The graded centers of derived discrete algebras

TL;DR

This paper characterizes the graded centers of derived categories of derived discrete algebras, showing their nontriviality depends on the algebra's global dimension and relating nilpotent parts to Auslander--Reiten translation and suspension functor.

Contribution

It provides a complete description of the graded centers for derived discrete algebras and links their structure to algebraic properties like global dimension.

Findings

Reduced part of the graded center is nontrivial iff the algebra has infinite global dimension.

Nilpotent part is governed by objects where Auslander--Reiten translation equals a power of suspension.

The structure of graded centers is explicitly characterized for derived discrete algebras.

Abstract

We describe in the paper the graded centers of the derived categories of the derived discrete algebras. In particular, we prove that if $A$ is a derived discrete algebra, then the reduced part of the graded center of the derived category of $A$ is nontrivial if and only if $A$ has infinite global dimension. Moreover, we show that the nilpotent part of the graded center is controlled by the objects for which the Auslander--Reiten translation coincides with a power of the suspension functor.