A set of objects classically generates the full subcategory of compact objects iff it generates the whole category

Question

Sorry in advance if my question doesn't have the level of this community.

I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated categories.

As long as I was figuring out the definitions of "classically generated", "generated" and "compactly generated", I came up with the Theorem 2.1.2 of this paper which says that in a compactly generated category $\mathcal{C}$, a set of objects $\mathcal{E}\subset \mathcal{C}^{c}$ classically generates $\mathcal{C}^{c}$ if and only if it generates $\mathcal{C}$. For the proof I started to read the reffering paper by Neeman and successfully finished and understood it. But there are some problems of connecting the information of the 2 papers.

1. First of all, while theorem 2.1.2 uses the notion of "generates" that Bondal and Van Den Bergh define, in Neeman's paper there is nowhere such a notion and maybe instead of this, he uses the notion of the smallest localising triangulated subcategory that contains a set of objects. I think that there is the following connection which I unfortunately can't prove $$\mathcal{R}\ \text {is the smallest localising triangulated subcategory that contains a set}\ R\subset \mathcal{C}^{c}\ \Leftrightarrow $$$$R^{\perp}=0 $$ where $R^{\perp}$ is defined as in Bondal's paper as a subcategory of $\mathcal{R}$ and $\mathcal{R}$ is the smallest localising triangulated subcategory that contains a set $R\subset \mathcal{C}^{c}$ as in Neeman's paper. I strongly think that this is true under the hypothesis that $\mathcal{R}\subset \mathcal{C}$ where $\mathcal{C}$ is compactly generated.
2. If the 1. is true then I construct the proof of Theorem 2.1.2 in the following way: by lemma 2.2 in Neeman's Paper(in the main proof of theorem 2.1) we clearly have the implication $\mathcal{E}$ generates $\mathcal{C}$ $\implies$ $\mathcal{E}\subset \mathcal{C}^{c}$ classically generates $\mathcal{C}^{c}$. For the other direction we note that if the smallest localising triangulated subcategory that contains $\mathcal{C}^{c}$ is the whole $C$, then $\mathcal{C}$ consists of coproducts of objects in $\mathcal{C}^{c}$. Then it is easy, since $Hom(M,\bigoplus E_{i})\cong \bigoplus Hom(M,E_{i})$.

As it seems there is a problem of combining these two papers and any help will be accepted.

Edit: To be honest I hadn't found this post that has an answer from Leonid Positselski, so I reform my question: as the proof is for well-generated categories (a generalised notion of compact objects) and uses the Brown representability theorem for triangulated categories I was wondering if there is a simpler proof for compact objects without using it. Maybe the first proof before the introduction of well-generatedness by Krause and others.

Answer 1

I agree that the various notions of 'generates' can be confusing. I think the following result may clarify what you are after (this can be found in Lemma 2.2.1 of 'Stable model categories are categories of modules' by Schwede and Shipley).

Let $\mathcal{C}$ be a triangulated category with infinite coproducts and let $\mathcal{P}$ be a set of compact objects. Then the following are equivalent:

(i) The smallest localizing subcategory of $\mathcal{C}$ containing $\mathcal{P}$ is $\mathcal{C}$.

(ii) An object $X \in \mathcal{C}$ is trivial if and only if $[P,X]_* = 0$ for all $P \in \mathcal{P}$ (in the language of the question, $\mathcal{P}^{\perp} = 0$).

Finally, we also have the following, due to Thomason:

Suppose $\mathcal{C}$ is a compactly generated triangulated category, and $\mathcal{A}$ is a set of compact objects, then $\mathop{Loc(A)} \cap \mathcal{C}^{c} = \mathop{Thick(A)}$.

Here $\mathop{Loc}(\mathcal{A})$ denotes the smallest localizing subcategory of $\mathcal{C}$ containing $\mathcal{A}$, and $\mathop{Thick}(\mathcal{A})$ denotes the smallest thick (epaisse in Bondal--Van Den Bergh) subcategory of $\mathcal{C}^c$ containing $\mathcal{A}$.