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

Abstract

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 $A=(V/(\pi^a))^r$, $C=(V/(\pi^c))^s$ and $m=\min\{a,c\}$. Then \[ \Ext^1_V(C,A)\cong \operatorname{Mat}_{r\times s}(V/(\pi^m)), \] and the $\operatorname{Aut}(A)\times\operatorname{Aut}(C)$-orbits are classified by a Smith profile $0\leq\nu_1\leq\cdots\leq\nu_q\leq m$, $q=\min\{r,s\}$. The corresponding middle term is \[ \bigoplus_{i=1}^{q} \bigl(V/(\pi^{\nu_i})\oplus V/(\pi^{a+c-\nu_i})\bigr) \oplus (V/(\pi^a))^{r-q}\oplus(V/(\pi^c))^{s-q}, \] 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 $C4^{\ast}$ 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.