Two criteria for quasihomogeneity

Abstract:

Let $(R,\mathfrak {m}_R,\mathbb {k})$ be a one-dimensional complete local reduced $\mathbb {k}$-algebra over a field of characteristic zero. The ring $R$ is said to be quasihomogeneous if there exists a surjection $\Omega _R\twoheadrightarrow \mathfrak {m}$ where $\Omega _R$ denotes the module of differentials. We present two characterizations of quasihomogeniety of $R$ in the case when $R$ is a domain. The first one on the valuation semigroup of $R$ and the other on the trace ideal of the module $\Omega _R$.