Geometric Reductivity--A Quotient Space Approach

TL;DR

This paper provides a new proof that reductive algebraic groups are geometrically reductive by demonstrating the existence of certain quotient spaces, thereby deriving Haboush's Theorem from a geometric perspective.

Contribution

It introduces a quotient space approach to prove the geometric reductivity of reductive algebraic groups, offering an alternative proof to Haboush's Theorem.

Findings

Existence of quotients of semi-stable loci under reductive group actions.

Derivation of Haboush's Theorem from geometric quotient constructions.

New proof technique based on quotient space approach.

Abstract

We give another proof that a reductive algebraic group is geometrically reductive. We show that a quotient of the semi-stable locus (by a linear action of a reductive algebraic group on a projective scheme) exists, and from this Haboush's Theorem (Mumford's Conjecture) follows.