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Two Criteria For Quasihomogeneity

Sarasij Maitra, Vivek Mukundan

Abstract

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

Two Criteria For Quasihomogeneity

Abstract

Let be a one-dimensional complete local reduced -algebra over a field of characteristic zero. The ring is said to be quasihomogeneous if there exists a surjection where denotes the module of differentials. We present two characterizations of quasihomogeneity of in the situation when is a domain: the first one on the valuation semigroup of and the other on the trace ideal of the module .
Paper Structure (9 sections, 6 theorems, 19 equations)

This paper contains 9 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Universally Finite Module of Differentials
  4. Quasihomogeneous Rings
  5. Quasihomogeneity via order valuation on units
  6. Order Valuation of Units
  7. Quasihomogeneity via trace ideal of $\Omega_R$
  8. Fractional Ideal of Derivatives
  9. The $\operatorname{h}(\cdot)$ Invariant

Key Result

Theorem 1

Let $(R,\mathfrak{m},\mathbb{k})$ be a non-regular, complete, local one dimensional domain which is a $\mathbb{k}$-algebra with $\mathbb{k}$ algebraically closed of characteristic $0$. Let $\overline{R}=\mathbb{k}\llbracket {t} \rrbracket$ with the conductor of $R$ given by $\mathfrak{C}_R=(t^{c_R}) Then $R$ is quasihomogeneous if $o(\alpha_r)+a\geqslant c_R$.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Example 3.2
  • Example 3.3
  • ...and 8 more