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Suppose that $A$ is a commutative dg-algebra (say over base $k$) which is bounded in non-positive degree (with cochain complex conventions). There exists a quasi-free resolution of $A$. My question is, under what conditions on $A$ does $A$ admit a quasi-free resolution which is also finitely generated (as a graded commutative algebra). Can we completely characterize such algebras intrinsically?

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Compact objects in the homotopy category maybe? – Fernando Muro Jul 8 '14 at 15:03

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