General local cohomology modules in view of low points and high points
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 Ass_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 pSpec(R). Then, as a consequence, we prove that if (R,m) is a complete local ring, Φ is countable, and nN_0 is such that (Ass_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 Mod(R) into itself, which has the global vanishing property on Mod(R) and for an arbitrary Serre subcategory S and tN, we prove that R^iT(R)S for all i≥ t if and only if R^iT(M)S for any finitely generated R-module M and all i≥ t. Then we obtain some results on general local cohomology modules. , number=5, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Sadeghi, M. Y. and Ahmadi Amoli, Kh. and Chaghamirza М., year=2023, month=May, pages=698–711