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Consider a tangent bundle with even and odd parts $T_0 + T_1$, define a space $\Omega^{k,l}_{p,q}$ consisting of (p,q)-forms taking values in $\wedge^k T_0 \otimes \wedge^l T_1$, i.e. the space of $(p,q)$-forms which are also $(k,l)$-multivectors.

Now to compute a direct sum of $\lambda + \rho$, where $\lambda \in \Omega^{k,0}_{p,q}$ and $\rho \in \Omega^{k,1}_{0,q}$, we need $\lambda$ and $\rho$ to be over the same polynomial ring. Any tips?

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I think more context should be added and a more specific question. "Consider a tangent bundle" to me sounds like it should be of a smooth manifold, but then (p,q)-forms came up so I was thinking maybe it might need to be a complex manifold? Then you use "direct sum", but I think this just means "sum" since these are just elements? The notation is confusing too, since these are technically wedge powers of tangent bundles, but the word "form" and symbol $\Omega$ float around implying they are differential forms... –  Matt Dec 19 '11 at 3:42

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