  The notion of naive lifting of DG modules was introduced by the authors in
[16,17] for the purpose of studying problems in homological commutative algebra
that involve self-vanishing of Ext. Our goal in this paper is to deeply study
the naive lifting property using the tensor algebra of the shift of the
diagonal ideal (or, diagonal tensor algebra, as is phrased in the title of this
paper). Our main result provides several characterizations of naive liftability
of DG modules under certain Ext vanishing conditions. As an application, we
affirmatively answer [19, Question 4.10] under the same assumptions.
