Cyclic crossed-product envelopes and arity-detecting relative $K$-theory of automorphism-derived $n$-ary algebras

Abstract

Let $m=n-1$. For a unital associative algebra $A$, an automorphism $\sigma$, and $c\in A^\times$ satisfying $\sigma(c)=c$ and $\sigma^m=\operatorname{Ad}_c$, we study the Hossz\'u--Gluskin-type product $[a_0,\ldots,a_m]=a_0\sigma(a_1)\cdots\sigma^{m-1}(a_{m-1})ca_m$. We prove that its endpoint-anchored all-slot envelope is the cyclic crossed product generated by $A\otimes A^{\mathrm{op}}$ and $u$, with $ux=\sigma^e(x)u$ and $u^m=c\otimes(c^{-1})^{\mathrm{op}}$. For the finite-order case $c=1$ and permutation algebras $A=k^X$, this gives an orbit--stabilizer block decomposition. A general dimension-defect theorem then computes the homotopy fibre of the forgetful $K$-theory map. In characteristic zero, transitive $C_m$-sets of every divisor length $d\mid m$ have the same ordinary spectrum $\K(k)^m$ but boundary $\mathbb Z/d$. The regular action gives $\pi_{-1}\cong\mathbb Z/m$, recovering the arity; over finite fields, the remaining torsion is governed by multiplicative orders.