The Graded Center of a Triangulated Category

TL;DR

This paper investigates the structure of the graded center in triangulated categories with Serre functors, revealing new elements and their classifications, especially in the context of representation theory and cohomology of block algebras.

Contribution

It characterizes elements of the graded center supported on shift orbits, connecting them to known constructions and extending to cases with infinitely many orbits, using functorial methods.

Findings

Elements supported on a single shift orbit are of a known type.

Finite support in multiple orbits accounts for all such elements under certain conditions.

Infinite support in the stable module category reveals new elements not present in wild block algebras.

Abstract

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor We show that such natural transformations which have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.