We prove lifting results for DG modules that are akin to Auslander, Ding, and Solberg’s famous lifting results for modules. Introduction Convention. Throughout this paper, let R be a commutative noetherian ring. Hochster famously wrote that “life is really worth living” in a Cohen-Macaulay ring [7]. For instance, if R is Cohen-Macaulay and local with maximal regular sequence t, then R/(t) is ar...

We discuss Matijevic–Roberts type theorem on strong F -regularity, F -purity, and Cohen–Macaulay F -injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of F -purity of homomorphisms using Radu–André homomorphisms, and prove basic properties of it. We also discuss a...

In this paper we generalize the Macaulay-Buchberger basis theorem to the case, where the residue class polynomial ring over a Noetherian ring is not necessarily a free module. Recently, this theorem has been extended from polynomial rings over fields to rings, when residue class polynomial ring is free in (Francis & Dukkipati, 2014). As an application of this generalization we develop a theory ...