Appendix A. Tensor products, direct and inverse limits
FC13 — Matsumura, Commutative Ring Theory.
This chapter contains 10 definitions, 2 theorems, 2 examples, 1 counterexample, 4 constructions and 8 formulas.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition: bilinear map
For A-modules M,N,L, a map M×N→L is bilinear when it is A-linear in each argument separately; T_A(M,N;L) denotes the A-module of bilinear maps.
Definition/construction: tensor product
The tensor product M⊗_A N is characterized by a universal bilinear map M×N→M⊗_A N. It exists as the quotient of the free A-module on M×N by the bilinearity relations.
Formula: Formula 1 (tensor universal property)
Hom_A(M⊗_A N,L)≅T_A(M,N;L) naturally.
Construction/formula: multilinear tensor products and Formula 2
The r-fold tensor product represents r-multilinear maps. Tensor product is associative: (M⊗M′)⊗M″≅M⊗M′⊗M″≅M⊗(M′⊗M″).
Formulas: Formulas 3–5
There are canonical isomorphisms M⊗N≅N⊗M, M⊗A≅M, and (⊕_λ M_λ)⊗N≅⊕_λ(M_λ⊗N).
Formula: Formula 6 (tensor product of maps)
Linear maps f:M→M′ and g:N→N′ induce f⊗g:M⊗N→M′⊗N′ with (f⊗g)(Σx_i⊗y_i)=Σf(x_i)⊗g(y_i).
Formula: Formula 7 (tensor quotient for two surjections)
For exact 0→K→M→M′→0 and 0→L→N→N′→0, M′⊗N′ is (M⊗N) modulo the sum of the images K⊗N and M⊗L.
Formula: Formula 8 (right exactness of tensor product)
If M₁→M₂→M₃→0 is exact, then M₁⊗N→M₂⊗N→M₃⊗N→0 is exact.
Definition/lemma: split exactness and flat modules
Tensoring preserves split injections. An A-module N is flat precisely when −⊗_A N is exact.
Definition: two-sided A–B module and scalar action on tensor products
For an A–B-bimodule P, M⊗_A P is naturally a B-module by (Σy_i⊗x_i)b=Σy_i⊗(x_i b).
Formula: Formula 9 (Hom–tensor adjunction for bimodules)
For A–B-bimodule P, A-module M and B-module N, Hom_A(M,Hom_B(P,N))≅Hom_B(M⊗_A P,N).
Formula: Formula 10 (associativity under change of rings)
For compatible bimodules, (M⊗_A P)⊗_B N≅M⊗_A(P⊗_B N).
Definition: extension of scalars
For a ring homomorphism A→B and A-module M, M⊗_A B is the B-module obtained by extension of scalars, also denoted M_B.
Formula: Formula 11 (tensor product commutes with scalar extension)
For A-modules M,M′ and A→B, (M⊗_A B)⊗_B(M′⊗_A B)≅(M⊗_A M′)⊗_A B.
Construction: tensor product of A-algebras
For A-algebras B,C, the A-module B⊗_A C is an A-algebra with (b⊗c)(b′⊗c′)=bb′⊗cc′; it is generated by B⊗1 and 1⊗C.
Examples: quotient and polynomial base change
For an ideal a⊲A, B⊗_A(A/a)≅B/aB. Also B⊗_A A[X]≅B[X], and similarly in several variables.
Definition: directed set
A poset Λ is directed if any two elements admit a common upper bound; finite subsets ordered by inclusion are the standard non-total example.
Definition: direct system
A direct system indexed by a directed set consists of objects M_λ and transition maps f_{μλ}:M_λ→M_μ for λ≤μ satisfying identity and composition laws; the definition applies to sets, modules, rings, and categories.
Definition: morphism of direct systems
A morphism of direct systems with the same index set is a compatible family of maps commuting with all transition maps.
Definition/construction: direct limit
The direct limit lim→M_λ is characterized by the universal property for compatible maps out of the system. It exists as the quotient of the disjoint union by eventual equality; module and ring structures descend when present.
Theorem: Theorem A1
Tensor product commutes with direct limits: lim→(M_λ⊗_A N)≅(lim→M_λ)⊗_A N.
Theorem: Theorem A2
Direct limits of A-modules are exact: a levelwise exact sequence of three direct systems yields an exact sequence of their direct limits.
Construction: module as direct limit of finitely generated submodules
Every A-module is the direct limit of its finitely generated submodules ordered by inclusion.
Construction: ring as direct limit of Noetherian subrings
Every ring is a direct limit of subrings finitely generated over its minimal subring; these subrings are Noetherian.
Definition/construction: inverse system and inverse limit
Inverse systems and inverse limits are the arrow-reversed notions. The inverse limit is the compatible-subset of the direct product ∏M_λ and inherits module/ring structure.
Example: p-adic integers and I-adic completion
The inverse limit lim← Z/(p^n) is the ring of p-adic integers. More generally lim← A/I^n is the I-adic completion of A.
Counterexample: inverse limit is left exact but not exact
Inverse limit of modules is left exact in general but not right exact; the source gives an inverse system of exact rows whose inverse-limit map fails surjectivity.