General local cohomology modules in view of low points and high points
DOI:
https://doi.org/10.37863/umzh.v75i5.7008Keywords:
general local cohomology modules; local cohomology modules; Serre subcategory; associated primes; Faltings' Theorem; Local- global Principle.Abstract
UDC 512.5
Let R be a commutative Noetherian ring, let Φ be a system of ideals of R, let M be a finitely generated R-module, and let t be a nonnegative integer. We first show that a general local cohomology module HiΦ(M) is a finitely generated R-module for all i<t if and only if AssR(HiΦ(M)) is a finite set and HiΦp(Mp) is a finitely generated Rp-module for all i<t and all p∈Spec(R). Then, as a consequence, we prove that if (R,m) is a complete local ring, Φ is countable, and n∈N0 is such that (AssR(HhnΦ(M)Φ(M)))≥n is a finite set, then fnΦ(M)=hnΦ(M). In addition, we show that the properties of vanishing and finiteness of general local cohomology modules are equivalent on high points over an arbitrary Noetherian (not necessary local) ring. For each covariant R-linear functor T from Mod(R) into itself, which has the global vanishing property on Mod(R) and for an arbitrary Serre subcategory S and t∈N, we prove that RiT(R)∈S for all i≥t if and only if RiT(M)∈S for any finitely generated R-module M and all i≥t. Then we obtain some results on general local cohomology modules.
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