Tensor Products of Modules

Core Idea

The tensor product M ⊗_R N of two R-modules is the module that universally linearizes bilinear maps: every R-bilinear map M × N → P factors uniquely through M ⊗_R N. Tensor products perform "base change" (extending scalars from one ring to another), are right exact but not left exact in general, and are the algebraic mechanism behind many constructions in geometry and algebra.

Explainer

In linear algebra, given two vector spaces V and W over a field k, the tensor product V ⊗_k W constructs a new vector space whose elements represent "bilinear combinations" of elements from V and W. The dimension satisfies dim(V ⊗ W) = dim(V) · dim(W), and if {vᵢ} and {wⱼ} are bases, then {vᵢ ⊗ wⱼ} is a basis for V ⊗ W. The module-theoretic tensor product generalizes this to modules over any commutative ring, but the behavior is richer and more surprising because modules lack bases in general.

The tensor product M ⊗_R N is defined by a universal property: it is an R-module together with an R-bilinear map M × N → M ⊗_R N such that every R-bilinear map M × N → P factors uniquely through it. The elements m ⊗ n (called pure tensors) are generators, subject to the relations of bilinearity: (m₁ + m₂) ⊗ n = m₁ ⊗ n + m₂ ⊗ n, m ⊗ (n₁ + n₂) = m ⊗ n₁ + m ⊗ n₂, and r(m ⊗ n) = (rm) ⊗ n = m ⊗ (rn). A general element is a sum of pure tensors, and recognizing when such sums are zero is the main computational challenge.

The most important property of tensor products for commutative algebra is right exactness. If 0 → A → B → C → 0 is exact, then A ⊗ N → B ⊗ N → C ⊗ N → 0 is exact — the tensor product preserves surjections and cokernels. But the map A ⊗ N → B ⊗ N need not be injective: tensor products can kill elements. This failure of left exactness is precisely what the Tor functor measures, and modules N for which − ⊗ N is exact (preserves all short exact sequences) are called flat modules.

The computation ℤ/mℤ ⊗_ℤ ℤ/nℤ ≅ ℤ/gcd(m,n)ℤ illustrates the key features. When gcd(m,n) = 1, the tensor product vanishes — the two modules have "incompatible" structures. The general identity M ⊗_R R/I ≅ M/IM shows that tensoring with a quotient ring performs "reduction modulo I," which is the algebraic operation underlying restriction to a closed subvariety. In algebraic geometry, tensor products implement fiber products and base change — changing the ring over which a module is defined. These operations are ubiquitous: extending scalars from ℤ to ℚ (rationalization), from R to R/𝔭 (reduction modulo a prime), or from R to its completion are all instances of tensor product base change.