C4*模的精确序列稳定性与环境实现
Smith-Orbit Classification of Extensions and Exact-Sequence Asymmetry for C4*-Modules
AI总结：
本文研究C4*模的精确序列稳定性，提出稳定条件并探讨其在子模和商模中的应用，同时构建了适合求商敏感模理论的环境框架。
摘要：
C4*-模理论目前主要依赖分解方法，但缺乏系统的闭包理论。本文发展了C4*-模和强C4*-模的精确序列框架，识别出在分裂扩展、可接受核和余核下稳定的显式假设。文章还区分了正向和反向方向：闭包结果在求商提升和因子控制假设下成立，而反向结果表明这些假设通常无法去除。这产生了关于扩展稳定性及子模和商模过渡的具体障碍模式。此外，文章还旨在范畴论层面，识别出自然的环境设置，使得C4*-模构成一个扩展封闭子范畴，或至少适合求商敏感模理论的相对精确类。最后，证明了具体的环境验证定理：半单右模在任何环上提供了一个标准的精确环境，而在半单艺林环上，这扩展到完整的模范畴。
英文摘要：
Let $R$ be a ring and let a $C4^{\ast}$-module mean a module all of whose submodules are $C4$-modules. We first determine its asymmetric exact-sequence behavior: kernel closure is unconditional, while explicit commutative examples disprove split-extension and cokernel closure. We then classify the finite-length torsion objects over a Dedekind domain. A primary component is $C4$ exactly when it is homocyclic, and it is $C4^{\ast}$ exactly when it is cyclic or semisimple; in finite length $C4^{\ast}$ and strongly $C4^{\ast}$ coincide. The main result is a complete orbit calculation over a discrete valuation ring $V$. Put $A=(V/(π^a))^r$, $C=(V/(π^c))^s$ and $m=\min\{a,c\}$. Then \[ \Ext^1_V(C,A)\cong \operatorname{Mat}_{r\times s}(V/(π^m)), \] and the $\operatorname{Aut}(A)\times\operatorname{Aut}(C)$-orbits are classified by a Smith profile $0\leqν_1\leq\cdots\leqν_q\leq m$, $q=\min\{r,s\}$. The corresponding middle term is \[ \bigoplus_{i=1}^{q} \bigl(V/(π^{ν_i})\oplus V/(π^{a+c-ν_i})\bigr) \oplus (V/(π^a))^{r-q}\oplus(V/(π^c))^{s-q}, \] where $V/(π^0)=0$. This gives exact orbit-level criteria: the middle term is $C4$ precisely for the zero orbit with $a=c$, or for an invertible square orbit with $r=s$; it is $C4^{\ast}$ precisely for the zero orbit with $a=c=1$, or for a unit scalar orbit with $r=s=1$. We reformulate the Smith data as a valuation--rank profile, determine the complete geometry and additivity of the preservation loci, and globalize the result prime by prime. Consequently every extension between arbitrary finite-length torsion $C4^{\ast}$-modules over a Dedekind domain is decided by explicit local orbit invariants.
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