Theoretical development of homological algebra in exact categories and triangulated categories

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K18727
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Algebra, Geometry, and related fields
• Research Institution: Nagoya University (2019); Kagoshima University (2017-2018)
• Principal Investigator: Nakaoka Hiroyuki 名古屋大学, 多元数理科学研究科, 准教授 (90568677)
• Project Period (FY): 2017-06-30 – 2020-03-31
• Keywords: 完全圏 / 三角圏 / n-角圏 / n-完全圏

Outline of Final Research Achievements

The collaboration with Yann Palu, in which we have introduced the notion of an extriangulated category, was published after revision. In the sequel, a collaboration with Yu Liu on the heart of cotorsion pairs on extriangulated categories was published after revision.
In collaboration with Martin Herschend and Yu Liu, we have introduced the notion of an n-exangulated category as a higher version of extriangulated category. This manuscript was submitted to a journal, now under review. In collaboration with Osamu Iyama and Yann Palu, we wrote a preprint on the Auslander-Reiten theory in extriangulated categories.
In addition, I have submitted a preprint concerning derived invariants of gentle algebras to arXiv.

Free Research Field

代数学における圏論

Academic Significance and Societal Importance of the Research Achievements

ホモロジー代数の舞台として重要な完全圏・三角圏のクラスは定義としては互いにほぼ排他的であり、両者で定義される類似の概念が同じような性質を持つ場合であっても、別々に取り扱い、しばしば同じような議論を二度行う必要が生じていた。
本研究では、これらの二つの圏のクラスをExt1-関手の言葉で同時に扱う概念としてYann Paluと共に導入したextriangulated category を用いて、統一的なホモロジー代数の理論整備を目指している。相対ホモロジー代数、Auslander-Reiten理論といった完全圏・三角圏で知られる事柄を実際に統一的に扱うことができた。また、高次数版の定義も考察した。