Local cohomology modules via certain flat extension rings

Let ( R , m ) be a Noetherian local ring and M a finitely generated R-module. Let P ∈ Spec ( R ˆ ) and p = P ∩ R . Then the natural map R p → R ˆ P is a flat local homomorphism. In this paper, we provide some relations between the two sets of attached primes of the Artinian local cohomology modules H P R ˆ P i + r P ( M p ⊗ R p R ˆ P ) and H p R p i ( M p ) , where i ≥ 0 is an integer and r P = dim ⁡ ( R ˆ P / p R ˆ P ) . Then, we compute the dimension and multiplicity of H P R ˆ P i + r P ( M p ⊗ R p R ˆ P ) in terms of that of H p R p i ( M p ) respectively. As applications, we give connections between the Cohen–Macaulayness in dimension >s of M ˆ P and that of M p , for an integer s ≥ − 1 .