On the invariant theory for tame tilted algebras

Abstract

We show that a tilted algebra $A$ is tame if and only if for each generic root $d$ of $A$ and each indecomposable irreducible component $C$ of $mod\left(A,d\right)$, the field of rational invariants $k{\left(C\right)}^{GL\left(d\right)}$ is isomorphic to $k$ or $k\left(x\right)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $d$ of $A$ and each indecomposable irreducible component $C\subseteq mod\left(A,d\right)$, the moduli space $\mathcal{ℳ}{\left(C\right)}_{\theta }^{ss}$ is either a point or just ${\mathbb{P}}^1$ whenever $\theta$ is an integral weight for which $C_{\theta }^s\ne \varnothing$. We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space $\mathcal{ℳ}{\left(C\right)}_{\theta }^{ss}$ being smooth for each generic root $d$ of $A$, each indecomposable irreducible component $C\subseteq mod\left(A,d\right)$, and each integral weight $\theta$ for which $C_{\theta }^s\ne \varnothing$. 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.