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I don't think $F^{d-\bullet}$ will often be universal, even when $\mathcal{A}$ does have enough projectives. Let $A$ be any ring with finite global dimension that is not hereditary, so that there is …

In the example where $\mathcal{D}$ is the category of abelian groups and $\mathcal{C}$ is the category of finite abelian groups, take $F(X)=X\otimes_\mathbb{Z}\mathbb{Q}/\mathbb{Z}$. Then the restrict …

It's not true, without boundedness conditions, that a left exact functor always preserves quasi-isomorphisms between complexes of $F$-acyclic objects. As alluded to in the question, a chain map is a q …

Take $R=\mathbb{Z}_{(p)}$ for some prime $p$, with $x=p$, and $M=\mathbb{Q}\oplus R$. To show that this is a counterexample, the only nonobvious thing to show is that $\operatorname{Ext}^{1}_{R}(\math …

Not even for a Gorenstein ring. Let $R$ be $\mathbb{Z}_p$, the ring of $p$-adic integers. It is Gorenstein, and therefore its own dualizing module. Let $M$ be the direct sum of the two complexes $$\cd …