Čech and Local Duality

  • Peter SchenzelEmail author
  • Anne-Marie Simon
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter we provide some duality formulas for the Čech cohomology of an unbounded complex, which involve the general Matlis dual \({\check{C}}_{\underline{x}}^{\vee }\) of the Čech complex. When the sequence \(\underline{x}\) is a system of parameters of a Noetherian local ring our formulas provide a version of the Grothendieck Local Duality for Cohen–Macaulay or Gorenstein local rings. As a byproduct we obtain new characterizations of Gorenstein local rings in terms of local homology. As another byproduct there are some characterizations of \(\mathfrak {m}\)-torsion and \(\mathfrak {m}\)-pseudo complete modules over a Gorenstein local ring. When the sequence \(\underline{x}\) is a system of parameters of a complete Noetherian local ring, it turns out that the complex \({\check{C}}_{\underline{x}}^{\vee }\) is a bounded complex of injective modules with finitely generated cohomology. For that reason we start the chapter with an investigation of such complexes.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany
  2. 2.Service de Geometrie DifferentielleUniversité Libre de BruxellesBrusselsBelgium

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