Liftings and Quasi-Liftings of DG Modules

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 artinian and the natural epimorphism R → R/(t) is nice enough to allow for transfer of properties between the two rings. Thus, if one can prove a result for artinian local rings, then one can (often) prove a similar result for Cohen-Macaulay local rings by showing that the desired conclusion descends from R/(t) to R. When R is complete, then this is aided sometimes by the lifting result of Auslander, Ding, and Solberg [2, Propositions 1.7 and 2.6]. Theorem. Let t ∈ R be an R-regular sequence, and let M be a finitely generated R/(t)-module. Assume that R is local and (t)-adically complete. (a) If ExtR/(t)(M,M) = 0, then M is “liftable” to R, that is, there is a finitely generated R-module N such that R/(t)⊗RN ∼= M and Tor R i (R/(t), N) = 0 for all i > 1. (b) If ExtR/(t)(M,M) = 0, then M has at most one lift to R. In this paper, we are concerned with what happens when the sequence t is not Rregular. One would like a similar mechanism for reducing questions about arbitrary local rings to the artinian case. It is well known that the map R → R/(t) is not nice enough in general to guarantee good descent/lifting behavior. Our perspective in this matter is that this is not the right map to consider in general: the correct one is the natural map from R to the Koszul complex K = K(t). This perspective requires one to make some adjustments. For instance, K is a differential graded R-algebra, so not a commutative ring in the traditional sense. This may cause some consternation, but the payoff can be handsome. For instance, in [8] we use this perspective to answer 2000 Mathematics Subject Classification. 13C05, 13C60, 13D02, 13D07, 13J10.