On the invariant theory for tame tilted algebras

Chindris, Calin

doi:10.2140/ant.2013.7.193

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On the invariant theory for tame tilted algebras

Calin Chindris

Vol. 7 (2013), No. 1, 193–214

DOI: 10.2140/ant.2013.7.193

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 $mod (A, d)$ , the field of rational invariants $k {(C)}^{GL (d)}$ 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 $d$ of $A$ and each indecomposable irreducible component $C \subseteq mod (A, d)$ , the moduli space $ℳ {(C)}_{θ}^{s s}$ is either a point or just $ℙ^{1}$ whenever $θ$ is an integral weight for which $C_{θ}^{s} \neq \emptyset$ . We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space $ℳ {(C)}_{θ}^{s s}$ being smooth for each generic root $d$ of $A$ , each indecomposable irreducible component $C \subseteq mod (A, d)$ , and each integral weight $θ$ for which $C_{θ}^{s} \neq \emptyset$ . 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.

Keywords

exceptional sequences, moduli spaces, rational invariants, tame and wild algebras, tilting

Mathematical Subject Classification 2010

Primary: 16G10

Secondary: 16R30, 16G60, 16G20

Milestones

Received: 17 September 2011

Revised: 2 January 2012

Accepted: 20 February 2012

Published: 28 March 2013

Authors

Calin Chindris
Department of Mathematics University of Missouri Columbia, MO 65211 United States
http://www.math.missouri.edu/~chindrisc

Vol. 7, No. 1, 2013