Some Properties of Serre Subcategories in the Graded Local Cohomology Modules

Abstract EN

Let R=⊕n≥0Rn be a standard homogeneous Noetherian ring with local base ring (R0,m0) and let M be a finitely generated graded R-module.

Let HR+i(M) be the ith local cohomology module of M with respect to R+=⊕n>0Rn.

Let S be a Serre subcategory of the category of R-modules and let i be a nonnegative integer.

In this paper, if dimR0≤1, then we investigate some conditions under which the R-modules R0/m0 ⊗R0 HR+i(M),Γm0R(HR+i(M)) and Hm0R1(HR+i(M)) are in S for all i≥0.

Also, we prove that if dimR0≤2, then the graded R-module Hm01(HR+i(M)) is in S for all i≥0.

Finally, we prove that if n is the biggest integer such that Hai(M)∉S, then HR+i(M)/m0HR+i(M)∈S for all i≥n.