Width and Local Homology Dimension for Triangulated Categories
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsAll is as in the report!
Comments for author File: 
Comments.pdf
Author Response
[Comments 1] For all, there exists, implies, if and only if, instead the corresponding logical symbols;
[Response 1} In page 4 line 16, change ``if and only if " to ``$\Leftrightarrow$". Considering the typographic aesthetics of the article, the others were not changed.
[Comments 2] Check the names of all the journals in the list of references including the abbreviated of ones;
[Response 2] The names that appear in the references are checked and there are no errors.
[Comments 3] Add still recent paper in the list of references;
[Response 3] Added 7 references, which are:9,10,11,12,16,17,18.
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsReport on "Width and Local Homology Dimension for Triangulated Categories"
Author: Li Wang
Affiliation: Department of Mathematics, Lanzhou City University, Lanzhou 730070, China
Email: [email protected]
There are few suggestions:
1. Enhance the explanation of the theoretical framework underlying the concepts of width and local homology dimension. Make sure that the connection to known results is thoroughly discussed.
2. Ensure consistency in terminology and notation throughout the paper. For instance, clarify the distinction between different types of support and co support, and ensure that these terms are used consistently.
3. Provide clear definitions for all key terms and concepts at their first mention. This includes terms such as "compactly generated triangulated category," "Koszul object," and "local homology functor."
4. Reorganize sections for better flow. The current structure jumps between concepts and results, which may be confusing. A more logical progression from definitions to theorems and proofs would improve readability.
5. Ensure that all references are formatted according to the journal’s guidelines. Check for consistency in citation style.
6. Consider citing additional relevant work to provide a more comprehensive background. This could include recent advancements in the field or related studies.
Comments on the Quality of English Language
The English is fine
Author Response
[Comments 1] Enhance the explanation of the theoretical framework underlying the concepts of width and local homology dimension. Make sure that the connection to known results is thoroughly discussed;
[Response 1]So far, I haven't found any examples related to the known results. Because there are few articles on local homology functor and width, and the definitions of local homology functor and width in triangulated category are abstract, I have not found them at present.
[Comments 2] Ensure consistency in terminology and notation throughout the paper. For instance, clarify the distinction between different types of support and co support, and ensure that these terms are used consistently;
[Response 2] Page 2, ``(big)support" change to ``Support(big support)", follow the statement in reference [13]. Page 4, line 20, ``small cosupport" change to ``cosupport", follow the statement in reference [15].
At present, there are four definitions in the compactly generated triangulated category: support, cosupport, Support(big support) and Cosupport (big cosupport). So at that time, I wrote ``coupport" as ``small cosupport". In this paper, only cosupport is used.
[Comments 3] Provide clear definitions for all key terms and concepts at their first mention. This includes terms such as ``compactly generated triangulated category," ``Koszul object," and ``local homology functor."
[Response 3] Dear referee, ``compactly generated triangulated category" first appeared in the abstract. I have never seen an article about definitions written in the abstract, so I would like to ask, do you want to put all these definitions in the first mentioned point? (These definitions are all in the second part).
[Comments 4] Reorganize sections for better flow. The current structure jumps between concepts and results, which may be confusing. A more logical progression from definitions to theorems and proofs would improve readability;
[Response 4] Dear referee, I think the distribution of this chapter is reasonable. First, give the basic definition, then study the width in the next section, and finally study the homology dimension. I'm not sure how to change this distribution.
[Comments 5] Reorganize sections for better flow. The current structure jumps between concepts and results, which may be confusing. A more logical progression from definitions to theorems and proofs would improve readability;
[Response 5] Dear referee, I think the distribution of this chapter is reasonable. First, give the basic definition, then study the width in the next section, and finally study the homology dimension. I'm not sure how to change this distribution.
[Comments 6] Consider citing additional relevant work to provide a more comprehensive background. This could include recent advancements in the field or related studies;
[Response 6] Added 7 references, which are:9,10,11,12,16,17,18.
Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsFor a Noetherian graded-commutative ring $R$, its homogeneous ideal $a\subset R$ and for a compactly generated triangulated category $T$, two numerical invariants for objects of $T$ are defined, width and local homology dimension. After establishing some basic properties of these invariants, the author provides certain estimates for them, and relates the second invariant to algebraic properties of $R$.
These results are interesting for specialists in algebra and category theory.
The main deficiency of the paper is the absence of examples. It would gain a lot if the author could provide a couple of examples, both trivial and non-trivial ones.
I would also suggest to add some details into the Introduction section. Currently, after the motivation, the author describes the main results without defining the notation. The formulas defining the two invariants and the following text use a lot of notation defined much later. I understand that moving all definitions to the Introduction section would overload it, but the main ones should be mentioned here, and, at least, references to the future definitions should be inserted. For example, it should be mentioned that $a$ denotes a homogeneous ideal in $R$ (now $a$ is used in Theorem A, and is explained only in Theorem B).
Comments on the Quality of English Language
Some minor issues should be fixed. Here are some examples.
Page 2, lines 10-12: I don't understand the meaning of the phrase "To classify colocalized subcategories, Benson, Iyengar and Krause [5] developed local homology functor “Λ” and cosupport for triangulated categories, which building on their work of local cohomology and support." Maybe it should read "... which is based on their work ..."?
Page 2, Theorem B: "For any $X\neq 0$ be an object ..." should be "Let $X\neq 0$ be an object ..." or "For any non-zero object $X$ of ..."
Page 3, line 19: "Note that the subsets V(a) and Z(p) are specialization closed." What is "specialization closed"?
Page 10, line preceding Corollary 4.5: "It is complete the proof". As there is the sign of the proof end, it is better to skip this phrase at all. A similar problem at page 11, etc.
Page 14, Acknowledgement. Both Acknowledgement and the author should be in singular.
Author Response
[Comments 1:] The main deficiency of the paper is the absence of examples. It would gain a lot if the author could provide a couple of examples, both trivial and non-trivial ones;
[Response 1:] Sorry, I haven't found some examples so far. Because there are few articles on local homology functor and width, and the definitions of local homology functor and width in triangulated category are abstract, I have not found them at present.
[Comments 2:] I would also suggest to add some details into the Introduction section. Currently, after the motivation, the author describes the main results without defining the notation. The formulas defining the two invariants and the following text use a lot of notation defined much later. I understand that moving all definitions to the Introduction section would overload it, but the main ones should be mentioned here, and, at least, references to the future definitions should be inserted. For example, it should be mentioned that $a$ denotes a homogeneous ideal in $R$ (now $a$ is used in Theorem A, and is explained only in Theorem B);
[Response 2:] It has modified.
[Comments 3:] Page 2, lines 10-12: I don't understand the meaning of the phrase?``To classify colocalized subcategories, Benson, Iyengar and Krause [5] developed local homology functor ``$\Lambda$" and cosupport for triangulated categories, which building on their work of local cohomology and support.`` Maybe it should read "... which is based on their work ...";
[Response 3:] It has modified.
[Comments 4:] Page 2, Theorem B: ``For any $X\neq 0$ be an object ..." should be ``Let $X\neq 0$ be an object ..." or ``For any non-zero object $X$ of ..."?
[Response 4:] It has modified.
[Comments 5:] Page 3, line 19: ``Note that the subsets $\mathcal{V}(\mathfrak{a})$ and $\mathcal{Z}(\mathfrak{p})$ are specialization closed." What is ``specialization closed";
[Response 5:] On page 3, I added the definition of ``specialization closed".
[Comments 6:] Page 10, line preceding Corollary 4.5: ``It is complete the proof". As there is the sign of the proof end, it is better to skip this phrase at all. A similar problem at page 11, etc;
[Response 6:] Removed all ``It is complete the proof".
[Comments 7:] Page 14, Acknowledgement. Both Acknowledgement and the author should be in singular;
[Response 7:] ``Acknowledgements" change to ``Acknowledgement", ``authors" change to ``author".
Author Response File: Author Response.pdf
