Filter Rings Under Flat Base Change

Abstract

Let φ: (R, 𝔪) → (S, 𝔫) be a flat local homomorphism of rings. In this paper, we prove: (1) If dim S/𝔪S > 0, then S is a filter ring if and only if R and k(𝔭) ⊗R𝔭 S𝔮 are Cohen–Macaulay for all 𝔮 ∈ Spec (S) \ {𝔫} and 𝔭= 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. (2) If dim S/𝔪S = 0, then S is a filter ring if and only if R is a filter ring and k(𝔭) ⊗R𝔭 S𝔮 is Cohen–Macaulay for all 𝔮 ∈ Spec (S) \ {𝔫} and 𝔭 = 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. As an application, it is shown that for a k-algebra R and an algebraic field extension K of k, if K ⊗k R is locally equidimensional, then R is a locally filter ring if and only if K ⊗k R is a locally filter ring.