Let $X$ be a smooth projective scheme and $F$ a locally free sheaf of $\text{rank}(F) = r$ and $L$ a line bundle on $X$.
This week here during a seminar, there was heated discussion about a result. The person who was presenting the seminar said:
\begin{equation} (1) \bigwedge ^{i}(F \otimes L) \simeq (\bigwedge^{i} F) \otimes L \end{equation}
For me, the correct one is:
\begin{equation} (2) \bigwedge ^{i}(F \otimes L) \simeq (\bigwedge^{i} F )\otimes( L^{\otimes i} ) \end{equation}
In addition, the other people who attended this seminar agreed with isomorphism (1).
I have tried to argue that correct isomorphism is (2), but, I am told that the correct is isomorphism (1), because line bundles do not "see" the exterior power in this case. To me this is not true.
Only I, not agreeing with isormorphism (1), humbly come to this forum to ask: Is the isomorphism (1) really correct?
If not, how can I rigorously prove that ismorphism (2) is correct?
Remark: I started studying these tools a little while ago, getting ready for the master's degree, so I think because I am still a graduate student, they didn't believe my argument.
Thank you very much.