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Tachikawa's second conjecture, derived recollements, and gendo-symmetric algebras

Part of: Homological methods Rings and algebras arising under various constructions Representation theory of rings and algebras Higher algebraic $K$-theory

Published online by Cambridge University Press:  27 December 2024

Hongxing Chen ,
Ming Fang  and
Changchang Xi
Hongxing Chen
Affiliation:
School of Mathematical Sciences & Academy for Multidisciplinary Studies, Capital Normal University, 100048 Beijing, P. R. China chenhx@cnu.edu.cn
Ming Fang
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190 Beijing, P. R. China fming@amss.ac.cn School of Mathematical Sciences, University of Chinese Academy of Sciences, 100049 Beijing, P. R. China
Changchang Xi
Affiliation:
School of Mathematical Sciences, Capital Normal University, 100048 Beijing P. R. China xicc@cnu.edu.cn School of Mathematics and Statistics, Shaanxi Normal University, 710119 Xi'an, Shaanxi, P. R. China
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Abstract

Tachikawa's second conjecture for symmetric algebras is shown to be equivalent to indecomposable symmetric algebras not having any nontrivial stratifying ideals. The conjecture is also shown to be equivalent to the supremum of stratified ratios being less than $1$, when taken over all indecomposable symmetric algebras. An explicit construction provides a series of counterexamples to Tachikawa's second conjecture from each (potentially existing) gendo-symmetric algebra that is a counterexample to Nakayama's conjecture. The results are based on establishing recollements of derived categories and on constructing new series of algebras.

Keywords

Auslander algebragendo-symmetric algebramirror-reflective algebraNakayama conjecturerecollementsymmetric algebraTachikawa's second conjecture

MSC classification

Primary: 16G10: Representations of Artinian rings 16E35: Derived categories 19D50: Computations of higher $K$-theory of rings
Secondary: 16E10: Homological dimension 16S70: Extensions of rings by ideals

Information

Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 11 , November 2024 , pp. 2704 - 2737
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Copyright
© The Author(s), 2024. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

In memory of Professor Hiroyuki Tachikawa (1930–2022)

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