Abstract
Let R denote a commutative Noetherian ring, \(\mathfrak {a}\) an ideal of R, and M an \(\mathfrak {a}\)-cofinite R-module. The purpose of this article is to show that for a positive integer t, the R-module \(H_\mathfrak {a}^i(M)\) is finitely generated for all \(i<t\) if and only if the \(R_{\mathfrak {p}}\)-module \(H_{\mathfrak {a}R_{\mathfrak {p}}}^i(M_{{\mathfrak {p}}})\) is finitely generated for all \(i<t\) and all \(\mathfrak {p}\in \mathrm {Spec}(R)\). As a consequence, we provide a generalization and short proof of Faltings’ local–global principle for finiteness dimensions; i.e., \(f_\mathfrak {a}(M)=\mathrm{{inf}}\{f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{{\mathfrak {p}}}) | \ \mathfrak {p}\in \mathrm {Spec}(R)\}.\)
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Acknowledgements
The authors are deeply grateful to the referee for a very careful reading of the manuscript and many valuable suggestions in improving the quality of the paper and for drawing the authors’ attention to Theorem 2.6. We also would like to thank Professor Kamal Bahmanpour for his reading of the first draft and valuable discussions. Finally, we would like to thank the Institute for Research in Fundamental Sciences (IPM) for the financial support.
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This research was in part supported by a Grant from IPM.
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Abdi, L., Naghipour, R. & Sedghi, M. Faltings’ local–global principle for finiteness dimension of cofinite modules. Arch. Math. 112, 33–39 (2019). https://doi.org/10.1007/s00013-018-1247-0
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DOI: https://doi.org/10.1007/s00013-018-1247-0