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THE GRADED CENTER OF A TRIANGULATED CATEGORY

Part of: Representation theory of groups Representation theory of rings and algebras Abelian categories

Published online by Cambridge University Press:  26 September 2016

JON F. CARLSON  and
PETER WEBB
JON F. CARLSON*
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA email [email protected]
PETER WEBB
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA email [email protected]
*
✉ [email protected]
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Abstract

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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 that 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.

Keywords

Auslander–Reiten trianglegraded centerSerre functorstable module category

MSC classification

Primary: 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
Secondary: 18E30: Derived categories, triangulated categories 20C20: Modular representations and characters
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 1 , February 2017 , pp. 74 - 95
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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