Cyclically valued rings and formal power series

Leloup, Gérard

Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 37-60.

Rings of formal power series $k\left[\right[C\left]\right]$ with exponents in a cyclically ordered group $C$ were defined in [2]. Now, there exists a “valuation” on $k\left[\right[C\left]\right]$ : for every $\sigma$ in $k\left[\right[C\left]\right]$ and $c$ in $C$, we let $v(c,\sigma )$ be the first element of the support of $\sigma$ which is greater than or equal to $c$. Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by “twisting” the multiplication in $k\left[\right[C\left]\right]$. We prove that a cyclically valued ring is a subring of a power series ring $k\left[\right[C,\theta \left]\right]$ with twisted multiplication if and only if there exist invertible monomials of every degree, and the support of every element is well-ordered. We also give a criterion for being isomorphic to a power series ring with twisted multiplication. Next, by the way of quotients of cyclic valuations, it follows that any power series ring $k\left[\right[C,\theta \left]\right]$ with twisted multiplication is isomorphic to a $R^{\prime }\left[[C^{\prime },{\theta }^{\prime }]\right]$, where $C^{\prime }$ is a subgroup of the cyclically ordered group of all roots of $1$ in the field of complex numbers, and $R^{\prime }\simeq k\left[[H,\theta ]\right]$, with $H$ a totally ordered group. We define a valuation $v(ϵ,·)$ which is closer to the usual valuations because, with the topology defined by $v(a,·)$, a cyclically valued ring is a topological ring if and only if $a=ϵ$ and the cyclically ordered group is indeed a totally ordered one.

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