Coverings and Truncations of Graded Selfinjective Algebras

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.