Rank additivity for quasi-tilted algebras of canonical type

Given the category ${\displaystyle cohsym\left\{X\right\}}$ of coherent sheaves over a weighted projective line ${\displaystyle sym\left\{X\right\}=sym\left\{X\right\}(und\left\{\lambda \right\},und\left\{p\right\})}$ (of any representation type), the endomorphism ring ${\displaystyle mit\Sigma =End\left(cal\left\{T\right\}\right)}$ 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 ${\displaystyle cohsym\left\{X\right\}}$ (Example 4.3).