Bounds on the global dimension of certain piecewise hereditary   categories
  We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables.   We use this to show that the global dimension of a finite dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight. 
Sefi Ladkani