The graded center of a triangulated category

Jon F. Carlson, Peter Webb

Research output: Contribution to journalArticlepeer-review

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 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.

Original languageEnglish (US)
Pages (from-to)74-95
Number of pages22
JournalJournal of the Australian Mathematical Society
Volume102
Issue number1
DOIs
StatePublished - Feb 1 2017

Keywords

  • Auslander-Reiten triangle
  • Serre functor
  • graded center
  • stable module category

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