Vol. 7, No. 1, 2013

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ISSN: 1944-7833 (e-only)
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On the invariant theory for tame tilted algebras

Calin Chindris

Vol. 7 (2013), No. 1, 193–214
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 mod(A,d), the moduli space (C)θss is either a point or just 1 whenever θ is an integral weight for which Cθs. We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space (C)θss being smooth for each generic root d of A, each indecomposable irreducible component C mod(A,d), and each integral weight θ for which Cθs. 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