发表机构
School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究了导出解环层次与相反大有限维数的比较，证明了Artin代数中导出解环层次不超过子导出解环层次，并构造了反例表明两者差异可任意大，即使对单项式代数也如此。
AI 中文摘要
Guo和Igusa引入了导出和子导出解环层次，作为解环层次的细化，并询问这两个不变量如何比较，导出解环层次是否总是等于相反的大有限维数，以及如果不相等，它们的差异能有多大。我们回答了这些问题。首先，我们证明每个Artin代数$A$满足$\mathrm{ddell}(A)\leq\mathrm{sub\mbox{-}ddell}(A)$。这给出了普通次数比较问题的肯定回答，而一个显式的八维代数$C$对每个$k\geq 2$满足$k\mbox{-}\mathrm{sub\mbox{-}ddell}(C)=0<1=k\mbox{-}\mathrm{ddell}(C)$。然后，我们计算了对于每个$k\geq 1$，Barrios--Lanzilotta--Mata的单项式族的所有$k$-导出和$k$-子导出解环层次。特别地，该族对每个$k\geq 1$满足$k\mbox{-}\mathrm{ddell}(A_{n,s})=\mathrm{Findim}(A_{n,s}^{\mathrm{op}})$，因此它没有提供后续文献中建议的反例。相比之下，我们在任意域上构造了一个十一维单项式代数$\Lambda$，使得$\mathrm{Findim}(\Lambda^{\mathrm{op}})=1<2=\mathrm{ddell}(\Lambda)=\mathrm{dell}(\Lambda)$。还给出了一个十维非单项式变体。最后，我们构造了一个有限维单项式代数族$B_r$，满足$\mathrm{Findim}(B_r^{\mathrm{op}})=1$和$\mathrm{ddell}(B_r)=2r+1$。因此，即使在对相反大有限维数固定的单项式代数中，差异$\mathrm{ddell}(A)-\mathrm{Findim}(A^{\mathrm{op}})$也是无界的。
英文摘要
Guo and Igusa introduced the derived and sub-derived delooping levels as refinements of the delooping level and asked how these two invariants compare, whether the derived delooping level always coincides with the opposite big finitistic dimension, and, if not, how large their difference can be. We answer these questions. First, we prove that every Artin algebra $A$ satisfies $\operatorname{ddell} A\leq \operatorname{sub-ddell} A$. This gives a positive answer to the ordinary-degree comparison problem, whereas a single explicit eight-dimensional algebra $C$ satisfies $k\text{-}\operatorname{sub-ddell} C=0<1=k\text{-}\operatorname{ddell} C$ for $k\geq 2$. We then compute, for every $k\geq 1$, all $k$-derived and $k$-sub-derived delooping levels of the monomial family of Barrios--Lanzilotta--Mata. In particular, that family satisfies $k\text{-}\operatorname{ddell} A_{n,s}=\operatorname{Findim}(A_{n,s}^{\mathrm{op}})$ for $k\geq 1$. In ordinary degree, this recovers an equality previously announced by Lam. We also construct a two-vertex, eleven-dimensional monomial algebra $Λ$, over an arbitrary field, such that $\operatorname{Findim}(Λ^{\mathrm{op}})=1<2=\operatorname{ddell}Λ=\operatorname{dell}Λ$. The same strict inequality is obtained by Gao, Liu and Xu using different algebras. A ten-dimensional non-monomial variant with a commutative corner algebra is also given. Finally, we construct a family of finite-dimensional monomial algebras $B_r$ satisfying $\operatorname{Findim}(B_r^{\mathrm{op}})=1$ and $\operatorname{ddell} B_r=2r+1$. Thus the difference $\operatorname{ddell} A-\operatorname{Findim}(A^{\mathrm{op}})$ is unbounded even among monomial algebras with fixed opposite big finitistic dimension.
Comments26 pages
