Hello!

Let S be a commutative local Noetherian base ring and A be a dg-S-algebra.

Suppose M is a bounded above dg-A-module with bounded cohomology. Does there exist a bounded dg-A-module isomorphic to M in the derived category of dg-A-modules?

Moreover: If M is S-free, can N be chosen to be S-free, too? For this it seems reasonable to assume that S is regular, otherwise this even fails for A = S.

I would be happy to have an answer already in case where S is regular and A is both S-free and bounded.

What is known about these things?

Thank you,

Hanno