Complements and Applications

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


Here we will add some extensions and applications to the previous chapters. We present various aspects of local homology and cohomology, some more properties of the completion and torsion functors. We start with the composites of the derived functors of the completion and the torsion. Their interplay with the Hom-functors and the tensor product is also considered. In the second section there are results about duality and adjointness linking local homology and local cohomology with respect to an ideal generated by a weakly pro-regular sequence. It also provides a first version of local duality. In the third section we prove results about the endomorphism complex of \({\check{C}}_{\underline{x}}\) and related objects. Then the Mayer–Vietoris sequences for local homology and local cohomology are proved. We also consider Mayer–Vietoris sequences for Čech homology and local cohomology. There are also applications of the classes \(\mathcal {C}_{\mathfrak {a}}\) and \(\mathcal {B}_{\mathfrak {a}}\) to unbounded complexes. Homologically complete and cohomologically torsion complexes are studied in Sect. 9.6. This is completed in Sect. 9.7 by studying the cosupport. In the final section there are change of rings theorems.

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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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