We introduce a general version of the singular compactness theorem which makes it possible to show that being a $$\Sigma $$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.

Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules

We introduce a general version of singular compactness theorem which makes it possible to show that being a $\Sigma$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.

Finiteness conditions and cotorsion pairs

In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we further assume our triangulated category to be algebraic, it follows that the heart of any nondegenerate compactly generated t-structure is a Grothendieck category.

pdf/torsion-pairs-in-silting-theory-2heinyi4i8.pdf

Torsion pairs in silting theory

We characterize Ding modules and complexes over Ding-Chen rings. We show that, over a Ding-Chen ring $R$, the Ding projective (respectively, Ding injective, respectively, Ding flat) $R$-modules coincide with the Gorenstein projective (respectively, Gorenstein injective, respectively, Gorenstein flat) modules, which, in turn, are noth\-ing more than modules appearing as a cycle of an exact complex of projective (respectively, injective, respectively, flat) modules. We prove a similar characterization for chain complexes of $R$-modules: a complex~$X$ is Ding projective (respectively, Ding injective, respectively, Ding flat) if and only if each component $X_n$ is Ding projective (respectively, Ding injective, respectively, Ding flat). Along the way, we generalize some results of Stovicek and Bravo, Gillespie and Hovey to obtain other interesting corollaries. For example, we show that, over any Noetherian ring, any exact chain complex with Gorenstein injective components must have all cotorsion cycle modules, that is, $Ext ^1_R(F,Z_nI) = 0$ for any such complex $I$ and flat module $F$. On the other hand, over any coherent ring, the cycles of any exact complex $P$ with projective components must satisfy $Ext ^1_R(Z_nP,A) = 0$ for any absolutely pure module~$A$.

On Ding injective, Ding projective and Ding flat modules and complexes

For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each other and with the usual stable module category. Our stable module categories are homotopy categories of Quillen model structures on the category of R-modules. These model categories involve generalizations of Gorenstein projective and injective modules that we derive by replacing finitely presented modules by modules of type FP-infinity. Along the way, we extend the perfect duality between injective left modules and flat right modules that holds over Noetherian rings to general rings by considering weaker notions of injectivity and flatness.

The stable module category of a general ring

Absolutely clean and level R-modules were introduced in [2] and used to show how Gorenstein homological algebra can be extended to an arbitrary ring R. This led to the notion of Gorenstein AC-injective and Gorenstein AC-projective R-modules. Here we study these concepts in the category of chain complexes of R-modules. We define, characterize and deduce properties of absolutely clean, level, Gorenstein AC-injective, and Gorenstein AC-projective chain complexes. We show that the category Ch(R) of chain complexes has a cofibrantly generated model structure where every object is cofibrant and the fibrant objects are exactly the Gorenstein AC-injective chain complexes.

Absolutely Clean, Level, and Gorenstein AC-Injective Complexes

We prove that, for any n ≥ 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat ...

FPn-Injective, FPn-Flat Covers and Preenvelopes, and Gorenstein AC-Flat Covers

We investigate conditions when the t-structure of Happel–Reiten–Smalo associated to a torsion pair is a compactly generated t-structure. The concept of a tCG torsion pair is introduced and for any ...

tCG torsion pairs

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Daniel Bravo

Author Tools

Chat about Author