More General Algebraic Contexts
Let R be a commutative ring and let M be a finite free module over R. Then contraction operates on the full (mixed) tensor algebra of M in exactly the same way as it does in the case of vector spaces over a field. (The key fact is that the natural pairing is still perfect in this case.)
More generally, let OX be a sheaf of commutative rings over a topological space X, e.g. OX could be the structure sheaf of a complex manifold, analytic space, or scheme. Let M be a locally free sheaf of modules over OX of finite rank. Then the dual of M is still well-behaved and contraction operations make sense in this context.
Read more about this topic: Tensor Contraction
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