[Paper Review] Quasi-monomial actions and some 4-dimensional rationality problems

Akinari Hoshi, Ming-chang Kang+1 more|arXiv (Cornell University)|Jan 6, 2012

Advanced Algebra and Geometry16 references4 citations

TL;DR

This paper introduces the concept of quasi-monomial group actions on rational function fields, generalizing monomial actions, and proves that the fixed field $ k(x_1,x_2,x_3,x_4)^G $ is rational over $ k $ when the action is decomposable. The key contribution is establishing rationality for purely quasi-monomial actions in dimension four by leveraging lattice decomposition and retract rationality techniques.

ABSTRACT

Let $G$ be a finite group acting on $k(x_1,...,x_n)$, the rational function field of $n$ variables over a field $k$. The action is called a purely monomial action if $σ...x_j=\prod_{1\le i\le n} x_i^{a_{ij}}$ for all $σ\in G$, for $1\le j\le n$ where $(a_{ij})_{1\le i,j\le n} \in GL_n(\bm{Z})$. The main question is that, under what situations, the fixed field $k(x_1,...,x_n)^G$ is rational (= purely transcendental) over $k$. This rationality problem has been studied by Hajja, Kang, Hoshi, Rikuna when $n\le 3$. In this paper we will prove that $k(x_1,x_2,x_3,x_4)^G$ is rational over $k$ provided that the purely monomial action is decomposable. To prove this result, we introduce a new notion, the quasi-monomial action, which is a generalization of previous notions of multiplicative group actions. Moreover, we determine the rationality problem of purely quasi-monomial actions of $K(x, y)^G$ over $k$ where $k= K^G$.

Motivation & Objective

• To generalize monomial group actions on rational function fields to the broader class of quasi-monomial actions.
• To investigate the rationality problem for purely quasi-monomial actions in dimension four, particularly when the action is decomposable.
• To determine conditions under which the fixed field $ K(x_1,x_2,x_3,x_4)^G $ is rational over $ k $, where $ k = K^G $.
• To extend known results on rationality of algebraic tori in dimensions two and three to dimension four via lattice decomposition techniques.

Proposed method

• Introduce the notion of quasi-monomial actions, where group elements act via twisted monomials with coefficients in a finite Galois extension $ K $ of $ k $.
• Define purely quasi-monomial actions as a special case where coefficients $ c_j(\sigma) = 1 $, reducing to monomial actions when $ k = K $.
• Use the anti-equivalence between algebraic tori and $ \mathbb{Z}[G] $-lattices to translate the rationality problem into a lattice-theoretic question.
• Apply the decomposition of the $ G $-lattice $ M = M_1 \oplus M_2 $ to analyze the fixed field $ K(M)^G $ as a free composite of $ K(M_1)^G $ and $ K(M_2)^G $.
• Leverage Saltman’s theory of flabby resolutions and retract rationality to determine when $ K(M)^G $ is rational over $ k $, especially when $ \text{rank}_{\mathbb{Z}} M_i \leq 3 $.
• Use Yamasaki’s result to show that certain fixed fields are not retract rational, thereby establishing non-rationality in exceptional cases.

Experimental results

Research questions

• RQ1 Under what conditions is the fixed field $ K(x_1,x_2,x_3,x_4)^G $ rational over $ k $ when $ G $ acts via quasi-monomial $ k $-automorphisms?
• RQ2 When is the fixed field $ K(x_1,x_2,x_3,x_4)^G $ rational over $ k $ for a decomposable quasi-monomial action?
• RQ3 Can the rationality of $ K(M)^G $ be reduced to the rationality of its components $ K(M_1)^G $ and $ K(M_2)^G $ when $ M = M_1 \oplus M_2 $?
• RQ4 Are there cases where $ K(M)^G $ is not retract rational, even if $ K(M_1)^G $ and $ K(M_2)^G $ are rational?
• RQ5 How does the structure of the $ G $-lattice $ M $, especially its decomposition, affect the rationality of the fixed field?

Key findings

• The fixed field $ k(x_1,x_2,x_3,x_4)^G $ is rational over $ k $ if the purely quasi-monomial action is decomposable.
• For a finite group $ G $ with $ G \simeq \text{Gal}(K/k) $, $ K(M)^G $ is retract $ k $-rational if and only if $ K(M_1)^G $ and $ K(M_2)^G $ are retract $ k $-rational.
• If $ K(M_1)^G $ and $ K(M_2)^G $ are $ k $-rational, then $ K(M)^G $ is also $ k $-rational, provided $ M = M_1 \oplus M_2 $ as $ \mathbb{Z}[G] $-modules.
• When $ \text{rank}_{\mathbb{Z}} M_i \leq 3 $, $ K(M)^G $ is $ k $-rational if and only if $ K(M_1)^G $ and $ K(M_2)^G $ are $ k $-rational.
• The fixed field $ k(x_1,x_2,x_3,x_4)^{\langle \tau \rangle} $ is not retract rational over $ k(x_4) $, and thus not retract $ k $-rational, as shown via Yamasaki’s lemma.
• The rationality of $ K(M)^G $ is equivalent to the invertibility of the flabby class $ [M]^{fl} $, and $ [M_1 \oplus M_2]^{fl} = [M_1]^{fl} \oplus [M_2]^{fl} $, enabling decomposition-based analysis.