Whitney sum of vector bundles

Question

I would like to know how to establish that $ \bigwedge^k ( E \oplus F ) = \bigoplus_{p+q=k} \bigwedge^p E \otimes \bigwedge^q F $ such that $ E $ and $ F $ are two vector bundles. Thanks a lot.

Answer 1

For vector spaces $E,F$ over a field (more generally: modules over a commutative ring) there is a canonical isomorphism $\wedge^k(E \oplus F) \cong \oplus_{p+q=k} \wedge^p(E) \otimes \wedge^q(F)$. For a proof, verify that both sides are quotients of $(E \oplus F)^{\otimes k} \cong \oplus_{p+q=k} E^{\otimes p} \otimes F^{\otimes q}$ w.r.t. the same subspace. You can also consult N. Bourbaki, Algebra 1, Chapter III, §7.8, Prop. 11. For vector bundles, just glue these isomorphisms together. This works basically since they commute with restriction maps.

(Alternatively, you can repeat the proof sketched above for arbitrary sheaves of modules on a ringed space (we even don't have to assume that they are vector bundles, i.e. locally trivial), thus giving a direct and global proof. Actually this works even more general, the isomorphism holds in every (sufficiently nice) cocomplete symmetric monoidal category.)