Let $E$ be an elliptic curve over a field $K$ and $\ell \neq \operatorname{char}(K)$ be a prime, assume that $F := \operatorname{End}_K(E) \otimes_{\Bbb Z} \Bbb Q$ is a quadratic field.
In this answer, it can be read "The two dimensional $\mathbb{Q}_{\ell}$-vector space $V_{\ell}:= T_{\ell}(E) \otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_{\ell}$ is a rank one free module over $F_{\ell} = F \otimes_{\Bbb Q} \Bbb Q_{\ell}$."
Why is it true? In particular, why is it a free module? It is true if $F_{\ell}$ is a field, but in general it is only a $\Bbb Q_{\ell}$-algebra. We may use my previous question.