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Local Cohomology and Local Homology

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

Abstract

Let R denote a commutative ring and \(\mathfrak {a}\) an ideal of R. The \(\mathfrak {a}\)-torsion functor and the \(\mathfrak {a}\)-completion functor, defined respectively by \(\varGamma _{\mathfrak {a}}(M):= \{m\in M \mid \mathfrak {a}^t m=0 \text { for some } t\in \mathbb {N}\}\) and \(\varLambda ^{\mathfrak a}(M):= \varprojlim (R/\mathfrak {a}^t \otimes _R M)\) for any R-module M, extends naturally to complexes. In this chapter we first recall that \(\mathrm{L} \varLambda ^{\mathfrak a}\) and \(\mathrm{R} \varGamma _{\mathfrak a}\) are well defined in the derived category and fix some notations. Then we investigate when \(\mathrm{L} \varLambda ^{\mathfrak a}(X)\) and \(\mathrm{R} \varGamma _{\mathfrak a}(X)\) vanish. For complexes homologically-bounded on the good size we obtain more precise results, previously known when the ring is Noetherian, possibly new in the present generality. We also provide a description of \(\mathrm{L} \varLambda ^{\mathfrak a}(X)\) and \(\mathrm{R} \varGamma _{\mathfrak a}(X)\) in terms of microscope and telescope. These descriptions do not refer to the resolutions of X, which could be an advantage.

Copyright information

© 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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