, by Proposition 3.10, there is a triangle
in
in
Our main result is the following.
Theorem 3.14
Let
There are two triangles
and
in
Proof
with all
with all
for
4 Additive quotient categories and
ξ
-cellular towers with respect to a resolving subcategory
In this section, we will further study objects having finite resolution dimension with respect to a resolving subcategory
4.1 Adjoint pairs
Suppose that
Lemma 4.1
Let
Proof
It suffices to show that
in
Lemma 4.2
Let
in
There exists a morphism
If
Proof
Apply Proposition 3.10.
Suppose
Note that
By Lemma 4.2, there exists a well-defined additive functor
which maps an object
Clearly, we have
where
Now we show that the additive functor
Theorem 4.3
Let
Proof
Let
in
is natural in both
Indeed, since
Now, assume that
Note that
Corollary 4.4
Let
There is a triangle
in
Proof
The assertion
is isomorphic. Since
Next, we compare additive quotients
Lemma 4.5
Let
in
There exists a morphism
If
Proof
Since
Suppose
Note that
By Lemma 4.5, there exists a well-defined additive functor
which maps an object
Clearly, we have
where
Now we show that the additive functor
Theorem 4.6
Let
Proof
Let
in
is natural in both
Indeed, since
Now, assume that
in
It follows that
Corollary 4.7
Let
There is a triangle
in
Proof
The assertion
is isomorphic. Since
4.2 A characterization of finite resolution dimension via
ξ
-cellular towers
For
in
in
in
Applying cobase change for the triangle (13) along the morphism
where the triangle
is in
where the triangle
Continuing in this manner, we obtain the following commutative diagram: 
where all the horizontal triangles are in
Set
which we call the
According to the above construction, one can obtain the following result by Proposition 3.3.
Theorem 4.8
Let
For each
5 Applications
In this section, we will construct a new resolving subcategory from a given resolving subcategory, which generalizes the notion of
Definition 5.1
Let
in
are corresponding triangles in
Remark 5.2
Since
If
is a
Evidently,
with
We have the following result.
Theorem 5.3
Let
Proof
Let
in
are corresponding triangles in
As a similar argument to the proof of [18, Theorem 4.3(1)], we obtain that
Lemma 5.4
Let
Proof
Let
in
where the two isomorphisms follow from the assumption that
It is obvious that
As an application of Theorem 3.14, we have:
Proposition 5.5
Let
There are two triangles
and
in
Immediately, we have:
Corollary 5.6
Let
There are two triangles
and
in
Remark 5.7
As in Corollary 5.6,
Following Theorems 4.8 and 5.3, we have the following result, which is a generalization of [15, Proposition 5.1].
Proposition 5.8
Let
For each
Acknowledgements
The authors thank the reviewers for their suggestions.
-
Funding information: This work was supported by the NSF of China (11901341, 11971225, 12001168), Henan University of Engineering (DKJ2019010) and the Key Research Project of Education Department of Henan Province (21A110006), the project ZR2019QA015 supported by Shandong Provincial Natural Science Foundation, the project funded by China Postdoctoral Science Foundation (2020M682141), and the Young Talents Invitation Program of Shandong Province.
-
Conflict of interest: The authors state no conflict of interest.
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© 2021 Xin Ma and Tiwei Zhao, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
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- The concept of cone b-Banach space and fixed point theorems
- Complete consistency for the estimator of nonparametric regression model based on m-END errors
- A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
- Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
- Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
- A new characterization of the automorphism groups of Mathieu groups
- The role of w-tilting modules in relative Gorenstein (co)homology
- Primitive and decomposable elements in homology of ΩΣℂP∞
- The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
- Classification of f-biharmonic submanifolds in Lorentz space forms
- Some new results on the weaving of K-g-frames in Hilbert spaces
- Matrix representation of a cross product and related curl-based differential operators in all space dimensions
- Global optimization and applications to a variational inequality problem
- Functional equations related to higher derivations in semiprime rings
- A partial order on transformation semigroups with restricted range that preserve double direction equivalence
- On multi-step methods for singular fractional q-integro-differential equations
- Compact perturbations of operators with property (t)
- Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
- Random attractors for stochastic plate equations with memory in unbounded domains
- On the convergence of two-step modulus-based matrix splitting iteration method
- On the separation method in stochastic reconstruction problem
- Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
- Structure of coincidence isometry groups
- Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
- Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
- Euler-type sums involving multiple harmonic sums and binomial coefficients
- Poly-falling factorial sequences and poly-rising factorial sequences
- Geometric approximations to transition densities of Jump-type Markov processes
- Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
- Bifurcations and exact traveling wave solutions for the regularized Schamel equation
- Almost factorizable weakly type B semigroups
- The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
- Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
- Epi-quasi normality
- Derivative and higher-order Cauchy integral formula of matrix functions
- Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
- Solutions to a multi-phase model of sea ice growth
- Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
- Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
- Review Article
- Semiprimeness of semigroup algebras
- Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
- Third-order differential equations with three-point boundary conditions
- Fractional calculus, zeta functions and Shannon entropy
- Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
- Synchronization of Caputo fractional neural networks with bounded time variable delays
- On quasilinear elliptic problems with finite or infinite potential wells
- Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
- On a fractional Schrödinger-Poisson system with strong singularity
- Parabolic inequalities in Orlicz spaces with data in L1
- Special Issue on Evolution Equations, Theory and Applications (Part II)
- Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
- Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
- On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
- Blow-up results of the positive solution for a class of degenerate parabolic equations
- Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
- On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
- General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
- On hyponormality on a weighted annulus
- Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
- Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
- Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
- Marangoni convection in layers of water-based nanofluids under the effect of rotation
- A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
- Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
- Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
- Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
- On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
- The Lp dual Minkowski problem about 0 < p < 1 and q > 0
