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Acyclicity of Complexes of Flat Modules

Published online by Cambridge University Press:  11 January 2016

Mitsuyasu Hashimoto
Mitsuyasu Hashimoto*
Affiliation:
Graduate School of Mathematics Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan, hasimoto@math.nagoya-u.ac.jp
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Abstract

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Let R be a noetherian commutative ring, and

a complex of flat R-modules. We prove that if is acyclic for every ρ ϵ Spec R, then is acyclic, and H0() is R-flat. It follows that if is a (possibly unbounded) complex of flat R-modules and is exact for every ρ ϵ Spec R, then is exact for every R-complex . If, moreover, is a complex of projective R-modules, then it is null-homotopic (follows from Neeman’s theorem).

Keywords

13C1113C10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 192 , 2008 , pp. 111 - 118
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[1] Enochs, E. E., Minimal pure injective resolutions of flat modules, J. Algebra, 105 (1987), 351–364.Google Scholar
[2] Hashimoto, M., Auslander-Buchweitz Approximations of Equivariant Modules, London Mathematical Society Lecture Note Series 282, Cambridge, 2000.Google Scholar
[3] Neeman, A., The homotopy category of flat modules, preprint.Google Scholar
[4] Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math., 65 (1988), 121–154.Google Scholar