# coherent analytic cohomology vanishes for q > 2dim

Given a complex-analytic manifold of dimension $d$, why does the cohomology of coherent sheaves vanish in dimension $> 2d$ (without using GAGA)?

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Shouldn't the cohomology of coherent sheaves vanish above $d$? This should be somewhere in "Coherent Analytic Sheaves", by Grauert and Remmert, but I don't have the book here so I can't check. –  Angelo Jun 30 '10 at 14:03
Angelo, that is a very good point. I'm certain it isn't in the CAS book (I read it quite thoroughly, and there's no discussion of higher sheaf cohomology in that book apart from the chapter on Grauert's Higher Direct Image Theorem), so a better place to try is G&R's other book Theory of Stein Spaces. It would be nice to also have the result without smoothness. –  BCnrd Jun 30 '10 at 15:03
BCnrd the required vanishing follows from Andreotti-Grauert.See corollary4.15 page 428 of Demailly's book on CAG. –  Mohan Ramachandran Jun 30 '10 at 19:45

By the local analytic "Noether normalization" (really Weierstrass Preparation), twice the analytic dimension equals the "topological dimension" in the sense of dimension theory as in Engelking's marvelous book "General topology" for separable metric spaces. (For separable metric spaces, various notions of topological dimension are proved to agree; all done in that book. For opens in a real Euclidean space it recovers the expected "dimension"!) That book shows open covers of separable metric spaces have refinements whose $(n+1)$-fold overlaps are empty for $n$ beyond the topological dimension (in one of the various equivalent senses of dimension: the "covering" dimension!). Now use equality of Cech and derived functor cohomology for paracompact Hausdorff spaces to conclude.