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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, 픞 an ideal of R and r a non-negative integer such that is in dimension < 2 for all i<r and for some positive integer t, then for any minimax submodule N of , the R-module is finitely generated. As a consequence, it follows that the associated primes of are finite. This generalizes the main results of Brodmann-Lashgari [7] and Quy [24]. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I a proper ideal of R, and M a finitely generated R-module. We prove that there is a local-global principle for the Artinianness of the local cohomology module , i.e., for any integer n > 0, is Artinian for all i < n if and only if is Artinian for all i < n and all prime ideals 𝔭, which deduces one interesting property of filter depth. [ABSTRACT FROM PUBLISHER]

Using sheaves of special groups, we show that a general local-global principle holds for every reduced special group whose associated space of orderings only has a finite number of accumulation points. We also compute the behaviour of the Boolean hull functor applied to sheaves of special groups. [ABSTRACT FROM AUTHOR]

Let K be a function field with an A-algebra structure. The ring A arises in the definition of the Drinfeld module φ over K. By E(K) we denote K together with the A-module structure induced on it by φ. For any principal prime ideal (a)⊂A, we study the question whether an element x∈E(K) which is an a-fold in E(Kν) for every place ν of K, is an a-fold in E(K). In particular, we study the groupS(a,K)≔kerE(K)/aE(K)→∏lower limit ν E(Kν)/aE(Kν)for Drinfeld modules of rank 2. We show that this finite group is trivial in many cases, but can become arbitrarily large. [Copyright &y& Elsevier]