TITLE: Proving Two Complexes are Not Quasi-Isomorphic
QUESTION [8 upvotes]: In Richard Thomas' paper "Derived Categories for the Working Mathematician" he mentions (page 6) that the two complexes
$$
\begin{align*}
C^\bullet&= \mathbb{C}[x,y]^{\oplus 2}\xrightarrow{(x,y)}\mathbb{C}[x,y]\\
D^\bullet&= \mathbb{C}[x,y]\xrightarrow{0}\mathbb{C}
\end{align*}
$$
have the same cohomology but are not quasi-isomorphic. Proving that $C^\bullet$ and $D^\bullet$ have the same cohomology is straightforward, but I'm having trouble proving that they are not quasi-isomorphic. Any suggestions?
Thanks in advance!

REPLY [6 votes]: As was pointed out by Andrea in the comments to the other answer, the other answer does not answer the question properly.
An argument that the two complexes cannot be quasi-isomorphic in the general sense is sketched on page 2 of:
https://www.math.upenn.edu/~tpantev/rtg09bc/secnotes/ainfnity.davidovich.pdf