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Generally speaking a unique lifting does not exist and I believe it is open as to what the possible liftings can be. As an example of the non-uniqueness consider a slight variant of the particular ca …

The answer is in general no - $K(R\text{-}\mathrm{Mod})$ can fail to be well generated even when $R$ is artinian. As you mention $K(R\text{-}\mathrm{Mod})$ is compactly generated if $R$ is of finite r …

I would imagine it is false in general that given a triangulated category $T$ the category $T\otimes R$ is also triangulated. The following is a concrete counterexample. Consider $D^b(\mathbb{Z})$ an …

It is true in a Heller triangulated category aka $\infty$-triangulated category (although strictly speaking one only needs a 3-triangulation for octahedra) that any morphism between the bases of octah …

The title pretty much sums it up - but let me give a little bit of background first. In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) …

The category of sheaves of $\mathbb{Q}$ vector spaces on $M$ is a Grothendieck abelian category. It follows that the derived category of such, $D(M)$ in your notation, is a well generated triangulated …

The problem with respect to applications of the abelianization is that the abelian categories one produces are almost uniformly horrible. More precisely they are just too big to deal with. So using th …

The two notions are related in the sense that they share a common generalization, namely the notion of torsion pair on a pre-triangulated category (this term has at least two meanings, here we mean a …

To address Hanno's question about checking that composition gives a graded-commutative ring structure on $End^{*}(\mathbb{Z}) = \oplus_i [\mathbb{Z}, \Omega^{-i} \mathbb{Z}]$ suppose first that $a \st …

There is a similar question here. It is asked in the particular case of the derived category of abelian groups and a counterexample is given by Tyler Lawson. There is also an answer of mine which all …

This question has kind of been bothering me since I started thinking about it - I am far from an expert on KK-theory but I thought I'd throw something out there and maybe someone else will see it and …

One example is the following - suppose that $M$ is a non-compact connected manifold of dimension $\geq 1$. Then the unbounded derived category of chain complexes of sheaves of abelian groups on $M$ ha …

So if I understand correctly the question you wanted to ask was: Is it true that a triangle $$X \stackrel{u}{\to} Y \stackrel{v}{\to} Z \stackrel{w}{\to} \Sigma X$$ is split if and only if one of $u$, …

I just wanted to point out that this failure is quite standard rather than pathological. As a starting point it can go wrong more generally than Tyler points out. For instance there exist triangles wh …

For triangulated categories the notion of Bousfield localization is a "special case" of the notion of Verdier quotient. As you observe (and is shown in Lemma 3.1 of the paper you mention) any Bousfiel …