On the invariant theory for tame tilted algebras

Abstract

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 M(C) ss θ is either a point or just P 1 whenever θ is an integral weight for which C s θ = ∅. We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space M(C) ss θ being smooth for each generic root d of A, each indecomposable irreducible component C ⊆ mod(A, d), and each integral weight θ for which C s θ = ∅. 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.

On the invariant theory for acyclic gentle algebras

“…The other implication (1) =⇒ (2) of the conjecture above has been verified in [15] for quasi-tilted algebras when the irreducible components involved are isotropic. Since gentle algebras are tame, Theorem 1(2) proves this implication over regular irreducible components for triangular gentle algebras.…”

“…(2) for any irreducible component C ⊂ mod(A, d) and any weight θ such that C ss θ = ∅, M(C) ss θ is a product of projective spaces. 24 The implication (2) =⇒ (1) of this conjecture has been verified for the class of quasitilted algebras and of strongly simply connected algebras (see [15]).…”

“…A description of the tameness of quasi-tilted algebras in terms of the invariant theory of the algebras in question has been found in [14,15]. In this paper, we continue this line of inquiry for the class of triangular gentle algebras, which are known to be tame.…”

“…A θ-semi-stable irreducible component C ⊆ mod(A, d) is said to be θ-well-behaved if,whenever d is the dimension vector of a factor of a Jordan-Hölder filtration in mod(A) ss θ of a generic A-module in C and C 1 ,C 2 ⊆ mod(A, d ) are two distinct irreducible components, C s 1,θ ∩ C s 2,θ = ∅. (We should point out that this notion is slightly more general than the one in[15, Section 3.3].) It follows from the work of Bobinski and Skowroński in[6] that for a tame quasi-tilted algebra, any θ-semi-stable irreducible component is θ-well-behaved.…”

Moduli Spaces of Modules of Schur-Tame Algebras

“…In all the examples that we looked at these moduli spaces are nothing else but P 1 's. On the other hand, it is known that any wild (equivalently, strictly wild) quasi-tilted or strongly simply connected algebra has a singular moduli space of modules (see [13]). So, it is natural to ask whether one always encounters singular moduli spaces of modules for strictly wild algebras.…”