I'm not sure this question is research level question. Sorry in advance.

**Hypothesis**

- $k$ is a commutative ring.
- $A$ is an augmented $k$-algebra.
- $A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$. It is naturally augmented $k$-algebra.

**assumptions**

- $k$ (as left $A$-module) is quasi-isomorphic to a perfect complex. $k\in \mathbf{Perf}(A)$.
- $k$ (as left $A^e$-module) is quasi-isomorphic to a perfect complex. $k\in \mathbf{Perf}(A^e)$.
- $A$ (viewed as left $A^e$-module in a standard way) is quasi-isomorphic to a perfect complex. $A\in \mathbf{Perf}(A^e)$.

**Question**
Let $\langle A\rangle$ be the thick subcategory of the category of perfect complexes $\mathbf{Perf}(A^e)$ generated by the left $A^e$-module $A$ (where $A$ is viewed as $A^e$-module in standard way). Is it clear that $k$ (viewed as $A^e$-module via the augmentation $A^e\rightarrow k$ ) is an object of $\langle A\rangle$ ?