A cohomological characterization of Alexander schemes |
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Authors: | Shun-Ichi Kimura |
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Institution: | (1) Hiroshima University, Faculty of Science, Department of Mathematics, East Hiroshima City, 739 Japan (e-mail: kimura@math.sci.hiroshima-u.ac.jp), JP |
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Abstract: | Vistoli defined Alexander schemes in 19], which behave like smooth varieties from the viewpoint of intersection theory with
Q-coefficients. In this paper, we will affirmatively answer Vistoli’s conjecture that Alexander property is Zariski local.
The main tool is the abelian category of bivariant sheaves, and we will spend most of our time for proving basic properties
of this category. We show that a scheme is Alexander if and only if all the first cohomology groups of bivariant sheaves vanish,
which is an analogy of Serre’s theorem, which says that a scheme is affine if and only if all the first cohomology groups
of quasi-coherent sheaves vanish. Serre’s theorem implies that the union of affine closed subschemes is again affine. Mimicking
the proof line by line, we will prove that the union of Alexander open subschemes is again Alexander.
Oblatum 1-XII-1997 & 14-XII-1998 / Published online: 10 May 1999 |
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Keywords: | Mathematics Subject Classification (1991): 14C |
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