# Can compact generators detect zero morphisms?

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.

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".
• 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. – N.B. Feb 5 at 17:52
• 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? – N.B. Feb 5 at 18:02