Faltings’ local–global principle for the in dimension <n of local cohomology modules

ABSTRACT

The concept of Faltings’ local–global principle for the in dimension <n of local cohomology modules over a Noetherian ring R is introduced, and it is shown that this principle holds at levels 1, 2. We also establish the same principle at all levels over an arbitrary Noetherian ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. [8]. Moreover, as a generalization of Raghavan’s result, we show that the Faltings’ local–global principle for the in dimension <n of local cohomology modules holds at all levels r∈ℕ whenever the ring R is a homomorphic image of a Noetherian Gorenstein ring. Finally, it is shown that if M is a finitely generated R-module, 𝔞 an ideal of R and r a non-negative integer such that ${𝔞}^t{H}_{𝔞}^i\left(M\right)$ is in dimension < 2 for all i<r and for some positive integer t, then for any minimax submodule N of $H_{𝔞}^r\left(M\right)$, the R-module ${Hom}_R(R∕𝔞,H_{𝔞}^r(M)∕N)$ is finitely generated. As a consequence, it follows that the associated primes of $H_{𝔞}^r\left(M\right)∕N$ are finite. This generalizes the main results of Brodmann-Lashgari [7] and Quy [24].

KEYWORDS:

• Associated primes
• cofinite module
• Gorenstein ring
• in dimension <n module
• local cohomology
• local–global principle
• minimax module
• Noetherian module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D45
• 14B15
• 13E05

Acknowledgments

The authors are deeply grateful to the referee for his/her careful reading of the paper and valuable suggestions. Also, we would like to thank Professor Kamran Divaani-Aazar for his reading of the first draft and useful discussions. Finally, we would like to thank from the Institute for Research in Fundamental Sciences (IPM), for the financial support.