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Semisimple derivations, rational slice and kernels over affine domains

Luis Cid

Abstract

Let k be an algebraically closed field of characteristic zero and let B be a finitely generated k-domain. We study semisimple derivations on B, with special emphasis on those whose eigenvalues are integers. For such derivations, after passing to the field of fractions and choosing a rational slice s with D(s) = s, we describe the kernel of D explicitly in terms of semi-invariant generators. We also obtain descriptions of the kernel on suitable localizations of B and on B itself by intersection. Several basic properties of semisimple derivations and their behavior under conjugation are also discussed

Semisimple derivations, rational slice and kernels over affine domains

Abstract

Let k be an algebraically closed field of characteristic zero and let B be a finitely generated k-domain. We study semisimple derivations on B, with special emphasis on those whose eigenvalues are integers. For such derivations, after passing to the field of fractions and choosing a rational slice s with D(s) = s, we describe the kernel of D explicitly in terms of semi-invariant generators. We also obtain descriptions of the kernel on suitable localizations of B and on B itself by intersection. Several basic properties of semisimple derivations and their behavior under conjugation are also discussed
Paper Structure (4 sections, 14 theorems, 100 equations)

This paper contains 4 sections, 14 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on locally finite and semisimple derivations
  3. Semi-invariants and weight decomposition
  4. Rational slices and the kernel over the field of fractions

Key Result

Lemma 2.4

Let $D\in \operatorname{Der}_k(B)$ and let $S\subseteq B$ be a multiplicative subset such that $0\notin S$. Then $D$ extends uniquely to a derivation given by In particular, $D$ extends uniquely to a derivation of $K=\operatorname{Frac}(B)$. By abuse of notation, we denote all these extensions again by $D$.

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 27 more