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A Quick Proof of the Hartshorne-Lichtenbaum Vanishing Theorem

In: Algebraic Geometry and its Applications

Author

Listed:
  • Markus Brodmann
  • Craig Huneke

Abstract

In this note we give a short and elementary proof of the Hartshorne-Lichtenbaum vanishing theorem [8], 3.1 (which we refer to below as “HLVT”). Other authors have given nice proofs of this result, cf. [9], III, 3.1 and [4]. The reason we present another proof is threefold. First, it is an important and basic result, for example HLVT plays a crucial role in the vanishing theorems for local cohomology in [7], and in the simple proof of Falting’s connectedness theorem in [2]. Secondly we feel our proof does represent a true simplification, and thirdly the method of proof is probably useful in other contexts and is certainly suggestive of other vanishing theorems. Often a different proof is necessary for a variety of reasons, e. g. G. Lyubeznik [11] recently needed to give a new proof of HLVT (using a local Bertini theorem) to extend it to étale cohomology.

Suggested Citation

  • Markus Brodmann & Craig Huneke, 1994. "A Quick Proof of the Hartshorne-Lichtenbaum Vanishing Theorem," Springer Books, in: Chandrajit L. Bajaj (ed.), Algebraic Geometry and its Applications, chapter 18, pages 305-308, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2628-4_18
    DOI: 10.1007/978-1-4612-2628-4_18
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