Smallness condition for augmented algebras

No.

Let $k$ be a field, and let $A$ be the algebra of upper triangular $2times 2$ matrices over $k$, with augmentation map $pmatrix{a&b\0&c}mapsto a$.

$A$ and $A^e$ have finite global dimension, so all modules have finite projective dimension, and are therefore quasi-isomorphic to perfect complexes.

Let $mathcal{D}=mathcal{D}(A^e)$ be the derived category of $A^e$-modules. The category $$k^perp={Xinmathcal{D}mid operatorname{Hom}_{mathcal{D}}(k,X[t])=0text{ for all $tinmathbb{Z}$}}$$ is a thick subcategory of $mathcal{D}$. I claim (proof below) that $Ain k^{perp}$. So $langle Aranglesubseteq k^{perp}$ for all $tinmathbb{Z}$. But clearly $knotin k^{perp}$.

Proof of claim: Let $e_{11}=pmatrix{1&0\0&0}$ and $e_{22}=pmatrix{0&0\0&1}$, idempotent elements of $A$. Then $k=Ae_{11}$ is projective as a left $A$-module, and as a right $A$-module $k$ has a projective resolution $$0to e_{22}Ato e_{11}Ato kto0,$$ where the first nonzero map is $pmatrix{0&0\0&c}mapstopmatrix{0&c\0&0}$. So as an $A^e$-module, $k$ has a projective resolution $$0to Ae_{11}otimes_k e_{22}Ato Ae_{11}otimes_k e_{11}Ato kto0.$$ Applying the functor $operatorname{Hom}_{A^e}(-,A)$ to the projective terms of this resolution gives the map $e_{11}Ae_{11}to e_{11}Ae_{22}$ where $pmatrix{a&0\0&0}mapstopmatrix{0&a\0&0}$. The kernel and cokernel of this map are both zero, so $operatorname{Ext}^t(k,A)=0$ for all $tgeq0$.