We show that a tilted algebra
is tame if and only if for each generic root
of
and each indecomposable
irreducible component
of
, the field of
rational invariants
is isomorphic to
or
.
Next, we show that the tame tilted algebras are precisely those tilted algebras
with the property that
for each generic root
of
and each indecomposable irreducible component
, the moduli space
is either a point
or just
whenever
is an integral
weight for which
.
We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space
being smooth for
each generic root
of
, each indecomposable
irreducible component
,
and each integral weight
for which
.
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
implies the
tameness of
.
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
