This should be obvious but I'm not seeing it:


The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\\_)$ preserves coproducts and the localizing subcategory $\langle A\rangle$ of $\mathfrak T$ generated by $A$ is all of $\mathfrak T$.)


For instance, $\mathfrak T$ could be the derived category of a ring or a ring spectrum.


Let $\langle A\rangle\_{\aleph\_1}\subseteq\mathfrak T$ be the $\aleph\_1$-localizing subcategory of $\mathfrak T$ (that is, the smallest triangulated subcategory containing $A$ and being closed under *countable* coproducts).


Certainly, if $B$ belongs to $\langle A\rangle\_{\aleph\_1}$, then $\mathfrak T\_\*(A,B)$ is a countably generated module over $\mathfrak T\_\*(A,A)$.


Question: Does the converse hold, that is, do we have
$$
\langle A\rangle\_{\aleph\_1}=\{B\in\mathrm{Ob}(\mathfrak T)\mid\text{$\mathfrak T\_\*(A,B)$ is countably generated over $\mathfrak T\_\*(A,A)$}\}\;?
$$



Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?
I don't think what you say is true. Let $k$ be a field, $V$ a $k$-vector space of uncountable dimension and $R=k\oplus V$ the $k$-algebra where $V$ is a square-zero ideal. Consider $\mathfrak T=D(R)$ the derived category of $R$, and $A=R$. Take a non-trivial vector $0\neq v\in V$. The complex 
$$B=\cdots\rightarrow 0\rightarrow R\stackrel{v}\longrightarrow R\rightarrow 0\rightarrow \cdots$$
is in $\langle A\rangle\_{\aleph\_1}$, in fact it is in $\langle A\rangle\_{\aleph\_0}$. In this case $\mathfrak T\_\*(A,A)=R$ concentrated in degree $0$ and $\mathfrak T\_\*(A,B)=R/(v)\oplus V[1]$, which is not countably generated since $\dim\_kV$ is uncountable.


What you claim is true under some transfinite coherence hypothesis, e.g. it is proven in the literature under the hypothesis that $\mathfrak T\_\*(A,A)$ is countable.


