Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings

Abstract

Let \((R,\mathfrak {m})\) be a Noetherian local ring, M a non-zero finitely generated R-module, and let I be an ideal of R. In this paper, we establish some new properties of local cohomology modules \(\mathrm {H}^{i}_{I}(M)\), i ≥ 0. In particular, we show that if R is catenary, M an equidimensional R-module of dimension d, and \(x_{1},x_{2},\dots ,x_{t}\) is an I-filter regular sequence on M, then \((0:_{\mathrm {H}^{d-j}_{I}(\frac {M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})} x_{i})\) is I-cofinite for all \(i = 1,2,\dots ,t\) and all ijt if and only if \(\mathrm {H}^{d-j}_{I}(\frac {M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})\) is I-cofinite for all \(i = 1,2,\dots ,t\) and all ijt. Also we study the cofiniteness of local cohomology modules over homomorphic image of Cohen-Macaulay rings and we show that \(\frac {\mathrm {H}^{\mathcal {W}(I,M)}_{I}(M)}{I\mathrm {H}^{\mathcal {W}(I,M)}_{I}(M)}\) has finite support, where

$$\mathcal{W}(I,M) := \text{Max} \{i : \mathrm{H}^{i}_{I}(M) \text{~is not weakly Laskerian}\}. $$

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Acknowledgements

The authors are deeply grateful to the referee for his/her careful reading of the paper and valuable suggestions.

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Correspondence to Alireza Nazari.

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Farokhi, A., Nazari, A. Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings. Acta Math Vietnam 43, 565–574 (2018). https://doi.org/10.1007/s40306-018-0246-3

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Keywords

  • Weakly Laskerian modules
  • Cofinite modules
  • Local cohomology

Mathematics Subject Classification (2010)

  • 13D45
  • 14B15
  • 13E05