Editing Tensor contraction (section)

== More general algebraic contexts ==

Let ''R'' be a [[commutative ring]] and let ''M'' be a finite free [[module (mathematics)|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 canonical pairing is still perfect in this case.)

More generally, let ''O''<sub>X</sub> be a [[sheaf (mathematics)|sheaf]] of commutative rings over a [[topological space]] ''X'', e.g. ''O''<sub>X</sub> could be the [[structure sheaf]] of a [[complex manifold]], [[analytic space]], or [[scheme (mathematics)|scheme]]. Let ''M'' be a [[locally free sheaf]] of modules over ''O''<sub>X</sub> of finite rank. Then the dual of ''M'' is still well-behaved<ref name="hartshorne"/> and contraction operations make sense in this context.