Module Tensor Product

The module tensor product between -modules and over a commutative ring is a generalization of the vector space tensor product. The familiar formulas hold, but now is any element of ,

(1)

(2)

(3)

This generalizes the definition of a tensor product for vector spaces since a vector space is a module over its scalar field. For a finite-rank real or complex vector bundle over a compact Hausdorff space, its continuous sections form a finitely generated projective module over the ring of continuous scalar-valued functions (Swan 1962). The generalization applies to the tensor products of these section modules. A complex group representation of a group is a module over the group algebra . However, the representation tensor product is the vector space tensor product over , with the diagonal group action . It is generally different from a module tensor product over . For a noncommutative ring , a module tensor product is defined for a right -module and a left -module .

There are some interesting possibilities for the tensor product of modules that do not occur in the case of vector spaces. It is possible for to be identically zero. For example, the tensor product of and as modules over the integers, , has no nonzero elements. It is enough to see that . Notice that . Then

(4)

since in and in . In general, it is easier to show that elements are zero than to show they are not zero.

Another interesting property of tensor products is that if is a surjection, then so is the induced map for any other module . But if is an injection, then may fail to be an injection.

For example, , with is an injection, but , with , is not an injection. In , we have .

The failure of the tensor product to preserve injections is measured by Tor.

Another way to think of the tensor product is in terms of its universal property: Any bilinear map from factors uniquely through the natural bilinear map by a module homomorphism.