I have a very brief question. If you have a bunch of line bundles $L_1,\dots,L_p$ over a scheme $S$, why does $\bigwedge^p(L_1\oplus\cdots\oplus L_p)\cong L_1\otimes\cdots\otimes L_p$, and can't find a proof of this isomorphism, which seems to be taken for granted.
2026-04-01 23:33:47.1775086427
Why does $\bigwedge^p(L_1\oplus\cdots\oplus L_p)\cong L_1\otimes\cdots\otimes L_p$?
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Clearly both sides are line bundles. The transition functions of the right hand side line bundle are the products of the transition functions of the line bundles $L_1, \dotsc, L_p$. The transition functions of the left hand side are the determinants of transition matrices of the vector bundle $L_1 \oplus \dotsb \oplus L_p$, which are given by diagonal matrices with the transition functions on the diagonal. Hence the determinants are again the products of the transition functions.
So the line bundles are the same.