General local cohomology modules in view of low points and high points

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 H Φ i ( M ) is a finitely generated R -module for all i < t if and only if A s s R ( H Φ i ( M ) ) is a finite set and H Φ 𝔭 i ( M 𝔭 ) is a finitely generated R 𝔭 -module for all i < t and all 𝔭 ∈ S p e c ( R ) . Then, as a consequence, we prove that if ( R , 𝔪 ) is a complete local ring, Φ is countable, and n ∈ ℕ 0 is such that ( A s s R ( H Φ h Φ n ( M ) ( M ) ) ) ≥ n is a finite set, then f Φ n ( M ) = h Φ n ( 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 M o d ( R ) into itself, which has the global vanishing property on M o d ( R ) and for an arbitrary Serre subcategory 𝒮 and t ∈ ℕ , we prove that ℛ i T ( R ) ∈ 𝒮 for all i ≥ t if and only if ℛ i T ( M ) ∈ 𝒮 for any finitely generated R -module M and all i ≥ t . Then we obtain some results on general local cohomology modules.