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Let $T$ be a triangulated category admitting arbitrary coproducts with a set of compact generators $\mathcal{G}$. From the definition of compact generators we have that for any object $X$ of $T$ $[\Sigma^nZ,X]=0$ for all $n \in \mathbb{Z}$ and all $Z \in \mathcal{G}$ then $X=0$, equivalently $\mathcal{G}$ detects isomorphisms i.e. if a map $f \colon X \rightarrow Y$ has the property that $[\Sigma^nZ,f]$ is an isomorphism $\forall n \in \mathbb{Z}$ and $\forall \in \mathcal{G}$ then $f$ is an isomorphism.

But what if we have $[\Sigma^nZ,f]=0$? Can we deduce that $f$ is the zero map? I never found this claim anywhere and I could not prove it. I tried to show that the full subcategory of $T$ generated by $\{ A \in T : [A,f]=0 \}$ is a localizing subcategory but I have difficulties proving that is triangulated. Basically I get in a situation similar to the 5 lemma but where the two vertical maps on the left and on the right are zero and I cannot conclude the central one is zero from the exactness of the long horizontal sequences.

Thus I am induced to think that the claim is false: can you provide an explicit counterexample?

Also what if I add more structures or properties to $T$? For example I could consider a closed tensor triangulated category such that the elements of $\mathcal{G}$ are strongly dualizable or such that every cohomology functor is representable.

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If you allow yourself to choose the set $\mathcal{G}$ of compact generators then there are very simple examples. For example, take $T=D(\text{Mod }R)$ for some ring $R$, and $\mathcal{G}=\{\Sigma^nR\mid n\in\mathbb{Z}\}$, and let $f:M\to\Sigma N$ be a map representing a nonzero element of $\text{Ext}^1_R(M,N)$.

But even if you take $\mathcal{G}$ to be the class of all compact objects, then there are examples even in quite normal triangulated categories, that have been quite extensively studied under the name "phantom maps".

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  • $\begingroup$ I guessed that the argument was related to phantom maps. I read about it only the lecture notes by Lurie on chromatic homotopy theory. Are there recommended references for this topic? I found the paper "Phantom maps and homology theories" by Christensen and Strickland, also "Axiomatic stable homotopy theory" has some sections dedicated to to phantom maps. Is there any other text you would recommend? $\endgroup$ – N.B. Feb 1 at 13:34
  • $\begingroup$ @N.B. There are some papers by Gnacadja, who did a PhD on phantom maps in stable module categories of finite group algebras, that are a lot less general than those you mention, but might be useful if you want to see concrete examples in a relatively simple setting, and have any kind of background in representation theory. $\endgroup$ – Jeremy Rickard Feb 2 at 11:29
  • $\begingroup$ Thanks for the answer. Can I ask you if the following intuition is correct? In the paper I cited and also in your example we see a correlation between phantom maps and Ext terms. Unravelling your example: an element $f \in Ext^1(M,N)$ stands for a Yoneda class of extensions $0 \rightarrow N \rightarrow F \rightarrow M \rightarrow 0$ which in the derived category $D(R)$ "corresponds" to an exact triangle $\Sigma^-1 M \rightarrow N \rightarrow F \rightarrow M$ where the first map $\Sigma^-1 M \rightarrow N$ is phantom. I said "corresponds" because the phantom map is not uniquely determined. $\endgroup$ – N.B. Feb 5 at 17:52
  • $\begingroup$ But the point is that the phantom map stands for the impossibility of the triangulated structure of $D(R)$ to pass to an abelian structure where we encode information in short exact sequences. I do not know if there is a precise formalisation of this fact: maybe there is an encolement of triangulated categories where $D(R)$ is the central term and on the right we have some sort of category with morphisms the phantom maps? $\endgroup$ – N.B. Feb 5 at 18:02
  • $\begingroup$ @N.B. I’m rather busy right now, but I’ll think about what you’ve asked when I get some time. By “encolement” do you mean “recollement”? $\endgroup$ – Jeremy Rickard Feb 5 at 19:43

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