Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 5, pp 1781–1802 | Cite as

\((A,\delta )\)-modules, Hochschild homology and higher derivations

  • Abhishek BanerjeeEmail author
  • Surjeet Kour


In this paper, we develop the theory of modules over \((A,\delta )\), where A is an algebra and \(\delta :A\longrightarrow A\) is a derivation. Our approach is heavily influenced by Lie derivative operators in noncommutative geometry, which make the Hochschild homologies \(HH_\bullet (A)\) of A into a module over \((A,\delta )\). We also consider modules over \((A,\Delta )\), where \(\Delta =\{\Delta ^n\}_{n\ge 0}\) is a higher derivation on A. Further, we obtain a Cartan homotopy formula for an arbitrary higher derivation on A.


\(A, \delta \)-modules Hochschild homology Higher derivations 

Mathematics Subject Classification

13N15 16W25 



We are grateful to the referee for important comments and suggestions and especially for pointing out the fact that \((A,\delta )\)-modules may be seen as modules over the skew polynomial ring \(A[x;\delta ]\). In this respect, the reference to [8] was given to us by the referee.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Department of MathematicsIndian Institute of TechnologyDelhiIndia

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