The K\"ahler different of a 0-dimensional scheme

Given a 0-dimensional scheme $\mathbb{X}$ in the projective $n$-space ${\mathbb{P}}_K^n$ over a field $K$, we are interested in studying the K"ahler different of $\mathbb{X}$ and its applications. Using the K"ahler different, we characterize the generic position and Cayley-Bacharach properties of $\mathbb{X}$ in several certain cases. When $\mathbb{X}$ is in generic position, we prove a generalized version of the Ap'{e}ry-Gorenstein-Samuel theorem about arithmetically Gorenstein schemes. We also characterize 0-dimensional complete intersections in terms of the K"ahler different and the Cayley-Bacharach property.