12.12.2 Tensor Extension Target Domain

Tensor Extension Target Domain is the module or algebra produced by a tensor domain extension operation, namely the pair (V ⊗_R S, S) consisting of the extended object V ⊗_R S together with the new base ring S it is defined over, obtained by applying scalar extension to a source domain (V, R) along a ring homomorphism R → S. The target domain is the destination of the extension: it is the object in which computations are meant to be carried out after the operation completes, and every element, basis, and structural feature it possesses is derived from, but not identical to, the corresponding feature of the source domain.

Formation of the Target Domain

The Underlying Construction

The target domain's underlying set is built from formal symbols v ⊗ s for v ∈ V, s ∈ S, subject to the relations that make the pairing R-bilinear and that fold the R-action on V into the S-action via the homomorphism R → S. Every element of the target domain is, after simplification, a finite sum Σ vₖ ⊗ sₖ.

The S-Module Structure

The target domain is equipped with addition inherited termwise from the formal sums and a scalar multiplication by elements of S acting on the second tensor factor: s′ · (v ⊗ s) = v ⊗ (s′s). This is what makes V ⊗_R S an S-module rather than merely an abelian group, and it is this S-module structure — not any residual R-module structure — that characterizes the target domain going forward.

Features Inherited from the Source Domain

Basis Transport

If the source domain V is free over R with basis {eᵢ}, the target domain is free over S with basis {eᵢ ⊗ 1}, indexed by the same set. The target domain therefore has the same rank as the source domain, even though its elements are now S-linear combinations rather than R-linear combinations.

Dimension and Cardinality

When R and S are fields and V is finite-dimensional, dim_S(V ⊗_R S) = dim_R(V), since the basis transport above is a bijection on index sets. The target domain never gains or loses basis elements purely from the extension step; only the scalars available for forming linear combinations change.

Transported Multilinear Structure

Any R-multilinear form, bilinear pairing, or algebra multiplication carried by the source domain extends to a corresponding S-multilinear structure on the target domain, defined on simple tensors by acting on the V-factors and multiplying the S-factors, then extended by S-linearity. The target domain's structural richness is thus a direct image of the source domain's structure, transported along R → S.

Diagram of the Target Domain's Composition

Distinguishing the Target Domain from the Source Domain

No Independent Prior Existence

The target domain does not exist prior to the extension operation; it is produced by it. This distinguishes it from the source domain, which is given data. Referring to "the target domain" only makes sense in the context of a specific extension operation applied to a specific source domain along a specific homomorphism R → S.

Loss of the Original R-Module Identity

Although V ⊗_R S can be restricted back to an R-module by viewing S as an R-module via R → S, this restriction is generally not isomorphic to the original source domain V — its underlying R-module structure is typically larger, since S may be a much bigger R-module than R itself. The target domain should therefore be treated as a genuinely new object over S, not as a relabeling of the source domain.

Sensitivity to the Choice of Homomorphism

The same source domain (V, R) produces different target domains depending on which homomorphism R → S is used for the extension. The target domain is a function of both the source domain and the extending homomorphism jointly, so it cannot be described completely by reference to the source domain alone.

Uses of the Target Domain

Computations Requiring a Larger Scalar Supply

The target domain is used whenever a computation on V requires scalars from S that are unavailable in R — for example, requiring square roots, additional roots of unity, or a field closure not present in R. The extension operation exists precisely to make such computations possible without altering the combinatorial structure encoded in the source domain's basis.

Base for Further Extensions

The target domain (V ⊗_R S, S) can itself serve as the source domain of a subsequent extension along a further homomorphism S → T, and the composite of the two extensions agrees, up to canonical isomorphism, with a single extension along the composite homomorphism R → T. This composability is what allows scalar extension to be applied iteratively across a chain of base rings.