# Piecewise Hereditary Incidence Algebras

## TL;DR

This paper studies piecewise hereditary incidence algebras (PHI), exploring their global dimension, connectivity, and Hochschild cohomology, and provides bounds and conditions for their algebraic properties.

## Contribution

It characterizes PHI algebras, solves the Skowroński problem for non-wild types, and establishes bounds on their global dimension.

## Key findings

- PHI algebras have bounded strong global dimension.
- Trivial Hochschild cohomology HH^1 iff the algebra is simply connected.
- Upper bound of three for the strong global dimension of sincere piecewise hereditary algebras.

## Abstract

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.