TY - JOUR
T1 - Acyclicity of complexes of flat modules
AU - Hashimoto, Mitsuyasu
N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2008
Y1 - 2008
N2 - Let R be a noetherian commutative ring, and F : · · · → F2 → F1 → F0 → 0 a complex of flat R-modules. We prove that if κ(p{fraktur}) ⊗R F{double-struck} is acyclic for every p{fraktur} ∈ Spec R, then F{double-struck} is acyclic, and H0(F{double-struck}) is R-flat. It follows that if F{double-struck} is a (possibly unbounded) complex of flat R-modules and κ(p{fraktur}) ⊗R F{double-struck} is exact for every p{fraktur} ∈ Spec R, then G{double-struck}⊗•RF{double-struck} is exact for every R-complex G{double-struck}. If, moreover, F{double-struck} is a complex of projective R-modules, then it is null-homotopic (follows from Neeman's theorem).
AB - Let R be a noetherian commutative ring, and F : · · · → F2 → F1 → F0 → 0 a complex of flat R-modules. We prove that if κ(p{fraktur}) ⊗R F{double-struck} is acyclic for every p{fraktur} ∈ Spec R, then F{double-struck} is acyclic, and H0(F{double-struck}) is R-flat. It follows that if F{double-struck} is a (possibly unbounded) complex of flat R-modules and κ(p{fraktur}) ⊗R F{double-struck} is exact for every p{fraktur} ∈ Spec R, then G{double-struck}⊗•RF{double-struck} is exact for every R-complex G{double-struck}. If, moreover, F{double-struck} is a complex of projective R-modules, then it is null-homotopic (follows from Neeman's theorem).
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U2 - 10.1017/s002776300002599x
DO - 10.1017/s002776300002599x
M3 - Article
AN - SCOPUS:67449139070
VL - 192
SP - 111
EP - 118
JO - Nagoya Mathematical Journal
JF - Nagoya Mathematical Journal
SN - 0027-7630
ER -