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!
Thank you in anticipation