Piecewise hereditary incidence algebras

Abstract

Let $K\Delta$ be the incidence algebra associated with a finite poset $\Delta$ 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 explore the simply connectedness of PHI algebras, and we give a positive answer to the so-called Skowro\'nski problem for $K\Delta$ a PHI algebra of type $\mathcal{H}$, with connected quiver of tilting objects $\mathcal{K}_{D^b (\mathcal{H})}$: the group $HH^1(K\Delta)$ 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 algebra; furthermore, we extend this result to sincere algebras proving that the strong global dimension of a sincere piecewise hereditary algebra is less or equal to three. If $A$ is a representation-infinite quasi-tilted algebra of quiver-sheaf type with a sincere indecomposable module $M$ (thus a special type of PHI algebra), then $M$ is exceptional, which makes it possible to construct a PHI algebra of wild type as the form of one-point extension algebra $K\Delta[M]$ of some PHI algebra $K\Delta$ by the canonical sincere $K\Delta$-module $M$.