导出去环层数的次数平移导出不变性
Degree-shifted derived invariance of derived delooping levels
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中文总结 AI 辅助
该研究解决了导出去环层数有限性的导出不变性开放问题，证明其具有次数平移不变性，还引入整体导出去环层数并证明其∞版本有限性是导出不变量。
中文摘要 AI 辅助
Gélinas将Artin代数的去环层数定义为一种同调不变量，它给出了反代数的大有限维数的上界。一个自然的问题是，这类不变量是否在导出等价下保持不变。Chen最近证明了经典去环层数及其子导出变体的有限性不具有导出不变性，但Guo和Igusa提出的更精细的导出去环层数的情况仍未解决。在本文中，我们证明了导出去环层数的有限性在导出等价下具有次数平移不变性：若两个代数通过宽度为$k_T$的倾斜复形实现导出等价，则一侧$(k+k_T)$-导出去环层数的有限性蕴含另一侧$k$-导出去环层数的有限性。由此可得，导出$\boldsymbol{\text{∞}}$-去环层数的有限性在导出等价下是不变的。此外，我们引入了整体导出去环层数，并证明在导出等价下，其$\boldsymbol{\text{∞}}$版本的变化量不超过$k_T$。特别地，它的有限性是一种导出不变量。
英文摘要
Gélinas introduced the delooping level of an Artin algebra as a homological invariant that bounds the big finitistic dimension of the opposite algebra. A natural question is whether such invariants are preserved under derived equivalences. Chen recently showed that the finiteness of the classical delooping level and its sub-derived variant is not derived-invariant, but the case of the finer derived delooping level of Guo and Igusa remained open. In this paper, we prove that the finiteness of the derived delooping level is \emph{degree-shift invariant} under derived equivalences: if two algebras are derived equivalent via a tilting complex of width $k_T$, then finiteness of the $(k+k_T)$-derived delooping level on one side implies finiteness of the $k$-derived delooping level on the other. Consequently, the finiteness of the derived $\infty$-delooping level is invariant under derived equivalences. Furthermore, we introduce the global derived delooping level and prove that, under a derived equivalence, its $\infty$-version changes by at most $k_T$. In particular, its finiteness is a derived invariant.
发表机构
- Zhejiang Normal University（浙江师范大学）
- Nanjing Forestry University（南京林业大学）
- Jiangsu Normal University（江苏师范大学）
机构由 AI 辅助整理，请以论文原文为准。
