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On large cohomological dimension and tautness

Part of: Homology and cohomology theories Special properties

Published online by Cambridge University Press:  09 April 2009

Satya Deo ,
Subhash Muttepawar  and
Mohan Singh
Satya Deo
Affiliation:
Department of Mathematics University of JammuJammu-180001, India
Subhash Muttepawar
Affiliation:
Department of Mathematics University of JammuJammu-180001, India
Mohan Singh
Affiliation:
Department of Mathematics University of JammuJammu-180001, India
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Abstract

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We prove that for a non-discrete space X, the inequality DimL(X) ≥ dimL(X) + 1 always holds if (i) X is paracompact and each point is Gδ, or (ii) X is a completely paracompact Morita k-space. Consequently, if X is a non-discrete completely paracompact space in which each point is a Gδ-set or it is also a Morita k-space then, the equality DimL(X) = dimL(X) + 1 always holds. We apply this equality to show that for such a space X there exists a point x ∈ X and a family ϕ of supports on X such that {x} is not ϕ-taut with respect to sheaf cohomology. This generalizes a corresponding known result for Rn. We also discuss the usual sum theorems for this large cohomological dimension; the finite sum theorem for closed sets is proved, and for all others, counter examples are given. Subject to a small modification, however, all of the sum theorems hold for a large class of spaces.

Keywords

sheaf cohomologycohomological dimensionlarge cohomological dimensionip-tautnesssum theoremsMorita k-space.

MSC classification

Secondary: 54F45: Dimension theory 55N30: Sheaf cohomology
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 37 , Issue 3 , December 1984 , pp. 391 - 404
Copyright
Copyright © Australian Mathematical Society 1984

References

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