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No, not always. In Positselski, Leonid; Schnürer, Olaf M., Unbounded derived categories of small and big modules: is the natural functor fully faithful?, J. Pure Appl. Algebra 225, No. 11, Article ID …

Here's a counterexample that appears in nature. Fix a prime $p$ and a field $k$ of characteristic $p$, and let $G=C_{p^{n}}$ be a cyclic group of order $p^{n}$ (where $n\geq1$ if $p$ is odd, and $n\ge …

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 …

Let $A=k[x]$ and $B=k[x]/(x^2)$, let $X$ be the complex $\hskip{.1in}\dots\to 0 \to B\stackrel{x}{\to} B\to 0\to \dots$, and let $Y$ be $\hskip{.1in}\dots\to 0\to k\stackrel{0}{\to}k\to 0\to\dots$. Th …

Possibly the stack example in pbelman's answer is of this form, but an elementary way to construct examples is by taking infinite products. Let $\{\mathcal{C}_i\}_{i\in I}$ be an infinite collection …

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 …

There's a simple counterexample in Section 8 of Rickard, Jeremy, Morita theory for derived categories, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989). ZBL0642.16034. Let kQ be the path alge …

Yes. More generally, if $\mathcal{T}$ is a triangulated category and $\mathcal{S}$ is a thick subcategory, then any morphism $\varphi:M\to N$ of $\mathcal{T}$ that becomes zero in $\mathcal{T}/\mathca …

No. For example, let $A$ be any object of $\mathcal{A}$, let $X$ be the complex $$\dots\to0\to A\oplus A\stackrel{\begin{pmatrix}1&0\end{pmatrix}}{\longrightarrow}A\to0\to\dots$$ and $\alpha:X\to X$ t …

No, not in general. For example, let $R=k[x]/(x^2)$ for a field $k$ and let $X$ be the object $$\dots\stackrel{x}{\to}R\stackrel{x}{\to}R\stackrel{x}{\to}R\to0\to0\to\dots$$ of the derived category o …

In almost all cases I know of where people have proved derived equivalences between blocks of finite groups, the proof hasn't really gone that way (i.e., finding a virtual bimodule and refining it to …

I'm afraid this is not very close to the case that you say you're most interested in ... maybe you want $F$ to preserve products? But let $A=k[x]/(x^2)$, and let $\mathscr{A}$ be the category $\operat …

Yes. Let $$\tau^{\leq0}A^\bullet:= \cdots\to A^{-2}\to A^{-1}\to\ker(d^0)\to0\to\cdots$$ be the usual truncation of $A^\bullet$. Then the mapping cone of the inclusion map $\tau^{\leq0}A^\bullet\to A^ …

As pointed out in nikola karabatic's answer, a decomposition of M as a direct sum of two-term complexes induces decompositions $H^i=H^i_a\oplus H^i_b$ for each $i$. These have the property that $\xi_i …

I don't know the group of smallest order for which the conjecture has not been verified. But certainly it is known to be true for all groups of order less than 200. There are general results that deal …