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Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 1109–1155 | Cite as

Gorenstein homological algebra and universal coefficient theorems

  • Ivo Dell’Ambrogio
  • Greg Stevenson
  • Jan Šťovíček
Article
  • 294 Downloads

Abstract

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories.

Keywords

Gorenstein homological algebra Triangulated category Universal coefficient theorem Kasparov’s KK-theory 

Mathematics Subject Classification

Primary 16E65 Secondary 18E30 19K35 46L80 

Notes

Acknowledgements

We are very grateful to Ralf Meyer for asking us the interesting questions that have prompted our collaboration. Thanks are also due to him for first noticing that the Gorenstein closed condition of Theorem 8.6 is not only sufficient but also necessary. Another source of inspiration was a talk given in Bielefeld by Claus Ringel where he presented the results of [61] (see Example 9.21). The paper was finished during the research program IRTATCA (Interactions between Representation Theory, Algebraic Topology and Commutative Algebra) at Centre de Recerca Matemàtica in Barcelona. Our thanks also belong to the organizers for the hospitality and the creative environment there.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Ivo Dell’Ambrogio
    • 2
  • Greg Stevenson
    • 1
  • Jan Šťovíček
    • 3
  1. 1.Fakultät für Mathematik, BIREP GruppeUniversität BielefeldBielefeldGermany
  2. 2.Laboratoire de Mathématiques Paul PainlevéUniversité de Lille 1Villeneuve-d’Ascq CedexFrance
  3. 3.Department of AlgebraCharles University in PraguePraha 8Czech Republic

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