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Coverings and Truncations of Graded Selfinjective Algebras

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