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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 …

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 …

This follows by applying Theorem 3.8 of Beligiannis' 2000 paper to the opposite categories. $\mathcal{I}$ is a full additive subcategory of $\mathcal{A}$, closed under direct summands. It is covariant …

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 …

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 …

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^ …

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 …

For every $M$, $M\oplus R$ is virtually small, so your question is equivalent to the question: Does every finitely generated $R$-module embed in a finitely generated module of finite projective dimen …

There is such a sequence, but it's not very interesting. Given an element of $\text{Hom}_{D^b(\mathcal{A})}(E,F[i])$, then in the same way you describe, this gives a distinguished triangle $$F\to Z_{i …

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 …

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 …

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 …

No. Let $k$ be a field, and let $A$ be the algebra of upper triangular $2\times 2$ matrices over $k$, with augmentation map $\pmatrix{a&b\\0&c}\mapsto a$. $A$ and $A^e$ have finite global dimension, s …

Let $G$ be infinite cyclic, generated by $x$. Let $M_\bullet$ be a free resolution of the $\mathbb{Z}[G]$-module $U=\mathbb{Z}/3\mathbb{Z}$ with $x$ acting by multiplication by $-1$. For example, tak …

Up to shifts, every indecomposable object is of one of the forms described in the question. I don't know an explicit reference, but here's a sketch of a proof. By induction on the length, it's not h …