12.12.1 Tensor Extension Source Domain

Tensor Extension Source Domain is the original module, vector space, or ring over which a tensor object is defined before an extension-of-scalars (or extension-of-base) operation is applied to transport that object into a larger ambient structure. In an extension operation of the form V ↦ V ⊗_R S, where R is a base ring and S is an R-algebra, the source domain designates the pair (V, R) — the module V together with the base ring R it is originally defined over — as distinguished from the target domain (V ⊗_R S, S) produced by the extension. Identifying the source domain correctly is the first and logically prior step of any tensor domain extension, since every subsequent compatibility check, basis transport, and universal-property argument is stated relative to it.

Role Within the Extension Operation

Source Domain as the Operation's Input Data

A tensor domain extension operation is specified by three pieces of data: the source domain (V, R), a ring homomorphism R → S identifying the target base ring, and the extension construction itself, V ⊗_R S. The source domain is not incidental context; it is the operand of the operation. Changing the source domain while holding the homomorphism R → S fixed produces a different extended object, so the operation is only well-posed once the source domain has been fixed.

Distinguishing Source from Target

The source domain supplies the module V and the ring R relative to which V's linear structure, basis, and any bilinear or multilinear forms on V are originally expressed. The target domain inherits an S-module structure but no longer carries an independent identity apart from what the extension construction assigns it; every property of the target domain traces back to a corresponding property of the source domain plus the behavior of the homomorphism R → S.

Structural Requirements on the Source Domain

Module Structure

The source domain V must be a well-defined R-module (or R-vector space when R is a field) before extension is meaningful. This requires an addition operation on V and a scalar multiplication R × V → V satisfying the standard module axioms — distributivity over module and ring addition, compatibility of ring multiplication with scalar action, and unit action. Without these, the tensor product V ⊗_R S cannot be formed, since the tensor product construction itself is built from R-bilinear maps out of V.

Basis or Generating Data Relative to R

When V is free over R with basis {eᵢ}, this basis is a property of the source domain, expressed entirely in terms of R-linear combinations. The extension operation transports this same index set {eᵢ} to a basis (or generating set) {eᵢ ⊗ 1} of V ⊗_R S over S, so the source domain's basis data is the template from which the target domain's basis data is produced, not an independent construction.

Compatibility of Any Attached Forms

If the source domain carries additional structure — a bilinear form, a covariant or contravariant tensor, an algebra multiplication — that structure must be expressed using only R-multilinear maps for the extension to transport it faithfully. A form defined on V using operations not expressible over R cannot be unambiguously extended, since the extension functor (-) ⊗_R S only acts correctly on R-multilinear data.

Diagram of Source and Target Domains

Consequences of a Misidentified Source Domain

Failure of the Universal Property

The extension V ⊗_R S is characterized by a universal property relative to R-bilinear maps out of V × S. If the source domain is misidentified — for instance, if V is mistakenly treated as already an S-module before the homomorphism R → S is applied — the universal property no longer matches the intended construction, and maps that should factor uniquely through the extension may fail to exist or fail to be unique.

Basis Mismatch After Extension

Because the basis {eᵢ ⊗ 1} of the target domain is defined directly from the basis {eᵢ} of the source domain, an incorrect source domain produces an incorrect or ill-defined target basis. Any coordinate computation, matrix representation, or dimension count performed after the extension inherits this error, so verifying the source domain is the point at which such errors are least costly to catch.

Loss of Naturality

Tensor domain extension is required to behave naturally with respect to R-linear maps between source domains: for an R-linear map f : V → V′, the induced map f ⊗ id_S : V ⊗_R S → V′ ⊗_R S must commute with the extension of any structure carried by f. This naturality is stated entirely in terms of the source domains V and V′; an ambiguous or shifting source domain breaks the commutative square that naturality asserts.

Relation to the Broader Extension Operation

Source Domain as a Reusable Object

The same source domain (V, R) can serve as the input to multiple distinct extension operations, one for each choice of target ring S and homomorphism R → S. The source domain itself carries no information about which extension will be performed; it is characterized purely by its own R-module structure, independent of any particular extension.

Recovering the Source Domain from the Target

Under mild conditions — for example, when R → S is faithfully flat — the source domain can be recovered up to isomorphism from the target domain together with the descent data attached to R → S. This descent perspective confirms that the source domain is not lost during extension but is encoded, in a precise technical sense, inside the target domain and its accompanying structure maps.