Character idempotent ($e_{\chi }=|\Delta {|}^{-1}{\sum }_{\delta \in \Delta }\chi (\delta {)}^{-1}\delta$)

If the order of a finite abelian group $\Delta$ is invertible in the coefficient ring and all its character values belong to that ring, its character idempotents are orthogonal and sum to one. They decompose any module into character components. In a cyclotomic tower with odd $p$, the Teichmüller character provides these components integrally over ${\mathbb{Z}}_p$.