Realizability of modules over Tate cohomology

David John Benson, H. Krause, S. Schwede

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology gamma(G) is an element of HH3,-1 (H) over cap*(G,k) with the following property. Given a graded (H) over cap*(G,k)-module X, the image of gamma(G) in Ext((H) over cap*(G,k))(3,-1) (X,X) vanishes if and only if X is isomorphic to a direct summand of (H) over cap*(G, M) for some kG-module M.

The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra A, there is also a canonical element of Hochschild cohomology HH3,-1 H*(A) which is a predecessor for these obstructions.

Original languageEnglish
Pages (from-to)3621-3668
Number of pages47
JournalTransactions of the American Mathematical Society
Volume356
Issue number9
DOIs
Publication statusPublished - Jan 2004

Keywords

  • HOMOLOGICAL FINITENESS CONDITIONS
  • UNIVERSAL TODA BRACKETS
  • BROWN REPRESENTABILITY
  • INFINITE GROUPS
  • HOMOTOPY PAIRS
  • CATEGORIES
  • COMPLEXITY
  • VARIETIES
  • ALGEBRAS

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