Smith-Orbit Classification of Extensions and Exact-Sequence Asymmetry for C4*-Modules

Abstract

Let $R$ be a ring and let a $C{4}^{\ast }$-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 $C{4}^{\ast }$ exactly when it is cyclic or semisimple; in finite length $C{4}^{\ast }$ and strongly $C{4}^{\ast }$ coincide. The main result is a complete orbit calculation over a discrete valuation ring $V$. Put $A=(V/({\pi }^a){)}^r$, $C=(V/({\pi }^c){)}^s$ and $m=\min \{a,c\}$. Then \[ \Ext^1_V(C,A)\cong \operatorname{Mat}_{r\times s}(V/(π^m)), \] and the $\mathrm{Aut}(A)\times \mathrm{Aut}(C)$-orbits are classified by a Smith profile $0\le {\nu }_1\le \cdots \le {\nu }_q\le m$, $q=\min \{r,s\}$. 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 $V/({\pi }^0)=0$. 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 $C{4}^{\ast }$ 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 $C{4}^{\ast }$-modules over a Dedekind domain is decided by explicit local orbit invariants.