On the invariant theory for tame tilted algebras

TL;DR

This paper characterizes tame tilted algebras using invariant theory and moduli space properties, establishing a connection between algebraic tameness and geometric smoothness of associated moduli spaces.

Contribution

It provides a new invariant-theoretic criterion for tameness of tilted algebras and links this to the geometric smoothness of their moduli spaces, extending understanding of algebraic and geometric properties.

Findings

Tame tilted algebras have invariant fields isomorphic to k or k(x).

Tame tilted algebras correspond to moduli spaces being points or projective lines.

Smoothness of moduli spaces implies the algebra is tame.

Abstract

We show that a tilted algebra $A$ is tame if and only if for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C$ of $\module(A,\dd)$, the field of rational invariants $k(C)^{\GL(\dd)}$ is isomorphic to $k$ or $k(x)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C \subseteq \module(A,\dd)$, the moduli space $\M(C)^{ss}_{\theta}$ is either a point or just $\mathbb P^1$ whenever $\theta$ is an integral weight for which $C^s_{\theta}\neq \emptyset$. We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space $\M(C)^{ss}_{\theta}$ being smooth for each generic root $\dd$ of $A$, each indecomposable irreducible component $C \subseteq \module(A,\dd)$, and each integral weight $\theta$ for which $C^s_{\theta} \neq \emptyset$. As a consequence of this latter description, we show that the smoothness of the various moduli spaces of modules for a strongly simply connected algebra $A$ implies the tameness of $A$. Along the way, we explain how moduli spaces of modules for finite-dimensional algebras behave with respect to tilting functors, and to theta-stable decompositions.