  Let $\Lambda$ be a graded self-injective algebra. We describe its smash
product $\Lambda# k\mathbb Z^*$ with the group $\mathbb Z$, its Beilinson
algebra and their relationship. Starting with $\Lambda$, we construct algebras
with finite global dimension, called $\tau$-slice algebras, we show that their
trivial extensions are all isomorphic, and their repetitive algebras are the
same $\Lambda# k\mathbb Z^*$. There exist $\tau$-mutations similar to the BGP
reflections for the $\tau$-slice algebras. We also recover Iyama's absolute
$n$-complete algebra as truncation of the Koszul dual of certain self-injective
algebra.
