Piecewise Hereditary Incidence Algebras

Let $K\Delta$ be the incidence algebra associated with a finite poset $(\Delta,\preceq)$ over the algebraically closed field $K$. We present a study of incidence algebras $K\Delta$ that are piecewise hereditary, which we denominate PHI algebras. We investigate the strong global dimension, the simply conectedeness and the one-point extension algebras over a PHI algebras. We also give a positive answer to the so-called Skowro\'nski problem for $K\Delta$ a PHI algebra which is not of wild quiver type. That is for this kind of algebra we show that $HH^1(K\Delta)$ is trivial if, and only if, $K\Delta$ is a simply connected algebra. We determine an upper bound for the strong global dimension of PHI algebras; furthermore, we extend this result to sincere algebras proving that the strong global dimension of a sincere piecewise hereditary algebra is less or equal than three.