5.17.1 Tensor Product Scalar Field Compatibility

Tensor Product Scalar Field Compatibility is the requirement that the two vector spaces V and W entering a tensor product construction be defined over the identical scalar field F, ensuring that the homogeneity relations of the canonical map, which move scalars freely between the two factors, refer to one and the same set of scalars on both sides. Without this compatibility, the very statement of bilinearity, and the resulting construction of V ⊗ W, ceases to be meaningful, since there would be no shared notion of scalar multiplication linking the two spaces.

Why a Common Field Is Required

The Homogeneity Relation Presupposes a Shared Field

The defining homogeneity identity:

only makes sense if the scalar c can act on both v ∈ V and w ∈ W. This requires c to belong to a single field F over which both V and W are vector spaces, since scalar multiplication is only defined between a vector space and its own designated field of scalars.

Consequence of Mismatched Fields

If V were a vector space over a field F and W a vector space over a genuinely different field F' with no natural relationship between them, there would be no way to interpret an element of F acting on W, or vice versa, and the tensor product construction described for vector spaces over a common field would not directly apply.

The Role of the Common Field in the Construction

Free Module Built Over the Shared Field

The free module F(V × W) used in the quotient construction is itself defined as a vector space over F, the shared scalar field. Its basis elements are the pairs (v, w), and its scalars, used to form linear combinations of these pairs, are drawn from F.

The Relation Submodule Also Depends on the Shared Field

The homogeneity generators of the relation submodule R, namely (cu, w) − c(u, w) and (u, cw) − c(u, w), are parameterized by scalars c ∈ F, meaning the entire relation structure, and hence the resulting tensor product V ⊗ W, is built relative to this one field throughout.

Field Compatibility Across Different Contexts

Tensor Products of Real and Complex Vector Spaces

When working with vector spaces over the real numbers ℝ and wishing to combine them with complex vector spaces over ℂ, direct tensor product formation in the sense described here is not immediately available; instead, one of the spaces must first be reinterpreted consistently, for example by treating a real vector space as embedded in its complexification, before a tensor product over a common field such as ℂ can be formed.

Extension of Scalars as a Related but Distinct Construction

The related but separate technique of extension of scalars addresses the situation of combining a vector space over a smaller field with a larger field by first enlarging the smaller vector space's scalar field, after which an ordinary tensor product over the common, larger field becomes well-defined; this differs from tensor product scalar field compatibility, which addresses the requirement to have matching fields in the first place rather than the technique of producing one.

Diagram of the Compatibility Requirement

Consequences for the Resulting Tensor Product

Inherited Field Structure

Once compatibility is verified, V ⊗ W is itself naturally a vector space over the same shared field F, with scalar multiplication satisfying the homogeneity identities inherited directly from the construction, and with dimension dim(V) · dim(W) counted over this common field.

Necessity for the Universal Property to Hold as Stated

The universal property of the tensor product, factoring bilinear maps β: V × W → Z uniquely through linear maps V ⊗ W → Z, is stated for Z also a vector space over the same field F; scalar field compatibility across all three spaces V, W, and Z is implicitly assumed throughout the standard formulation of this universal property.

Broader Significance

A Prerequisite Rarely Stated Explicitly

Because most introductory treatments of tensor products fix a single field F from the outset and work exclusively with vector spaces over it, scalar field compatibility is often left implicit; making it explicit clarifies exactly which hypothesis is silently assumed whenever the tensor product of two vector spaces is discussed without further qualification.

Analogous Compatibility Requirements Elsewhere

The same type of compatibility requirement, that two structures share a common base object before a combined construction can be formed, reappears throughout algebra, for instance in requiring two modules to be defined over the same ring before their tensor product over that ring can be constructed, generalizing the field-based compatibility described here to the ring-based setting.