# Homotopy flat DG-modules

A right DG-module $F$ over an associative DG-algebra (or DG-category) $A$ is said to be homotopy flat (h-flat for brevity) if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups $F\otimes_A M$ is acyclic. Homotopy projective and injective DG-modules are defined in the similar way, e.g., a left DG-module $P$ over $A$ is h-projective if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups $Hom_A(P,M)$ is acyclic. Homotopy adjusted DG-modules are used to define derived functors on the derived categories of DG-modules.

My question is: can one describe the class of h-flat DG-modules in any way alternative to the definition? Here are examples of the kind of description I have in mind:

1. A DG-module is h-projective if and only if it is homotopy equivalent to a DG-module obtained from the free DG-module $A$ over $A$ using the operations of shift, cone, and infinite direct sum. (Similarly, a DG-module is h-injective if and only if it is homotopy equivalent to a DG-module obtained from the cofree DG-module $Hom_{\mathbb Z}(A,\mathbb Q/\mathbb Z)$ using the operations of shift, cone, and infinite product.)

2. A module over a noncommutative ring is flat if and only if it is a filtered inductive limit of projective (or even projective and finitely generated, or free and finitely generated) modules (the Govorov-Lazard theorem).

The class of h-flat DG-modules is closed with respect to the operations of shift, cone, filtered inductive limit, and passage to a homotopy equivalent DG-module. It also contains the free DG-module $A$ over $A$. Can all h-flat DG-modules be obtained from that one DG-module using these four operations?

References: 1. Keller's paper "Deriving DG-categories"; 2. my preprint arXiv:0905.2621, section 1.

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