发表机构
School of Mathematics and Statistics, Guizhou University（贵州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究定义了模的积分轮廓与有限维复代数的积分系数环，证明积分轮廓是模的完全不变量，还得出积分系数环的Morita不变性及代数整体维数、有限维数有限的等价条件等结果。
AI 中文摘要
我们定义了所有模的积分轮廓，并对所有有限维复代数$A$引入积分系数环${^{\mathscr{P}}}\mkern-6.5mu\text{\\&}\mkern-5.5mu{_{\mathscr{I}}}(A)$。模的积分轮廓是带参数的矩阵。我们给出了模的分类定理，具体而言：(1) 两个模$M$与$N$同构当且仅当它们的积分轮廓相似，即$M\cong N$当且仅当$\displaystyle \int M \sim \int N$，这意味着积分轮廓是有限维模的完全不变量。此外，本文还证明了以下结果：(2) 引入代数的中心积分，证明其与代数的中心同构；(3) 对某些特殊模给出了描述；(4) 带有相容正交固定嵌入系统的$A$的积分系数环具有Morita不变性；(5) $A$的整体维数有限当且仅当$\mathrm{top}(A)$的嵌入积分轮廓属于${^{\mathscr{P}}}\mkern-6.5mu\text{\\&}\mkern-5.5mu{_{\mathscr{I}}}(A)[x]$；(6) $A$的有限维数有限当且仅当每个模$M$的嵌入积分轮廓属于${^{\mathscr{P}}}\mkern-6.5mu\text{\\&}\mkern-5.5mu{_{\mathscr{I}}}(A)[x]$时，其次数不超过某个固定正整数$d\in\mathbb{N}^+$。
英文摘要
We define the integral profiles of all modules and introduce integral coefficient rings ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)$ for all finite-dimensional complex algebras $A$. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules $M\cong N$ are isomorphic if and only if their integral profiles are similar, i.e., $M\cong N$ if and only if $\displaystyle \int M \sim \int N$. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of $A$ (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of $A$ is finite if and only if the embedded integral profile of $\mathrm{top}(A)$ lies in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$; (6) the finitistic dimension of $A$ is finite if and only if each embedded integral profile of $M$ lying in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$ implies that its degree is less than or equal to a fixing integer $d\in\mathbb{N}^+$; (7) we provide two sufficient conditions, such that if a finite-dimensional complex algebra $A$ satisfies one of them, then its finitistic dimension is finite.
Comments107 pages
