On the Vanishing of H n (X, F for an n-Dimensional Variety

Abstract

Let X be an irreducible algebraic variety of dimension n. Then the cohomology group Hn (X, 5:) = 0 for all coherent sheaves F if and only if X is nonproper [=not complete]. This fact was conjectured by S. Lichtenbaum and proved by A. Grothendieck, in the more general form of the theorem stated below, by means of a delicate argument, which requires an examination both of the residue map and of the relation between local and global duality, [1]. This note gives a more elementary proof of this theorem. To prove the sufficiency, we reduce to the case X is normal. Here we construct an open affine subset U of X whose complement Y is again irreducible and nonproper, and we consider the canonical exact sequence