Cleft Extensions of Rings and Their Impact on Singularity Categories and Gorenstein Homological Properties
This paper investigates the homological properties of cleft extensions of rings, demonstrating their impact on Gorensteinness, singularity categories, and big singularity categories, ultimately leading to equivalences under specific conditions.
The Stabilized Bounded N-Derived Category of an Exact Category: Relating Singularity Categories and Stable Categories of N-Complexes
This research paper generalizes Buchweitz's Theorem, which relates singularity categories and stable categories of modules over Gorenstein rings, to the setting of N-complexes over exact categories.
Explicit Resolutions of Unbounded Complexes: A Counterexample and the Necessity of Roos' (Ab.4˚)-k Axiom
Several proposed methods for explicitly constructing resolutions of unbounded complexes fail in general, highlighting the necessity of Roos' (Ab.4˚)-k axiom for such constructions to hold.
Delooping Levels and Their Relationship to Other Homological Invariants in Artin Algebras
This research paper introduces the concept of global delooping level (Dell) for Artin algebras, investigates its relationship to other homological invariants like finitistic dimension and Igusa-Todorov functions, and demonstrates its application in specific cases like Gorenstein and truncated path algebras.
전역 차원과 특이점 범주의 일반화에 관하여
이 논문은 고전적인 특이점 범주를 일반화한 n-특이점 범주를 소개하고, 환의 n-전역 차원과의 관계를 규명합니다. 특히, 환의 n-특이점 범주가 사라지는 것과 환의 n-전역 차원이 유한한 것이 동치임을 보이고, recollement라는 개념을 사용하여 n-특이점 범주의 소멸성을 특징짓습니다.
Functorial Languages in Homological Algebra and Their Application to the Lower Central Series of Groups
This paper introduces new functorial languages, extending the fr-language, to compute higher limits of functors in homological algebra, particularly focusing on applications to the lower central series of groups and their homology.
Bounding the Finitistic Dimensions of DG-Modules over Commutative DG-Rings
This paper establishes bounds on the projective dimension of DG-modules over commutative noetherian DG-rings with finite amplitude, effectively answering Bass's questions regarding finitistic dimensions in this context.
Rouquier Dimension Does Not Imply Finite Global Dimension, But It Does Imply Finite Weak Global Dimension for Coherent Rings
Contrary to a previous claim, a ring having finite Rouquier dimension for its category of perfect complexes does not guarantee finite global dimension. However, for coherent rings, it is equivalent to having finite weak global dimension.
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