TITLE: Thick subcategory of unbounded derived category
QUESTION [0 upvotes]: Let R and S be two rings. It's a well known result that the subcategory of all compact objects of $D(R-Mod)$ which we denote by $D(R-Mod)^c$ is $K^b(R-proj)$ where $R-proj$ denotes the subcategory of all finitely generated projective R-modules. Since the subcategory of compact objects is thick,we deduce that the smallest thick subcategory containing $R$ is just $K^b(R-proj)$. 
Now assume we have a module $T$ which satisfies $Ext^i(T,T)=0$ for $i>0$,then $K^b(add \ T)$ is a triangulated subcategory of $D(R-Mod)$. My question is whether this subcategory is thick.

REPLY [1 votes]: Yes, it is. (Or more pedantically, the full subcategory of objects isomorphic to objects of $K^b(\text{add }T)$ is thick.)
I'll use right modules.
First, by induction on the length of complexes $X$ and $Y$ in $K^b(\text{add }T)$, an argument involving long exact sequences of $\text{Hom}$-groups and the five lemma shows that the natural map
$$\text{Hom}_{K^b(\text{add }T)}(X,Y)\to\text{Hom}_{D(\text{Mod} R)}(X,Y)$$
is an isomorphism: i.e., that the natural functor $K^b(\text{add }T)\to D(\text{Mod }R)$ is a full embedding. This is needed to prove that $K^b(\text{add }T)$ is a triangulated subcategory, so presumably you're familiar with this.
Next, let $E=\text{End}_R(T)$. Then the functor
$$\text{Hom}_R(T,-):\text{Mod }R\to\text{Mod }E$$
restricts to an equivalence of categories
$$\text{add }T\to\text{proj }E,$$
where $\text{proj }E$ is the category of finitely generated projective $E$-modules. 
Hence $K^b(\text{add }T)$ is equivalent to $K^b(\text{proj }E)$.
To prove that $K^b(\text{add }T)$ is a thick subcategory, it suffices to prove that its idempotents split (i.e., if an object $X$ has an idempotent endomorphism $e$, then it decomposes as $X=\text{im}(e)\oplus\ker(e)$). But this is true of the equivalent category $K^b(\text{proj }E)$, since it is a thick subcategory of $D(\text{Mod }E)$, which has split idempotents, and hence it is true of $K^b(\text{add }T)$.
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