Quotients by Connected Solvable Groups

TL;DR

This paper introduces the concept of excellent quotients for actions of connected solvable groups on affine schemes, providing an algorithm to compute such quotients without Gr"obner bases in polynomial cases.

Contribution

It defines excellent quotients, proves their existence for connected solvable group actions, and offers an efficient algorithm for their computation, especially over polynomial rings.

Findings

Existence of excellent quotients for connected solvable group actions.

Algorithm for computing quotients without Gr"obner bases in polynomial cases.

Quotients are complete intersections in polynomial ring scenarios.

Abstract

This paper introduces the notion of an excellent quotient, which is stronger than a universal geometric quotient. The main result is that for an action of a connected solvable group $G$ on an affine scheme Spec$(R)$ there exists a semi-invariant $f$ such that Spec$(R_f) \to$ Spec$((R_f)^G)$ is an excellent quotient. The paper contains an algorithm for computing $f$ and $(R_f)^G$. If $R$ is a polynomial ring over a field, the algorithm requires no Gr\"obner basis computations, and it also computes a presentation of $(R_f)^G$. In this case, $(R_f)^G$ is a complete intersection. The existence of an excellent quotient extends to actions on quasi-affine schemes.