Rank additivity for quasi-tilted algebras of canonical type

Abstract

Given the category coh X of coherent sheaves over a weighted projective line X = X(λ, p) (of any representation type), the endomorphism ring Σ = End(T ) of an arbitrary tilting sheaf-which is by definition an almost concealed canonical algebra-is shown to satisfy a rank additivity property (Theorem 3.2). Moreover, this property extends to the representationinfinite quasi-tilted algebras of canonical type (Theorem 4.2). Finally, it is demonstrated that rank additivity does not generalize to the case of tilting complexes over coh X (Example 4.3).

Constructing tilted algebras from cluster-tilted algebras

“…Let α : b → a be an arrow in an admissible tilted set S, and assume that there is another arrow β from b to a. We claim that β belongs to S. To see this, recall that by [Hu,2.4 Second, notice that Proposition 3.7 relies only on Theorem 3.5 (b), and thus holds in this generality.…”

“…Let α : b → a be an arrow in an admissible tilted set S, and assume that there is another arrow β from b to a. We claim that β belongs to S. To see this, recall that by [Hu,2.4], only one of the spaces Ext i C (S a , S b ) can be non-zero for i = 0, 1, 2, where S a , S b are the simple C-modules at the vertices a and b for the tilted algebra C = B/ S . Note that the arrow α in S corresponds to a minimal relation in Ext 2 C (S a , S b ) = 0.…”

Graded mutation in cluster categories coming from hereditary categories with a tilting object

“…injectives). Note that every almost complete tilting object in H * has exactly two complements (see [Hü3,Corollary 0.4] for the coh X case, and Corollary 3.7 for the general case).…”