(adsbygoogle = window.adsbygoogle || []).push({});

Hi,

I am edging my way towards Dolbeault cohomology on a complex manifold and one of the constructions involves taking the kth exterior product of a direct sum (the decomposition of the cotangent bundle into holomorphic and antiholomorphic subspaces). This relies on a theorem from multilinear algebra that says that the result is the direct sum of tensor products of exterior products of the subspaces (sorry but I do not have the Latex to set down the formula).

My problem is that when coordinates are subsequently used, the tensor product metamorphoses into a wedge product and I cannot convince myself that these are equivalent. I have not found a proof of the multilinear algebra theorem so I am not clear why a tensor product is required in the first place.

I appreciate that this is a rather detailed question of technique but it won't leave me alone!

Any suggestions?

Thank you in anticipation

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Exterior product of direct sum

**Physics Forums | Science Articles, Homework Help, Discussion**